Subgroups — #15688866

Order 3008.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094021	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	—	2	1	no	—
5	C₂×D₁₈₈	—	2	1	no	—
6	C₂×D₁₈₈	C₄	1	2	yes	—
7	C₂×C₈×C₄₇	—	2	1	no	—
8	C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₈	—	2	1	no	—
10	C₂²×D₄₇	—	2	2	no	—
11	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
14	D₁₈₈	—	4	2	no	—
15	D₁₈₈	—	4	1	no	—
16	D₁₈₈	—	4	1	no	—
17	D₁₈₈	—	4	1	no	—
18	D₁₈₈	—	4	1	no	—
19	C₈×C₄₇	—	4	1	no	—
20	C₄₇⋊C₈	—	4	1	no	—
21	C₂²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₄₇	—	4	2	no	—
23	C₂×D₄₇	—	4	2	no	—
24	C₂×D₄₇	—	4	2	no	—
25	C₄×C₄₇	—	2	1	no	—
26	C₄×C₄₇	D₈	1	1	yes	characteristic
27	C₄×C₄₇	D₈	1	1	yes	characteristic
28	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
29	C₄×C₄₇	SD₁₆	1	1	yes	characteristic
30	C₂×C₄₇	#100	1	1	yes	characteristic
31	C₂×C₄₇	#102	1	1	yes	characteristic
32	C₂×C₄₇	#102	1	1	yes	characteristic
33	D₄₇	—	8	2	no	—
34	#331	—	47	1	no	maximal, Sylow
35	C₄₇	#331	1	1	yes	characteristic, complemented, Sylow
36	#127	—	47	1	no	—
37	C₄⋊C₈	—	47	1	no	—
38	C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
39	C₂×D₄	—	47	2	no	—
40	C₂×D₄	—	94	1	no	—
41	C₂×D₄	—	94	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	94	1	no	—
44	C₄²	C₂×D₄₇	1	1	yes	characteristic
45	C₂³	—	94	2	no	—
46	D₄	—	188	2	no	—
47	D₄	—	188	1	no	—
48	D₄	—	188	1	no	—
49	D₄	—	188	1	no	—
50	D₄	—	188	1	no	—
51	C₂×C₄	C₄×D₄₇	1	1	yes	characteristic
52	C₂×C₄	D₁₈₈	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	188	1	no	—
56	C₂²	—	188	2	no	—
57	C₂²	—	188	2	no	—
58	C₂²	—	188	2	no	—
59	C₂²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#10625325	1	1	yes	characteristic
62	C₄	D₃₇₆	1	1	yes	characteristic
63	C₄	#10625334	1	1	yes	characteristic
64	C₄	#10625336	1	1	yes	characteristic
65	C₂	—	376	2	no	—
66	C₂	#13093940	1	1	yes	characteristic
67	C₂	#13093944	1	1	yes	characteristic
68	C₂	#13093946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄
Frattini	C₄²	C₂×D₄₇
Fitting	C₄⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²⋊C₄
prime radical	#13094021	C₂
cosocle	C₄²×C₄₇	C₂²
hypercenter	C₄⋊C₈	D₄₇
hypocenter	C₄₇	#331
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ 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
35. C₄₇	#331	semidirect
38. 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
#13094021	2
C₄⋊(C₄₇⋊C₈)	2
C₄⋊C₈×C₄₇	2
#331	47