Subgroups — #26103259

Order 12032.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	2	yes	maximal
2	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	—
5	C₄×C₄₇⋊C₁₆	—	2	2	no	—
6	C₄₇×#431	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#322	—	2	2	no	—
9	C₂×C₄₇⋊C₁₆	—	2	4	no	—
10	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₄₇	C₈	1	4	yes	—
13	C₂²⋊C₈×C₄₇	—	2	2	no	—
14	C₄×C₈×C₄₇	—	2	2	no	—
15	C₄×C₈×C₄₇	D₄	1	2	yes	—
16	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
17	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
20	C₂×C₈×C₄₇	—	2	4	no	—
21	C₂×C₈×C₄₇	—	4	2	no	—
22	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
24	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
26	C₄²×C₄₇	C₂×C₈	1	2	yes	—
27	C₄₇⋊C₁₆	—	4	4	no	—
28	C₂³×C₄₇	#97	1	1	yes	characteristic
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	—	2	4	no	—
31	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
32	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
34	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
35	C₈×C₄₇	—	2	2	no	—
36	C₈×C₄₇	—	2	2	no	—
37	C₈×C₄₇	—	4	4	no	—
38	#7198	—	47	1	no	maximal, Sylow
39	C₂²×C₄₇	—	2	1	no	—
40	C₂²×C₄₇	—	2	2	no	—
41	C₂²×C₄₇	#336	1	1	yes	characteristic
42	C₄×C₄₇	—	2	1	no	—
43	C₄×C₄₇	—	2	2	no	—
44	C₄×C₄₇	#344	1	2	yes	—
45	C₄×C₄₇	#350	1	4	yes	—
46	#2162	—	47	2	no	—
47	#1452	D₄₇	1	1	yes	characteristic
48	C₂×C₄₇	—	4	1	no	—
49	C₂×C₄₇	#1281	1	1	yes	characteristic
50	C₂×C₄₇	#1379	1	2	yes	—
51	#431	Dic₄₇	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
53	C₄⋊C₁₆	—	94	2	no	—
54	C₂²⋊C₁₆	—	47	4	no	—
55	C₄×C₁₆	—	94	2	no	—
56	#322	—	2	2	no	—
57	C₄₇	#7198	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	C₄×D₄₇	1	2	yes	—
59	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
60	C₂×C₁₆	—	94	4	no	—
61	C₄⋊C₈	C₄₇⋊C₈	1	4	yes	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	Dic₉₄	1	1	yes	characteristic
65	C₄×C₈	D₁₈₈	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
67	C₂²×C₄	#10625329	1	2	yes	—
68	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
72	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
73	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
74	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
75	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
76	C₁₆	—	188	4	no	—
77	C₂³	#13093935	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
81	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
83	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#15688878	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	#15688894	1	2	yes	—
93	C₄	#15688907	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	#20241233	1	1	yes	characteristic
96	C₂	#20241477	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1452	C₂
socle	C₂²×C₄₇	#336
prime radical	C₂³×C₄₇	#97
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#7198
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

Complements

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

N	H ≅ G/N	type
57. C₄₇	#7198	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
C₄₇×#1452	2
#7198	47