Subgroups — #26225040

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689525	C₂	1	1	yes	maximal, characteristic
2	#20246129	C₂	1	2	yes	maximal
3	#20244357	C₂	1	2	yes	maximal
4	#20244131	C₂	1	2	yes	maximal
5	#20244121	C₂	1	2	yes	maximal
6	#20243548	C₂	1	1	yes	maximal, characteristic
7	#20243285	C₂	1	2	yes	maximal
8	C₄×#13093958	C₂	1	2	yes	maximal
9	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094016	C₂²	1	2	yes	—
11	#15689525	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689021	C₂²	1	4	yes	—
17	#15689020	—	2	4	no	—
18	#15689010	—	2	4	no	—
19	#15688993	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
21	#15688886	C₂²	1	4	yes	—
22	#15688886	C₂²	1	4	yes	—
23	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#376	C₂²	1	1	yes	characteristic
30	#13094016	—	2	4	no	—
31	#13094016	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	4	no	—
36	C₂×Dic₄₇⋊C₄	—	2	4	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	#13093958	—	2	4	no	—
39	#13093958	—	2	4	no	—
40	#13093958	C₂×C₄	1	8	yes	—
41	#13093956	C₂×C₄	1	8	yes	—
42	C₄₇⋊#95	—	2	4	no	—
43	C₄₇⋊#95	—	2	4	no	—
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#97×C₄₇	—	2	2	no	—
52	#97×C₄₇	—	2	2	no	—
53	#97×C₄₇	Q₈	1	4	yes	—
54	C₄×C₈×C₄₇	—	2	2	no	—
55	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₄₇	Q₈	1	4	yes	—
57	#95×C₄₇	D₄	1	2	yes	—
58	#95×C₄₇	D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	1	no	—
67	C₂×C₈×C₄₇	—	2	1	no	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₄₇	#41	1	2	yes	—
77	C₂×C₈×C₄₇	#41	1	2	yes	—
78	C₄²×C₄₇	—	2	1	no	—
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	8	4	no	—
88	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
89	Dic₄₇⋊C₄	—	2	16	no	—
90	Dic₄₇⋊C₄	—	8	4	no	—
91	C₄×Dic₄₇	—	8	4	no	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₈×C₄₇	—	2	2	no	—
116	C₈×C₄₇	—	2	2	no	—
117	C₈×C₄₇	C₄×Q₈	1	4	yes	—
118	C₈×C₄₇	C₄×Q₈	1	4	yes	—
119	#21723	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₄₇	#532	1	1	yes	characteristic
123	C₂²×C₄₇	#576	1	2	yes	—
124	C₂²×C₄₇	#578	1	2	yes	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	4	1	no	—
128	C₄×C₄₇	—	4	1	no	—
129	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₄₇	#533	1	1	yes	characteristic
132	C₄×C₄₇	#533	1	1	yes	characteristic
133	Dic₄₇	—	4	8	no	—
134	Dic₄₇	—	8	4	no	—
135	C₂×#499	—	47	1	no	—
136	#2330	—	47	2	no	—
137	#1992	—	47	2	no	—
138	#1944	—	47	2	no	—
139	#1942	—	47	2	no	—
140	#1845	—	47	1	no	—
141	#1776	—	47	2	no	—
142	C₄×#107	—	47	2	no	—
143	#1726	D₄₇	1	1	yes	characteristic
144	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₄₇	#2951	1	2	yes	—
146	C₂×C₄₇	#3164	1	2	yes	—
147	C₂×C₄₇	#3176	1	1	yes	characteristic
148	C₂×C₄₇	#3177	1	1	yes	characteristic
149	C₂×#125	—	47	2	no	—
150	#499	—	47	8	no	—
151	C₂×#107	—	47	2	no	—
152	C₂×#107	—	47	2	no	—
153	C₂×#107	—	47	2	no	—
154	C₂×#106	—	47	2	no	—
155	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
161	#389	—	47	4	no	—
162	#389	—	94	4	no	—
163	#387	—	94	4	no	—
164	#384	—	47	2	no	—
165	C₄×C₄⋊C₄	—	47	4	no	—
166	#376	C₂×D₄₇	1	1	yes	characteristic
167	#340	—	47	4	no	—
168	#340	—	47	4	no	—
169	C₄₇	#21723	1	1	yes	characteristic, complemented, Sylow
170	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	#125	—	47	8	no	—
175	#125	—	94	4	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄⋊C₄	—	47	4	no	—
178	C₂×C₄⋊C₄	—	94	4	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	—	94	4	no	—
181	C₂×C₄²	—	94	4	no	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	#107	—	47	8	no	—
186	#107	—	94	4	no	—
187	#107	—	94	4	no	—
188	#106	—	47	8	no	—
189	#97	—	2	2	no	—
190	#97	—	2	2	no	—
191	#97	Dic₉₄	1	4	yes	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	Dic₉₄	1	4	yes	—
194	C₄×C₈	C₄×D₄₇	1	4	yes	—
195	#95	—	94	4	no	—
196	#95	—	94	4	no	—
197	#95	D₁₈₈	1	2	yes	—
198	#95	D₁₈₈	1	2	yes	—
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
211	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
212	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
213	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
216	C₂×C₈	#10625350	1	2	yes	—
217	C₂×C₈	#10625350	1	2	yes	—
218	C₄⋊C₄	—	47	8	no	—
219	C₄⋊C₄	—	94	16	no	—
220	C₄⋊C₄	—	94	4	no	—
221	C₄⋊C₄	—	94	4	no	—
222	C₄⋊C₄	—	94	4	no	—
223	C₄⋊C₄	—	376	4	no	—
224	C₄⋊C₄	—	376	4	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	376	4	no	—
228	C₄²	C₂×Dic₉₄	1	2	yes	—
229	C₄²	C₂×Dic₉₄	1	2	yes	—
230	C₄²	C₂×C₄×D₄₇	1	2	yes	—
231	C₄²	#10625350	1	2	yes	—
232	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	94	8	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	—	188	4	no	—
247	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
248	C₂×C₄	#13094024	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
251	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	C₄×Dic₉₄	1	4	yes	—
258	C₈	C₄×Dic₉₄	1	4	yes	—
259	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
260	C₂²	C₂×#13094024	1	1	yes	characteristic
261	C₂²	#15689673	1	1	yes	characteristic
262	C₂²	#15689928	1	2	yes	—
263	C₂²	#15689940	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	376	4	no	—
269	C₄	—	188	8	no	—
270	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
271	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
272	C₄	#15689680	1	1	yes	characteristic
273	C₄	#15689680	1	1	yes	characteristic
274	C₂	C₄×#13094024	1	1	yes	characteristic
275	C₂	#20252265	1	2	yes	—
276	C₂	#20253927	1	2	yes	—
277	C₂	#20254038	1	1	yes	characteristic
278	C₂	#20254047	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21723
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
11. #15689525	C₄	semidirect
169. C₄₇	#21723	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689525	2
#20246129	2
#20244357	2
#20244131	2
#20244121	2
#20243548	2
#20243285	2
C₄×#13093958	2
C₄₇×#1726	2
#21723	47