Subgroups — #26204082

Order 12032.

690 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	#20253974	C₂	1	1	yes	maximal, characteristic
2	#20253970	C₂	1	2	yes	maximal
3	#20253898	C₂	1	2	yes	maximal
4	#20243068	C₂	1	2	yes	maximal, complemented
5	#20243067	C₂	1	2	yes	maximal, complemented
6	#20243064	C₂	1	2	yes	maximal, complemented
7	#20243063	C₂	1	2	yes	maximal, complemented
8	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	2	yes	—
11	#15689528	C₂²	1	1	yes	characteristic
12	#15689526	C₂²	1	2	yes	—
13	#15689526	C₂²	1	2	yes	—
14	#15689526	C₂²	1	2	yes	—
15	#15689522	—	2	4	no	—
16	#15689519	C₂²	1	2	yes	—
17	#15689519	C₂²	1	2	yes	—
18	#15689503	C₂²	1	1	yes	characteristic
19	#15689502	C₂²	1	2	yes	—
20	#15689475	C₂²	1	2	yes	—
21	#15689474	C₂²	1	2	yes	—
22	#15689473	C₂²	1	2	yes	—
23	#15689364	—	2	2	no	—
24	#15689362	—	2	4	no	—
25	C₄×Dic₁₈₈	—	2	2	no	—
26	C₄×#10625337	—	2	4	no	—
27	#15689159	—	2	4	no	—
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	#15688845	C₂²	1	2	yes	—
31	#15688845	C₂²	1	2	yes	—
32	C₄₇×#533	C₂²	1	2	yes	—
33	C₄₇×#500	C₂²	1	1	yes	characteristic
34	C₄₇×#494	C₂²	1	1	yes	characteristic
35	C₄₇×#487	C₂²	1	2	yes	—
36	C₄₇×#467	—	2	2	no	—
37	C₄×Q₁₆×C₄₇	—	2	2	no	—
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₈₈	—	2	2	no	—
40	C₂×#10625337	—	2	4	no	—
41	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
42	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
43	#13094024	C₂³	1	2	yes	—
44	#13094024	C₂³	1	2	yes	—
45	#13094016	—	2	2	no	—
46	#13094003	—	2	2	no	—
47	#13094003	—	2	2	no	—
48	C₄×Dic₉₄	—	2	2	no	—
49	C₄×Dic₉₄	—	2	2	no	—
50	#13093983	—	2	2	no	—
51	#13093958	—	2	2	no	—
52	#13093957	—	2	4	no	—
53	#13093957	—	2	4	no	—
54	#13093957	—	2	4	no	—
55	#13093955	—	2	4	no	—
56	#13093955	—	2	4	no	—
57	#13093949	—	2	4	no	—
58	#13093948	—	2	4	no	—
59	#13093947	—	2	4	no	—
60	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
61	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
62	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
63	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
64	C₂×Q₁₆×C₄₇	—	2	2	no	—
65	C₂²×C₈×C₄₇	D₄	1	2	yes	—
66	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
68	#125×C₄₇	—	2	2	no	—
69	C₂²⋊Q₈×C₄₇	—	2	2	no	—
70	C₂²⋊Q₈×C₄₇	—	2	2	no	—
71	C₄×Q₈×C₄₇	—	2	2	no	—
72	C₄×Q₈×C₄₇	—	2	2	no	—
73	#117×C₄₇	—	2	2	no	—
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	—	2	2	no	—
76	#103×C₄₇	—	2	4	no	—
77	C₄×C₈×C₄₇	D₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×Dic₉₄	—	2	2	no	—
87	Q₁₆×C₄₇	—	4	4	no	—
88	C₂×C₈×C₄₇	—	4	2	no	—
89	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	4	2	no	—
96	C₄⋊C₄×C₄₇	—	4	2	no	—
97	C₂²⋊C₄×C₄₇	—	4	2	no	—
98	C₄²×C₄₇	—	4	2	no	—
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊Dic₄₇	—	4	2	no	—
104	#10625337	—	4	8	no	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	Dic₄₇⋊C₄	—	4	2	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	C₄×Dic₄₇	—	4	2	no	—
112	C₂×C₄₇⋊C₈	—	4	2	no	—
113	C₂×C₄₇⋊C₈	Q₁₆	1	8	yes	complemented
114	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
115	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
116	Dic₁₈₈	—	4	4	no	—
117	C₂³×C₄₇	#127	1	1	yes	characteristic
118	Q₈×C₄₇	—	4	4	no	—
119	Q₈×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	Dic₉₄	—	4	4	no	—
134	Dic₉₄	—	4	4	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₄₇⋊C₈	—	2	4	no	—
138	C₄₇⋊C₈	C₂×Q₁₆	1	8	yes	—
139	#19618	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	#494	1	2	yes	—
143	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	8	2	no	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	#494	1	2	yes	—
149	C₄×C₄₇	C₂²×Q₁₆	1	2	yes	—
150	C₄×C₄₇	#575	1	2	yes	—
151	C₄×C₄₇	#575	1	2	yes	—
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	#3170	—	47	1	no	—
155	#3170	—	47	2	no	—
156	#3170	D₄₇	1	1	yes	characteristic
157	#3160	—	47	2	no	—
158	#1723	—	47	2	no	—
159	#1723	—	47	2	no	—
160	#1723	—	47	2	no	—
161	#1723	—	47	2	no	—
162	C₂×#322	—	47	1	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
165	C₂×C₄₇	#3150	1	1	yes	characteristic
166	C₂×C₄₇	#3150	1	1	yes	characteristic
167	#533	—	47	2	no	—
168	#533	C₂×D₄₇	1	2	yes	—
169	#500	—	47	2	no	—
170	#500	—	47	2	no	—
171	#500	—	47	2	no	—
172	#500	—	47	1	no	—
173	#500	C₂×D₄₇	1	1	yes	characteristic
174	#499	—	94	4	no	—
175	#498	—	47	2	no	—
176	#498	—	47	2	no	—
177	#494	—	47	2	no	—
178	#494	—	47	1	no	—
179	#494	C₂×D₄₇	1	1	yes	characteristic
180	#487	—	47	2	no	—
181	#487	—	47	2	no	—
182	#487	—	47	2	no	—
183	#487	C₂×D₄₇	1	2	yes	—
184	#467	—	2	2	no	—
185	#467	—	94	4	no	—
186	#467	—	94	2	no	—
187	C₄×Q₁₆	—	2	2	no	—
188	C₄×Q₁₆	—	94	4	no	—
189	C₄×Q₁₆	—	94	2	no	—
190	#427	—	94	4	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	C₂×C₄×C₈	—	47	1	no	—
193	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
194	#322	—	47	2	no	—
195	#322	—	47	2	no	—
196	C₄₇	#19618	1	1	yes	characteristic, complemented, Sylow
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	94	4	no	—
199	C₂×Q₁₆	—	94	2	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
203	C₄⋊Q₈	—	47	2	no	—
204	C₄⋊Q₈	—	47	2	no	—
205	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
206	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
207	#125	—	2	2	no	—
208	#125	—	94	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	94	2	no	—
212	C₂²⋊Q₈	—	94	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	94	2	no	—
216	C₄×Q₈	—	94	2	no	—
217	#117	—	2	2	no	—
218	#117	—	94	2	no	—
219	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
220	#107	—	2	2	no	—
221	#107	—	94	4	no	—
222	#107	—	94	4	no	—
223	#107	—	94	4	no	—
224	#107	—	94	2	no	—
225	#106	—	94	4	no	—
226	#106	—	94	4	no	—
227	#103	—	2	4	no	—
228	#103	—	94	4	no	—
229	#103	—	94	4	no	—
230	#103	—	94	4	no	—
231	C₄×C₈	—	47	2	no	—
232	C₄×C₈	—	47	2	no	—
233	C₄×C₈	—	47	2	no	—
234	C₄×C₈	—	47	2	no	—
235	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
236	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	94	2	no	—
241	C₂×Q₈	—	94	2	no	—
242	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
245	Q₁₆	—	4	4	no	—
246	Q₁₆	—	188	4	no	—
247	Q₁₆	—	188	8	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	47	8	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	188	2	no	—
253	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	94	2	no	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	94	2	no	—
264	C₄⋊C₄	—	94	2	no	—
265	C₄⋊C₄	—	188	2	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	188	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	188	2	no	—
271	C₄²	D₄×D₄₇	1	1	yes	characteristic
272	C₄²	D₄×D₄₇	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₄²	C₂³×D₄₇	1	1	yes	characteristic
275	C₂³	#13094020	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	188	4	no	—
279	Q₈	—	188	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	#13094020	1	1	yes	characteristic
287	C₂×C₄	#13094020	1	1	yes	characteristic
288	C₂×C₄	#13094020	1	1	yes	characteristic
289	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
290	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	47	8	no	—
296	C₈	—	94	4	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	#15689501	1	2	yes	—
300	C₂²	C₂×#13094020	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	376	2	no	—
306	C₄	—	376	2	no	—
307	C₄	#15689501	1	2	yes	—
308	C₄	C₂×Q₁₆×D₄₇	1	2	yes	—
309	C₄	#15689920	1	2	yes	—
310	C₄	#15689921	1	2	yes	—
311	C₂	—	2	2	no	—
312	C₂	C₂×#15689501	1	1	yes	characteristic
313	C₂	#20253831	1	1	yes	characteristic
314	C₂	#20253833	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	C₂³×C₄₇	#127
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#19618
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #20243068	C₂	semidirect
5. #20243067	C₂	semidirect
6. #20243064	C₂	semidirect
7. #20243063	C₂	semidirect
113. C₂×C₄₇⋊C₈	Q₁₆	semidirect
196. C₄₇	#19618	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20253970	2
#20253898	2
#20243068	2
#20243067	2
#20243064	2
#20243063	2
C₂×#15688845	2
C₄₇×#3170	2
#19618	47