Subgroups — C₂₃×#18520 #19166146

Order 5888.

990 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	1	no	—
17	C₂₃×#559	—	2	1	no	—
18	C₂₃×#559	C₂²	1	1	yes	characteristic
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#556	—	2	1	no	—
21	C₂₃×#556	C₂²	1	1	yes	characteristic
22	C₂₃×#553	—	2	2	no	—
23	C₂₃×#553	—	2	1	no	—
24	C₂₃×#550	C₂²	1	1	yes	characteristic
25	C₂₃×#548	—	2	1	no	—
26	C₂₃×#532	—	2	1	no	—
27	C₂₃×#532	C₂²	1	1	yes	characteristic
28	C₂₃×#532	C₂²	1	1	yes	characteristic
29	C₂₃×#529	—	2	1	no	—
30	C₂₃×#529	—	2	2	no	—
31	C₄×#41×C₂₃	—	2	1	no	—
32	C₄×#41×C₂₃	—	2	1	no	—
33	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#500	C₂²	1	1	yes	characteristic
36	C₂₃×#493	C₂²	1	1	yes	characteristic
37	C₂₃×#486	C₂²	1	1	yes	characteristic
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#481	—	2	1	no	—
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#475	C₂²	1	1	yes	characteristic
42	C₂₃×#466	—	2	2	no	—
43	C₂₃×#464	—	2	1	no	—
44	C₂₃×#463	—	2	1	no	—
45	C₂₃×#462	C₂²	1	1	yes	characteristic
46	C₂₃×#461	C₂²	1	1	yes	characteristic
47	C₂₃×#460	C₂²	1	1	yes	characteristic
48	C₂₃×#460	C₂²	1	1	yes	characteristic
49	C₂₃×#460	C₂²	1	1	yes	characteristic
50	C₂₃×#459	C₂²	1	1	yes	characteristic
51	C₂₃×#448	—	2	1	no	—
52	C₄×SD₁₆×C₂₃	—	2	1	no	—
53	C₄×SD₁₆×C₂₃	—	2	1	no	—
54	C₄×D₈×C₂₃	—	2	2	no	—
55	C₂₃×#431	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×#416	—	2	1	no	—
58	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×#331	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
75	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
76	C₂₃×#135	—	4	4	no	—
77	C₂×Q₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	1	no	—
82	C₂×D₈×C₂₃	—	2	1	no	—
83	C₂×D₈×C₂₃	—	2	2	no	—
84	C₂²×C₈×C₂₃	—	2	1	no	—
85	C₂²×C₈×C₂₃	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	—	2	1	no	—
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×#127	—	2	1	no	—
90	C₂₃×#127	C₂³	1	1	yes	characteristic
91	C₂₃×#127	C₂³	1	1	yes	characteristic
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	2	no	—
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	1	no	—
97	C₂₃×#124	—	2	2	no	—
98	C₂₃×#124	—	2	1	no	—
99	C₂₃×#124	—	2	1	no	—
100	C₂₃×#124	—	2	1	no	—
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#123	—	2	2	no	—
104	C₂₃×#123	—	2	2	no	—
105	C₂₃×#123	—	2	2	no	—
106	C₂₃×C₂²⋊Q₈	—	2	2	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#121	—	2	2	no	—
110	C₂₃×#121	—	2	2	no	—
111	C₂₃×#121	—	2	1	no	—
112	C₂₃×#121	—	2	1	no	—
113	C₂₃×#121	—	2	1	no	—
114	C₂₃×#121	—	2	1	no	—
115	C₂₃×#121	D₄	1	2	yes	—
116	C₄×Q₈×C₂₃	—	2	1	no	—
117	C₄×Q₈×C₂₃	—	2	1	no	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	—	2	1	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
122	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
124	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
125	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×D₄×C₂₃	—	2	2	no	—
128	C₄×D₄×C₂₃	—	2	2	no	—
129	C₄×D₄×C₂₃	—	2	2	no	—
130	C₄×D₄×C₂₃	—	2	2	no	—
131	C₄×D₄×C₂₃	—	2	2	no	—
132	C₄×D₄×C₂₃	—	2	1	no	—
133	C₄×D₄×C₂₃	—	2	1	no	—
134	C₄×D₄×C₂₃	—	2	1	no	—
135	C₄×D₄×C₂₃	—	2	1	no	—
136	C₄×D₄×C₂₃	—	2	1	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
141	C₄×D₄×C₂₃	D₄	1	2	yes	—
142	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
143	C₂₃×#117	—	2	2	no	—
144	C₂₃×#117	—	2	2	no	—
145	C₂₃×#117	—	2	1	no	—
146	C₂₃×#117	—	2	1	no	—
147	C₂₃×#117	D₄	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
150	C₂×C₄²×C₂₃	—	2	2	no	—
151	C₂×C₄²×C₂₃	—	2	1	no	—
152	C₂×C₄²×C₂₃	—	2	1	no	—
153	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
154	C₂₃×#107	—	2	2	no	—
155	C₂₃×#106	—	2	1	no	—
156	C₂₃×#106	—	2	1	no	—
157	C₂₃×C₄⋊C₈	—	2	1	no	—
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#102	—	2	2	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	2	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×C₂²⋊C₈	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	2	2	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	4	2	no	—
182	#41×C₂₃	—	4	2	no	—
183	#41×C₂₃	—	4	2	no	—
184	#41×C₂₃	—	4	2	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	2	2	no	—
188	C₂×Q₈×C₂₃	—	2	1	no	—
189	C₂×Q₈×C₂₃	—	4	1	no	—
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂×D₄×C₂₃	—	2	2	no	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	—	2	2	no	—
196	C₂×D₄×C₂₃	—	2	2	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	2	1	no	—
201	C₂×D₄×C₂₃	—	2	1	no	—
202	C₂×D₄×C₂₃	—	4	1	no	—
203	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
205	C₂²×C₄×C₂₃	—	2	2	no	—
206	C₂²×C₄×C₂₃	—	2	1	no	—
207	C₂²×C₄×C₂₃	—	2	2	no	—
208	C₂²×C₄×C₂₃	—	2	1	no	—
209	C₂²×C₄×C₂₃	—	2	1	no	—
210	C₂²×C₄×C₂₃	—	2	2	no	—
211	C₂²×C₄×C₂₃	—	2	1	no	—
212	C₂²×C₄×C₂₃	—	2	1	no	—
213	C₂²×C₄×C₂₃	—	4	1	no	—
214	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
217	Q₁₆×C₂₃	—	4	2	no	—
218	SD₁₆×C₂₃	—	4	2	no	—
219	SD₁₆×C₂₃	—	4	2	no	—
220	SD₁₆×C₂₃	—	4	2	no	—
221	SD₁₆×C₂₃	—	4	2	no	—
222	D₈×C₂₃	—	4	4	no	—
223	D₈×C₂₃	—	4	2	no	—
224	C₂×C₈×C₂₃	—	2	1	no	—
225	C₂×C₈×C₂₃	—	2	1	no	—
226	C₂×C₈×C₂₃	—	2	1	no	—
227	C₂×C₈×C₂₃	—	2	1	no	—
228	C₂×C₈×C₂₃	—	2	1	no	—
229	C₂×C₈×C₂₃	—	2	1	no	—
230	C₂×C₈×C₂₃	—	4	2	no	—
231	C₂×C₈×C₂₃	—	4	2	no	—
232	C₄⋊C₄×C₂₃	—	2	2	no	—
233	C₄⋊C₄×C₂₃	—	2	2	no	—
234	C₄⋊C₄×C₂₃	—	2	2	no	—
235	C₄⋊C₄×C₂₃	—	2	2	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	1	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	—	2	1	no	—
246	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
247	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
248	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
249	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
251	C₂²⋊C₄×C₂₃	—	2	2	no	—
252	C₂²⋊C₄×C₂₃	—	2	2	no	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	2	no	—
256	C₂²⋊C₄×C₂₃	—	2	2	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	2	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	2	2	no	—
261	C₂²⋊C₄×C₂₃	—	2	1	no	—
262	C₂²⋊C₄×C₂₃	—	4	1	no	—
263	C₂²⋊C₄×C₂₃	—	4	1	no	—
264	C₄²×C₂₃	—	2	2	no	—
265	C₄²×C₂₃	—	2	2	no	—
266	C₄²×C₂₃	—	2	1	no	—
267	C₄²×C₂₃	—	2	1	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	—	2	1	no	—
271	C₄²×C₂₃	—	2	1	no	—
272	C₄²×C₂₃	—	2	1	no	—
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
275	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
276	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
277	#18520	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂³×C₂₃	—	2	1	no	—
279	C₂³×C₂₃	—	2	2	no	—
280	C₂³×C₂₃	—	4	1	no	—
281	C₂³×C₂₃	#120	1	1	yes	characteristic
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	2	1	no	—
284	Q₈×C₂₃	—	2	1	no	—
285	Q₈×C₂₃	—	2	1	no	—
286	Q₈×C₂₃	—	2	1	no	—
287	Q₈×C₂₃	—	4	2	no	—
288	Q₈×C₂₃	—	4	1	no	—
289	Q₈×C₂₃	—	4	1	no	—
290	Q₈×C₂₃	—	4	1	no	—
291	Q₈×C₂₃	—	4	1	no	—
292	Q₈×C₂₃	#135	1	2	yes	complemented
293	D₄×C₂₃	—	2	1	no	—
294	D₄×C₂₃	—	2	1	no	—
295	D₄×C₂₃	—	4	2	no	—
296	D₄×C₂₃	—	4	4	no	—
297	D₄×C₂₃	—	4	2	no	—
298	D₄×C₂₃	—	4	2	no	—
299	D₄×C₂₃	—	4	4	no	—
300	D₄×C₂₃	—	4	1	no	—
301	D₄×C₂₃	—	4	1	no	—
302	D₄×C₂₃	—	4	1	no	—
303	D₄×C₂₃	—	4	1	no	—
304	D₄×C₂₃	—	4	1	no	—
305	D₄×C₂₃	—	4	1	no	—
306	D₄×C₂₃	—	4	1	no	—
307	D₄×C₂₃	—	4	1	no	—
308	D₄×C₂₃	—	8	2	no	—
309	D₄×C₂₃	—	8	2	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	1	no	—
312	C₂×C₄×C₂₃	—	2	1	no	—
313	C₂×C₄×C₂₃	—	2	1	no	—
314	C₂×C₄×C₂₃	—	2	1	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	2	1	no	—
317	C₂×C₄×C₂₃	—	2	1	no	—
318	C₂×C₄×C₂₃	—	2	2	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	4	1	no	—
321	C₂×C₄×C₂₃	—	4	2	no	—
322	C₂×C₄×C₂₃	—	4	4	no	—
323	C₂×C₄×C₂₃	—	4	1	no	—
324	C₂×C₄×C₂₃	—	4	1	no	—
325	C₂×C₄×C₂₃	—	4	2	no	—
326	C₂×C₄×C₂₃	—	4	4	no	—
327	C₂×C₄×C₂₃	—	4	1	no	—
328	C₂×C₄×C₂₃	—	8	1	no	—
329	C₂×C₄×C₂₃	—	8	1	no	—
330	C₂×C₄×C₂₃	—	8	2	no	—
331	C₂×C₄×C₂₃	—	8	2	no	—
332	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
333	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
334	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
335	C₂×C₄×C₂₃	#135	1	2	yes	—
336	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
337	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
339	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
340	C₈×C₂₃	—	2	2	no	—
341	C₈×C₂₃	—	4	1	no	—
342	C₈×C₂₃	—	4	1	no	—
343	C₈×C₂₃	—	4	1	no	—
344	C₈×C₂₃	—	4	1	no	—
345	C₈×C₂₃	—	4	1	no	—
346	#3480	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#3169	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#3103	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#3035	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#1628	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
356	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
357	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
358	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
359	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
360	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
361	C₂²×C₂₃	—	2	1	no	—
362	C₂²×C₂₃	—	2	1	no	—
363	C₂²×C₂₃	—	4	1	no	—
364	C₂²×C₂₃	—	4	2	no	—
365	C₂²×C₂₃	—	4	2	no	—
366	C₂²×C₂₃	—	4	1	no	—
367	C₂²×C₂₃	—	4	2	no	—
368	C₂²×C₂₃	—	4	1	no	—
369	C₂²×C₂₃	—	8	1	no	—
370	C₂²×C₂₃	—	8	1	no	—
371	C₂²×C₂₃	—	8	1	no	—
372	C₂²×C₂₃	#449	1	2	yes	—
373	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
374	C₄×C₂₃	—	2	1	no	—
375	C₄×C₂₃	—	2	2	no	—
376	C₄×C₂₃	—	2	1	no	—
377	C₄×C₂₃	—	2	1	no	—
378	C₄×C₂₃	—	4	1	no	—
379	C₄×C₂₃	—	4	2	no	—
380	C₄×C₂₃	—	4	1	no	—
381	C₄×C₂₃	—	4	1	no	—
382	C₄×C₂₃	—	4	1	no	—
383	C₄×C₂₃	—	4	1	no	—
384	C₄×C₂₃	—	4	1	no	—
385	C₄×C₂₃	—	8	1	no	—
386	C₄×C₂₃	#449	1	2	yes	—
387	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
388	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
389	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₄×C₂₃	#575	1	1	yes	characteristic
391	C₂×#135	—	2	1	no	—
392	#559	—	2	1	no	—
393	#559	C₂²×C₂₃	1	1	yes	characteristic
394	#557	C₂²×C₂₃	1	1	yes	characteristic
395	#556	—	2	1	no	—
396	#556	C₂²×C₂₃	1	1	yes	characteristic
397	#553	—	2	2	no	—
398	#553	—	2	1	no	—
399	#550	C₂²×C₂₃	1	1	yes	characteristic
400	#548	—	2	1	no	—
401	#532	—	2	1	no	—
402	#532	C₂²×C₂₃	1	1	yes	characteristic
403	#532	C₂²×C₂₃	1	1	yes	characteristic
404	#529	—	2	1	no	—
405	#529	—	2	2	no	—
406	C₄×#41	—	2	1	no	—
407	C₄×#41	—	2	1	no	—
408	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
409	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
410	#500	C₂²×C₂₃	1	1	yes	characteristic
411	#493	C₂²×C₂₃	1	1	yes	characteristic
412	#486	C₂²×C₂₃	1	1	yes	characteristic
413	#486	C₂²×C₂₃	1	1	yes	characteristic
414	#481	—	2	1	no	—
415	#475	C₂²×C₂₃	1	1	yes	characteristic
416	#475	C₂²×C₂₃	1	1	yes	characteristic
417	#466	—	2	2	no	—
418	#464	—	2	1	no	—
419	#463	—	2	1	no	—
420	#462	C₂²×C₂₃	1	1	yes	characteristic
421	#461	C₂²×C₂₃	1	1	yes	characteristic
422	#460	C₂²×C₂₃	1	1	yes	characteristic
423	#460	C₂²×C₂₃	1	1	yes	characteristic
424	#460	C₂²×C₂₃	1	1	yes	characteristic
425	#459	C₂²×C₂₃	1	1	yes	characteristic
426	#448	—	2	1	no	—
427	C₄×SD₁₆	—	2	1	no	—
428	C₄×SD₁₆	—	2	1	no	—
429	C₄×D₈	—	2	2	no	—
430	#431	C₂²×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
432	#416	—	2	1	no	—
433	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
434	#332	C₂²×C₂₃	1	1	yes	characteristic
435	#332	C₂²×C₂₃	1	1	yes	characteristic
436	#331	C₂²×C₂₃	1	1	yes	characteristic
437	#331	C₂²×C₂₃	1	1	yes	characteristic
438	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
439	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
440	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
441	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
442	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
443	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
444	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
445	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
446	C₂×C₂₃	—	2	2	no	—
447	C₂×C₂₃	—	8	1	no	—
448	C₂×C₂₃	—	8	2	no	—
449	C₂×C₂₃	—	16	1	no	—
450	C₂×C₂₃	C₂×#449	1	1	yes	characteristic
451	C₂×C₂₃	#3002	1	1	yes	characteristic
452	C₂×C₂₃	#3018	1	1	yes	characteristic
453	C₂×#41	—	2	1	no	—
454	C₂×#41	—	2	1	no	—
455	C₂×#41	—	2	1	no	—
456	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
457	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
458	#135	—	4	4	no	—
459	C₂×Q₁₆	—	2	1	no	—
460	C₂×SD₁₆	—	2	1	no	—
461	C₂×SD₁₆	—	2	1	no	—
462	C₂×SD₁₆	—	2	1	no	—
463	C₂×SD₁₆	—	2	1	no	—
464	C₂×D₈	—	2	1	no	—
465	C₂×D₈	—	2	2	no	—
466	C₂²×C₈	—	2	1	no	—
467	C₂²×C₈	—	2	1	no	—
468	C₄⋊Q₈	—	2	1	no	—
469	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
470	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
471	#127	—	2	1	no	—
472	#127	C₂³×C₂₃	1	1	yes	characteristic
473	#127	C₂³×C₂₃	1	1	yes	characteristic
474	#126	—	2	2	no	—
475	#126	—	2	2	no	—
476	#126	—	2	2	no	—
477	#125	—	2	2	no	—
478	#125	—	2	1	no	—
479	#124	—	2	2	no	—
480	#124	—	2	1	no	—
481	#124	—	2	1	no	—
482	#124	—	2	1	no	—
483	#124	—	2	1	no	—
484	#124	—	2	1	no	—
485	#123	—	2	2	no	—
486	#123	—	2	2	no	—
487	#123	—	2	2	no	—
488	C₂²⋊Q₈	—	2	2	no	—
489	C₂²⋊Q₈	—	2	1	no	—
490	#121	—	2	2	no	—
491	#121	—	2	2	no	—
492	#121	—	2	2	no	—
493	#121	—	2	1	no	—
494	#121	—	2	1	no	—
495	#121	—	2	1	no	—
496	#121	—	2	1	no	—
497	#121	D₄×C₂₃	1	2	yes	—
498	C₄×Q₈	—	2	1	no	—
499	C₄×Q₈	—	2	1	no	—
500	C₄×Q₈	—	2	1	no	—
501	C₄×Q₈	—	2	1	no	—
502	C₄×Q₈	—	2	1	no	—
503	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
504	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
505	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
506	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
507	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
508	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	—	2	2	no	—
511	C₄×D₄	—	2	2	no	—
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	—	2	2	no	—
514	C₄×D₄	—	2	1	no	—
515	C₄×D₄	—	2	1	no	—
516	C₄×D₄	—	2	1	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	—	2	1	no	—
519	C₄×D₄	—	2	1	no	—
520	C₄×D₄	—	2	1	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
523	C₄×D₄	D₄×C₂₃	1	2	yes	—
524	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
525	#117	—	2	2	no	—
526	#117	—	2	2	no	—
527	#117	—	2	1	no	—
528	#117	—	2	1	no	—
529	#117	D₄×C₂₃	1	1	yes	characteristic
530	C₂×C₄⋊C₄	—	2	1	no	—
531	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₄²	—	2	2	no	—
533	C₂×C₄²	—	2	1	no	—
534	C₂×C₄²	—	2	1	no	—
535	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
536	#107	—	2	2	no	—
537	#106	—	2	1	no	—
538	#106	—	2	1	no	—
539	C₄⋊C₈	—	2	1	no	—
540	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
541	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
542	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
543	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
544	#103	—	2	1	no	—
545	#103	—	2	1	no	—
546	#103	—	2	1	no	—
547	#103	—	2	1	no	—
548	#102	—	2	2	no	—
549	#102	—	2	1	no	—
550	#102	—	2	1	no	—
551	#102	—	2	2	no	—
552	#102	—	2	1	no	—
553	#102	—	2	1	no	—
554	C₂²⋊C₈	—	2	1	no	—
555	C₄×C₈	—	2	1	no	—
556	C₄×C₈	—	2	1	no	—
557	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
558	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
559	C₂₃	#18520	1	1	yes	characteristic, direct product factor, Sylow
560	#41	—	2	2	no	—
561	#41	—	2	2	no	—
562	#41	—	4	2	no	—
563	#41	—	4	2	no	—
564	#41	—	4	2	no	—
565	#41	—	4	2	no	—
566	#41	—	4	2	no	—
567	#41	—	4	2	no	—
568	C₂×Q₈	—	2	1	no	—
569	C₂×Q₈	—	2	1	no	—
570	C₂×Q₈	—	2	2	no	—
571	C₂×Q₈	—	2	1	no	—
572	C₂×Q₈	—	4	1	no	—
573	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
574	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
575	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
576	C₂×D₄	—	2	2	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	2	2	no	—
579	C₂×D₄	—	2	2	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	2	1	no	—
583	C₂×D₄	—	2	1	no	—
584	C₂×D₄	—	2	1	no	—
585	C₂×D₄	—	4	1	no	—
586	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
587	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
588	C₂²×C₄	—	2	2	no	—
589	C₂²×C₄	—	2	1	no	—
590	C₂²×C₄	—	2	2	no	—
591	C₂²×C₄	—	2	1	no	—
592	C₂²×C₄	—	2	1	no	—
593	C₂²×C₄	—	2	2	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	1	no	—
596	C₂²×C₄	—	4	1	no	—
597	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
598	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
600	Q₁₆	—	4	2	no	—
601	SD₁₆	—	4	2	no	—
602	SD₁₆	—	4	2	no	—
603	SD₁₆	—	4	2	no	—
604	SD₁₆	—	4	2	no	—
605	D₈	—	4	4	no	—
606	D₈	—	4	2	no	—
607	C₂×C₈	—	2	1	no	—
608	C₂×C₈	—	2	1	no	—
609	C₂×C₈	—	2	1	no	—
610	C₂×C₈	—	2	1	no	—
611	C₂×C₈	—	2	1	no	—
612	C₂×C₈	—	2	1	no	—
613	C₂×C₈	—	4	2	no	—
614	C₂×C₈	—	4	2	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	2	no	—
617	C₄⋊C₄	—	2	2	no	—
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	2	1	no	—
626	C₄⋊C₄	—	2	1	no	—
627	C₄⋊C₄	—	2	1	no	—
628	C₄⋊C₄	—	2	1	no	—
629	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
630	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
631	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
632	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
633	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	2	no	—
638	C₂²⋊C₄	—	2	2	no	—
639	C₂²⋊C₄	—	2	2	no	—
640	C₂²⋊C₄	—	2	1	no	—
641	C₂²⋊C₄	—	2	2	no	—
642	C₂²⋊C₄	—	2	1	no	—
643	C₂²⋊C₄	—	2	2	no	—
644	C₂²⋊C₄	—	2	1	no	—
645	C₂²⋊C₄	—	4	1	no	—
646	C₂²⋊C₄	—	4	1	no	—
647	C₄²	—	2	2	no	—
648	C₄²	—	2	2	no	—
649	C₄²	—	2	1	no	—
650	C₄²	—	2	1	no	—
651	C₄²	—	2	1	no	—
652	C₄²	—	2	1	no	—
653	C₄²	—	2	1	no	—
654	C₄²	—	2	1	no	—
655	C₄²	—	2	1	no	—
656	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
657	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
658	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
659	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
660	C₂³	—	2	1	no	—
661	C₂³	—	2	2	no	—
662	C₂³	—	4	1	no	—
663	C₂³	C₂₃×#120	1	1	yes	characteristic
664	Q₈	—	2	1	no	—
665	Q₈	—	2	1	no	—
666	Q₈	—	2	1	no	—
667	Q₈	—	2	1	no	—
668	Q₈	—	2	1	no	—
669	Q₈	—	4	2	no	—
670	Q₈	—	4	1	no	—
671	Q₈	—	4	1	no	—
672	Q₈	—	4	1	no	—
673	Q₈	—	4	1	no	—
674	Q₈	C₂₃×#135	1	2	yes	complemented
675	D₄	—	2	1	no	—
676	D₄	—	2	1	no	—
677	D₄	—	4	2	no	—
678	D₄	—	4	4	no	—
679	D₄	—	4	2	no	—
680	D₄	—	4	2	no	—
681	D₄	—	4	4	no	—
682	D₄	—	4	1	no	—
683	D₄	—	4	1	no	—
684	D₄	—	4	1	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	4	1	no	—
688	D₄	—	4	1	no	—
689	D₄	—	4	1	no	—
690	D₄	—	8	2	no	—
691	D₄	—	8	2	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	2	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	2	no	—
704	C₂×C₄	—	4	4	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	4	2	no	—
708	C₂×C₄	—	4	4	no	—
709	C₂×C₄	—	4	1	no	—
710	C₂×C₄	—	8	1	no	—
711	C₂×C₄	—	8	1	no	—
712	C₂×C₄	—	8	2	no	—
713	C₂×C₄	—	8	2	no	—
714	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
715	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
716	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
717	C₂×C₄	C₂₃×#135	1	2	yes	—
718	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
719	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
720	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
721	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
722	C₈	—	2	2	no	—
723	C₈	—	4	1	no	—
724	C₈	—	4	1	no	—
725	C₈	—	4	1	no	—
726	C₈	—	4	1	no	—
727	C₈	—	4	1	no	—
728	C₂²	—	2	1	no	—
729	C₂²	—	2	1	no	—
730	C₂²	—	4	1	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	4	2	no	—
733	C₂²	—	4	1	no	—
734	C₂²	—	4	2	no	—
735	C₂²	—	4	1	no	—
736	C₂²	—	8	1	no	—
737	C₂²	—	8	1	no	—
738	C₂²	—	8	1	no	—
739	C₂²	C₂₃×#449	1	2	yes	—
740	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
741	C₄	—	2	1	no	—
742	C₄	—	2	2	no	—
743	C₄	—	2	1	no	—
744	C₄	—	2	1	no	—
745	C₄	—	4	1	no	—
746	C₄	—	4	2	no	—
747	C₄	—	4	1	no	—
748	C₄	—	4	1	no	—
749	C₄	—	4	1	no	—
750	C₄	—	4	1	no	—
751	C₄	—	4	1	no	—
752	C₄	—	8	1	no	—
753	C₄	C₂₃×#449	1	2	yes	—
754	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
755	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
756	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
757	C₄	C₂₃×#575	1	1	yes	characteristic
758	C₂	—	2	2	no	—
759	C₂	—	8	1	no	—
760	C₂	—	8	2	no	—
761	C₂	—	16	1	no	—
762	C₂	C₂×C₂₃×#449	1	1	yes	characteristic
763	C₂	C₂₃×#3002	1	1	yes	characteristic
764	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3480	C₂	semidirect
2. C₂₃×#3169	C₂	semidirect
3. C₂₃×#3103	C₂	semidirect
4. C₂₃×#3035	C₂	semidirect
5. C₂₃×#1650	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1642	C₂	semidirect
8. C₂₃×#1640	C₂	semidirect
9. C₂₃×#1628	C₂	semidirect
10. C₂₃×#1626	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1620	C₂	semidirect
13. C₂₃×#1540	C₂	semidirect
14. C₂₃×#1476	C₂	semidirect
15. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
277. #18520	C₂₃	direct (N × H)
292. Q₈×C₂₃	#135	semidirect
346. #3480	C₂×C₂₃	semidirect
347. #3169	C₂×C₂₃	semidirect
348. #3103	C₂×C₂₃	semidirect
349. #3035	C₂×C₂₃	semidirect
350. #1650	C₂×C₂₃	semidirect
351. #1648	C₂×C₂₃	semidirect
352. #1642	C₂×C₂₃	semidirect
353. #1640	C₂×C₂₃	semidirect
354. #1628	C₂×C₂₃	semidirect
355. #1626	C₂×C₂₃	semidirect
356. #1621	C₂×C₂₃	semidirect
357. #1620	C₂×C₂₃	semidirect
358. #1540	C₂×C₂₃	semidirect
359. #1476	C₂×C₂₃	semidirect
360. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
559. C₂₃	#18520	direct (N × H)
674. Q₈	C₂₃×#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3480	2
C₂₃×#3169	2
C₂₃×#3103	2
C₂₃×#3035	2
C₂₃×#1650	2
C₂₃×#1648	2
C₂₃×#1642	2
C₂₃×#1640	2
C₂₃×#1628	2
C₂₃×#1626	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1540	2
C₂₃×#1476	2
C₂×C₂₃×Q₈⋊C₈	2
#18520	23

Subgroups — C23×#18520 #19166146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18520 #19166146