Subgroups — #26316166

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251766	C₂	1	1	yes	maximal, characteristic
2	#20244529	C₂	1	4	yes	maximal
3	#20244209	C₂	1	2	yes	maximal
4	#20243523	C₂	1	2	yes	maximal
5	#20243403	C₂	1	4	yes	maximal
6	#20243241	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094024	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093948	C₂²	1	8	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	#15689029	—	2	2	no	—
13	#15689029	—	2	2	no	—
14	#15688989	C₂²	1	2	yes	—
15	#15688989	C₂²	1	2	yes	—
16	#15688885	C₂²	1	8	yes	—
17	#15688878	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#391	C₂²	1	2	yes	—
21	C₄₇×#389	—	2	2	no	—
22	C₄₇×#384	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13094024	—	4	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	#13093948	—	4	8	no	—
32	#13093935	—	2	4	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₇	—	2	2	no	—
38	C₄⋊Q₈×C₄₇	Q₈	1	4	yes	complemented
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×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	#95×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂×Dic₉₄	—	4	4	no	—
64	C₂×Dic₉₄	—	4	4	no	—
65	C₂×Dic₉₄	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₄⋊C₄×C₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	4	4	no	—
69	C₄⋊C₄×C₄₇	—	4	4	no	—
70	C₄²×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	—	4	4	no	—
74	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₇	#41	1	2	yes	—
76	C₄²×C₄₇	#41	1	4	yes	—
77	C₄⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	4	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₄₇	—	2	8	no	—
84	Q₈×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
97	C₂×C₄×C₄₇	#124	1	2	yes	—
98	C₂×C₄×C₄₇	#124	1	2	yes	—
99	C₂×C₄×C₄₇	#137	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	4	4	no	—
107	Dic₉₄	—	8	8	no	—
108	C₄₇⋊C₈	—	4	4	no	—
109	#29147	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
112	C₂²×C₄₇	#535	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#137	1	2	yes	—
114	C₂²×C₄₇	#575	1	2	yes	—
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	4	4	no	—
120	C₄×C₄₇	—	8	2	no	—
121	C₄×C₄₇	#554	1	2	yes	—
122	C₄×C₄₇	#555	1	2	yes	—
123	Dic₄₇	—	8	4	no	—
124	#2862	—	47	1	no	—
125	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
126	#2024	—	47	4	no	—
127	#1960	—	47	2	no	—
128	#1836	—	47	2	no	—
129	#1805	—	47	4	no	—
130	#1767	—	47	1	no	—
131	C₂×C₄₇	#2563	1	1	yes	characteristic
132	C₂×C₄₇	#3078	1	2	yes	—
133	C₂×C₄₇	#3135	1	1	yes	characteristic
134	C₂×C₄₇	#3139	1	1	yes	characteristic
135	C₂×C₄₇	#3215	1	2	yes	—
136	C₂×C₄⋊Q₈	—	47	2	no	—
137	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
139	C₂×C₄⋊C₈	—	47	2	no	—
140	C₂×#103	—	47	8	no	—
141	C₂×#97	—	47	2	no	—
142	#391	—	94	2	no	—
143	#391	—	94	2	no	—
144	#391	C₂×D₄₇	1	2	yes	—
145	#389	—	2	2	no	—
146	#384	C₂×D₄₇	1	1	yes	characteristic
147	#382	—	47	2	no	—
148	#382	—	47	2	no	—
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₄³	C₂×D₄₇	1	1	yes	characteristic
151	#340	—	47	8	no	—
152	#336	—	47	2	no	—
153	C₄₇	#29147	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	94	2	no	—
155	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
156	C₂²×C₈	—	47	4	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	188	4	no	—
159	C₄⋊Q₈	Dic₉₄	1	4	yes	complemented
160	C₄×Q₈	—	2	8	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	47	4	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₄⋊C₈	—	94	4	no	—
174	#103	—	188	8	no	—
175	#97	—	94	4	no	—
176	#95	—	2	4	no	—
177	#95	—	94	4	no	—
178	#95	—	94	2	no	—
179	#95	—	94	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	188	4	no	—
184	C₂×Q₈	—	188	4	no	—
185	C₂×Q₈	—	188	4	no	—
186	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	94	4	no	—
190	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	—	94	4	no	—
197	C₂×C₈	—	94	4	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	94	8	no	—
203	C₄⋊C₄	—	188	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	C₂×Dic₉₄	1	2	yes	—
209	C₄²	#10625350	1	2	yes	—
210	C₄²	#10625352	1	4	yes	—
211	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	Q₈	—	4	4	no	—
214	Q₈	—	376	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	188	4	no	—
227	C₂×C₄	—	188	4	no	—
228	C₂×C₄	—	188	4	no	—
229	C₂×C₄	#13094003	1	4	yes	—
230	C₂×C₄	#13094011	1	2	yes	—
231	C₂×C₄	#13094011	1	2	yes	—
232	C₂×C₄	#13094055	1	4	yes	—
233	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
234	C₂×C₄	C₂×#10625352	1	2	yes	—
235	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
236	C₂×C₄	#13094070	1	2	yes	—
237	C₈	—	188	4	no	—
238	C₂²	C₂×#13094003	1	1	yes	characteristic
239	C₂²	C₂×#13094011	1	1	yes	characteristic
240	C₂²	#15689687	1	1	yes	characteristic
241	C₂²	C₂×#13094055	1	2	yes	—
242	C₂²	#15689921	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	376	4	no	—
250	C₄	#15689830	1	2	yes	—
251	C₄	#15689838	1	2	yes	—
252	C₂	#20248561	1	1	yes	characteristic
253	C₂	#20253193	1	2	yes	—
254	C₂	#20253733	1	1	yes	characteristic
255	C₂	#20253768	1	1	yes	characteristic
256	C₂	#20254387	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#29147
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊Q₈×C₄₇	Q₈	semidirect
153. C₄₇	#29147	semidirect
159. C₄⋊Q₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20244529	2
#20244209	2
#20243523	2
#20243403	2
#20243241	2
C₄×C₄⋊Q₈×C₄₇	2
#29147	47