Subgroups — #15683546

Order 2944.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#67292	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092744	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092740	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092700	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092696	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092615	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092611	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092565	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092561	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092530	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092527	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092515	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67292	C₂²	1	1	yes	characteristic
17	C₂×#67288	—	2	1	no	—
18	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×#67274	C₂²	1	1	yes	characteristic
23	#10625111	C₂²	1	1	yes	characteristic
24	#10625109	C₂²	1	1	yes	characteristic
25	C₂×#67272	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₂₃	—	2	1	no	—
27	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
28	#10625066	C₂²	1	1	yes	characteristic
29	#10625064	—	2	1	no	—
30	#10625064	C₂²	1	1	yes	characteristic
31	#10625063	—	2	1	no	—
32	#10625063	C₂²	1	1	yes	characteristic
33	#10625059	C₂²	1	1	yes	characteristic
34	#10625056	—	2	1	no	—
35	#10625055	—	2	1	no	—
36	C₄×D₉₂	—	2	1	no	—
37	C₄×D₉₂	C₂²	1	1	yes	characteristic
38	C₄×D₉₂	C₂²	1	1	yes	characteristic
39	C₄²×D₂₃	—	2	1	no	—
40	C₄²×D₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×#103	C₂²	1	1	yes	characteristic
43	C₂₃×#102	C₂²	1	1	yes	characteristic
44	C₂₃×#97	C₂²	1	1	yes	characteristic
45	#10625017	C₂²	1	1	yes	characteristic
46	#10625014	C₂²	1	1	yes	characteristic
47	#10625003	C₂²	1	1	yes	characteristic
48	#10625003	C₂²	1	1	yes	characteristic
49	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#10624999	C₂²	1	1	yes	characteristic
52	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
54	#10624991	C₂²	1	1	yes	characteristic
55	#10624991	C₂²	1	1	yes	characteristic
56	#10624990	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	#10624985	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂×C₂²⋊D₂₃	—	2	1	no	—
63	C₂×C₂²⋊D₂₃	—	2	1	no	—
64	C₂×C₂²⋊D₂₃	—	2	1	no	—
65	#67292	—	2	2	no	—
66	#67292	—	2	2	no	—
67	#67288	—	4	4	no	—
68	C₂×D₉₂	—	2	1	no	—
69	C₂×D₉₂	—	2	1	no	—
70	C₂×D₉₂	C₂³	1	1	yes	characteristic
71	C₂×D₉₂	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₂₃	—	2	1	no	—
73	C₂×C₄×D₂₃	—	2	1	no	—
74	C₂×C₄×D₂₃	—	2	1	no	—
75	C₂×C₄×D₂₃	—	2	1	no	—
76	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
77	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
78	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₄₆	—	2	1	no	—
80	SD₁₆×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	C₂³	1	1	yes	characteristic
89	C₂²⋊Dic₂₃	—	2	1	no	—
90	#67274	—	2	2	no	—
91	#67272	—	2	2	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	(C₂×D₂₃)⋊C₄	—	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	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	—	2	1	no	—
103	C₄×Dic₂₃	D₄	1	1	yes	characteristic
104	C₄×Dic₂₃	D₄	1	1	yes	characteristic
105	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
108	D₁₈₄	—	2	2	no	—
109	C₂³×C₂₃	—	2	1	no	—
110	C₂²×D₂₃	—	2	1	no	—
111	C₂²×D₂₃	—	2	1	no	—
112	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₂₃	—	2	1	no	—
114	Q₈×C₂₃	#41	1	2	yes	complemented
115	D₄×C₂₃	—	2	1	no	—
116	D₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₂₃	—	4	2	no	—
125	C₂²⋊D₂₃	—	4	2	no	—
126	C₂²⋊D₂₃	—	4	2	no	—
127	C₂×Dic₂₃	—	2	1	no	—
128	C₂×Dic₂₃	—	2	1	no	—
129	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
132	D₉₂	—	2	4	no	—
133	D₉₂	—	2	1	no	—
134	D₉₂	—	2	1	no	—
135	D₉₂	—	2	1	no	—
136	D₉₂	—	4	2	no	—
137	D₉₂	#41	1	2	yes	complemented
138	C₄×D₂₃	—	2	4	no	—
139	C₄×D₂₃	—	2	1	no	—
140	C₄×D₂₃	—	2	1	no	—
141	C₄×D₂₃	—	2	2	no	—
142	C₄×D₂₃	—	4	1	no	—
143	C₄×D₂₃	—	4	2	no	—
144	C₄×D₂₃	—	4	2	no	—
145	C₄×D₂₃	—	4	2	no	—
146	Dic₄₆	—	4	2	no	—
147	C₈×C₂₃	—	2	1	no	—
148	C₈×C₂₃	—	2	1	no	—
149	C₂₃⋊C₈	—	2	1	no	—
150	C₂₃⋊C₈	—	2	1	no	—
151	#3115	—	23	1	no	maximal, Sylow
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	—	4	1	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₂₃	—	2	1	no	—
157	C₂×D₂₃	—	2	2	no	—
158	C₂×D₂₃	—	2	1	no	—
159	C₂×D₂₃	—	2	1	no	—
160	C₂×D₂₃	—	4	1	no	—
161	C₂×D₂₃	—	4	1	no	—
162	C₂×D₂₃	—	4	1	no	—
163	C₂×D₂₃	—	4	1	no	—
164	C₂×D₂₃	—	4	1	no	—
165	C₄×C₂₃	—	2	2	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	—	4	1	no	—
170	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
171	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
172	Dic₂₃	—	2	1	no	—
173	Dic₂₃	—	2	1	no	—
174	Dic₂₃	—	4	1	no	—
175	Dic₂₃	—	4	1	no	—
176	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	—
177	#532	—	23	1	no	—
178	C₄×#41	—	23	1	no	—
179	#482	—	23	1	no	—
180	#482	—	23	1	no	—
181	#475	—	23	1	no	—
182	#474	—	23	1	no	—
183	#460	—	23	1	no	—
184	#459	—	23	1	no	—
185	#449	—	23	1	no	—
186	#449	—	23	1	no	—
187	#442	—	23	1	no	—
188	#442	—	23	1	no	—
189	#440	—	23	1	no	—
190	#440	D₂₃	1	1	yes	characteristic, complemented
191	#432	—	23	1	no	—
192	C₂×C₂₃	—	8	1	no	—
193	C₂×C₂₃	#525	1	1	yes	characteristic
194	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂×C₂₃	#575	1	1	yes	characteristic
196	D₂₃	—	4	1	no	—
197	D₂₃	—	4	2	no	—
198	D₂₃	—	8	1	no	—
199	C₂×#41	—	23	1	no	—
200	C₂×#41	—	46	1	no	—
201	C₂×SD₁₆	—	23	1	no	—
202	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
203	C₂×D₈	—	23	1	no	—
204	C₂×D₈	—	23	1	no	—
205	C₄⋊Q₈	—	23	1	no	—
206	#127	—	23	1	no	—
207	#124	—	46	1	no	—
208	#121	—	23	1	no	—
209	#121	—	23	1	no	—
210	#121	—	46	1	no	—
211	C₄×Q₈	—	23	1	no	—
212	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
213	C₄×D₄	—	23	1	no	—
214	C₄×D₄	—	23	1	no	—
215	C₄×D₄	—	23	1	no	—
216	C₄×D₄	—	46	1	no	—
217	C₄×D₄	—	46	1	no	—
218	C₄×D₄	—	46	1	no	—
219	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
220	#117	—	23	1	no	—
221	#117	—	46	1	no	—
222	C₂×C₄²	—	23	1	no	—
223	C₂×C₄²	—	46	1	no	—
224	#107	—	23	1	no	—
225	#107	C₂×D₂₃	1	1	yes	characteristic
226	#106	—	23	1	no	—
227	#106	—	23	1	no	—
228	C₄⋊C₈	—	23	1	no	—
229	C₄⋊C₈	—	23	1	no	—
230	#103	—	23	1	no	—
231	#103	C₂×D₂₃	1	1	yes	characteristic
232	#102	—	23	1	no	—
233	#102	—	23	1	no	—
234	#102	—	23	1	no	—
235	#102	—	23	1	no	—
236	#102	—	23	1	no	—
237	#102	C₂×D₂₃	1	1	yes	characteristic
238	C₂²⋊C₈	—	23	1	no	—
239	C₂²⋊C₈	—	23	1	no	—
240	#97	—	23	1	no	—
241	#97	C₂×D₂₃	1	1	yes	characteristic
242	C₂₃	#3115	1	1	yes	characteristic, complemented, Sylow
243	#41	—	46	2	no	—
244	#41	—	46	2	no	—
245	#41	—	92	4	no	—
246	C₂×Q₈	—	46	1	no	—
247	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
248	C₂×D₄	—	23	1	no	—
249	C₂×D₄	—	23	1	no	—
250	C₂×D₄	—	46	1	no	—
251	C₂×D₄	—	46	1	no	—
252	C₂×D₄	—	46	1	no	—
253	C₂×D₄	—	46	1	no	—
254	C₂×D₄	—	46	1	no	—
255	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	23	1	no	—
258	C₂²×C₄	—	23	1	no	—
259	C₂²×C₄	—	23	1	no	—
260	C₂²×C₄	—	46	1	no	—
261	C₂²×C₄	—	46	1	no	—
262	C₂²×C₄	—	46	1	no	—
263	C₂²×C₄	—	46	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	46	2	no	—
266	D₈	—	46	2	no	—
267	D₈	—	46	2	no	—
268	C₂×C₈	—	23	1	no	—
269	C₂×C₈	—	23	1	no	—
270	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	23	1	no	—
274	C₄⋊C₄	—	23	1	no	—
275	C₄⋊C₄	—	46	1	no	—
276	C₄⋊C₄	—	46	1	no	—
277	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	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	—	23	1	no	—
288	C₄²	—	23	1	no	—
289	C₄²	—	23	1	no	—
290	C₄²	—	46	1	no	—
291	C₄²	—	46	1	no	—
292	C₄²	C₂²×D₂₃	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	23	1	no	—
295	C₂³	—	46	1	no	—
296	C₂³	—	46	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	92	2	no	—
299	Q₈	#67288	1	2	yes	complemented
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	23	2	no	—
303	D₄	—	46	4	no	—
304	D₄	—	46	1	no	—
305	D₄	—	46	1	no	—
306	D₄	—	46	1	no	—
307	D₄	—	92	2	no	—
308	D₄	—	92	2	no	—
309	D₄	—	92	2	no	—
310	D₄	—	92	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	23	1	no	—
316	C₂×C₄	—	23	1	no	—
317	C₂×C₄	—	23	1	no	—
318	C₂×C₄	—	46	2	no	—
319	C₂×C₄	—	46	4	no	—
320	C₂×C₄	—	46	1	no	—
321	C₂×C₄	—	46	1	no	—
322	C₂×C₄	—	46	1	no	—
323	C₂×C₄	—	46	1	no	—
324	C₂×C₄	—	92	1	no	—
325	C₂×C₄	—	92	2	no	—
326	C₂×C₄	—	92	2	no	—
327	C₂×C₄	—	92	2	no	—
328	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
329	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	46	1	no	—
334	C₈	—	46	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	46	2	no	—
339	C₂²	—	46	1	no	—
340	C₂²	—	46	1	no	—
341	C₂²	—	46	1	no	—
342	C₂²	—	92	1	no	—
343	C₂²	—	92	1	no	—
344	C₂²	—	92	1	no	—
345	C₂²	—	92	1	no	—
346	C₂²	—	92	1	no	—
347	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	23	2	no	—
354	C₄	—	46	1	no	—
355	C₄	—	46	1	no	—
356	C₄	—	92	1	no	—
357	C₄	—	92	1	no	—
358	C₄	C₂×#67288	1	1	yes	characteristic
359	C₄	#41×D₂₃	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	92	1	no	—
362	C₂	—	92	2	no	—
363	C₂	—	184	1	no	—
364	C₂	#13092926	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
366	C₂	#13093216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#440	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#440	D₂₃
hypocenter	C₂₃	#3115
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #13092964	C₂	semidirect
2. C₄×#67292	C₂	semidirect
3. #13092744	C₂	semidirect
4. #13092740	C₂	semidirect
5. #13092700	C₂	semidirect
6. #13092696	C₂	semidirect
7. #13092615	C₂	semidirect
8. #13092611	C₂	semidirect
9. #13092565	C₂	semidirect
10. #13092561	C₂	semidirect
11. #13092530	C₂	semidirect
12. #13092527	C₂	semidirect
13. #13092515	C₂	semidirect
14. #13092472	C₂	semidirect
15. C₂₃×#440	C₂	semidirect
114. Q₈×C₂₃	#41	semidirect
137. D₉₂	#41	semidirect
190. #440	D₂₃	semidirect
242. C₂₃	#3115	semidirect
299. Q₈	#67288	semidirect

Maximal subgroups

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

subgroup	index
#13092964	2
C₄×#67292	2
#13092744	2
#13092740	2
#13092700	2
#13092696	2
#13092615	2
#13092611	2
#13092565	2
#13092561	2
#13092530	2
#13092527	2
#13092515	2
#13092472	2
C₂₃×#440	2
#3115	23