Subgroups — #26371817

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253969	C₂	1	2	yes	maximal
2	#20245662	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245639	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245614	C₂	1	2	yes	maximal, complemented
5	#20245479	C₂	1	1	yes	maximal, characteristic
6	#20245477	C₂	1	2	yes	maximal
7	Q₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245246	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245244	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2216	C₂	1	1	yes	maximal, characteristic
12	#15689675	—	2	2	no	—
13	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
14	C₂×#13093970	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#15689527	C₂²	1	2	yes	—
17	#15689501	C₂²	1	2	yes	—
18	#15689472	—	2	2	no	—
19	#15689364	C₂²	1	2	yes	—
20	#15689362	—	2	2	no	—
21	Q₁₆×Dic₄₇	C₂²	1	2	yes	—
22	#15689217	C₂²	1	2	yes	—
23	#15689215	—	2	2	no	—
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688949	C₂²	1	1	yes	characteristic
26	#15688948	C₂²	1	2	yes	—
27	#15688928	C₂²	1	2	yes	—
28	#15688927	C₂²	1	2	yes	—
29	#15688926	C₂²	1	2	yes	—
30	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#467	C₂²	1	2	yes	—
37	C₄₇×#366	C₂²	1	1	yes	characteristic
38	C₄₇×#358	C₂²	1	2	yes	—
39	C₂×Dic₁₈₈	C₂³	1	2	yes	—
40	C₂×#10625337	—	2	2	no	—
41	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	#13094023	—	2	2	no	—
43	#13094022	—	2	2	no	—
44	#13094014	—	4	2	no	—
45	#13094003	—	2	2	no	—
46	#13094001	—	4	2	no	—
47	Q₈×Dic₄₇	—	2	2	no	—
48	#13093989	—	2	2	no	—
49	#13093989	—	4	2	no	—
50	#13093980	—	4	2	no	—
51	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
52	Dic₃₇₆	—	2	2	no	—
53	#13093970	—	2	4	no	—
54	C₂×C₄₇⋊C₁₆	—	2	2	no	—
55	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#13093958	C₂³	1	2	yes	—
58	#13093957	—	2	2	no	—
59	#13093949	—	2	2	no	—
60	#13093948	—	2	2	no	—
61	#13093947	—	2	2	no	—
62	C₈×Dic₄₇	D₄	1	1	yes	characteristic
63	C₈×Dic₄₇	D₄	1	1	yes	characteristic
64	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
65	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₄₇	—	2	2	no	—
69	Q₃₂×C₄₇	—	2	2	no	—
70	C₂×C₁₆×C₄₇	—	2	2	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
73	#107×C₄₇	C₂³	1	2	yes	—
74	#103×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
78	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
79	C₂×Dic₉₄	—	2	2	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	Q₁₆×C₄₇	—	2	2	no	—
82	Q₁₆×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	1	no	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	—	4	2	no	—
89	C₂²⋊C₄×C₄₇	—	4	2	no	—
90	C₂²⋊Dic₄₇	—	4	2	no	—
91	#10625337	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	Dic₄₇⋊C₄	—	4	2	no	—
95	Dic₄₇⋊C₄	—	4	2	no	—
96	Dic₄₇⋊C₄	—	4	2	no	—
97	Dic₄₇⋊C₄	—	4	2	no	—
98	Dic₄₇⋊C₄	—	4	2	no	—
99	C₄×Dic₄₇	—	4	2	no	—
100	C₄×Dic₄₇	—	4	2	no	—
101	C₄×Dic₄₇	D₈	1	1	yes	characteristic
102	C₄×Dic₄₇	D₈	1	1	yes	characteristic
103	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	Dic₁₈₈	—	2	2	no	—
109	Dic₁₈₈	—	2	2	no	—
110	C₁₆×C₄₇	—	2	1	no	—
111	C₁₆×C₄₇	#41	1	2	yes	—
112	C₄₇⋊C₁₆	—	2	1	no	—
113	C₄₇⋊C₁₆	#41	1	2	yes	—
114	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₄₇	—	4	2	no	—
116	Q₈×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	Q₃₂	1	4	yes	complemented
127	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
131	Dic₉₄	—	4	2	no	—
132	Dic₉₄	—	4	2	no	—
133	Dic₉₄	—	8	4	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₄₇⋊C₈	—	2	1	no	—
138	C₄₇⋊C₈	#135	1	2	yes	—
139	#33734	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	—	8	2	no	—
147	C₄×C₄₇	#532	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
149	Dic₄₇	—	2	1	no	—
150	Dic₄₇	—	8	2	no	—
151	Dic₄₇	—	8	2	no	—
152	Dic₄₇	C₂×Q₃₂	1	2	yes	—
153	Dic₄₇	C₂×Q₃₂	1	2	yes	—
154	Dic₄₇	#508	1	2	yes	—
155	#3170	—	47	2	no	—
156	#2252	—	47	1	no	—
157	#2247	—	47	1	no	—
158	#2241	—	47	2	no	—
159	#2216	—	47	1	no	—
160	#2216	—	47	2	no	—
161	#2216	D₄₇	1	1	yes	characteristic
162	C₄×Q₃₂	—	47	1	no	—
163	C₄×Q₃₂	—	47	1	no	—
164	C₄×Q₃₂	—	47	2	no	—
165	C₂×C₄×C₁₆	—	47	1	no	—
166	C₂×C₄₇	—	2	2	no	—
167	C₂×C₄₇	#3169	1	1	yes	characteristic
168	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
170	#533	—	94	2	no	—
171	C₂×Q₃₂	—	47	2	no	—
172	C₂×Q₃₂	—	47	1	no	—
173	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
174	C₂²×C₁₆	—	47	1	no	—
175	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
176	#500	—	47	2	no	—
177	#494	—	47	2	no	—
178	#487	—	94	2	no	—
179	#467	—	47	2	no	—
180	#467	—	94	2	no	—
181	#467	C₂×D₄₇	1	2	yes	—
182	C₄×Q₁₆	—	47	2	no	—
183	C₄×Q₁₆	—	47	2	no	—
184	C₄×Q₁₆	—	94	2	no	—
185	C₂×C₄×C₈	—	47	1	no	—
186	#366	—	47	1	no	—
187	#366	—	47	2	no	—
188	#366	C₂×D₄₇	1	1	yes	characteristic
189	#358	—	47	2	no	—
190	#358	—	47	2	no	—
191	#358	—	47	2	no	—
192	#358	C₂×D₄₇	1	2	yes	—
193	C₄×C₁₆	—	47	1	no	—
194	C₄×C₁₆	—	47	1	no	—
195	C₄×C₁₆	—	47	1	no	—
196	C₄×C₁₆	—	47	1	no	—
197	C₄₇	#33734	1	1	yes	characteristic, complemented, Sylow
198	C₂×Q₁₆	—	47	2	no	—
199	C₂×Q₁₆	—	94	2	no	—
200	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
201	C₂²×C₈	—	47	1	no	—
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	94	2	no	—
204	C₄⋊Q₈	—	94	2	no	—
205	#125	—	188	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	94	2	no	—
208	C₂²⋊Q₈	—	188	2	no	—
209	C₄×Q₈	—	94	2	no	—
210	C₄×Q₈	—	94	2	no	—
211	C₄×Q₈	—	188	2	no	—
212	#117	—	188	2	no	—
213	C₂×C₄²	—	47	1	no	—
214	Q₃₂	—	2	2	no	—
215	Q₃₂	—	94	4	no	—
216	Q₃₂	—	94	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	47	1	no	—
219	C₂×C₁₆	—	47	1	no	—
220	C₂×C₁₆	—	94	2	no	—
221	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	47	2	no	—
224	#107	—	94	2	no	—
225	#107	C₂²×D₄₇	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	94	2	no	—
228	#103	—	94	2	no	—
229	#103	—	94	2	no	—
230	C₄×C₈	—	47	1	no	—
231	C₄×C₈	—	47	1	no	—
232	C₄×C₈	—	47	1	no	—
233	C₄×C₈	—	47	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	94	2	no	—
236	C₂×Q₈	—	188	2	no	—
237	C₂²×C₄	—	47	1	no	—
238	C₂²×C₄	—	47	1	no	—
239	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	94	2	no	—
243	Q₁₆	—	94	2	no	—
244	Q₁₆	—	188	4	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	47	1	no	—
248	C₂×C₈	—	47	1	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	188	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	—	188	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	188	2	no	—
263	C₄²	—	47	1	no	—
264	C₄²	—	47	1	no	—
265	C₄²	—	47	1	no	—
266	C₄²	—	47	1	no	—
267	C₄²	—	188	2	no	—
268	C₄²	—	188	2	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	47	2	no	—
271	C₁₆	—	94	1	no	—
272	C₁₆	#10625352	1	2	yes	—
273	C₂³	D₈×D₄₇	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	376	4	no	—
277	Q₈	—	188	2	no	—
278	Q₈	—	188	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	47	4	no	—
284	C₂×C₄	—	47	1	no	—
285	C₂×C₄	—	47	1	no	—
286	C₂×C₄	—	47	1	no	—
287	C₂×C₄	—	47	1	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	47	2	no	—
295	C₈	—	94	1	no	—
296	C₈	#41×D₄₇	1	1	yes	characteristic
297	C₈	C₂×#10625352	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	Q₃₂×D₄₇	1	2	yes	—
301	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	47	2	no	—
306	C₄	—	47	2	no	—
307	C₄	—	47	2	no	—
308	C₄	—	94	1	no	—
309	C₄	—	376	2	no	—
310	C₄	—	376	2	no	—
311	C₄	#15689668	1	1	yes	characteristic
312	C₄	#135×D₄₇	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#20253962	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₄₇	1	1	yes	characteristic
316	C₂	#508×D₄₇	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×#2216	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#2216	D₄₇
hypocenter	C₄₇	#33734
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #20245662	C₂	semidirect
3. #20245639	C₂	semidirect
4. #20245614	C₂	semidirect
7. Q₃₂×Dic₄₇	C₂	semidirect
8. #20245246	C₂	semidirect
9. #20245244	C₂	semidirect
126. C₂×Dic₄₇	Q₃₂	semidirect
197. C₄₇	#33734	semidirect

Maximal subgroups

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

subgroup	index
#20253969	2
#20245662	2
#20245639	2
#20245614	2
#20245479	2
#20245477	2
Q₃₂×Dic₄₇	2
#20245246	2
#20245244	2
C₂×C₁₆×Dic₄₇	2
C₄₇×#2216	2
#33734	47