Subgroups — C₂₃×#18575 #19166201

Order 5888.

894 conjugacy classes of subgroups.

The subgroup lattice has 558 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
2	C₂₃×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3107	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1645	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1645	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1644	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1643	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
11	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#554	—	2	2	no	—
15	C₂₃×#554	C₂²	1	2	yes	—
16	D₄×Q₈×C₂₃	—	2	2	no	—
17	D₄×Q₈×C₂₃	—	2	1	no	—
18	C₂₃×#544	—	2	2	no	—
19	C₂₃×#544	—	2	1	no	—
20	C₂₃×#533	—	2	1	no	—
21	C₂₃×#533	C₂²	1	1	yes	characteristic
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#520	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#491	—	2	2	no	—
28	C₂₃×#487	C₂²	1	2	yes	—
29	C₂₃×#487	C₂²	1	2	yes	—
30	C₂₃×#485	—	2	1	no	—
31	C₂₃×#475	—	2	1	no	—
32	C₂₃×#475	C₂²	1	2	yes	—
33	C₂₃×#475	C₂²	1	1	yes	characteristic
34	C₂₃×#474	C₂²	1	1	yes	characteristic
35	C₂₃×#470	—	2	2	no	—
36	C₂₃×#462	—	2	1	no	—
37	C₂₃×#462	C₂²	1	2	yes	—
38	C₂₃×#462	C₂²	1	1	yes	characteristic
39	C₂₃×#461	C₂²	1	1	yes	characteristic
40	C₂₃×#452	—	2	1	no	—
41	C₄×Q₁₆×C₂₃	—	2	1	no	—
42	C₄×SD₁₆×C₂₃	—	2	1	no	—
43	C₂₃×#431	C₂²	1	1	yes	characteristic
44	C₂₃×#423	C₂²	1	1	yes	characteristic
45	C₂₃×#419	—	2	2	no	—
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	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₂₃	—	2	2	no	—
55	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
56	C₂×Q₁₆×C₂₃	—	2	1	no	—
57	C₂×Q₁₆×C₂₃	—	2	2	no	—
58	C₂×SD₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	2	1	no	—
60	C₂×M₁₆×C₂₃	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	2	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	4	no	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	—	2	1	no	—
67	C₂₃×C₄⋊Q₈	—	2	1	no	—
68	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
70	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
71	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×#125	—	2	4	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#125	—	2	1	no	—
78	C₂₃×#125	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	—	2	2	no	—
80	C₂₃×C₂²⋊Q₈	—	2	4	no	—
81	C₂₃×C₂²⋊Q₈	—	2	4	no	—
82	C₂₃×C₂²⋊Q₈	—	2	4	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₂₃	—	2	4	no	—
98	C₄×D₄×C₂₃	—	2	2	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
102	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂₃×#117	—	2	1	no	—
107	C₂₃×#117	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂₃×#107	—	2	1	no	—
115	C₂₃×#106	—	2	1	no	—
116	C₂₃×#106	—	2	2	no	—
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×#103	—	2	4	no	—
122	C₂₃×#103	—	2	4	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	2	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	—	2	1	no	—
127	C₂₃×#103	—	2	2	no	—
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×#97	—	2	1	no	—
131	C₄×C₈×C₂₃	—	2	1	no	—
132	C₄×C₈×C₂₃	C₂³	1	2	yes	—
133	#41×C₂₃	—	2	2	no	—
134	#41×C₂₃	—	2	2	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	4	4	no	—
141	C₂×Q₈×C₂₃	—	8	4	no	—
142	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	Q₁₆×C₂₃	—	4	2	no	—
156	Q₁₆×C₂₃	—	8	2	no	—
157	SD₁₆×C₂₃	—	4	2	no	—
158	M₁₆×C₂₃	—	4	2	no	—
159	M₁₆×C₂₃	—	8	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	2	no	—
162	C₂×C₈×C₂₃	—	2	2	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	4	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	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	1	no	—
183	C₄⋊C₄×C₂₃	—	4	1	no	—
184	C₄⋊C₄×C₂₃	—	4	1	no	—
185	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	2	no	—
190	C₂²⋊C₄×C₂₃	—	2	1	no	—
191	C₂²⋊C₄×C₂₃	—	2	1	no	—
192	C₂²⋊C₄×C₂₃	—	2	1	no	—
193	C₂²⋊C₄×C₂₃	—	4	1	no	—
194	C₄²×C₂₃	—	2	4	no	—
195	C₄²×C₂₃	—	2	1	no	—
196	C₄²×C₂₃	—	2	1	no	—
197	C₄²×C₂₃	—	2	1	no	—
198	C₄²×C₂₃	—	2	2	no	—
199	C₄²×C₂₃	—	4	1	no	—
200	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
204	#18575	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₂₃	—	2	1	no	—
206	C₂³×C₂₃	#120	1	1	yes	characteristic
207	Q₈×C₂₃	—	2	1	no	—
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	4	4	no	—
211	Q₈×C₂₃	—	4	4	no	—
212	Q₈×C₂₃	—	4	2	no	—
213	Q₈×C₂₃	—	4	1	no	—
214	Q₈×C₂₃	—	4	1	no	—
215	Q₈×C₂₃	—	4	1	no	—
216	Q₈×C₂₃	—	4	1	no	—
217	Q₈×C₂₃	—	8	4	no	—
218	Q₈×C₂₃	#137	1	2	yes	—
219	D₄×C₂₃	—	2	4	no	—
220	D₄×C₂₃	—	2	1	no	—
221	D₄×C₂₃	#137	1	2	yes	—
222	C₂×C₄×C₂₃	—	2	4	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	4	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	8	2	no	—
240	C₂×C₄×C₂₃	—	8	4	no	—
241	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
242	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
243	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	C₈×C₂₃	—	4	2	no	—
248	C₈×C₂₃	—	4	1	no	—
249	C₈×C₂₃	—	4	1	no	—
250	C₈×C₂₃	—	4	2	no	—
251	#3503	C₂×C₂₃	1	1	yes	characteristic
252	#3255	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#3107	C₂×C₂₃	1	1	yes	characteristic
255	#1645	C₂×C₂₃	1	2	yes	complemented
256	#1645	C₂×C₂₃	1	2	yes	complemented
257	#1644	C₂×C₂₃	1	2	yes	complemented
258	#1643	C₂×C₂₃	1	2	yes	complemented
259	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1489	C₂×C₂₃	1	1	yes	characteristic
261	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
262	C₂²×C₂₃	—	2	1	no	—
263	C₂²×C₂₃	—	2	2	no	—
264	C₂²×C₂₃	—	2	1	no	—
265	C₂²×C₂₃	—	2	1	no	—
266	C₂²×C₂₃	—	4	1	no	—
267	C₂²×C₂₃	—	4	1	no	—
268	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
269	C₄×C₂₃	—	2	1	no	—
270	C₄×C₂₃	—	2	2	no	—
271	C₄×C₂₃	—	2	1	no	—
272	C₄×C₂₃	—	2	1	no	—
273	C₄×C₂₃	—	4	1	no	—
274	C₄×C₂₃	—	4	2	no	—
275	C₄×C₂₃	—	4	1	no	—
276	C₄×C₂₃	—	4	4	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	8	1	no	—
279	C₄×C₂₃	—	8	1	no	—
280	C₄×C₂₃	#455	1	2	yes	—
281	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
282	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
283	C₄×C₂₃	#577	1	1	yes	characteristic
284	C₄×C₂₃	#578	1	1	yes	characteristic
285	Q₈²	C₂²×C₂₃	1	2	yes	—
286	#557	—	2	1	no	—
287	#554	—	2	2	no	—
288	#554	C₂²×C₂₃	1	2	yes	—
289	D₄×Q₈	—	2	2	no	—
290	D₄×Q₈	—	2	1	no	—
291	#544	—	2	2	no	—
292	#544	—	2	1	no	—
293	#533	—	2	1	no	—
294	#533	C₂²×C₂₃	1	1	yes	characteristic
295	#533	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₄⋊Q₈	—	2	1	no	—
297	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
298	#520	—	2	1	no	—
299	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
300	#491	—	2	2	no	—
301	#487	C₂²×C₂₃	1	2	yes	—
302	#487	C₂²×C₂₃	1	2	yes	—
303	#485	—	2	1	no	—
304	#475	—	2	1	no	—
305	#475	C₂²×C₂₃	1	1	yes	characteristic
306	#475	C₂²×C₂₃	1	2	yes	—
307	#474	C₂²×C₂₃	1	1	yes	characteristic
308	#470	—	2	2	no	—
309	#462	—	2	1	no	—
310	#462	C₂²×C₂₃	1	1	yes	characteristic
311	#462	C₂²×C₂₃	1	2	yes	—
312	#461	C₂²×C₂₃	1	1	yes	characteristic
313	#452	—	2	1	no	—
314	C₄×Q₁₆	—	2	1	no	—
315	C₄×SD₁₆	—	2	1	no	—
316	#431	C₂²×C₂₃	1	1	yes	characteristic
317	#423	C₂²×C₂₃	1	1	yes	characteristic
318	#419	—	2	2	no	—
319	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
320	#333	C₂²×C₂₃	1	2	yes	—
321	#333	C₂²×C₂₃	1	2	yes	—
322	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
323	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
324	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
325	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
326	C₂×C₂₃	—	4	1	no	—
327	C₂×C₂₃	—	4	2	no	—
328	C₂×C₂₃	#3004	1	1	yes	characteristic
329	C₂×C₂₃	#3013	1	1	yes	characteristic
330	C₂×C₂₃	C₂×#455	1	1	yes	characteristic
331	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
332	C₂²×Q₈	—	2	2	no	—
333	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
334	C₂×Q₁₆	—	2	1	no	—
335	C₂×Q₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×M₁₆	—	2	1	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	4	no	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	—	2	2	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
347	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
348	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
349	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
352	#125	—	2	2	no	—
353	#125	—	2	4	no	—
354	#125	—	2	2	no	—
355	#125	—	2	1	no	—
356	#125	—	2	1	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	4	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	C₂²⋊Q₈	—	2	1	no	—
364	C₂²⋊Q₈	—	2	1	no	—
365	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	—	2	1	no	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
373	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
374	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
380	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
381	#117	—	2	2	no	—
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	—	2	1	no	—
385	#117	D₄×C₂₃	1	1	yes	characteristic
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
392	#107	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	2	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
397	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
399	#103	—	2	4	no	—
400	#103	—	2	4	no	—
401	#103	—	2	1	no	—
402	#103	—	2	2	no	—
403	#103	—	2	1	no	—
404	#103	—	2	1	no	—
405	#103	—	2	2	no	—
406	#102	—	2	1	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	#97	—	2	1	no	—
409	C₄×C₈	—	2	1	no	—
410	C₄×C₈	C₂³×C₂₃	1	2	yes	—
411	C₂₃	#18575	1	1	yes	characteristic, direct product factor, Sylow
412	#41	—	2	2	no	—
413	#41	—	2	2	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	4	4	no	—
420	C₂×Q₈	—	8	4	no	—
421	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
422	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
434	Q₁₆	—	4	2	no	—
435	Q₁₆	—	8	2	no	—
436	SD₁₆	—	4	2	no	—
437	M₁₆	—	4	2	no	—
438	M₁₆	—	8	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
465	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	1	no	—
470	C₂²⋊C₄	—	2	1	no	—
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	4	1	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	4	1	no	—
479	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
483	C₂³	—	2	1	no	—
484	C₂³	C₂₃×#120	1	1	yes	characteristic
485	Q₈	—	2	1	no	—
486	Q₈	—	2	2	no	—
487	Q₈	—	2	2	no	—
488	Q₈	—	4	4	no	—
489	Q₈	—	4	4	no	—
490	Q₈	—	4	2	no	—
491	Q₈	—	4	1	no	—
492	Q₈	—	4	1	no	—
493	Q₈	—	4	1	no	—
494	Q₈	—	4	1	no	—
495	Q₈	—	8	4	no	—
496	Q₈	C₂₃×#137	1	2	yes	—
497	D₄	—	2	4	no	—
498	D₄	—	2	1	no	—
499	D₄	C₂₃×#137	1	2	yes	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	8	2	no	—
518	C₂×C₄	—	8	4	no	—
519	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
520	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
521	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
525	C₈	—	4	2	no	—
526	C₈	—	4	1	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	2	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
536	C₄	—	2	1	no	—
537	C₄	—	2	2	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	4	1	no	—
541	C₄	—	4	2	no	—
542	C₄	—	4	1	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	2	no	—
545	C₄	—	8	1	no	—
546	C₄	—	8	1	no	—
547	C₄	C₂₃×#455	1	2	yes	—
548	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
549	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
550	C₄	C₂₃×#577	1	1	yes	characteristic
551	C₄	C₂₃×#578	1	1	yes	characteristic
552	C₂	—	4	1	no	—
553	C₂	—	4	2	no	—
554	C₂	C₂₃×#3004	1	1	yes	characteristic
555	C₂	C₂₃×#3013	1	1	yes	characteristic
556	C₂	C₂×C₂₃×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
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
2. C₂₃×#3255	C₂	semidirect
3. C₂₃×#3139	C₂	semidirect
5. C₂₃×#1645	C₂	semidirect
6. C₂₃×#1645	C₂	semidirect
7. C₂₃×#1644	C₂	semidirect
8. C₂₃×#1643	C₂	semidirect
9. C₂₃×#1548	C₂	semidirect
11. C₂₃×#1485	C₂	semidirect
204. #18575	C₂₃	direct (N × H)
252. #3255	C₂×C₂₃	semidirect
253. #3139	C₂×C₂₃	semidirect
255. #1645	C₂×C₂₃	semidirect
256. #1645	C₂×C₂₃	semidirect
257. #1644	C₂×C₂₃	semidirect
258. #1643	C₂×C₂₃	semidirect
259. #1548	C₂×C₂₃	semidirect
261. #1485	C₂×C₂₃	semidirect
411. C₂₃	#18575	direct (N × H)

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×#3255	2
C₂₃×#3139	2
C₂₃×#3107	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1643	2
C₂₃×#1548	2
C₂₃×#1489	2
C₂₃×#1485	2
#18575	23

Subgroups — C23×#18575 #19166201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18575 #19166201