Subgroups — #26371455

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254047	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252241	C₂	1	1	yes	maximal, characteristic
3	#20245597	C₂	1	2	yes	maximal, complemented
4	#20245546	C₂	1	2	yes	maximal, complemented
5	#20245434	C₂	1	2	yes	maximal, complemented
6	#20245319	C₂	1	2	yes	maximal
7	#20245254	C₂	1	2	yes	maximal, complemented
8	C₄×#13093969	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689735	—	2	1	no	—
12	#15689597	—	2	1	no	—
13	C₂×#13093969	C₂²	1	2	yes	—
14	C₂×#13093969	C₂²	1	2	yes	—
15	#15689525	C₂²	1	1	yes	characteristic, complemented
16	#15689503	C₂²	1	1	yes	characteristic, complemented
17	#15689492	—	2	1	no	—
18	#15689381	—	2	1	no	—
19	#15689364	C₂²	1	2	yes	—
20	#15689236	—	2	1	no	—
21	#15689236	—	2	1	no	—
22	#15689231	—	2	1	no	—
23	C₄×Dic₁₈₈	C₂²	1	2	yes	complemented
24	C₄×D₃₇₆	C₂²	1	2	yes	—
25	C₄×D₃₇₆	C₂²	1	2	yes	—
26	#15689161	C₂²	1	1	yes	characteristic
27	#15689129	—	2	1	no	—
28	#15688951	C₂²	1	2	yes	complemented
29	#15688949	C₂²	1	2	yes	complemented
30	#15688928	C₂²	1	2	yes	complemented
31	#15688928	C₂²	1	2	yes	complemented
32	#15688925	C₂²	1	2	yes	—
33	#15688925	C₂²	1	2	yes	—
34	C₄₇×#405	C₂²	1	1	yes	characteristic
35	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
36	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₄₇×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	—	2	1	no	—
41	C₂×Dic₁₈₈	C₂³	1	2	yes	—
42	C₂×D₃₇₆	—	2	1	no	—
43	C₂×D₃₇₆	—	2	4	no	—
44	C₂×D₃₇₆	C₂³	1	2	yes	—
45	#13094024	—	2	1	no	—
46	#13094017	—	4	2	no	—
47	#13094016	—	2	1	no	—
48	#13094014	—	4	2	no	—
49	#13094003	—	2	2	no	—
50	C₄×Dic₉₄	—	2	2	no	—
51	C₄×D₁₈₈	—	2	2	no	—
52	C₄×D₁₈₈	—	2	2	no	—
53	#13093986	—	4	2	no	—
54	#13093983	—	2	1	no	—
55	C₂²⋊C₄×D₄₇	—	4	2	no	—
56	#13093969	—	2	4	no	—
57	#13093969	—	2	4	no	—
58	#13093958	C₂³	1	2	yes	—
59	#13093958	C₂³	1	2	yes	—
60	#13093956	—	2	1	no	—
61	#13093956	—	2	1	no	—
62	#13093949	—	2	2	no	—
63	#13093949	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	#13093946	—	2	2	no	—
66	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#117×C₄₇	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
70	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
71	#97×C₄₇	D₄	1	1	yes	characteristic
72	#97×C₄₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂³×D₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×D₁₈₈	—	2	4	no	—
78	C₂×D₁₈₈	—	2	2	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×C₄×D₄₇	—	4	2	no	—
82	C₂×C₄×D₄₇	—	4	2	no	—
83	C₂×Dic₉₄	—	2	2	no	—
84	M₁₆×C₄₇	—	4	1	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
91	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₄₇	—	4	1	no	—
95	C₂²⋊Dic₄₇	—	4	2	no	—
96	(C₂×D₄₇)⋊C₄	—	4	2	no	—
97	(C₂×D₄₇)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	Dic₄₇⋊C₄	—	4	2	no	—
102	Dic₄₇⋊C₄	—	4	2	no	—
103	Dic₄₇⋊C₄	—	4	2	no	—
104	C₄×Dic₄₇	—	4	1	no	—
105	C₄×Dic₄₇	—	4	2	no	—
106	Dic₁₈₈	—	2	2	no	—
107	Dic₁₈₈	—	2	2	no	—
108	Dic₁₈₈	—	4	2	no	—
109	D₃₇₆	—	2	4	no	—
110	D₃₇₆	—	2	2	no	—
111	D₃₇₆	#41	1	4	yes	—
112	C₁₆×C₄₇	—	2	2	no	—
113	C₁₆×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₇	—	4	2	no	—
116	C₂²×D₄₇	—	4	4	no	—
117	C₂²×D₄₇	—	16	2	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	D₁₈₈	—	2	4	no	—
128	D₁₈₈	—	4	4	no	—
129	D₁₈₈	—	4	2	no	—
130	C₄×D₄₇	—	16	2	no	—
131	C₄×D₄₇	—	16	2	no	—
132	Dic₉₄	—	4	2	no	—
133	Dic₉₄	—	4	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	C₂×#41	1	2	yes	—
138	#33683	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₇	—	4	4	no	—
143	C₂×D₄₇	—	8	2	no	—
144	C₂×D₄₇	—	8	2	no	—
145	C₂×D₄₇	—	16	4	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	#525	1	1	yes	characteristic
150	C₄×C₄₇	#575	1	1	yes	characteristic
151	Dic₄₇	—	8	2	no	—
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	#3177	—	47	1	no	—
155	#2948	—	47	1	no	—
156	#2237	—	47	2	no	—
157	#2228	—	47	2	no	—
158	#2208	—	47	2	no	—
159	#2187	—	47	2	no	—
160	#2175	—	47	2	no	—
161	C₄×SD₃₂	—	47	2	no	—
162	#2164	D₄₇	1	1	yes	characteristic, complemented
163	C₂×C₄₇	—	4	1	no	—
164	C₂×C₄₇	#3100	1	1	yes	characteristic
165	C₂×C₄₇	#3416	1	2	yes	—
166	D₄₇	—	8	4	no	—
167	C₂²×D₈	—	47	1	no	—
168	#541	—	94	1	no	—
169	#518	—	94	1	no	—
170	C₂×SD₃₂	—	47	2	no	—
171	C₂×SD₃₂	—	47	2	no	—
172	#499	—	47	1	no	—
173	#494	—	47	1	no	—
174	#491	—	94	1	no	—
175	#470	—	94	1	no	—
176	#467	—	47	2	no	—
177	#442	—	94	1	no	—
178	#442	—	94	1	no	—
179	#441	—	94	1	no	—
180	C₄×Q₁₆	—	47	2	no	—
181	C₄×D₈	—	47	2	no	—
182	C₄×D₈	—	47	2	no	—
183	#427	—	47	1	no	—
184	#418	—	94	1	no	—
185	#405	C₂×D₄₇	1	1	yes	characteristic
186	#367	—	47	2	no	—
187	#366	—	47	2	no	—
188	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	complemented
189	#358	—	47	2	no	—
190	#358	—	47	2	no	—
191	#357	—	47	2	no	—
192	#357	—	47	2	no	—
193	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
194	#346	C₂×D₄₇	1	1	yes	characteristic, complemented
195	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
196	C₄₇	#33683	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	94	1	no	—
198	C₂×Q₁₆	—	47	2	no	—
199	C₂×Q₁₆	—	94	1	no	—
200	C₂×D₈	—	47	2	no	—
201	C₂×D₈	—	94	4	no	—
202	C₂×D₈	—	94	1	no	—
203	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	94	1	no	—
206	#126	—	188	2	no	—
207	#125	—	94	1	no	—
208	#125	—	188	2	no	—
209	C₂²⋊Q₈	—	94	2	no	—
210	C₄×Q₈	—	94	2	no	—
211	C₄×D₄	—	94	2	no	—
212	C₄×D₄	—	94	2	no	—
213	C₄×D₄	—	188	2	no	—
214	#117	—	94	1	no	—
215	#117	D₁₈₈	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	188	2	no	—
217	SD₃₂	—	94	4	no	—
218	SD₃₂	—	94	4	no	—
219	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
220	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
221	#107	—	47	2	no	—
222	#107	—	47	2	no	—
223	#106	—	94	1	no	—
224	#106	—	94	1	no	—
225	#103	—	94	2	no	—
226	#103	—	94	2	no	—
227	#102	—	94	2	no	—
228	#102	—	94	2	no	—
229	#97	D₁₈₈	1	1	yes	characteristic
230	#97	D₁₈₈	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂⁴	—	188	1	no	—
234	C₂×Q₈	—	94	2	no	—
235	C₂×D₄	—	94	4	no	—
236	C₂×D₄	—	94	2	no	—
237	C₂×D₄	—	188	2	no	—
238	C₂×D₄	—	188	1	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
242	Q₁₆	—	94	2	no	—
243	Q₁₆	—	94	2	no	—
244	Q₁₆	—	188	2	no	—
245	D₈	—	47	4	no	—
246	D₈	—	94	4	no	—
247	D₈	—	94	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	—	188	2	no	—
259	C₄⋊C₄	—	188	2	no	—
260	C₄⋊C₄	D₃₇₆	1	2	yes	—
261	C₂²⋊C₄	—	188	1	no	—
262	C₂²⋊C₄	—	188	2	no	—
263	C₂²⋊C₄	—	188	2	no	—
264	C₂²⋊C₄	—	188	2	no	—
265	C₂²⋊C₄	D₃₇₆	1	2	yes	—
266	C₄²	—	188	1	no	—
267	C₄²	—	188	2	no	—
268	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
269	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₂³	—	188	4	no	—
273	C₂³	—	188	2	no	—
274	C₂³	—	752	2	no	—
275	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
276	Q₈	—	188	2	no	—
277	Q₈	—	188	2	no	—
278	D₄	—	94	4	no	—
279	D₄	—	188	4	no	—
280	D₄	—	188	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	752	2	no	—
287	C₂×C₄	—	752	2	no	—
288	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
289	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
290	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
291	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#41×D₄₇	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	376	2	no	—
299	C₂²	—	376	2	no	—
300	C₂²	—	188	4	no	—
301	C₂²	—	752	4	no	—
302	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	376	2	no	—
307	C₄	—	376	2	no	—
308	C₄	—	376	2	no	—
309	C₄	#15689635	1	1	yes	characteristic
310	C₄	#15689923	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	376	4	no	—
313	C₂	#20253389	1	1	yes	characteristic
314	C₂	#20256536	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2164	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	#2164	D₄₇
hypocenter	C₄₇	#33683
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2164

Complements

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

N	H ≅ G/N	type
1. #20254047	C₂	semidirect
3. #20245597	C₂	semidirect
4. #20245546	C₂	semidirect
5. #20245434	C₂	semidirect
7. #20245254	C₂	semidirect
8. C₄×#13093969	C₂	semidirect
9. C₄₇×#2164	C₂	semidirect
15. #15689525	C₂²	semidirect
16. #15689503	C₂²	semidirect
23. C₄×Dic₁₈₈	C₂²	semidirect
28. #15688951	C₂²	semidirect
29. #15688949	C₂²	semidirect
30. #15688928	C₂²	semidirect
31. #15688928	C₂²	semidirect
35. C₄₇×C₄⋊C₁₆	C₂²	semidirect
37. C₄₇×#346	C₂²	semidirect
38. C₄×C₁₆×C₄₇	C₂²	semidirect
162. #2164	D₄₇	semidirect
188. C₄⋊C₁₆	C₂×D₄₇	semidirect
194. #346	C₂×D₄₇	semidirect
195. C₄×C₁₆	C₂×D₄₇	semidirect
196. C₄₇	#33683	semidirect

Maximal subgroups

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

subgroup	index
#20254047	2
#20252241	2
#20245597	2
#20245546	2
#20245434	2
#20245319	2
#20245254	2
C₄×#13093969	2
C₄₇×#2164	2
#33683	47