Subgroups — C₂₃×#18519 #19166145

Order 5888.

914 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3076	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1649	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic
15	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#557	—	2	1	no	—
18	C₂₃×#557	C₂²	1	1	yes	characteristic
19	C₂₃×#554	—	2	2	no	—
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#554	—	2	1	no	—
22	C₂₃×#554	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₂₃	—	2	1	no	—
24	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#533	—	2	2	no	—
26	C₂₃×#533	—	2	1	no	—
27	C₂₃×#533	—	2	1	no	—
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#533	C₂²	1	1	yes	characteristic
30	C₄×#41×C₂₃	—	2	1	no	—
31	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
33	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#494	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#484	—	2	1	no	—
39	C₂₃×#479	—	2	1	no	—
40	C₂₃×#478	—	2	1	no	—
41	C₂₃×#477	C₂²	1	1	yes	characteristic
42	C₂₃×#477	C₂²	1	1	yes	characteristic
43	C₂₃×#477	C₂²	1	1	yes	characteristic
44	C₂₃×#476	C₂²	1	1	yes	characteristic
45	C₂₃×#475	C₂²	1	1	yes	characteristic
46	C₂₃×#474	C₂²	1	1	yes	characteristic
47	C₂₃×#467	—	2	2	no	—
48	C₂₃×#462	C₂²	1	1	yes	characteristic
49	C₂₃×#462	C₂²	1	1	yes	characteristic
50	C₂₃×#451	—	2	1	no	—
51	C₄×Q₁₆×C₂₃	—	2	2	no	—
52	C₄×Q₁₆×C₂₃	—	2	1	no	—
53	C₄×Q₁₆×C₂₃	—	2	1	no	—
54	C₂₃×#431	C₂²	1	1	yes	characteristic
55	C₂₃×#427	—	2	1	no	—
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×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	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	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₂₃	—	2	1	no	—
73	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×Q₁₆×C₂₃	—	2	2	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂²×C₈×C₂₃	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×#125	—	2	2	no	—
90	C₂₃×#125	—	2	2	no	—
91	C₂₃×#125	—	2	2	no	—
92	C₂₃×#125	—	2	1	no	—
93	C₂₃×#125	—	2	1	no	—
94	C₂₃×#125	—	2	1	no	—
95	C₂₃×#125	—	2	1	no	—
96	C₂₃×C₂²⋊Q₈	—	2	2	no	—
97	C₂₃×C₂²⋊Q₈	—	2	2	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	—	2	2	no	—
100	C₂₃×C₂²⋊Q₈	—	2	2	no	—
101	C₂₃×C₂²⋊Q₈	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	2	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	—	2	1	no	—
105	C₂₃×C₂²⋊Q₈	—	2	2	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
109	C₄×Q₈×C₂₃	—	2	2	no	—
110	C₄×Q₈×C₂₃	—	2	2	no	—
111	C₄×Q₈×C₂₃	—	2	2	no	—
112	C₄×Q₈×C₂₃	—	2	1	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
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	D₄	1	2	yes	—
125	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
128	C₄×D₄×C₂₃	—	2	2	no	—
129	C₄×D₄×C₂₃	—	2	2	no	—
130	C₄×D₄×C₂₃	—	2	1	no	—
131	C₄×D₄×C₂₃	—	2	1	no	—
132	C₄×D₄×C₂₃	—	2	1	no	—
133	C₄×D₄×C₂₃	—	2	1	no	—
134	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
135	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
136	C₂₃×#117	—	2	2	no	—
137	C₂₃×#117	—	2	1	no	—
138	C₂₃×#117	—	2	1	no	—
139	C₂₃×#117	—	2	1	no	—
140	C₂₃×#117	D₄	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
145	C₂×C₄²×C₂₃	—	2	2	no	—
146	C₂×C₄²×C₂₃	—	2	1	no	—
147	C₂×C₄²×C₂₃	—	2	1	no	—
148	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
149	C₂₃×#107	—	2	1	no	—
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	2	no	—
152	C₂₃×#107	—	2	1	no	—
153	C₂₃×#107	—	2	1	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×#106	—	2	1	no	—
156	C₂₃×C₄⋊C₈	—	2	1	no	—
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	2	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#103	—	2	1	no	—
168	C₂₃×#103	—	2	2	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×C₂²⋊C₈	—	2	1	no	—
172	C₄×C₈×C₂₃	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	2	2	no	—
177	#41×C₂₃	Q₁₆	1	4	yes	complemented
178	C₂×Q₈×C₂₃	—	2	2	no	—
179	C₂×Q₈×C₂₃	—	2	2	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	4	1	no	—
186	C₂×Q₈×C₂₃	—	4	2	no	—
187	C₂×Q₈×C₂₃	—	4	2	no	—
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
192	C₂×D₄×C₂₃	—	2	1	no	—
193	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₂₃	—	2	1	no	—
195	C₂²×C₄×C₂₃	—	2	2	no	—
196	C₂²×C₄×C₂₃	—	2	1	no	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	1	no	—
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	Q₁₆×C₂₃	—	4	4	no	—
205	Q₁₆×C₂₃	—	4	2	no	—
206	Q₁₆×C₂₃	—	4	2	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	2	1	no	—
213	C₂×C₈×C₂₃	—	4	2	no	—
214	C₂×C₈×C₂₃	—	4	2	no	—
215	C₄⋊C₄×C₂₃	—	2	2	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	2	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	1	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	—	4	1	no	—
240	C₄⋊C₄×C₂₃	—	4	1	no	—
241	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
245	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
246	C₂²⋊C₄×C₂₃	—	2	2	no	—
247	C₂²⋊C₄×C₂₃	—	2	2	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	4	1	no	—
253	C₄²×C₂₃	—	2	2	no	—
254	C₄²×C₂₃	—	2	2	no	—
255	C₄²×C₂₃	—	2	1	no	—
256	C₄²×C₂₃	—	2	1	no	—
257	C₄²×C₂₃	—	2	2	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	1	no	—
260	C₄²×C₂₃	—	2	1	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	2	1	no	—
263	C₄²×C₂₃	—	4	1	no	—
264	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
268	#18519	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂³×C₂₃	—	2	1	no	—
270	C₂³×C₂₃	#120	1	1	yes	characteristic
271	Q₈×C₂₃	—	2	1	no	—
272	Q₈×C₂₃	—	2	1	no	—
273	Q₈×C₂₃	—	2	1	no	—
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	4	2	no	—
276	Q₈×C₂₃	—	4	2	no	—
277	Q₈×C₂₃	—	4	4	no	—
278	Q₈×C₂₃	—	4	2	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	4	1	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	—	8	2	no	—
284	Q₈×C₂₃	C₂×Q₁₆	1	2	yes	—
285	Q₈×C₂₃	#135	1	2	yes	—
286	D₄×C₂₃	—	2	4	no	—
287	D₄×C₂₃	—	2	1	no	—
288	D₄×C₂₃	C₂×Q₁₆	1	2	yes	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	1	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	2	4	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	2	2	no	—
297	C₂×C₄×C₂₃	—	2	2	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	4	2	no	—
303	C₂×C₄×C₂₃	—	4	4	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	8	1	no	—
309	C₂×C₄×C₂₃	—	8	1	no	—
310	C₂×C₄×C₂₃	—	8	2	no	—
311	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
312	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
313	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
314	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
315	C₂×C₄×C₂₃	#137	1	2	yes	—
316	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
319	C₈×C₂₃	—	2	2	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	C₈×C₂₃	—	4	1	no	—
323	C₈×C₂₃	—	4	1	no	—
324	C₈×C₂₃	—	4	1	no	—
325	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic
326	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#3104	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#3076	C₂×C₂₃	1	1	yes	characteristic
329	#1649	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1542	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1476	C₂×C₂₃	1	1	yes	characteristic
339	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
340	C₂²×C₂₃	—	2	1	no	—
341	C₂²×C₂₃	—	2	1	no	—
342	C₂²×C₂₃	—	2	2	no	—
343	C₂²×C₂₃	—	4	1	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	#451	1	2	yes	—
346	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
347	C₄×C₂₃	—	2	2	no	—
348	C₄×C₂₃	—	2	1	no	—
349	C₄×C₂₃	—	2	2	no	—
350	C₄×C₂₃	—	2	1	no	—
351	C₄×C₂₃	—	2	1	no	—
352	C₄×C₂₃	—	4	1	no	—
353	C₄×C₂₃	—	4	2	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	1	no	—
356	C₄×C₂₃	—	4	2	no	—
357	C₄×C₂₃	—	4	1	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	8	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	#451	1	2	yes	—
362	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
363	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
364	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
365	C₄×C₂₃	#575	1	1	yes	characteristic
366	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
367	#557	—	2	1	no	—
368	#557	C₂²×C₂₃	1	1	yes	characteristic
369	#554	—	2	2	no	—
370	#554	—	2	1	no	—
371	#554	—	2	1	no	—
372	#554	C₂²×C₂₃	1	1	yes	characteristic
373	D₄×Q₈	—	2	1	no	—
374	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
375	#533	—	2	2	no	—
376	#533	—	2	1	no	—
377	#533	—	2	1	no	—
378	#533	C₂²×C₂₃	1	1	yes	characteristic
379	#533	C₂²×C₂₃	1	1	yes	characteristic
380	C₄×#41	—	2	1	no	—
381	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₄×Q₈	—	2	1	no	—
383	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
384	#500	C₂²×C₂₃	1	1	yes	characteristic
385	#494	C₂²×C₂₃	1	1	yes	characteristic
386	#487	C₂²×C₂₃	1	1	yes	characteristic
387	#487	C₂²×C₂₃	1	1	yes	characteristic
388	#484	—	2	1	no	—
389	#479	—	2	1	no	—
390	#478	—	2	1	no	—
391	#477	C₂²×C₂₃	1	1	yes	characteristic
392	#477	C₂²×C₂₃	1	1	yes	characteristic
393	#477	C₂²×C₂₃	1	1	yes	characteristic
394	#476	C₂²×C₂₃	1	1	yes	characteristic
395	#475	C₂²×C₂₃	1	1	yes	characteristic
396	#474	C₂²×C₂₃	1	1	yes	characteristic
397	#467	—	2	2	no	—
398	#462	C₂²×C₂₃	1	1	yes	characteristic
399	#462	C₂²×C₂₃	1	1	yes	characteristic
400	#451	—	2	1	no	—
401	C₄×Q₁₆	—	2	1	no	—
402	C₄×Q₁₆	—	2	1	no	—
403	C₄×Q₁₆	—	2	2	no	—
404	#431	C₂²×C₂₃	1	1	yes	characteristic
405	#427	—	2	1	no	—
406	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
407	#416	—	2	1	no	—
408	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
409	#333	C₂²×C₂₃	1	1	yes	characteristic
410	#333	C₂²×C₂₃	1	1	yes	characteristic
411	#332	C₂²×C₂₃	1	1	yes	characteristic
412	#332	C₂²×C₂₃	1	1	yes	characteristic
413	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
414	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
415	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
416	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
417	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
418	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
421	C₂×C₂₃	—	2	2	no	—
422	C₂×C₂₃	—	4	2	no	—
423	C₂×C₂₃	C₂×#451	1	1	yes	characteristic
424	C₂×C₂₃	#3002	1	1	yes	characteristic
425	C₂×C₂₃	#3018	1	1	yes	characteristic
426	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
427	C₂²×Q₈	—	2	1	no	—
428	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
429	C₂×Q₁₆	—	2	1	no	—
430	C₂×Q₁₆	—	2	1	no	—
431	C₂×Q₁₆	—	2	2	no	—
432	C₂²×C₈	—	2	1	no	—
433	C₂²×C₈	—	2	1	no	—
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	2	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
444	#125	—	2	2	no	—
445	#125	—	2	2	no	—
446	#125	—	2	2	no	—
447	#125	—	2	1	no	—
448	#125	—	2	1	no	—
449	#125	—	2	1	no	—
450	#125	—	2	1	no	—
451	C₂²⋊Q₈	—	2	2	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	2	no	—
456	C₂²⋊Q₈	—	2	1	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	1	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
464	C₄×Q₈	—	2	2	no	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	2	no	—
467	C₄×Q₈	—	2	1	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	—	2	1	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
477	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
478	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
479	C₄×Q₈	D₄×C₂₃	1	2	yes	—
480	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
481	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
483	C₄×D₄	—	2	2	no	—
484	C₄×D₄	—	2	2	no	—
485	C₄×D₄	—	2	1	no	—
486	C₄×D₄	—	2	1	no	—
487	C₄×D₄	—	2	1	no	—
488	C₄×D₄	—	2	1	no	—
489	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
490	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
491	#117	—	2	2	no	—
492	#117	—	2	1	no	—
493	#117	—	2	1	no	—
494	#117	—	2	1	no	—
495	#117	D₄×C₂₃	1	1	yes	characteristic
496	C₂×C₄⋊C₄	—	2	2	no	—
497	C₂×C₄⋊C₄	—	2	1	no	—
498	C₂×C₄⋊C₄	—	2	1	no	—
499	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
500	C₂×C₄²	—	2	2	no	—
501	C₂×C₄²	—	2	1	no	—
502	C₂×C₄²	—	2	1	no	—
503	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
504	#107	—	2	1	no	—
505	#107	—	2	1	no	—
506	#107	—	2	2	no	—
507	#107	—	2	1	no	—
508	#107	—	2	1	no	—
509	#106	—	2	1	no	—
510	#106	—	2	1	no	—
511	C₄⋊C₈	—	2	1	no	—
512	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
513	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
514	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
515	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
516	#103	—	2	1	no	—
517	#103	—	2	1	no	—
518	#103	—	2	1	no	—
519	#103	—	2	1	no	—
520	#103	—	2	2	no	—
521	#103	—	2	1	no	—
522	#103	—	2	1	no	—
523	#103	—	2	2	no	—
524	#102	—	2	1	no	—
525	#102	—	2	1	no	—
526	C₂²⋊C₈	—	2	1	no	—
527	C₄×C₈	—	2	1	no	—
528	C₄×C₈	—	2	1	no	—
529	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
530	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
531	C₂₃	#18519	1	1	yes	characteristic, direct product factor, Sylow
532	#41	—	2	2	no	—
533	#41	Q₁₆×C₂₃	1	4	yes	complemented
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	2	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	4	1	no	—
542	C₂×Q₈	—	4	2	no	—
543	C₂×Q₈	—	4	2	no	—
544	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
546	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
560	Q₁₆	—	4	4	no	—
561	Q₁₆	—	4	2	no	—
562	Q₁₆	—	4	2	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	4	2	no	—
570	C₂×C₈	—	4	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	4	1	no	—
596	C₄⋊C₄	—	4	1	no	—
597	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
598	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
600	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
601	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	1	no	—
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	4	1	no	—
609	C₄²	—	2	2	no	—
610	C₄²	—	2	2	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	2	no	—
613	C₄²	—	2	1	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	2	1	no	—
616	C₄²	—	2	1	no	—
617	C₄²	—	2	1	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	4	1	no	—
620	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
621	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
622	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
623	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
624	C₂³	—	2	1	no	—
625	C₂³	C₂₃×#120	1	1	yes	characteristic
626	Q₈	—	2	1	no	—
627	Q₈	—	2	1	no	—
628	Q₈	—	2	1	no	—
629	Q₈	—	2	1	no	—
630	Q₈	—	4	2	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	4	no	—
633	Q₈	—	4	2	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	8	2	no	—
639	Q₈	C₂×Q₁₆×C₂₃	1	2	yes	—
640	Q₈	C₂₃×#135	1	2	yes	—
641	D₄	—	2	4	no	—
642	D₄	—	2	1	no	—
643	D₄	C₂×Q₁₆×C₂₃	1	2	yes	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	4	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	4	4	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	8	1	no	—
664	C₂×C₄	—	8	1	no	—
665	C₂×C₄	—	8	2	no	—
666	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
667	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
668	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
669	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
670	C₂×C₄	C₂₃×#137	1	2	yes	—
671	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
672	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
674	C₈	—	2	2	no	—
675	C₈	—	4	1	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₈	—	4	1	no	—
680	C₂²	—	2	1	no	—
681	C₂²	—	2	1	no	—
682	C₂²	—	2	2	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	C₂₃×#451	1	2	yes	—
686	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
687	C₄	—	2	2	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	2	no	—
690	C₄	—	2	1	no	—
691	C₄	—	2	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	C₂₃×#451	1	2	yes	—
702	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
703	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
704	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
705	C₄	C₂₃×#575	1	1	yes	characteristic
706	C₂	—	2	2	no	—
707	C₂	—	4	2	no	—
708	C₂	C₂×C₂₃×#451	1	1	yes	characteristic
709	C₂	C₂₃×#3002	1	1	yes	characteristic
710	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
prime radical	C₂×#41×C₂₃	D₄
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
2. C₂₃×#3170	C₂	semidirect
3. C₂₃×#3104	C₂	semidirect
5. C₂₃×#1649	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1645	C₂	semidirect
8. C₂₃×#1644	C₂	semidirect
9. C₂₃×#1637	C₂	semidirect
10. C₂₃×#1636	C₂	semidirect
11. C₂₃×#1633	C₂	semidirect
12. C₂₃×#1632	C₂	semidirect
13. C₂₃×#1542	C₂	semidirect
15. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
177. #41×C₂₃	Q₁₆	semidirect
268. #18519	C₂₃	direct (N × H)
326. #3170	C₂×C₂₃	semidirect
327. #3104	C₂×C₂₃	semidirect
329. #1649	C₂×C₂₃	semidirect
330. #1648	C₂×C₂₃	semidirect
331. #1645	C₂×C₂₃	semidirect
332. #1644	C₂×C₂₃	semidirect
333. #1637	C₂×C₂₃	semidirect
334. #1636	C₂×C₂₃	semidirect
335. #1633	C₂×C₂₃	semidirect
336. #1632	C₂×C₂₃	semidirect
337. #1542	C₂×C₂₃	semidirect
339. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
531. C₂₃	#18519	direct (N × H)
533. #41	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₂₃	2
C₂₃×#3170	2
C₂₃×#3104	2
C₂₃×#3076	2
C₂₃×#1649	2
C₂₃×#1648	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1637	2
C₂₃×#1636	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1542	2
C₂₃×#1476	2
C₂×C₂₃×Q₈⋊C₈	2
#18519	23

Subgroups — C23×#18519 #19166145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18519 #19166145