Subgroups — #49088491

Order 35744.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₁₁₁₇	C₂	1	2	yes	maximal, complemented
2	C₂²⋊Dic₁₁₁₇	C₂	1	2	yes	maximal
3	C₄×Dic₁₁₁₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₁₁₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₁₁₇	C₂²	1	4	yes	—
6	C₂³×C₁₁₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₁₁₇	C₄	1	4	yes	complemented
8	C₂×C₄×C₁₁₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁₁₇	—	2	2	no	—
11	C₂²×C₁₁₁₇	C₂×C₄	1	2	yes	—
12	C₂²×C₁₁₁₇	C₂³	1	1	yes	characteristic
13	C₄×C₁₁₁₇	C₂×C₄	1	2	yes	—
14	C₄×C₁₁₁₇	C₂×C₄	1	2	yes	—
15	Dic₁₁₁₇	—	2	4	no	—
16	C₂×C₁₁₁₇	—	2	2	no	—
17	C₂×C₁₁₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₁₁₁₇	#41	1	2	yes	—
19	C₁₁₁₇	#117	1	1	yes	characteristic, complemented, Sylow
20	#117	—	1117	1	no	maximal, Sylow
21	C₂²×C₄	D₁₁₁₇	1	1	yes	characteristic
22	C₄⋊C₄	—	1117	2	no	—
23	C₂²⋊C₄	—	1117	2	no	—
24	C₄²	—	1117	2	no	—
25	C₂³	C₂×D₁₁₁₇	1	1	yes	characteristic
26	C₂×C₄	—	1117	4	no	—
27	C₂×C₄	Dic₁₁₁₇	1	4	yes	complemented
28	C₂×C₄	C₂×D₁₁₁₇	1	1	yes	characteristic
29	C₂×C₄	C₂×D₁₁₁₇	1	1	yes	characteristic
30	C₂²	—	2	2	no	—
31	C₂²	C₂×Dic₁₁₁₇	1	2	yes	—
32	C₂²	C₂²×D₁₁₁₇	1	1	yes	characteristic
33	C₄	—	2234	4	no	—
34	C₄	C₂×Dic₁₁₁₇	1	2	yes	—
35	C₄	C₂×Dic₁₁₁₇	1	2	yes	—
36	C₂	—	2	2	no	—
37	C₂	C₂²×Dic₁₁₁₇	1	1	yes	characteristic
38	C₂	#31821910	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₁₇
commutator	C₂×C₁₁₁₇	C₂²×C₄
Frattini	C₂²	C₂²×D₁₁₁₇
Fitting	C₂²×C₄×C₁₁₁₇	C₂
socle, cosocle	C₂²×C₁₁₁₇	C₂³
prime radical	C₂³×C₁₁₁₇	C₂²
hypercenter	C₂²×C₄	D₁₁₁₇
hypocenter	C₁₁₁₇	#117
LCS	G ▷ C₂×C₁₁₁₇ ▷ C₁₁₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊Dic₁₁₁₇	C₂	semidirect
3. C₄×Dic₁₁₁₇	C₂	semidirect
7. C₂×C₄×C₁₁₁₇	C₄	semidirect
19. C₁₁₁₇	#117	semidirect
27. C₂×C₄	Dic₁₁₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₁₁₁₇	2
C₂²⋊Dic₁₁₁₇	2
C₄×Dic₁₁₁₇	2
C₂²×C₄×C₁₁₁₇	2
#117	1117