Subgroups — #26371333

Order 12032.

542 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	#20253974	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20245586	C₂	1	2	yes	maximal
4	#20245531	C₂	1	2	yes	maximal, complemented
5	#20245439	C₂	1	2	yes	maximal, complemented
6	#20245324	C₂	1	2	yes	maximal
7	#20245260	C₂	1	4	yes	maximal, complemented
8	C₄₇×#2163	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	#15689680	—	2	1	no	—
11	C₂×C₄×Dic₉₄	—	2	1	no	—
12	C₂×Dic₃₇₆	C₂²	1	4	yes	—
13	#15689528	C₂²	1	1	yes	characteristic
14	#15689503	C₂²	1	1	yes	characteristic
15	#15689474	—	2	1	no	—
16	#15689364	—	2	1	no	—
17	#15689364	C₂²	1	2	yes	—
18	C₄×Dic₁₈₈	—	2	1	no	—
19	C₄×Dic₁₈₈	—	2	1	no	—
20	C₄×Dic₁₈₈	—	2	1	no	—
21	C₄×Dic₁₈₈	C₂²	1	2	yes	—
22	C₄×Dic₁₈₈	C₂²	1	2	yes	—
23	C₄×Dic₁₈₈	C₂²	1	2	yes	—
24	C₂×#13093958	C₂²	1	1	yes	characteristic
25	C₂×#13093949	—	2	1	no	—
26	#15688951	C₂²	1	4	yes	—
27	#15688928	C₂²	1	4	yes	—
28	#15688928	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₄₇×#346	C₂²	1	2	yes	—
33	C₂²×Dic₉₄	—	2	1	no	—
34	C₂×Dic₁₈₈	—	2	1	no	—
35	C₂×Dic₁₈₈	—	2	4	no	—
36	C₂×Dic₁₈₈	—	2	1	no	—
37	C₂×Dic₁₈₈	C₂³	1	2	yes	—
38	C₂×Dic₁₈₈	C₂³	1	2	yes	—
39	#13094024	—	2	1	no	—
40	#13094024	—	2	1	no	—
41	#13094016	—	4	1	no	—
42	#13094003	—	2	2	no	—
43	#13094003	—	4	1	no	—
44	C₄×Dic₉₄	—	2	2	no	—
45	C₄×Dic₉₄	—	2	2	no	—
46	C₄×Dic₉₄	—	2	2	no	—
47	C₄×Dic₉₄	—	4	1	no	—
48	C₄×Dic₉₄	—	4	1	no	—
49	C₄×Dic₉₄	—	4	1	no	—
50	#13093983	—	4	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇⋊C₄	—	4	1	no	—
53	C₂×C₄×Dic₄₇	—	4	1	no	—
54	Dic₃₇₆	—	2	8	no	—
55	#13093958	—	2	1	no	—
56	#13093958	—	2	1	no	—
57	#13093958	C₂³	1	2	yes	—
58	#13093958	C₂³	1	2	yes	—
59	#13093949	—	2	2	no	—
60	#13093949	—	2	2	no	—
61	#13093949	—	2	2	no	—
62	#13093949	—	2	2	no	—
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
66	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
67	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	4	1	no	—
75	C₂²×Dic₄₇	—	4	1	no	—
76	C₂×Dic₉₄	—	2	4	no	—
77	C₂×Dic₉₄	—	2	2	no	—
78	C₂×Dic₉₄	—	2	2	no	—
79	C₂×Dic₉₄	—	4	2	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₄²×C₄₇	D₈	1	1	yes	characteristic
89	C₄²×C₄₇	D₈	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	4	1	no	—
96	C₄⋊Dic₄₇	—	4	1	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄⋊Dic₄₇	—	4	1	no	—
99	Dic₄₇⋊C₄	—	4	2	no	—
100	Dic₄₇⋊C₄	—	4	2	no	—
101	Dic₄₇⋊C₄	—	4	2	no	—
102	Dic₄₇⋊C₄	—	4	2	no	—
103	C₄×Dic₄₇	—	4	2	no	—
104	C₄×Dic₄₇	—	4	2	no	—
105	C₄×Dic₄₇	—	4	2	no	—
106	Dic₁₈₈	—	2	4	no	—
107	Dic₁₈₈	—	2	2	no	—
108	Dic₁₈₈	—	2	4	no	—
109	Dic₁₈₈	—	4	2	no	—
110	Dic₁₈₈	#41	1	4	yes	—
111	C₁₆×C₄₇	—	2	4	no	—
112	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	8	2	no	—
129	Dic₉₄	—	2	4	no	—
130	Dic₉₄	—	4	4	no	—
131	Dic₉₄	—	4	2	no	—
132	Dic₉₄	—	4	2	no	—
133	Dic₉₄	—	4	2	no	—
134	Dic₉₄	—	8	2	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	C₂×#41	1	2	yes	—
139	#33673	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	#510	1	2	yes	—
143	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	#510	1	2	yes	—
149	C₄×C₄₇	#525	1	1	yes	characteristic
150	C₄×C₄₇	#575	1	1	yes	characteristic
151	Dic₄₇	—	4	4	no	—
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	Dic₄₇	—	8	2	no	—
155	#3170	—	47	1	no	—
156	C₂×C₄×Q₁₆	—	47	1	no	—
157	#2235	—	47	2	no	—
158	#2225	—	47	2	no	—
159	#2209	—	47	2	no	—
160	#2188	—	47	2	no	—
161	#2176	—	47	4	no	—
162	#2163	D₄₇	1	1	yes	characteristic
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	#3100	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
167	C₂²×Q₁₆	—	47	1	no	—
168	#533	—	94	1	no	—
169	C₂×C₄×Q₈	—	94	1	no	—
170	C₂×Q₃₂	—	47	4	no	—
171	#500	—	47	1	no	—
172	#494	—	47	1	no	—
173	#487	—	94	1	no	—
174	#467	—	47	2	no	—
175	#467	—	94	1	no	—
176	C₄×Q₁₆	—	47	2	no	—
177	C₄×Q₁₆	—	47	2	no	—
178	C₄×Q₁₆	—	47	2	no	—
179	C₄×Q₁₆	—	94	1	no	—
180	C₄×Q₁₆	—	94	1	no	—
181	C₄×Q₁₆	—	94	1	no	—
182	C₂×#107	—	47	1	no	—
183	C₂×#103	—	94	1	no	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#367	—	47	4	no	—
186	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
187	#358	—	47	4	no	—
188	#358	—	47	4	no	—
189	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
190	#346	C₂×D₄₇	1	2	yes	—
191	C₄₇	#33673	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	94	1	no	—
193	C₂×Q₁₆	—	47	2	no	—
194	C₂×Q₁₆	—	47	2	no	—
195	C₂×Q₁₆	—	94	4	no	—
196	C₂×Q₁₆	—	94	1	no	—
197	C₂×Q₁₆	—	94	1	no	—
198	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₄⋊Q₈	—	94	1	no	—
201	C₄⋊Q₈	—	94	1	no	—
202	#125	—	188	1	no	—
203	C₂²⋊Q₈	—	94	2	no	—
204	C₂²⋊Q₈	—	188	1	no	—
205	C₄×Q₈	—	94	2	no	—
206	C₄×Q₈	—	94	2	no	—
207	C₄×Q₈	—	94	2	no	—
208	C₄×Q₈	—	188	1	no	—
209	C₄×Q₈	—	188	1	no	—
210	C₄×Q₈	—	188	1	no	—
211	#117	—	188	1	no	—
212	C₂×C₄⋊C₄	—	94	1	no	—
213	C₂×C₄⋊C₄	—	188	1	no	—
214	C₂×C₄²	—	188	1	no	—
215	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
216	Q₃₂	—	94	8	no	—
217	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
218	#107	—	47	2	no	—
219	#107	—	47	2	no	—
220	#107	—	94	1	no	—
221	#107	—	94	1	no	—
222	#103	—	94	2	no	—
223	#103	—	94	2	no	—
224	#103	—	94	2	no	—
225	#103	—	94	2	no	—
226	C₄×C₈	D₁₈₈	1	1	yes	characteristic
227	C₄×C₈	D₁₈₈	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	C₂×Q₈	—	94	4	no	—
231	C₂×Q₈	—	94	2	no	—
232	C₂×Q₈	—	94	2	no	—
233	C₂×Q₈	—	188	2	no	—
234	C₂×Q₈	—	188	1	no	—
235	C₂×Q₈	—	188	1	no	—
236	C₂²×C₄	—	188	1	no	—
237	C₂²×C₄	—	188	1	no	—
238	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
239	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	Q₁₆	—	47	4	no	—
242	Q₁₆	—	94	4	no	—
243	Q₁₆	—	94	2	no	—
244	Q₁₆	—	94	4	no	—
245	Q₁₆	—	188	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	188	2	no	—
255	C₄⋊C₄	—	188	2	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	—	188	1	no	—
259	C₄⋊C₄	—	188	1	no	—
260	C₄⋊C₄	—	188	1	no	—
261	C₄⋊C₄	—	188	1	no	—
262	C₂²⋊C₄	—	188	2	no	—
263	C₄²	—	188	2	no	—
264	C₄²	—	188	2	no	—
265	C₄²	—	188	2	no	—
266	C₄²	D₃₇₆	1	1	yes	characteristic
267	C₄²	D₃₇₆	1	1	yes	characteristic
268	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
269	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
270	C₁₆	—	2	4	no	—
271	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
272	Q₈	—	94	4	no	—
273	Q₈	—	376	2	no	—
274	Q₈	—	188	4	no	—
275	Q₈	—	188	2	no	—
276	Q₈	—	188	2	no	—
277	Q₈	—	188	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	376	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
289	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
290	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
291	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
292	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#41×D₄₇	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#15689566	1	2	yes	—
301	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	376	2	no	—
307	C₄	—	376	2	no	—
308	C₄	—	376	2	no	—
309	C₄	—	188	4	no	—
310	C₄	#15689566	1	2	yes	—
311	C₄	#15689635	1	1	yes	characteristic
312	C₄	#15689923	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#20253389	1	1	yes	characteristic
315	C₂	C₂×#15689566	1	1	yes	characteristic
316	C₂	C₂×#15689566	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2163	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#2163	D₄₇
hypocenter	C₄₇	#33673
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
4. #20245531	C₂	semidirect
5. #20245439	C₂	semidirect
7. #20245260	C₂	semidirect
191. C₄₇	#33673	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
C₂×C₄×Dic₁₈₈	2
#20245586	2
#20245531	2
#20245439	2
#20245324	2
#20245260	2
C₄₇×#2163	2
#33673	47