Subgroups — #15688850

Order 3008.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₈₈	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄₇	—	2	1	no	—
6	C₂×C₈×C₄₇	—	2	1	no	—
7	C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	(C₂×D₄₇)⋊C₄	—	2	1	no	—
9	C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄₇⋊C₈	—	2	1	no	—
11	C₂²×D₄₇	—	2	1	no	—
12	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂×Dic₄₇	—	2	1	no	—
16	D₁₈₈	—	2	1	no	—
17	D₁₈₈	C₈	1	2	yes	complemented
18	C₄×D₄₇	—	4	1	no	—
19	C₄×D₄₇	—	4	1	no	—
20	C₈×C₄₇	—	4	1	no	—
21	C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₇	—	2	2	no	—
24	C₂×D₄₇	—	4	1	no	—
25	C₂×D₄₇	—	4	1	no	—
26	C₄×C₄₇	—	2	1	no	—
27	C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₄₇	M₁₆	1	1	yes	characteristic
29	C₄×C₄₇	D₈	1	1	yes	characteristic
30	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
31	Dic₄₇	—	4	1	no	—
32	C₂×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₄₇	#102	1	1	yes	characteristic
34	C₂×C₄₇	#104	1	1	yes	characteristic
35	D₄₇	—	4	2	no	—
36	D₄⋊C₈	—	47	1	no	maximal, Sylow
37	C₄₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₄	—	47	1	no	—
39	C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
40	C₄×C₈	—	47	1	no	—
41	C₂×D₄	—	47	1	no	—
42	C₂²×C₄	—	94	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	94	1	no	—
45	C₄⋊C₄	—	47	1	no	—
46	C₂²⋊C₄	—	94	1	no	—
47	C₄²	C₂×D₄₇	1	1	yes	characteristic
48	C₂³	—	94	1	no	—
49	D₄	—	47	2	no	—
50	D₄	—	94	1	no	—
51	C₂×C₄	—	94	1	no	—
52	C₂×C₄	—	188	1	no	—
53	C₂×C₄	—	188	1	no	—
54	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
55	C₂×C₄	D₁₈₈	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	94	2	no	—
59	C₂²	—	94	2	no	—
60	C₂²	—	188	1	no	—
61	C₂²	—	188	1	no	—
62	C₂²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	188	1	no	—
65	C₄	C₈×D₄₇	1	1	yes	characteristic
66	C₄	#10625324	1	1	yes	characteristic
67	C₄	#10625334	1	1	yes	characteristic
68	C₄	#10625336	1	1	yes	characteristic
69	C₂	—	188	2	no	—
70	C₂	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
71	C₂	#13093944	1	1	yes	characteristic
72	C₂	#13093950	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₈
Frattini	C₄²	C₂×D₄₇
Fitting	C₄⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²⋊C₄
prime radical	C₂×D₁₈₈	C₄
cosocle	C₄²×C₄₇	C₂²
hypercenter	C₄⋊C₈	D₄₇
hypocenter	C₄₇	D₄⋊C₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄₇⋊C₈	C₂	semidirect
3. C₄⋊C₈×C₄₇	C₂	semidirect
17. D₁₈₈	C₈	semidirect
37. C₄₇	D₄⋊C₈	semidirect
39. C₄⋊C₈	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₈₈	2
C₄×C₄₇⋊C₈	2
C₄⋊C₈×C₄₇	2
D₄⋊C₈	47