Subgroups — #26135892

Order 12032.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688950	C₂	1	2	yes	maximal
2	#20244121	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	2	yes	maximal
4	#20241466	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	1	yes	characteristic
8	C₂×#13093958	C₂²	1	1	yes	characteristic
9	#15689021	—	2	1	no	—
10	#15688950	—	2	4	no	—
11	#15688886	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
13	C₄₇×#500	—	2	4	no	—
14	C₂×#107×C₄₇	—	2	1	no	—
15	C₂×#107×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₇	—	2	1	no	—
18	C₂×C₄⋊Dic₄₇	—	2	1	no	—
19	C₂×Dic₄₇⋊C₄	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
23	#13093958	—	2	1	no	—
24	#13093958	—	2	2	no	—
25	#13093958	Q₈	1	2	yes	complemented
26	C₄₇⋊#95	—	4	1	no	—
27	C₂²×Q₈×C₄₇	—	4	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₄₇	—	2	4	no	—
31	C₄⋊Q₈×C₄₇	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
34	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
35	#107×C₄₇	—	2	4	no	—
36	#107×C₄₇	—	2	2	no	—
37	#107×C₄₇	D₄	1	4	yes	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	C₂×Q₈×C₄₇	—	4	4	no	—
40	C₂×Q₈×C₄₇	—	8	2	no	—
41	C₂×Q₈×C₄₇	—	8	2	no	—
42	C₂²×C₄×C₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₄₇	—	4	1	no	—
47	C₂²×Dic₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	—	2	1	no	—
49	C₂×C₈×C₄₇	—	2	1	no	—
50	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
51	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄⋊C₄×C₄₇	—	4	4	no	—
57	C₄⋊C₄×C₄₇	—	4	2	no	—
58	C₄²×C₄₇	—	2	1	no	—
59	C₄²×C₄₇	—	2	1	no	—
60	C₄⋊Dic₄₇	—	2	2	no	—
61	C₄⋊Dic₄₇	—	4	1	no	—
62	C₄⋊Dic₄₇	—	4	1	no	—
63	C₄⋊Dic₄₇	—	4	1	no	—
64	Dic₄₇⋊C₄	—	8	2	no	—
65	C₄₇⋊C₁₆	—	2	4	no	—
66	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₄₇	—	8	8	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	4	4	no	—
73	C₂×C₄×C₄₇	—	8	2	no	—
74	C₂×C₄×C₄₇	—	8	2	no	—
75	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
76	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
77	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	8	1	no	—
85	C₂×Dic₄₇	—	8	1	no	—
86	C₂×Dic₄₇	—	8	1	no	—
87	C₂×Dic₄₇	—	8	1	no	—
88	C₂×Dic₄₇	—	8	1	no	—
89	C₈×C₄₇	—	2	2	no	—
90	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
92	#13452	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	#447	1	1	yes	characteristic
94	C₂²×C₄₇	#474	1	1	yes	characteristic
95	C₂²×C₄₇	#500	1	1	yes	characteristic
96	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
98	C₂²×C₄₇	#509	1	1	yes	characteristic
99	C₂²×C₄₇	#510	1	1	yes	characteristic
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	8	4	no	—
103	C₄×C₄₇	#395	1	1	yes	characteristic
104	C₄×C₄₇	#451	1	1	yes	characteristic
105	C₄×C₄₇	#477	1	1	yes	characteristic
106	C₄×C₄₇	#501	1	1	yes	characteristic
107	Dic₄₇	—	8	2	no	—
108	Dic₄₇	—	16	1	no	—
109	C₂×#500	D₄₇	1	1	yes	characteristic
110	C₂×#367	—	47	2	no	—
111	#1942	—	47	1	no	—
112	#1380	—	47	2	no	—
113	#1374	—	47	1	no	—
114	C₂×C₄₇	#2057	1	1	yes	characteristic
115	C₂×C₄₇	#2185	1	1	yes	characteristic
116	C₂×C₄₇	#2187	1	1	yes	characteristic
117	C₂×C₄₇	#2225	1	1	yes	characteristic
118	C₂×C₄₇	#2226	1	1	yes	characteristic
119	C₂×C₄₇	#2254	1	1	yes	characteristic
120	C₂×C₄₇	#2255	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	—	2	1	no	—
122	C₂²×C₁₆	—	47	2	no	—
123	#500	—	2	4	no	—
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	47	1	no	—
126	C₂×#107	—	47	1	no	—
127	C₂×#107	C₂×D₄₇	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	#389	—	94	1	no	—
130	#367	—	94	4	no	—
131	#340	—	94	1	no	—
132	C₄₇	#13452	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	4	1	no	—
134	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	94	1	no	—
141	C₂×C₄⋊C₄	—	94	1	no	—
142	C₂×C₄⋊C₄	—	188	1	no	—
143	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₁₆	—	47	4	no	—
145	C₂×C₁₆	—	94	2	no	—
146	C₂×C₁₆	—	94	2	no	—
147	#107	—	2	4	no	—
148	#107	—	2	2	no	—
149	#107	—	47	2	no	—
150	#107	—	94	1	no	—
151	#107	—	94	2	no	—
152	#107	C₂²⋊D₄₇	1	4	yes	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	188	1	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	—	8	2	no	—
157	C₂×Q₈	—	8	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	188	1	no	—
160	C₂²×C₄	—	188	1	no	—
161	C₂²×C₄	#10625334	1	1	yes	characteristic
162	C₂²×C₄	#10625334	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	#10625337	1	2	yes	—
167	C₂×C₈	Q₈×D₄₇	1	2	yes	—
168	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
169	C₂×C₈	#10625354	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	94	2	no	—
175	C₄⋊C₄	—	376	2	no	—
176	C₄⋊C₄	—	188	1	no	—
177	C₄⋊C₄	—	188	1	no	—
178	C₄⋊C₄	—	188	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	94	4	no	—
182	C₂³	C₂×#10625334	1	1	yes	characteristic
183	Q₈	—	8	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	376	1	no	—
192	C₂×C₄	—	376	1	no	—
193	C₂×C₄	—	376	1	no	—
194	C₂×C₄	—	376	1	no	—
195	C₂×C₄	—	376	1	no	—
196	C₂×C₄	—	188	2	no	—
197	C₂×C₄	#13093967	1	2	yes	—
198	C₂×C₄	#13093968	1	2	yes	—
199	C₂×C₄	#13093993	1	1	yes	characteristic
200	C₂×C₄	#13094000	1	1	yes	characteristic
201	C₂×C₄	#13094022	1	1	yes	characteristic
202	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
203	C₂×C₄	#13094050	1	1	yes	characteristic
204	C₂×C₄	#13094055	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#13094000	1	2	yes	—
207	C₈	#13094022	1	2	yes	—
208	C₂²	#15689257	1	1	yes	characteristic
209	C₂²	#15689400	1	1	yes	characteristic
210	C₂²	#15689526	1	1	yes	characteristic
211	C₂²	C₂×#13093967	1	1	yes	characteristic
212	C₂²	C₂×#13093968	1	1	yes	characteristic
213	C₂²	#15689559	1	1	yes	characteristic
214	C₂²	#15689564	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	4	no	—
218	C₄	—	376	2	no	—
219	C₄	—	752	1	no	—
220	C₄	#15689042	1	1	yes	characteristic
221	C₄	#15689279	1	1	yes	characteristic
222	C₄	#15689415	1	1	yes	characteristic
223	C₄	#15689529	1	1	yes	characteristic
224	C₂	#20244713	1	1	yes	characteristic
225	C₂	#20245305	1	1	yes	characteristic
226	C₂	#20245317	1	1	yes	characteristic
227	C₂	#20245527	1	1	yes	characteristic
228	C₂	#20245532	1	1	yes	characteristic
229	C₂	#20245668	1	1	yes	characteristic
230	C₂	#20245671	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#13452
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
22. C₂×C₄₇⋊C₁₆	Q₈	semidirect
25. #13093958	Q₈	semidirect
132. C₄₇	#13452	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688950	2
#20244121	2
#20241479	2
#20241466	2
C₂×C₄₇×#500	2
#13452	47