Subgroups — #15683314

Order 2944.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092929	C₂	1	1	yes	maximal, characteristic
2	#117×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092761	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092760	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092754	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092749	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092744	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092743	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092737	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092732	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092453	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092414	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092411	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×(C₂×D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67288	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#117	C₂²	1	1	yes	characteristic
20	C₂₃×#117	C₂²	1	1	yes	characteristic
21	#10625097	C₂²	1	1	yes	characteristic
22	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
23	#10625068	C₂²	1	1	yes	characteristic
24	#10625068	C₂²	1	1	yes	characteristic
25	#10625066	C₂²	1	1	yes	characteristic
26	#10625066	C₂²	1	1	yes	characteristic
27	#10625063	C₂²	1	1	yes	characteristic
28	#10625063	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
31	#10625057	—	2	1	no	—
32	#10625053	—	2	1	no	—
33	C₂²⋊C₄×D₂₃	—	2	1	no	—
34	#10625048	—	2	1	no	—
35	C₄×D₉₂	—	2	1	no	—
36	C₄×D₉₂	—	2	1	no	—
37	#10625043	—	2	1	no	—
38	C₄²×D₂₃	—	2	1	no	—
39	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
43	#10625003	C₂²	1	1	yes	characteristic, complemented
44	#10625003	C₂²	1	1	yes	characteristic, complemented
45	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
49	#10624998	C₂²	1	1	yes	characteristic, complemented
50	#10624998	C₂²	1	1	yes	characteristic, complemented
51	#10624992	C₂²	1	1	yes	characteristic, complemented
52	#10624992	C₂²	1	1	yes	characteristic, complemented
53	#10624991	C₂²	1	1	yes	characteristic, complemented
54	#10624991	C₂²	1	1	yes	characteristic, complemented
55	#10624990	C₂²	1	1	yes	characteristic, complemented
56	#10624990	C₂²	1	1	yes	characteristic, complemented
57	#10624989	C₂²	1	1	yes	characteristic, complemented
58	#10624989	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₂₃	—	2	1	no	—
62	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
63	#67288	—	2	2	no	—
64	#67288	—	2	2	no	—
65	C₂×D₉₂	—	2	1	no	—
66	C₂×D₉₂	—	2	1	no	—
67	C₂×D₉₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₂₃	—	2	2	no	—
69	C₂×C₄×D₂₃	—	2	2	no	—
70	C₂×C₄×D₂₃	—	2	1	no	—
71	C₂×C₄×D₂₃	—	2	1	no	—
72	C₂×C₄×D₂₃	—	2	1	no	—
73	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
77	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	—	2	1	no	—
89	C₂²⋊Dic₂₃	—	2	1	no	—
90	(C₂×D₂₃)⋊C₄	—	2	1	no	—
91	(C₂×D₂₃)⋊C₄	—	2	1	no	—
92	(C₂×D₂₃)⋊C₄	—	2	1	no	—
93	(C₂×D₂₃)⋊C₄	—	2	1	no	—
94	(C₂×D₂₃)⋊C₄	—	2	1	no	—
95	(C₂×D₂₃)⋊C₄	—	2	1	no	—
96	C₄⋊Dic₂₃	—	2	1	no	—
97	C₄⋊Dic₂₃	—	2	1	no	—
98	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
100	Dic₂₃⋊C₄	—	2	1	no	—
101	Dic₂₃⋊C₄	—	2	1	no	—
102	C₄×Dic₂₃	—	2	1	no	—
103	C₄×Dic₂₃	—	2	1	no	—
104	C₄×Dic₂₃	—	2	1	no	—
105	C₂×C₂₃⋊C₈	—	2	2	no	—
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₂₃	—	2	2	no	—
110	C₂²×D₂₃	—	2	2	no	—
111	C₂²×D₂₃	—	2	1	no	—
112	C₂²×D₂₃	—	4	1	no	—
113	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#41	1	2	yes	—
121	C₂×C₄×C₂₃	#41	1	2	yes	—
122	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₂₃	—	4	2	no	—
124	C₂×Dic₂₃	—	2	2	no	—
125	C₂×Dic₂₃	—	2	1	no	—
126	C₂×Dic₂₃	—	2	1	no	—
127	C₂×Dic₂₃	—	2	1	no	—
128	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	D₉₂	—	2	1	no	—
131	D₉₂	—	2	1	no	—
132	D₉₂	—	4	2	no	—
133	D₉₂	—	4	2	no	—
134	C₄×D₂₃	—	2	2	no	—
135	C₄×D₂₃	—	2	2	no	—
136	C₄×D₂₃	—	2	1	no	—
137	C₄×D₂₃	—	2	1	no	—
138	C₄×D₂₃	—	2	1	no	—
139	C₄×D₂₃	—	2	1	no	—
140	C₄×D₂₃	—	4	2	no	—
141	C₄×D₂₃	—	4	2	no	—
142	C₄×D₂₃	—	4	2	no	—
143	Dic₄₆	—	2	1	no	—
144	Dic₄₆	—	2	1	no	—
145	C₈×C₂₃	—	2	1	no	—
146	C₈×C₂₃	—	2	1	no	—
147	C₂₃⋊C₈	—	2	1	no	—
148	C₂₃⋊C₈	—	2	1	no	—
149	#3091	—	23	1	no	maximal, Sylow
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	#123	1	2	yes	—
153	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₂₃	—	2	1	no	—
155	C₂×D₂₃	—	2	1	no	—
156	C₂×D₂₃	—	2	1	no	—
157	C₂×D₂₃	—	2	1	no	—
158	C₂×D₂₃	—	2	1	no	—
159	C₂×D₂₃	—	4	4	no	—
160	C₂×D₂₃	—	4	1	no	—
161	C₂×D₂₃	—	4	1	no	—
162	C₂×D₂₃	—	4	1	no	—
163	C₂×D₂₃	#135	1	2	yes	—
164	C₄×C₂₃	—	4	1	no	—
165	C₄×C₂₃	—	4	1	no	—
166	C₄×C₂₃	—	4	1	no	—
167	C₄×C₂₃	—	4	1	no	—
168	C₄×C₂₃	#123	1	2	yes	—
169	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
171	Dic₂₃	—	2	1	no	—
172	Dic₂₃	—	2	1	no	—
173	Dic₂₃	—	4	1	no	—
174	Dic₂₃	—	4	1	no	—
175	Dic₂₃	—	4	1	no	—
176	#525	—	23	1	no	—
177	C₂×#117	—	23	1	no	—
178	#485	—	23	1	no	—
179	#485	—	23	1	no	—
180	#484	—	23	1	no	—
181	#483	—	23	1	no	—
182	#482	—	23	1	no	—
183	#482	—	23	1	no	—
184	#481	—	23	1	no	—
185	#480	—	23	1	no	—
186	#427	—	23	1	no	—
187	#427	D₂₃	1	1	yes	characteristic, complemented
188	#416	—	23	1	no	—
189	#416	—	23	1	no	—
190	C₂×C₂²⋊C₈	—	23	1	no	—
191	C₂×C₂₃	—	2	2	no	—
192	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
193	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
194	C₂×C₂₃	#575	1	1	yes	characteristic
195	D₂₃	—	2	2	no	—
196	D₂₃	—	4	1	no	—
197	D₂₃	—	8	1	no	—
198	C₂×#41	—	23	1	no	—
199	C₂³×C₄	—	23	1	no	—
200	C₂²×C₈	—	23	1	no	—
201	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
202	#123	—	46	1	no	—
203	C₂²⋊Q₈	—	23	1	no	—
204	C₂²⋊Q₈	—	23	1	no	—
205	#121	—	23	1	no	—
206	#121	—	23	1	no	—
207	#120	—	46	1	no	—
208	C₄×D₄	—	46	1	no	—
209	C₄×D₄	—	46	1	no	—
210	#117	—	23	1	no	—
211	#117	—	23	1	no	—
212	#117	—	23	1	no	—
213	#117	—	46	1	no	—
214	#117	—	46	1	no	—
215	#117	C₂×D₂₃	1	1	yes	characteristic
216	#117	C₂×D₂₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	23	1	no	—
218	C₂×C₄⋊C₄	—	23	1	no	—
219	C₂×C₂²⋊C₄	—	46	1	no	—
220	C₂×C₄²	—	46	1	no	—
221	#107	—	23	1	no	—
222	#107	—	23	1	no	—
223	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
224	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
225	#106	—	23	1	no	—
226	#106	—	23	1	no	—
227	#106	C₂×D₂₃	1	1	yes	characteristic, complemented
228	#106	C₂×D₂₃	1	1	yes	characteristic, complemented
229	#103	—	23	1	no	—
230	#103	—	23	1	no	—
231	#103	—	23	1	no	—
232	#103	—	23	1	no	—
233	#102	—	23	1	no	—
234	#102	—	23	1	no	—
235	#102	—	23	1	no	—
236	#102	—	23	1	no	—
237	C₂²⋊C₈	—	23	1	no	—
238	C₂²⋊C₈	—	23	1	no	—
239	C₂²⋊C₈	—	23	1	no	—
240	C₂²⋊C₈	—	23	1	no	—
241	C₂₃	#3091	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	23	1	no	—
243	#41	—	46	2	no	—
244	#41	—	46	2	no	—
245	C₂×Q₈	—	23	1	no	—
246	C₂×D₄	—	23	1	no	—
247	C₂×D₄	—	46	1	no	—
248	C₂×D₄	—	46	1	no	—
249	C₂×D₄	—	46	1	no	—
250	C₂²×C₄	—	23	1	no	—
251	C₂²×C₄	—	23	1	no	—
252	C₂²×C₄	—	23	1	no	—
253	C₂²×C₄	—	23	1	no	—
254	C₂²×C₄	—	23	1	no	—
255	C₂²×C₄	—	46	1	no	—
256	C₂²×C₄	—	46	1	no	—
257	C₂²×C₄	—	46	2	no	—
258	C₂²×C₄	—	46	2	no	—
259	C₂²×C₄	—	46	1	no	—
260	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	23	1	no	—
263	C₂×C₈	—	23	1	no	—
264	C₂×C₈	—	46	2	no	—
265	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
267	C₄⋊C₄	—	23	1	no	—
268	C₄⋊C₄	—	23	1	no	—
269	C₄⋊C₄	—	46	1	no	—
270	C₄⋊C₄	—	46	1	no	—
271	C₄⋊C₄	—	46	1	no	—
272	C₄⋊C₄	—	46	1	no	—
273	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	46	1	no	—
280	C₂²⋊C₄	—	46	1	no	—
281	C₂²⋊C₄	—	46	1	no	—
282	C₂²⋊C₄	—	46	1	no	—
283	C₂²⋊C₄	—	46	1	no	—
284	C₂²⋊C₄	—	46	1	no	—
285	C₂²⋊C₄	—	46	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	46	1	no	—
289	C₄²	—	46	1	no	—
290	C₄²	—	46	1	no	—
291	C₂³	—	23	1	no	—
292	C₂³	—	23	1	no	—
293	C₂³	—	46	2	no	—
294	C₂³	—	46	2	no	—
295	C₂³	—	46	1	no	—
296	C₂³	—	92	1	no	—
297	C₂³	D₄×D₂₃	1	1	yes	characteristic
298	Q₈	—	46	1	no	—
299	Q₈	—	46	1	no	—
300	D₄	—	46	1	no	—
301	D₄	—	46	1	no	—
302	D₄	—	92	2	no	—
303	D₄	—	92	2	no	—
304	D₄	—	92	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	23	1	no	—
310	C₂×C₄	—	23	1	no	—
311	C₂×C₄	—	46	2	no	—
312	C₂×C₄	—	46	2	no	—
313	C₂×C₄	—	46	2	no	—
314	C₂×C₄	—	46	1	no	—
315	C₂×C₄	—	46	1	no	—
316	C₂×C₄	—	46	1	no	—
317	C₂×C₄	—	46	1	no	—
318	C₂×C₄	—	46	1	no	—
319	C₂×C₄	—	46	1	no	—
320	C₂×C₄	—	46	1	no	—
321	C₂×C₄	—	92	2	no	—
322	C₂×C₄	—	92	2	no	—
323	C₂×C₄	—	92	2	no	—
324	C₂×C₄	#67288	1	2	yes	—
325	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
326	C₂×C₄	#67292	1	2	yes	—
327	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	46	1	no	—
331	C₈	—	46	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	23	2	no	—
335	C₂²	—	46	1	no	—
336	C₂²	—	46	1	no	—
337	C₂²	—	46	1	no	—
338	C₂²	—	46	1	no	—
339	C₂²	—	46	1	no	—
340	C₂²	—	92	4	no	—
341	C₂²	—	92	1	no	—
342	C₂²	—	92	1	no	—
343	C₂²	—	92	1	no	—
344	C₂²	#10625065	1	2	yes	—
345	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	46	1	no	—
351	C₄	—	46	1	no	—
352	C₄	—	92	1	no	—
353	C₄	—	92	1	no	—
354	C₄	—	92	1	no	—
355	C₄	#10625065	1	2	yes	—
356	C₄	C₂×#67288	1	1	yes	characteristic
357	C₄	C₂×#67292	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	46	2	no	—
360	C₂	—	92	1	no	—
361	C₂	—	184	1	no	—
362	C₂	C₂×#10625065	1	1	yes	characteristic
363	C₂	#135×D₂₃	1	1	yes	characteristic
364	C₂	#13093214	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#13092929	C₂
hypercenter	#427	D₂₃
hypocenter	C₂₃	#3091
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
2. #117×D₂₃	C₂	semidirect
3. #13092761	C₂	semidirect
4. #13092760	C₂	semidirect
5. #13092754	C₂	semidirect
6. #13092749	C₂	semidirect
7. #13092744	C₂	semidirect
8. #13092743	C₂	semidirect
9. #13092737	C₂	semidirect
10. #13092732	C₂	semidirect
11. #13092453	C₂	semidirect
12. #13092414	C₂	semidirect
13. #13092411	C₂	semidirect
14. C₂×(C₂×D₂₃)⋊C₈	C₂	semidirect
15. C₂₃×#427	C₂	semidirect
39. C₂₃×#107	C₂²	semidirect
40. C₂₃×#107	C₂²	semidirect
41. C₂₃×#106	C₂²	semidirect
42. C₂₃×#106	C₂²	semidirect
43. #10625003	C₂²	semidirect
44. #10625003	C₂²	semidirect
49. #10624998	C₂²	semidirect
50. #10624998	C₂²	semidirect
51. #10624992	C₂²	semidirect
52. #10624992	C₂²	semidirect
53. #10624991	C₂²	semidirect
54. #10624991	C₂²	semidirect
55. #10624990	C₂²	semidirect
56. #10624990	C₂²	semidirect
57. #10624989	C₂²	semidirect
58. #10624989	C₂²	semidirect
187. #427	D₂₃	semidirect
223. #107	C₂×D₂₃	semidirect
224. #107	C₂×D₂₃	semidirect
227. #106	C₂×D₂₃	semidirect
228. #106	C₂×D₂₃	semidirect
241. C₂₃	#3091	semidirect

Maximal subgroups

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

subgroup	index
#13092929	2
#117×D₂₃	2
#13092761	2
#13092760	2
#13092754	2
#13092749	2
#13092744	2
#13092743	2
#13092737	2
#13092732	2
#13092453	2
#13092414	2
#13092411	2
C₂×(C₂×D₂₃)⋊C₈	2
C₂₃×#427	2
#3091	23