Subgroups — #15688860

Order 3008.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₄₇	—	2	1	no	—
7	C₄⋊C₄×C₄₇	—	2	1	no	—
8	C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊C₄×C₄₇	—	2	1	no	—
10	C₂×C₄₇⋊C₈	—	2	1	no	—
11	C₂³×C₄₇	D₄	1	1	yes	characteristic
12	Q₈×C₄₇	—	4	1	no	—
13	C₂×C₄×C₄₇	—	2	1	no	—
14	C₂×C₄×C₄₇	—	2	1	no	—
15	C₂×C₄×C₄₇	—	4	1	no	—
16	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂×Dic₄₇	—	2	2	no	—
19	C₂×Dic₄₇	—	4	1	no	—
20	C₂×Dic₄₇	—	4	1	no	—
21	C₄₇⋊C₈	—	4	1	no	—
22	C₂²×C₄₇	—	2	1	no	—
23	C₂²×C₄₇	—	2	1	no	—
24	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄₇	SD₁₆	1	1	yes	characteristic
26	C₂²×C₄₇	Q₁₆	1	1	yes	characteristic
27	C₄×C₄₇	—	2	1	no	—
28	C₄×C₄₇	—	2	1	no	—
29	C₄×C₄₇	—	4	1	no	—
30	C₄×C₄₇	—	4	1	no	—
31	Dic₄₇	—	4	2	no	—
32	C₂×C₄₇	—	2	1	no	—
33	C₂×C₄₇	—	2	1	no	—
34	C₂×C₄₇	#99	1	1	yes	characteristic
35	C₂×C₄₇	#103	1	1	yes	characteristic
36	C₂×C₄₇	#104	1	1	yes	characteristic
37	#328	—	47	1	no	maximal, Sylow
38	C₄₇	#328	1	1	yes	characteristic, complemented, Sylow
39	C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
40	C₂²⋊C₈	—	47	1	no	—
41	#95	—	47	1	no	—
42	C₂×Q₈	Dic₄₇	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	94	1	no	—
44	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
45	C₂×C₈	—	94	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	Dic₄₇	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	2	1	no	—
49	C₂³	C₂²⋊D₄₇	1	1	yes	characteristic
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	94	2	no	—
55	C₂×C₄	—	188	1	no	—
56	C₂×C₄	—	188	1	no	—
57	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
59	C₈	—	188	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	#10625336	1	1	yes	characteristic
63	C₂²	#10625337	1	1	yes	characteristic
64	C₂²	C₂²⋊Dic₄₇	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	4	1	no	—
69	C₄	—	188	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#13093939	1	1	yes	characteristic
73	C₂	#13093948	1	1	yes	characteristic
74	C₂	#13093951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₄₇
Fitting	C₂²⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²⋊C₄
prime radical	C₂³×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂²
hypercenter	C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#328
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
4. C₂×Q₈×C₄₇	C₄	semidirect
8. C₄⋊C₄×C₄₇	C₄	semidirect
38. C₄₇	#328	semidirect
42. C₂×Q₈	Dic₄₇	semidirect
47. C₄⋊C₄	Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₄₇⋊C₈)	2
C₄₇⋊#95	2
C₂²⋊Q₈×C₄₇	2
#328	47