Subgroups — #26374000

Order 12032.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256361	C₂	1	1	yes	maximal, characteristic, complemented
2	#20256257	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245628	C₂	1	2	yes	maximal, complemented
4	#20245543	C₂	1	2	yes	maximal, complemented
5	#20245532	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245422	C₂	1	2	yes	maximal, complemented
7	#20245420	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245296	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245261	C₂	1	2	yes	maximal, complemented
10	C₄×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689834	—	2	1	no	—
13	#15689793	—	2	1	no	—
14	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
15	C₂×#13093965	C₂²	1	2	yes	—
16	#15689524	C₂²	1	2	yes	—
17	#15689498	C₂²	1	2	yes	—
18	#15689426	—	2	2	no	—
19	#15689410	—	2	1	no	—
20	#15689316	—	2	2	no	—
21	#15689315	—	2	1	no	—
22	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
23	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
24	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
25	D₈×Dic₄₇	C₂²	1	2	yes	—
26	#15689205	C₂²	1	2	yes	—
27	#15688950	C₂²	1	2	yes	—
28	#15688949	C₂²	1	1	yes	characteristic
29	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
30	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
31	#15688925	C₂²	1	1	yes	characteristic
32	#15688925	C₂²	1	1	yes	characteristic
33	#15688924	C₂²	1	2	yes	—
34	#15688923	C₂²	1	2	yes	—
35	#15688898	C₂²	1	2	yes	—
36	C₄₇×#500	C₂²	1	1	yes	characteristic
37	C₄₇×#493	C₂²	1	1	yes	characteristic
38	C₄₇×#357	C₂²	1	2	yes	—
39	C₄₇×#357	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625352	—	4	1	no	—
42	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
43	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
44	C₂×#10625334	—	2	2	no	—
45	#13094020	—	2	2	no	—
46	#13094015	—	2	2	no	—
47	#13094004	—	4	1	no	—
48	#13094000	—	4	2	no	—
49	#13093996	—	4	2	no	—
50	#13093995	—	4	1	no	—
51	C₄×Dic₉₄	D₄	1	1	yes	characteristic
52	C₄×D₁₈₈	—	2	1	no	—
53	C₄×D₁₈₈	—	2	1	no	—
54	D₄×Dic₄₇	—	2	2	no	—
55	#13093986	—	2	2	no	—
56	C₄⋊C₄×D₄₇	—	4	1	no	—
57	D₇₅₂	—	2	2	no	—
58	#13093965	—	2	4	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
60	#13093958	C₂³	1	1	yes	characteristic
61	#13093958	C₂³	1	1	yes	characteristic
62	#13093955	—	2	2	no	—
63	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
64	Dic₄₇⋊C₈	D₄	1	2	yes	—
65	#13093946	—	2	1	no	—
66	#13093946	—	2	1	no	—
67	#13093945	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	C₈×Dic₄₇	C₂³	1	2	yes	—
70	C₂×D₈×C₄₇	—	2	1	no	—
71	C₂×D₈×C₄₇	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₄₇	—	2	1	no	—
73	#127×C₄₇	—	2	1	no	—
74	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	#107×C₄₇	—	2	1	no	—
77	#107×C₄₇	C₂³	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	4	1	no	—
82	C₂×D₄×C₄₇	—	4	1	no	—
83	C₂×C₂²⋊D₄₇	—	4	2	no	—
84	C₂×C₂²⋊D₄₇	—	4	2	no	—
85	C₂²×Dic₄₇	—	4	2	no	—
86	#10625352	—	8	4	no	—
87	C₂×D₁₈₈	—	2	1	no	—
88	C₂×D₁₈₈	—	2	1	no	—
89	C₂×C₄×D₄₇	—	4	1	no	—
90	C₂×C₄×D₄₇	—	4	1	no	—
91	C₂×C₄×D₄₇	—	4	2	no	—
92	C₂×Dic₉₄	D₈	1	1	yes	characteristic
93	D₈×C₄₇	—	2	2	no	—
94	D₈×C₄₇	—	2	2	no	—
95	D₈×C₄₇	—	8	1	no	—
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊Dic₄₇	—	4	2	no	—
102	#10625334	—	8	2	no	—
103	(C₂×D₄₇)⋊C₄	—	4	1	no	—
104	(C₂×D₄₇)⋊C₄	—	4	1	no	—
105	(C₂×D₄₇)⋊C₄	—	4	2	no	—
106	C₄⋊Dic₄₇	—	2	1	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	4	2	no	—
109	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
110	Dic₄₇⋊C₄	—	4	2	no	—
111	Dic₄₇⋊C₄	D₈	1	2	yes	—
112	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
113	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
114	D₃₇₆	—	2	1	no	—
115	D₃₇₆	—	2	1	no	—
116	D₃₇₆	—	2	1	no	—
117	D₃₇₆	#41	1	2	yes	complemented
118	C₁₆×C₄₇	—	2	1	no	—
119	C₁₆×C₄₇	—	2	1	no	—
120	C₄₇⋊C₁₆	—	2	2	no	—
121	C₂³×C₄₇	—	4	2	no	—
122	C₂²×D₄₇	—	4	1	no	—
123	C₂²×D₄₇	—	4	1	no	—
124	Q₈×C₄₇	—	16	1	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	4	2	no	—
127	D₄×C₄₇	—	8	2	no	—
128	D₄×C₄₇	—	16	1	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₄₇	—	8	4	no	—
134	C₂²⋊D₄₇	—	16	2	no	—
135	C₂×Dic₄₇	—	4	1	no	—
136	C₂×Dic₄₇	—	4	1	no	—
137	C₂×Dic₄₇	—	8	4	no	—
138	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
139	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
140	D₁₈₈	—	2	2	no	—
141	D₁₈₈	—	4	1	no	—
142	D₁₈₈	—	4	1	no	—
143	D₁₈₈	—	4	1	no	—
144	C₄×D₄₇	—	8	2	no	—
145	C₄×D₄₇	—	8	2	no	—
146	C₄×D₄₇	—	16	2	no	—
147	Dic₉₄	—	2	2	no	—
148	C₈×C₄₇	—	2	1	no	—
149	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₄₇⋊C₈	—	2	2	no	—
152	#33933	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	—	8	2	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₄₇	—	4	2	no	—
158	C₂×D₄₇	—	8	1	no	—
159	C₂×D₄₇	—	8	1	no	—
160	C₂×D₄₇	—	8	1	no	—
161	C₂×D₄₇	—	8	1	no	—
162	C₂×D₄₇	—	8	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#509	1	2	yes	—
166	C₄×C₄₇	#548	1	1	yes	characteristic
167	C₄×C₄₇	#577	1	1	yes	characteristic
168	Dic₄₇	—	2	2	no	—
169	Dic₄₇	—	2	2	no	—
170	Dic₄₇	—	8	1	no	—
171	Dic₄₇	—	8	1	no	—
172	#3391	—	47	1	no	—
173	#3380	—	47	1	no	—
174	#2244	—	47	2	no	—
175	#2240	D₄₇	1	1	yes	characteristic, complemented
176	#2228	—	47	2	no	—
177	#2226	—	47	1	no	—
178	#2206	—	47	1	no	—
179	#2206	—	47	2	no	—
180	#2183	—	47	1	no	—
181	#2177	—	47	2	no	—
182	C₄×D₁₆	—	47	1	no	—
183	C₂×C₄₇	—	16	2	no	—
184	C₂×C₄₇	#3358	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
186	C₂×C₄₇	#3415	1	1	yes	characteristic
187	D₄₇	—	8	2	no	—
188	D₄₇	—	16	1	no	—
189	#554	—	94	1	no	—
190	#550	—	94	1	no	—
191	C₂×D₁₆	—	47	2	no	—
192	C₂×D₁₆	—	47	1	no	—
193	#500	C₂×D₄₇	1	1	yes	characteristic
194	#499	—	47	2	no	—
195	#493	—	47	2	no	—
196	#493	C₂×D₄₇	1	1	yes	characteristic
197	#478	—	94	2	no	—
198	#476	—	94	1	no	—
199	#459	—	94	1	no	—
200	#459	—	94	2	no	—
201	C₈×Q₈	—	47	1	no	—
202	C₄×D₈	—	47	2	no	—
203	C₄×D₈	—	47	2	no	—
204	C₄×D₈	—	47	1	no	—
205	C₄×D₈	—	47	1	no	—
206	#367	—	47	2	no	—
207	#366	—	47	1	no	—
208	C₄⋊C₁₆	—	47	2	no	—
209	C₄⋊C₁₆	—	47	1	no	—
210	#357	—	47	2	no	—
211	#357	—	47	1	no	—
212	#357	—	47	1	no	—
213	#357	—	47	2	no	—
214	#357	C₂×D₄₇	1	2	yes	—
215	#357	C₂×D₄₇	1	2	yes	—
216	#346	—	47	2	no	—
217	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
218	C₄₇	#33933	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	188	1	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	47	1	no	—
222	C₂×D₈	—	47	1	no	—
223	C₂×D₈	—	94	2	no	—
224	C₂×D₈	C₂²×D₄₇	1	2	yes	—
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	2	1	no	—
227	#127	—	94	2	no	—
228	#125	—	94	2	no	—
229	C₂²⋊Q₈	—	188	1	no	—
230	C₂²⋊Q₈	—	188	2	no	—
231	#121	—	188	1	no	—
232	#121	—	188	2	no	—
233	C₄×Q₈	—	47	1	no	—
234	C₄×D₄	—	94	2	no	—
235	C₄×D₄	—	94	2	no	—
236	C₄×D₄	—	94	1	no	—
237	C₄×D₄	—	94	1	no	—
238	C₂×C₄⋊C₄	—	188	1	no	—
239	D₁₆	—	94	2	no	—
240	D₁₆	—	94	4	no	—
241	C₂×C₁₆	—	47	2	no	—
242	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
244	#107	—	2	1	no	—
245	#107	—	47	1	no	—
246	#107	—	47	1	no	—
247	#107	C₂²×D₄₇	1	2	yes	—
248	#106	—	94	2	no	—
249	C₄⋊C₈	—	47	2	no	—
250	C₄⋊C₈	—	47	1	no	—
251	#102	—	94	2	no	—
252	#102	—	94	2	no	—
253	#102	—	94	1	no	—
254	#102	—	94	1	no	—
255	C₄×C₈	—	47	2	no	—
256	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
257	#41	—	376	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	47	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	94	1	no	—
264	C₂×D₄	—	94	1	no	—
265	C₂×D₄	—	188	2	no	—
266	C₂×D₄	—	188	2	no	—
267	C₂²×C₄	—	188	2	no	—
268	C₂²×C₄	—	188	2	no	—
269	C₂²×C₄	—	188	1	no	—
270	C₂²×C₄	—	188	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	8	1	no	—
274	D₈	—	47	2	no	—
275	D₈	—	94	1	no	—
276	D₈	—	94	1	no	—
277	D₈	—	94	1	no	—
278	D₈	—	376	2	no	—
279	C₂×C₈	—	47	2	no	—
280	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	47	2	no	—
285	C₄⋊C₄	—	47	1	no	—
286	C₄⋊C₄	—	94	1	no	—
287	C₄⋊C₄	—	94	1	no	—
288	C₄⋊C₄	—	188	2	no	—
289	C₄⋊C₄	—	188	2	no	—
290	C₂²⋊C₄	—	188	2	no	—
291	C₂²⋊C₄	—	188	2	no	—
292	C₂²⋊C₄	—	188	1	no	—
293	C₂²⋊C₄	—	188	1	no	—
294	C₄²	—	47	2	no	—
295	C₄²	D₄×D₄₇	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	94	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	188	1	no	—
301	C₂³	—	188	1	no	—
302	Q₈	—	16	1	no	—
303	Q₈	—	94	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	8	2	no	—
307	D₄	—	16	1	no	—
308	D₄	—	94	2	no	—
309	D₄	—	376	4	no	—
310	D₄	—	188	1	no	—
311	D₄	—	188	1	no	—
312	D₄	—	188	1	no	—
313	D₄	—	752	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	47	2	no	—
316	C₂×C₄	—	47	2	no	—
317	C₂×C₄	—	376	4	no	—
318	C₂×C₄	—	376	2	no	—
319	C₂×C₄	—	376	2	no	—
320	C₂×C₄	—	188	1	no	—
321	C₂×C₄	—	188	1	no	—
322	C₂×C₄	—	752	2	no	—
323	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
324	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	94	2	no	—
328	C₈	#41×D₄₇	1	1	yes	characteristic
329	C₈	#13094071	1	1	yes	characteristic
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	376	1	no	—
334	C₂²	—	376	1	no	—
335	C₂²	—	376	1	no	—
336	C₂²	—	376	1	no	—
337	C₂²	—	376	1	no	—
338	C₂²	—	188	2	no	—
339	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	94	2	no	—
343	C₄	—	94	2	no	—
344	C₄	—	376	1	no	—
345	C₄	—	376	1	no	—
346	C₄	#15689560	1	2	yes	—
347	C₄	#15689781	1	1	yes	characteristic
348	C₄	#15689929	1	1	yes	characteristic
349	C₂	—	16	2	no	—
350	C₂	—	376	2	no	—
351	C₂	—	752	1	no	—
352	C₂	#20256020	1	1	yes	characteristic
353	C₂	C₂×#15689560	1	1	yes	characteristic
354	C₂	#20256525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2240	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#2240	D₄₇
hypocenter	C₄₇	#33933
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #20256361	C₂	semidirect
2. #20256257	C₂	semidirect
3. #20245628	C₂	semidirect
4. #20245543	C₂	semidirect
5. #20245532	C₂	semidirect
6. #20245422	C₂	semidirect
7. #20245420	C₂	semidirect
8. #20245296	C₂	semidirect
9. #20245261	C₂	semidirect
10. C₄×D₇₅₂	C₂	semidirect
11. C₄₇×#2240	C₂	semidirect
117. D₃₇₆	#41	semidirect
175. #2240	D₄₇	semidirect
218. C₄₇	#33933	semidirect

Maximal subgroups

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

subgroup	index
#20256361	2
#20256257	2
#20245628	2
#20245543	2
#20245532	2
#20245422	2
#20245420	2
#20245296	2
#20245261	2
C₄×D₇₅₂	2
C₄₇×#2240	2
#33933	47