Subgroups — #15689147

Order 3008.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625329	C₂	1	2	yes	maximal
2	C₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₄₇	C₄	1	2	yes	—
5	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
6	C₄²×C₄₇	C₄	1	1	yes	characteristic
7	C₄²×C₄₇	C₄	1	1	yes	characteristic
8	C₄²×C₄₇	C₂²	1	2	yes	—
9	#10625329	—	2	4	no	—
10	C₂×C₄₇⋊C₈	C₂²	1	4	yes	—
11	C₂³×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₄₇	—	2	4	no	—
13	C₂×C₄×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₄₇	D₄	1	2	yes	—
17	C₂×C₄×C₄₇	Q₈	1	2	yes	—
18	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₄₇⋊C₈	—	2	4	no	—
20	C₂²×C₄₇	—	2	2	no	—
21	C₂²×C₄₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₄₇	—	2	2	no	—
24	C₄×C₄₇	C₄⋊C₄	1	2	yes	—
25	C₄×C₄₇	M₁₆	1	4	yes	complemented
26	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄₇	—	2	2	no	—
29	C₂×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄₇	C₂×M₁₆	1	2	yes	—
31	#423	—	47	1	no	maximal, Sylow
32	C₄₇	#423	1	1	yes	characteristic, complemented, Sylow
33	C₂×M₁₆	—	47	2	no	—
34	C₂×C₄²	D₄₇	1	1	yes	characteristic
35	C₄⋊C₈	—	47	4	no	—
36	C₂²×C₄	Dic₄₇	1	2	yes	—
37	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
38	M₁₆	—	94	4	no	—
39	C₂×C₈	—	47	4	no	—
40	C₄²	Dic₄₇	1	1	yes	characteristic
41	C₄²	Dic₄₇	1	1	yes	characteristic
42	C₄²	C₂×D₄₇	1	2	yes	—
43	C₂³	C₂×Dic₄₇	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	Dic₉₄	1	2	yes	—
46	C₂×C₄	D₁₈₈	1	2	yes	—
47	C₂×C₄	C₂×Dic₄₇	1	2	yes	—
48	C₂×C₄	C₂×Dic₄₇	1	2	yes	—
49	C₂×C₄	C₂×Dic₄₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
51	C₈	—	94	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₄₇	1	2	yes	—
54	C₂²	C₂²×Dic₄₇	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	#10625329	1	4	yes	complemented
57	C₄	C₄⋊Dic₄₇	1	2	yes	—
58	C₄	C₂×Dic₉₄	1	1	yes	characteristic
59	C₄	C₂×D₁₈₈	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
62	C₂	C₂×#10625329	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄
Fitting	C₂×C₄²×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂³
hypercenter	C₂×C₄²	D₄₇
hypocenter	C₄₇	#423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
25. C₄×C₄₇	M₁₆	semidirect
32. C₄₇	#423	semidirect
56. C₄	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625329	2
C₄⋊(C₄₇⋊C₈)	2
C₂×C₄²×C₄₇	2
#423	47