Subgroups — #26225024

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#20246127	C₂	1	2	yes	maximal
3	#20244356	C₂	1	2	yes	maximal
4	#20244124	C₂	1	2	yes	maximal
5	#20244114	C₂	1	2	yes	maximal
6	#20243288	C₂	1	2	yes	maximal
7	#106×Dic₄₇	C₂	1	2	yes	maximal
8	#20243074	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1844	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×#13093956	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	1	yes	characteristic
15	C₂×#13093934	C₂²	1	1	yes	characteristic
16	C₂×#13093934	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15689019	C₂²	1	4	yes	—
20	#15689018	—	2	4	no	—
21	#15689004	—	2	4	no	—
22	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
23	#15688969	C₂²	1	1	yes	characteristic
24	#15688885	C₂²	1	4	yes	—
25	#15688885	C₂²	1	4	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂×#106×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#384	C₂²	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	#13094016	—	2	4	no	—
33	#13094016	C₂×C₄	1	8	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	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
40	#13093958	C₂×C₄	1	8	yes	—
41	#13093956	C₂×C₄	1	8	yes	—
42	#13093935	—	2	2	no	—
43	#13093934	—	2	2	no	—
44	#13093934	Q₈	1	4	yes	—
45	C₈×Dic₄₇	—	2	2	no	—
46	C₈×Dic₄₇	Q₈	1	4	yes	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	—	2	4	no	—
49	C₄₇⋊#95	D₄	1	4	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#106×C₄₇	—	2	4	no	—
56	#106×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₄₇	#41	1	2	yes	—
69	C₂×C₈×C₄₇	#41	1	2	yes	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	8	4	no	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	#41	1	2	yes	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
77	Dic₄₇⋊C₄	—	2	16	no	—
78	Dic₄₇⋊C₄	—	8	4	no	—
79	C₄×Dic₄₇	—	2	1	no	—
80	C₄×Dic₄₇	—	2	1	no	—
81	C₄×Dic₄₇	—	8	4	no	—
82	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
83	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	8	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	4	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
109	C₈×C₄₇	C₄×Q₈	1	4	yes	—
110	C₈×C₄₇	C₄×Q₈	1	4	yes	—
111	C₄₇⋊C₈	—	2	2	no	—
112	C₄₇⋊C₈	—	2	2	no	—
113	#21722	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	#532	1	1	yes	characteristic
117	C₂²×C₄₇	#577	1	4	yes	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	8	4	no	—
120	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₄₇	#533	1	1	yes	characteristic
123	C₄×C₄₇	#533	1	1	yes	characteristic
124	Dic₄₇	—	2	4	no	—
125	Dic₄₇	—	4	2	no	—
126	Dic₄₇	—	4	8	no	—
127	C₂×#499	—	47	1	no	—
128	#2330	—	47	2	no	—
129	#1992	—	47	2	no	—
130	#1943	—	47	2	no	—
131	#1941	—	47	2	no	—
132	#1844	D₄₇	1	1	yes	characteristic
133	#1776	—	47	2	no	—
134	C₄×#106	—	47	2	no	—
135	#1726	—	47	1	no	—
136	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄₇	#2952	1	2	yes	—
138	C₂×C₄₇	#3163	1	2	yes	—
139	C₂×C₄₇	#3174	1	2	yes	—
140	C₂×#125	—	47	2	no	—
141	#499	—	47	8	no	—
142	C₂×#107	—	47	2	no	—
143	C₂×#106	—	47	2	no	—
144	C₂×#106	C₂×D₄₇	1	2	yes	—
145	C₂×#106	C₂×D₄₇	1	2	yes	—
146	C₂×#97	—	47	1	no	—
147	C₂×#97	—	47	1	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×C₄×C₈	—	47	1	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	#389	—	47	4	no	—
153	#389	—	94	4	no	—
154	#387	—	94	4	no	—
155	#384	C₂×D₄₇	1	2	yes	—
156	C₄×C₄⋊C₄	—	47	4	no	—
157	#376	—	47	1	no	—
158	#340	—	47	4	no	—
159	#340	—	47	4	no	—
160	C₄₇	#21722	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₈	—	47	1	no	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
165	#125	—	47	8	no	—
166	#125	—	94	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₄⋊C₄	—	94	4	no	—
170	C₂×C₄⋊C₄	—	94	4	no	—
171	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	94	4	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#107	—	47	8	no	—
177	#106	—	2	4	no	—
178	#106	—	2	4	no	—
179	#106	—	47	8	no	—
180	#97	—	47	4	no	—
181	#97	—	94	2	no	—
182	#97	—	94	2	no	—
183	C₄×C₈	—	47	4	no	—
184	C₄×C₈	—	94	2	no	—
185	C₄×C₈	C₄×D₄₇	1	4	yes	—
186	#95	—	47	4	no	—
187	#95	—	94	4	no	—
188	#95	—	94	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	94	4	no	—
193	C₂²×C₄	D₄×D₄₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
195	C₂×C₈	—	47	2	no	—
196	C₂×C₈	—	47	2	no	—
197	C₂×C₈	—	94	1	no	—
198	C₂×C₈	—	94	1	no	—
199	C₂×C₈	—	94	1	no	—
200	C₂×C₈	—	94	1	no	—
201	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
203	C₂×C₈	#10625352	1	2	yes	—
204	C₂×C₈	#10625352	1	2	yes	—
205	C₂×C₈	Q₈×D₄₇	1	2	yes	—
206	C₂×C₈	Q₈×D₄₇	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	8	4	no	—
210	C₄⋊C₄	—	47	8	no	—
211	C₄⋊C₄	—	94	16	no	—
212	C₄⋊C₄	—	94	4	no	—
213	C₄⋊C₄	—	376	4	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	47	2	no	—
216	C₄²	—	94	1	no	—
217	C₄²	—	94	1	no	—
218	C₄²	—	376	4	no	—
219	C₄²	C₂×C₄×D₄₇	1	2	yes	—
220	C₄²	#10625354	1	2	yes	—
221	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	47	4	no	—
227	C₂×C₄	—	94	4	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	94	4	no	—
231	C₂×C₄	—	188	4	no	—
232	C₂×C₄	—	188	4	no	—
233	C₂×C₄	#13093986	1	4	yes	—
234	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
239	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
240	C₈	—	94	2	no	—
241	C₈	—	94	2	no	—
242	C₈	#13093989	1	4	yes	—
243	C₈	#13093989	1	4	yes	—
244	C₂²	C₂×#13093986	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
246	C₂²	#15689671	1	1	yes	characteristic
247	C₂²	#15689929	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	8	4	no	—
250	C₄	—	94	4	no	—
251	C₄	—	188	2	no	—
252	C₄	—	188	8	no	—
253	C₄	C₂×#13093989	1	1	yes	characteristic
254	C₄	C₂×#13093989	1	1	yes	characteristic
255	C₄	#15689678	1	1	yes	characteristic
256	C₄	#15689678	1	1	yes	characteristic
257	C₂	#20245955	1	1	yes	characteristic
258	C₂	#20252271	1	2	yes	—
259	C₂	#20253913	1	2	yes	—
260	C₂	#20254013	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#21722
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. #15689525	C₄	semidirect
160. C₄₇	#21722	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
#20246127	2
#20244356	2
#20244124	2
#20244114	2
#20243288	2
#106×Dic₄₇	2
#20243074	2
C₄₇×#1844	2
#21722	47