Subgroups — #26373999

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	#20256253	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245628	C₂	1	2	yes	maximal, complemented
4	#20245611	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245547	C₂	1	2	yes	maximal, complemented
6	#20245536	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245421	C₂	1	2	yes	maximal, complemented
8	#20245293	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245264	C₂	1	2	yes	maximal, complemented
10	C₄×#13093965	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2206	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689830	—	2	1	no	—
13	#15689793	—	2	1	no	—
14	C₂×#13093965	C₂²	1	1	yes	characteristic
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689524	C₂²	1	2	yes	—
17	#15689498	C₂²	1	2	yes	—
18	#15689421	—	2	2	no	—
19	#15689413	—	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	D₈×Dic₄₇	C₂²	1	2	yes	—
25	D₈×Dic₄₇	C₂²	1	2	yes	—
26	#15688951	C₂²	1	2	yes	—
27	#15688948	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
29	#15688925	C₂²	1	2	yes	—
30	#15688924	C₂²	1	1	yes	characteristic
31	#15688924	C₂²	1	2	yes	—
32	#15688924	C₂²	1	2	yes	—
33	#15688923	C₂²	1	1	yes	characteristic
34	#15688898	C₂²	1	2	yes	—
35	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#493	C₂²	1	1	yes	characteristic
38	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄₇×#357	C₂²	1	2	yes	—
41	C₂×#10625352	—	4	1	no	—
42	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
43	C₂×#10625334	—	2	2	no	—
44	#13094024	—	2	1	no	—
45	#13094020	—	2	2	no	—
46	#13094015	—	2	2	no	—
47	#13094003	—	4	1	no	—
48	#13094001	—	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	D₄×Dic₄₇	—	2	2	no	—
54	D₄×Dic₄₇	—	2	2	no	—
55	C₂×C₄⋊Dic₄₇	—	4	1	no	—
56	#13093965	—	2	2	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	#13093958	—	2	1	no	—
60	#13093958	C₂³	1	2	yes	—
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	#13093945	—	2	2	no	—
67	#13093945	—	2	2	no	—
68	C₈×Dic₄₇	C₂³	1	2	yes	—
69	C₂×D₈×C₄₇	—	2	1	no	—
70	C₂×D₈×C₄₇	C₂³	1	2	yes	—
71	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
72	#127×C₄₇	—	2	1	no	—
73	C₄×D₄×C₄₇	—	2	1	no	—
74	D₁₆×C₄₇	—	2	4	no	—
75	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	#102×C₄₇	—	2	1	no	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₇	—	2	1	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	4	1	no	—
82	C₂×D₄×C₄₇	—	4	1	no	—
83	C₂²×C₄×C₄₇	—	4	1	no	—
84	C₂×C₂²⋊D₄₇	—	4	2	no	—
85	C₂×C₂²⋊D₄₇	—	4	2	no	—
86	C₂²×Dic₄₇	—	4	2	no	—
87	C₂²×Dic₄₇	—	4	2	no	—
88	#10625352	—	8	4	no	—
89	C₂×D₁₈₈	—	2	1	no	—
90	C₂×C₄×D₄₇	—	4	1	no	—
91	C₂×Dic₉₄	—	4	1	no	—
92	C₂×Dic₉₄	D₈	1	1	yes	characteristic
93	D₈×C₄₇	—	2	2	no	—
94	D₈×C₄₇	—	2	1	no	—
95	D₈×C₄₇	—	2	2	no	—
96	D₈×C₄₇	—	8	1	no	—
97	D₈×C₄₇	#41	1	2	yes	complemented
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₄₇	—	2	1	no	—
101	C₂²⋊C₄×C₄₇	—	4	1	no	—
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂²⋊Dic₄₇	—	4	2	no	—
104	C₂²⋊Dic₄₇	—	4	2	no	—
105	#10625334	—	8	2	no	—
106	(C₂×D₄₇)⋊C₄	—	4	1	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	4	1	no	—
110	C₄⋊Dic₄₇	—	4	2	no	—
111	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
112	Dic₄₇⋊C₄	—	4	2	no	—
113	Dic₄₇⋊C₄	D₈	1	2	yes	—
114	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
115	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
116	D₃₇₆	—	2	1	no	—
117	D₃₇₆	—	2	1	no	—
118	C₁₆×C₄₇	—	2	2	no	—
119	C₄₇⋊C₁₆	—	2	1	no	—
120	C₄₇⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₇	—	4	1	no	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂²×D₄₇	—	4	1	no	—
124	D₄×C₄₇	—	2	2	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	4	1	no	—
127	D₄×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	8	2	no	—
129	D₄×C₄₇	—	16	1	no	—
130	C₂×C₄×C₄₇	—	4	1	no	—
131	C₂×C₄×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₄₇	—	8	4	no	—
136	C₂²⋊D₄₇	—	16	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	1	no	—
139	C₂×Dic₄₇	—	8	4	no	—
140	C₂×Dic₄₇	—	16	2	no	—
141	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
142	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
143	D₁₈₈	—	4	1	no	—
144	D₁₈₈	—	4	1	no	—
145	C₄×D₄₇	—	8	2	no	—
146	Dic₉₄	—	2	2	no	—
147	Dic₉₄	—	16	1	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	—	4	2	no	—
154	C₂²×C₄₇	—	8	1	no	—
155	C₂²×C₄₇	—	8	1	no	—
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	8	2	no	—
159	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
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	1	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	2	no	—
172	#3391	—	47	1	no	—
173	#3380	—	47	1	no	—
174	#2244	—	47	2	no	—
175	#2240	—	47	1	no	—
176	#2228	—	47	2	no	—
177	#2226	—	47	1	no	—
178	#2206	—	47	2	no	—
179	#2206	D₄₇	1	1	yes	characteristic, complemented
180	#2183	—	47	1	no	—
181	#2177	—	47	2	no	—
182	C₄×D₁₆	—	47	1	no	—
183	C₂×C₄₇	—	8	2	no	—
184	C₂×C₄₇	—	16	2	no	—
185	C₂×C₄₇	#3358	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
187	C₂×C₄₇	#3415	1	1	yes	characteristic
188	D₄₇	—	16	1	no	—
189	#554	—	94	1	no	—
190	#550	—	94	1	no	—
191	C₂×D₁₆	—	47	1	no	—
192	C₂×D₁₆	C₂×D₄₇	1	2	yes	—
193	#500	—	47	1	no	—
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	C₂×D₄₇	1	1	yes	characteristic
206	#367	—	47	2	no	—
207	#366	—	47	1	no	—
208	C₄⋊C₁₆	—	47	2	no	—
209	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
210	#357	—	47	1	no	—
211	#357	—	47	2	no	—
212	#357	—	47	2	no	—
213	#357	—	47	2	no	—
214	#357	—	47	1	no	—
215	#357	C₂×D₄₇	1	2	yes	—
216	#346	—	47	2	no	—
217	C₄×C₁₆	—	47	1	no	—
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	—	94	2	no	—
223	C₂×D₈	C₂²×D₄₇	1	2	yes	—
224	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	94	1	no	—
226	#127	—	2	1	no	—
227	#127	—	94	2	no	—
228	#125	—	94	2	no	—
229	C₂²⋊Q₈	—	188	2	no	—
230	C₂²⋊Q₈	—	188	1	no	—
231	#121	—	188	1	no	—
232	#121	—	188	2	no	—
233	C₄×Q₈	—	47	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	94	2	no	—
236	C₄×D₄	—	94	2	no	—
237	C₄×D₄	—	94	1	no	—
238	C₂×C₄⋊C₄	—	188	1	no	—
239	D₁₆	—	2	4	no	—
240	D₁₆	—	94	2	no	—
241	C₂×C₁₆	—	47	1	no	—
242	C₂×C₁₆	—	47	1	no	—
243	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
244	#107	—	47	2	no	—
245	#107	—	47	1	no	—
246	#107	—	94	1	no	—
247	#107	C₂²×D₄₇	1	1	yes	characteristic
248	#106	—	94	2	no	—
249	C₄⋊C₈	—	47	2	no	—
250	C₄⋊C₈	—	47	1	no	—
251	#102	—	2	1	no	—
252	#102	—	94	2	no	—
253	#102	—	94	2	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	—	47	1	no	—
259	C₂×Q₈	—	188	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	94	1	no	—
265	C₂×D₄	—	188	2	no	—
266	C₂×D₄	—	188	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	188	2	no	—
269	C₂²×C₄	—	188	2	no	—
270	C₂²×C₄	—	188	1	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	2	no	—
274	D₈	—	8	1	no	—
275	D₈	—	94	1	no	—
276	D₈	—	94	1	no	—
277	D₈	—	376	2	no	—
278	D₈	#10625352	1	2	yes	complemented
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	1	no	—
283	C₄⋊C₄	—	47	2	no	—
284	C₄⋊C₄	—	47	1	no	—
285	C₄⋊C₄	—	94	1	no	—
286	C₄⋊C₄	—	94	2	no	—
287	C₄⋊C₄	—	188	2	no	—
288	C₄⋊C₄	—	188	2	no	—
289	C₄⋊C₄	—	188	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	188	2	no	—
292	C₂²⋊C₄	—	188	2	no	—
293	C₂²⋊C₄	—	188	1	no	—
294	C₄²	—	47	2	no	—
295	C₄²	D₄×D₄₇	1	1	yes	characteristic
296	C₁₆	—	2	2	no	—
297	C₁₆	—	94	1	no	—
298	C₁₆	—	94	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	188	1	no	—
302	Q₈	—	94	2	no	—
303	Q₈	—	752	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	1	no	—
310	D₄	—	376	4	no	—
311	D₄	—	188	1	no	—
312	D₄	—	188	1	no	—
313	D₄	—	752	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	47	2	no	—
317	C₂×C₄	—	47	2	no	—
318	C₂×C₄	—	376	4	no	—
319	C₂×C₄	—	376	2	no	—
320	C₂×C₄	—	188	1	no	—
321	C₂×C₄	—	188	2	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	C₂×#10625352	1	1	yes	characteristic
329	C₈	#13094071	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	376	1	no	—
337	C₂²	—	376	1	no	—
338	C₂²	—	376	1	no	—
339	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	94	2	no	—
343	C₄	—	94	2	no	—
344	C₄	—	376	2	no	—
345	C₄	—	376	1	no	—
346	C₄	#15689557	1	2	yes	—
347	C₄	#15689779	1	1	yes	characteristic
348	C₄	#15689933	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	752	1	no	—
352	C₂	#20256018	1	1	yes	characteristic
353	C₂	C₂×#15689557	1	1	yes	characteristic
354	C₂	#20256528	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×#2206	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#2206	D₄₇
hypocenter	C₄₇	#33933
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2206

Complements

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

N	H ≅ G/N	type
1. #20256361	C₂	semidirect
2. #20256253	C₂	semidirect
3. #20245628	C₂	semidirect
4. #20245611	C₂	semidirect
5. #20245547	C₂	semidirect
6. #20245536	C₂	semidirect
7. #20245421	C₂	semidirect
8. #20245293	C₂	semidirect
9. #20245264	C₂	semidirect
10. C₄×#13093965	C₂	semidirect
11. C₄₇×#2206	C₂	semidirect
97. D₈×C₄₇	#41	semidirect
179. #2206	D₄₇	semidirect
218. C₄₇	#33933	semidirect
278. D₈	#10625352	semidirect

Maximal subgroups

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

subgroup	index
#20256361	2
#20256253	2
#20245628	2
#20245611	2
#20245547	2
#20245536	2
#20245421	2
#20245293	2
#20245264	2
C₄×#13093965	2
C₄₇×#2206	2
#33933	47