Subgroups — #26124726

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688878	C₂	1	3	yes	maximal
2	C₂×#15688878	C₂	1	3	yes	maximal
3	C₄×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	C₂²	1	3	yes	—
5	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
6	#15688878	—	2	6	no	—
7	#15688878	—	2	6	no	—
8	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
9	C₂×#95×C₄₇	C₄	1	3	yes	—
10	C₂×#95×C₄₇	C₄	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₈	—	2	3	no	—
12	C₂²×C₄₇⋊C₈	—	2	3	no	—
13	C₂²×C₄₇⋊C₈	—	2	3	no	—
14	C₂²×C₄₇⋊C₈	—	2	3	no	—
15	C₂²×C₄₇⋊C₈	—	2	3	no	—
16	C₂²×C₄₇⋊C₈	—	2	3	no	—
17	C₂²×C₄₇⋊C₈	—	2	3	no	—
18	C₂²×C₄₇⋊C₈	—	2	3	no	—
19	C₂²×C₄₇⋊C₈	D₄	1	6	yes	—
20	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	—	2	3	no	—
25	C₂×C₄²×C₄₇	—	2	3	no	—
26	C₂×C₄²×C₄₇	—	2	3	no	—
27	C₂×C₄²×C₄₇	—	2	3	no	—
28	C₂×C₄²×C₄₇	—	2	3	no	—
29	C₂×C₄²×C₄₇	—	2	3	no	—
30	#95×C₄₇	—	2	6	no	—
31	#95×C₄₇	C₈	1	4	yes	complemented
32	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	—	2	3	no	—
34	C₂²×C₄×C₄₇	—	2	3	no	—
35	C₂²×C₄×C₄₇	—	2	3	no	—
36	C₂²×C₄×C₄₇	—	2	3	no	—
37	C₂²×C₄×C₄₇	—	2	3	no	—
38	C₂²×C₄×C₄₇	—	2	3	no	—
39	C₂²×C₄×C₄₇	—	2	3	no	—
40	C₂²×C₄×C₄₇	—	2	3	no	—
41	C₂²×C₄×C₄₇	—	2	3	no	—
42	C₂²×C₄×C₄₇	—	2	3	no	—
43	C₂²×C₄×C₄₇	—	2	3	no	—
44	C₂²×C₄×C₄₇	—	2	3	no	—
45	C₂²×C₄×C₄₇	C₂×C₈	1	6	yes	—
46	C₂²×C₄×C₄₇	M₁₆	1	6	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	#41	1	3	yes	—
50	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	3	yes	—
52	C₂²×C₄×C₄₇	#41	1	3	yes	—
53	C₄²×C₄₇	—	4	3	no	—
54	C₄²×C₄₇	—	4	3	no	—
55	C₄²×C₄₇	—	4	3	no	—
56	C₄²×C₄₇	—	4	3	no	—
57	C₂×C₄₇⋊C₈	—	2	6	no	—
58	C₂×C₄₇⋊C₈	—	2	6	no	—
59	C₂×C₄₇⋊C₈	—	2	6	no	—
60	C₂×C₄₇⋊C₈	—	2	6	no	—
61	C₂×C₄₇⋊C₈	—	4	3	no	—
62	C₂×C₄₇⋊C₈	—	4	3	no	—
63	C₂×C₄₇⋊C₈	—	4	3	no	—
64	C₂×C₄₇⋊C₈	—	4	1	no	—
65	C₂×C₄₇⋊C₈	—	4	3	no	—
66	C₂×C₄₇⋊C₈	—	4	3	no	—
67	C₂³×C₄₇	#117	1	3	yes	—
68	C₂³×C₄₇	C₄×D₄	1	3	yes	—
69	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
70	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
71	C₂³×C₄₇	C₂×M₁₆	1	3	yes	—
72	C₂³×C₄₇	#131	1	3	yes	—
73	C₂³×C₄₇	#131	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	6	no	—
75	C₂×C₄×C₄₇	—	2	6	no	—
76	C₂×C₄×C₄₇	—	2	6	no	—
77	C₂×C₄×C₄₇	—	2	6	no	—
78	C₂×C₄×C₄₇	—	2	6	no	—
79	C₂×C₄×C₄₇	—	2	6	no	—
80	C₂×C₄×C₄₇	—	4	3	no	—
81	C₂×C₄×C₄₇	—	4	3	no	—
82	C₂×C₄×C₄₇	—	4	3	no	—
83	C₂×C₄×C₄₇	—	4	3	no	—
84	C₂×C₄×C₄₇	—	4	3	no	—
85	C₂×C₄×C₄₇	—	4	3	no	—
86	C₂×C₄×C₄₇	—	4	3	no	—
87	C₂×C₄×C₄₇	—	4	3	no	—
88	C₂×C₄×C₄₇	#121	1	3	yes	—
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
90	C₂×C₄×C₄₇	#123	1	3	yes	—
91	C₂×C₄×C₄₇	#123	1	3	yes	—
92	C₂×C₄×C₄₇	#124	1	3	yes	—
93	C₂×C₄×C₄₇	#125	1	3	yes	—
94	C₂×C₄×C₄₇	#126	1	3	yes	—
95	C₂×C₄×C₄₇	#126	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#126	1	3	yes	—
97	C₂×C₄×C₄₇	#126	1	3	yes	—
98	C₄₇⋊C₈	—	4	2	no	—
99	C₄₇⋊C₈	—	4	6	no	—
100	#11298	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	#383	1	1	yes	characteristic
102	C₂²×C₄₇	#387	1	3	yes	—
103	C₂²×C₄₇	#388	1	3	yes	—
104	C₂²×C₄₇	#431	1	3	yes	—
105	C₂²×C₄₇	#432	1	3	yes	—
106	C₂²×C₄₇	#433	1	3	yes	—
107	C₂²×C₄₇	#433	1	3	yes	—
108	C₂²×C₄₇	C₈×D₄	1	3	yes	—
109	C₂²×C₄₇	#435	1	3	yes	—
110	C₂²×C₄₇	#435	1	3	yes	—
111	C₂²×C₄₇	#436	1	3	yes	—
112	C₂²×C₄₇	C₈×Q₈	1	1	yes	characteristic
113	C₂²×C₄₇	#446	1	3	yes	—
114	C₄×C₄₇	—	4	6	no	—
115	C₄×C₄₇	—	4	6	no	—
116	C₄×C₄₇	#396	1	1	yes	characteristic
117	C₄×C₄₇	#397	1	3	yes	—
118	C₄×C₄₇	#400	1	3	yes	—
119	C₄×C₄₇	#401	1	1	yes	characteristic
120	C₂×#336	—	47	3	no	—
121	C₂×#336	—	47	3	no	—
122	C₄×#95	D₄₇	1	1	yes	characteristic
123	C₂×C₄₇	#1443	1	1	yes	characteristic
124	C₂×C₄₇	#1839	1	1	yes	characteristic
125	C₂×C₄₇	#1840	1	1	yes	characteristic
126	C₂×C₄₇	#1916	1	3	yes	—
127	C₂×C₄₇	#1917	1	3	yes	—
128	C₂×C₄₇	#1923	1	3	yes	—
129	C₂×C₄₇	#1924	1	3	yes	—
130	C₂³×C₈	—	47	3	no	—
131	C₂³×C₈	—	47	1	no	—
132	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
133	C₂×#95	Dic₄₇	1	3	yes	—
134	C₂×#95	Dic₄₇	1	1	yes	characteristic
135	#336	—	94	6	no	—
136	#336	—	94	6	no	—
137	C₄₇	#11298	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
139	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂²×C₈	—	47	6	no	—
142	C₂²×C₈	—	47	2	no	—
143	C₂²×C₈	—	94	3	no	—
144	C₂²×C₈	—	94	3	no	—
145	C₂²×C₈	—	94	3	no	—
146	C₂²×C₈	—	94	3	no	—
147	C₂²×C₈	—	94	3	no	—
148	C₂²×C₈	—	94	3	no	—
149	C₂²×C₈	—	94	3	no	—
150	C₂²×C₈	—	94	3	no	—
151	C₂×C₄²	—	2	3	no	—
152	C₂×C₄²	—	2	3	no	—
153	C₂×C₄²	—	2	3	no	—
154	C₂×C₄²	—	2	3	no	—
155	C₂×C₄²	—	2	3	no	—
156	C₂×C₄²	—	2	3	no	—
157	#95	—	2	6	no	—
158	#95	C₄₇⋊C₈	1	4	yes	complemented
159	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
160	C₂²×C₄	—	2	3	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	—	2	3	no	—
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	—	2	3	no	—
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	3	no	—
172	C₂²×C₄	C₂×C₄₇⋊C₈	1	6	yes	—
173	C₂²×C₄	#10625329	1	6	yes	—
174	C₂²×C₄	#10625350	1	3	yes	—
175	C₂²×C₄	#10625350	1	3	yes	—
176	C₂²×C₄	D₄×D₄₇	1	3	yes	—
177	C₂²×C₄	#10625352	1	3	yes	—
178	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
179	C₂²×C₄	#10625354	1	1	yes	characteristic
180	C₂×C₈	—	94	6	no	—
181	C₂×C₈	—	94	6	no	—
182	C₂×C₈	—	94	6	no	—
183	C₂×C₈	—	94	6	no	—
184	C₂×C₈	—	188	3	no	—
185	C₂×C₈	—	188	3	no	—
186	C₂×C₈	—	188	3	no	—
187	C₂×C₈	—	188	1	no	—
188	C₂×C₈	—	188	3	no	—
189	C₂×C₈	—	188	3	no	—
190	C₄²	—	4	3	no	—
191	C₄²	—	4	3	no	—
192	C₄²	—	4	3	no	—
193	C₄²	—	4	3	no	—
194	C₂³	#13093983	1	3	yes	—
195	C₂³	D₄×Dic₄₇	1	3	yes	—
196	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
197	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
198	C₂³	C₂×#10625329	1	3	yes	—
199	C₂³	#13094031	1	1	yes	characteristic
200	C₂³	#13094031	1	3	yes	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	—	4	3	no	—
210	C₂×C₄	—	4	3	no	—
211	C₂×C₄	—	4	3	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	—	4	3	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	#13093997	1	3	yes	—
216	C₂×C₄	#13094002	1	3	yes	—
217	C₂×C₄	#13094006	1	3	yes	—
218	C₂×C₄	#13094009	1	3	yes	—
219	C₂×C₄	#13094010	1	3	yes	—
220	C₂×C₄	#13094014	1	3	yes	—
221	C₂×C₄	#13094017	1	3	yes	—
222	C₂×C₄	#13094018	1	1	yes	characteristic
223	C₂×C₄	#13094018	1	3	yes	—
224	C₂×C₄	#13094019	1	3	yes	—
225	C₈	—	188	2	no	—
226	C₈	—	188	6	no	—
227	C₂²	#15688991	1	1	yes	characteristic
228	C₂²	#15689009	1	3	yes	—
229	C₂²	#15689016	1	3	yes	—
230	C₂²	#15689176	1	3	yes	—
231	C₂²	#15689180	1	3	yes	—
232	C₂²	#15689184	1	3	yes	—
233	C₂²	#15689184	1	3	yes	—
234	C₂²	D₄×C₄₇⋊C₈	1	3	yes	—
235	C₂²	#15689195	1	3	yes	—
236	C₂²	#15689195	1	3	yes	—
237	C₂²	#15689201	1	3	yes	—
238	C₂²	Q₈×C₄₇⋊C₈	1	1	yes	characteristic
239	C₂²	#15689254	1	3	yes	—
240	C₄	—	4	6	no	—
241	C₄	—	4	6	no	—
242	C₄	#15689047	1	1	yes	characteristic
243	C₄	#15689052	1	3	yes	—
244	C₄	#15689064	1	3	yes	—
245	C₄	C₄₇⋊#401	1	1	yes	characteristic
246	C₂	#20241650	1	1	yes	characteristic
247	C₂	#20243531	1	1	yes	characteristic
248	C₂	#20243533	1	1	yes	characteristic
249	C₂	#20243947	1	3	yes	—
250	C₂	#20243954	1	3	yes	—
251	C₂	#20243996	1	3	yes	—
252	C₂	#20244003	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Fitting	C₄×#95×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#95	D₄₇
hypocenter	C₄₇	#11298
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
31. #95×C₄₇	C₈	semidirect
137. C₄₇	#11298	semidirect
158. #95	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688878	2
C₄×#95×C₄₇	2
#11298	47