Subgroups — C₂₃×#18567 #19166193

Order 5888.

980 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3106	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1633	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1632	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1623	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1622	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	—	2	2	no	—
13	Q₈²×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#557	—	2	1	no	—
15	C₂₃×#554	—	2	2	no	—
16	C₂₃×#554	C₂²	1	2	yes	—
17	D₄×Q₈×C₂₃	—	2	2	no	—
18	D₄×Q₈×C₂₃	—	2	1	no	—
19	C₂₃×#544	—	2	2	no	—
20	C₂₃×#544	—	2	1	no	—
21	C₂₃×#533	—	2	1	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#532	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂₃×#520	—	2	1	no	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#497	—	2	2	no	—
29	C₂₃×#494	C₂²	1	2	yes	—
30	C₂₃×#492	C₂²	1	2	yes	—
31	C₂₃×#482	—	2	1	no	—
32	C₂₃×#477	C₂²	1	2	yes	—
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#476	C₂²	1	1	yes	characteristic
35	C₂₃×#474	—	2	1	no	—
36	C₂₃×#468	—	2	2	no	—
37	C₂₃×#461	—	2	1	no	—
38	C₂₃×#460	C₂²	1	2	yes	—
39	C₂₃×#460	C₂²	1	1	yes	characteristic
40	C₂₃×#459	C₂²	1	1	yes	characteristic
41	C₂₃×#449	—	2	1	no	—
42	C₄×SD₁₆×C₂₃	—	2	1	no	—
43	C₄×D₈×C₂₃	—	2	1	no	—
44	C₂₃×#431	C₂²	1	1	yes	characteristic
45	C₂₃×#423	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
47	C₂₃×#333	C₂²	1	2	yes	—
48	C₂₃×#333	C₂²	1	2	yes	—
49	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	—	2	1	no	—
54	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	2	no	—
56	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
57	C₂₃×#137	—	4	8	no	—
58	C₂×Q₁₆×C₂₃	—	2	4	no	—
59	C₂×SD₁₆×C₂₃	—	2	4	no	—
60	C₂×SD₁₆×C₂₃	—	2	1	no	—
61	C₂×SD₁₆×C₂₃	—	2	2	no	—
62	C₂×D₈×C₂₃	—	2	1	no	—
63	C₂×M₁₆×C₂₃	—	2	1	no	—
64	C₂×M₁₆×C₂₃	—	2	1	no	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	—	2	4	no	—
68	C₂₃×C₄⋊Q₈	—	2	2	no	—
69	C₂₃×C₄⋊Q₈	—	2	2	no	—
70	C₂₃×C₄⋊Q₈	—	2	1	no	—
71	C₂₃×C₄⋊Q₈	—	2	1	no	—
72	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
74	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₂₃×#127	C₂³	1	1	yes	characteristic
78	C₂₃×#125	—	2	4	no	—
79	C₂₃×#125	—	2	2	no	—
80	C₂₃×#125	—	2	2	no	—
81	C₂₃×#125	—	2	1	no	—
82	C₂₃×#124	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	—	2	4	no	—
85	C₂₃×C₂²⋊Q₈	—	2	4	no	—
86	C₂₃×C₂²⋊Q₈	—	2	4	no	—
87	C₂₃×C₂²⋊Q₈	—	2	2	no	—
88	C₂₃×C₂²⋊Q₈	—	2	1	no	—
89	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
90	C₂₃×#121	—	2	1	no	—
91	C₂₃×#121	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	2	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
98	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₂₃	—	2	4	no	—
100	C₄×D₄×C₂₃	—	2	2	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	—	2	1	no	—
105	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
106	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂₃×#117	—	2	2	no	—
108	C₂₃×#117	—	2	1	no	—
109	C₂₃×#117	—	2	1	no	—
110	C₂₃×#117	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	—	2	2	no	—
119	C₂₃×#106	—	2	1	no	—
120	C₂₃×C₄⋊C₈	—	2	1	no	—
121	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
122	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	2	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×#102	—	2	2	no	—
130	C₂₃×C₂²⋊C₈	—	2	1	no	—
131	C₂₃×#97	—	2	1	no	—
132	C₄×C₈×C₂₃	—	2	1	no	—
133	C₄×C₈×C₂₃	C₂³	1	2	yes	—
134	#41×C₂₃	—	2	2	no	—
135	#41×C₂₃	—	2	2	no	—
136	#41×C₂₃	—	4	4	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	—	2	2	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	4	4	no	—
142	C₂×Q₈×C₂₃	—	8	4	no	—
143	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
144	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	4	1	no	—
151	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	4	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	4	1	no	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	Q₁₆×C₂₃	—	4	8	no	—
162	SD₁₆×C₂₃	—	4	8	no	—
163	SD₁₆×C₂₃	—	4	2	no	—
164	SD₁₆×C₂₃	—	8	2	no	—
165	D₈×C₂₃	—	4	2	no	—
166	M₁₆×C₂₃	—	2	4	no	—
167	M₁₆×C₂₃	—	8	1	no	—
168	C₂×C₈×C₂₃	—	2	2	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	2	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	4	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	2	no	—
183	C₄⋊C₄×C₂₃	—	2	2	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²⋊C₄×C₂₃	—	2	4	no	—
193	C₂²⋊C₄×C₂₃	—	2	2	no	—
194	C₂²⋊C₄×C₂₃	—	2	1	no	—
195	C₂²⋊C₄×C₂₃	—	2	1	no	—
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	4	1	no	—
198	C₂²⋊C₄×C₂₃	—	4	1	no	—
199	C₄²×C₂₃	—	2	4	no	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	2	no	—
204	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
208	#18567	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₂₃	—	2	1	no	—
210	C₂³×C₂₃	—	4	1	no	—
211	C₂³×C₂₃	#120	1	1	yes	characteristic
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	2	2	no	—
214	Q₈×C₂₃	—	2	2	no	—
215	Q₈×C₂₃	—	4	4	no	—
216	Q₈×C₂₃	—	4	2	no	—
217	Q₈×C₂₃	—	4	4	no	—
218	Q₈×C₂₃	—	4	2	no	—
219	Q₈×C₂₃	—	8	4	no	—
220	Q₈×C₂₃	#137	1	2	yes	complemented
221	D₄×C₂₃	—	2	4	no	—
222	D₄×C₂₃	—	2	1	no	—
223	D₄×C₂₃	—	4	2	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	8	2	no	—
229	D₄×C₂₃	#137	1	2	yes	complemented
230	C₂×C₄×C₂₃	—	2	4	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	2	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	8	2	no	—
247	C₂×C₄×C₂₃	—	8	4	no	—
248	C₂×C₄×C₂₃	—	8	2	no	—
249	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
250	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
251	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
255	C₈×C₂₃	—	2	4	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	2	no	—
259	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3254	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#3106	C₂×C₂₃	1	1	yes	characteristic
263	#1633	C₂×C₂₃	1	2	yes	complemented
264	#1632	C₂×C₂₃	1	2	yes	complemented
265	#1623	C₂×C₂₃	1	2	yes	complemented
266	#1622	C₂×C₂₃	1	2	yes	complemented
267	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
270	C₂²×C₂₃	—	2	1	no	—
271	C₂²×C₂₃	—	2	2	no	—
272	C₂²×C₂₃	—	2	1	no	—
273	C₂²×C₂₃	—	2	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	8	1	no	—
277	C₂²×C₂₃	—	8	1	no	—
278	C₂²×C₂₃	—	8	1	no	—
279	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
280	C₄×C₂₃	—	2	1	no	—
281	C₄×C₂₃	—	2	2	no	—
282	C₄×C₂₃	—	2	1	no	—
283	C₄×C₂₃	—	2	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	2	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	4	no	—
288	C₄×C₂₃	—	4	2	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	#453	1	2	yes	—
291	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
292	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
293	C₄×C₂₃	#577	1	1	yes	characteristic
294	C₄×C₂₃	#578	1	1	yes	characteristic
295	C₂×#137	—	2	2	no	—
296	Q₈²	C₂²×C₂₃	1	2	yes	—
297	#557	—	2	1	no	—
298	#554	—	2	2	no	—
299	#554	C₂²×C₂₃	1	2	yes	—
300	D₄×Q₈	—	2	2	no	—
301	D₄×Q₈	—	2	1	no	—
302	#544	—	2	2	no	—
303	#544	—	2	1	no	—
304	#533	—	2	1	no	—
305	#533	C₂²×C₂₃	1	1	yes	characteristic
306	#532	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊Q₈	—	2	1	no	—
308	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
309	#520	—	2	1	no	—
310	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
311	#497	—	2	2	no	—
312	#494	C₂²×C₂₃	1	2	yes	—
313	#492	C₂²×C₂₃	1	2	yes	—
314	#482	—	2	1	no	—
315	#477	C₂²×C₂₃	1	1	yes	characteristic
316	#477	C₂²×C₂₃	1	2	yes	—
317	#476	C₂²×C₂₃	1	1	yes	characteristic
318	#474	—	2	1	no	—
319	#468	—	2	2	no	—
320	#461	—	2	1	no	—
321	#460	C₂²×C₂₃	1	1	yes	characteristic
322	#460	C₂²×C₂₃	1	2	yes	—
323	#459	C₂²×C₂₃	1	1	yes	characteristic
324	#449	—	2	1	no	—
325	C₄×SD₁₆	—	2	1	no	—
326	C₄×D₈	—	2	1	no	—
327	#431	C₂²×C₂₃	1	1	yes	characteristic
328	#423	C₂²×C₂₃	1	1	yes	characteristic
329	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
330	#333	C₂²×C₂₃	1	2	yes	—
331	#333	C₂²×C₂₃	1	2	yes	—
332	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
333	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
334	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
335	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
336	C₂×C₂₃	—	4	1	no	—
337	C₂×C₂₃	—	4	2	no	—
338	C₂×C₂₃	—	16	1	no	—
339	C₂×C₂₃	#3004	1	1	yes	characteristic
340	C₂×C₂₃	#3013	1	1	yes	characteristic
341	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
342	C₂×#41	—	2	1	no	—
343	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
344	C₂²×Q₈	—	2	2	no	—
345	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
346	#137	—	4	8	no	—
347	C₂×Q₁₆	—	2	4	no	—
348	C₂×SD₁₆	—	2	4	no	—
349	C₂×SD₁₆	—	2	1	no	—
350	C₂×SD₁₆	—	2	2	no	—
351	C₂×D₈	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	—	2	4	no	—
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	—	2	2	no	—
359	C₄⋊Q₈	—	2	1	no	—
360	C₄⋊Q₈	—	2	1	no	—
361	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
362	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
363	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
364	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
366	#127	C₂³×C₂₃	1	1	yes	characteristic
367	#125	—	2	2	no	—
368	#125	—	2	4	no	—
369	#125	—	2	2	no	—
370	#125	—	2	1	no	—
371	#124	—	2	1	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	—	2	4	no	—
374	C₂²⋊Q₈	—	2	4	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
379	#121	—	2	1	no	—
380	#121	—	2	1	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	—	2	2	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
386	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
395	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
396	#117	—	2	2	no	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	D₄×C₂₃	1	1	yes	characteristic
400	C₂×C₄⋊C₄	—	2	2	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
406	#107	—	2	1	no	—
407	#107	—	2	2	no	—
408	#106	—	2	1	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
411	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
413	#103	—	2	1	no	—
414	#103	—	2	2	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	2	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	#97	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	C₂³×C₂₃	1	2	yes	—
423	C₂₃	#18567	1	1	yes	characteristic, direct product factor, Sylow
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	4	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	4	4	no	—
432	C₂×Q₈	—	8	4	no	—
433	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
434	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
451	Q₁₆	—	4	8	no	—
452	SD₁₆	—	4	8	no	—
453	SD₁₆	—	4	2	no	—
454	SD₁₆	—	8	2	no	—
455	D₈	—	4	2	no	—
456	M₁₆	—	2	4	no	—
457	M₁₆	—	8	1	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
481	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	2	1	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	4	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	2	no	—
494	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
498	C₂³	—	2	1	no	—
499	C₂³	—	4	1	no	—
500	C₂³	C₂₃×#120	1	1	yes	characteristic
501	Q₈	—	2	1	no	—
502	Q₈	—	2	2	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	4	4	no	—
505	Q₈	—	4	2	no	—
506	Q₈	—	4	4	no	—
507	Q₈	—	4	2	no	—
508	Q₈	—	8	4	no	—
509	Q₈	C₂₃×#137	1	2	yes	complemented
510	D₄	—	2	4	no	—
511	D₄	—	2	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	1	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	1	no	—
517	D₄	—	8	2	no	—
518	D₄	C₂₃×#137	1	2	yes	complemented
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	8	2	no	—
536	C₂×C₄	—	8	4	no	—
537	C₂×C₄	—	8	2	no	—
538	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
539	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
540	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
544	C₈	—	2	4	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	2	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	1	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
558	C₄	—	2	1	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	1	no	—
562	C₄	—	4	1	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	4	no	—
566	C₄	—	4	2	no	—
567	C₄	—	8	1	no	—
568	C₄	C₂₃×#453	1	2	yes	—
569	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
570	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
571	C₄	C₂₃×#577	1	1	yes	characteristic
572	C₄	C₂₃×#578	1	1	yes	characteristic
573	C₂	—	4	1	no	—
574	C₂	—	4	2	no	—
575	C₂	—	16	1	no	—
576	C₂	C₂₃×#3004	1	1	yes	characteristic
577	C₂	C₂₃×#3013	1	1	yes	characteristic
578	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3106	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3503	C₂	semidirect
2. C₂₃×#3254	C₂	semidirect
3. C₂₃×#3152	C₂	semidirect
5. C₂₃×#1633	C₂	semidirect
6. C₂₃×#1632	C₂	semidirect
7. C₂₃×#1623	C₂	semidirect
8. C₂₃×#1622	C₂	semidirect
9. C₂₃×#1548	C₂	semidirect
10. C₂₃×#1489	C₂	semidirect
11. C₂₃×#1485	C₂	semidirect
208. #18567	C₂₃	direct (N × H)
220. Q₈×C₂₃	#137	semidirect
229. D₄×C₂₃	#137	semidirect
259. #3503	C₂×C₂₃	semidirect
260. #3254	C₂×C₂₃	semidirect
261. #3152	C₂×C₂₃	semidirect
263. #1633	C₂×C₂₃	semidirect
264. #1632	C₂×C₂₃	semidirect
265. #1623	C₂×C₂₃	semidirect
266. #1622	C₂×C₂₃	semidirect
267. #1548	C₂×C₂₃	semidirect
268. #1489	C₂×C₂₃	semidirect
269. #1485	C₂×C₂₃	semidirect
423. C₂₃	#18567	direct (N × H)
509. Q₈	C₂₃×#137	semidirect
518. D₄	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₂₃×#3254	2
C₂₃×#3152	2
C₂₃×#3106	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1548	2
C₂₃×#1489	2
C₂₃×#1485	2
#18567	23

Subgroups — C23×#18567 #19166193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18567 #19166193