Subgroups — #13094007

Order 1504.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₄₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	Dic₄₇⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇	—	2	2	no	—
9	C₂×C₄×C₄₇	C₂²	1	2	yes	—
10	C₂²⋊D₄₇	—	2	2	no	—
11	C₂×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇	C₂²	1	2	yes	—
13	C₂²×C₄₇	—	2	2	no	—
14	C₂²×C₄₇	D₄	1	2	yes	—
15	C₂²×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×D₄₇	—	2	1	no	—
17	C₂×D₄₇	—	2	2	no	—
18	C₄×C₄₇	—	2	2	no	—
19	Dic₄₇	—	2	1	no	—
20	Dic₄₇	—	2	2	no	—
21	C₂×C₄₇	—	2	2	no	—
22	C₂×C₄₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄₇	#41	1	2	yes	—
24	D₄₇	—	4	1	no	—
25	C₄₇	#123	1	1	yes	characteristic, complemented, Sylow
26	#123	—	47	1	no	maximal, Sylow
27	C₂×D₄	—	47	1	no	—
28	C₂²×C₄	D₄₇	1	1	yes	characteristic, complemented
29	C₄⋊C₄	—	47	2	no	—
30	C₂²⋊C₄	—	47	1	no	—
31	C₂²⋊C₄	—	47	2	no	—
32	C₂³	—	47	1	no	—
33	C₂³	C₂×D₄₇	1	1	yes	characteristic
34	D₄	—	94	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	47	2	no	—
37	C₂×C₄	—	47	1	no	—
38	C₂×C₄	C₂×D₄₇	1	2	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	—	94	2	no	—
41	C₂²	—	94	1	no	—
42	C₂²	C₂²⋊D₄₇	1	2	yes	—
43	C₂²	C₂²×D₄₇	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	94	2	no	—
46	C₄	—	94	1	no	—
47	C₂	—	2	2	no	—
48	C₂	—	188	1	no	—
49	C₂	#10625350	1	2	yes	—
50	C₂	C₂×C₂²⋊D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂³
Fitting	C₂²×C₄×C₄₇	C₂
prime radical	C₂×C₂²⋊D₄₇	C₂
hypercenter	C₂²×C₄	D₄₇
hypocenter	C₄₇	#123
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×C₄₇	C₂	semidirect
3. C₂²⋊Dic₄₇	C₂	semidirect
4. (C₂×D₄₇)⋊C₄	C₂	semidirect
5. Dic₄₇⋊C₄	C₂	semidirect
25. C₄₇	#123	semidirect
28. C₂²×C₄	D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇	2
C₂×C₂²⋊D₄₇	2
C₂²⋊Dic₄₇	2
(C₂×D₄₇)⋊C₄	2
Dic₄₇⋊C₄	2
#123	47