Subgroups — #17984150

Order 4384.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14555518	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587739	—	2	1	no	—
5	#11764047	—	2	1	no	—
6	C₂²×D₁₃₇	—	2	1	no	—
7	D₅₄₈	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊D₁₃₇	—	2	1	no	—
9	C₄×D₁₃₇	C₂²	1	1	yes	characteristic
10	D₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₁₃₇	—	2	1	no	—
12	C₂×D₁₃₇	—	2	1	no	—
13	C₂×D₁₃₇	—	4	1	no	—
14	C₂×D₁₃₇	—	4	1	no	—
15	C₂×D₁₃₇	D₄	1	1	yes	characteristic
16	#10587739	—	4	1	no	—
17	C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	Dic₁₃₇	D₄	1	1	yes	characteristic
19	C₂×C₁₃₇	—	4	1	no	—
20	C₂×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
21	D₁₃₇	—	4	1	no	—
22	D₁₃₇	D₈	1	1	yes	characteristic
23	D₁₃₇	SD₁₆	1	1	yes	characteristic
24	C₁₃₇	#102	1	1	yes	characteristic, complemented, Sylow
25	#102	—	137	1	no	maximal, Sylow
26	C₂×D₄	—	137	1	no	—
27	C₂×C₈	—	137	1	no	—
28	C₄⋊C₄	—	137	1	no	—
29	C₂³	—	274	1	no	—
30	D₄	—	274	1	no	—
31	D₄	—	137	1	no	—
32	D₄	#10587739	1	1	yes	characteristic, complemented
33	C₂×C₄	—	274	1	no	—
34	C₂×C₄	—	137	1	no	—
35	C₈	—	274	1	no	—
36	C₂²	—	2	1	no	—
37	C₂²	—	274	1	no	—
38	C₂²	—	548	1	no	—
39	C₂²	—	548	1	no	—
40	C₂²	—	137	1	no	—
41	C₄	—	548	1	no	—
42	C₄	—	137	1	no	—
43	C₄	C₂×#10587739	1	1	yes	characteristic
44	C₂	—	4	1	no	—
45	C₂	—	548	1	no	—
46	C₂	—	137	1	no	—
47	C₂	—	137	1	no	—
48	C₂	C₂²⋊#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₄
Frattini	C₄	C₂×#10587739
Fitting	D₄×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂²⋊C₄
prime radical	D₄×D₁₃₇	C₂
cosocle	C₄×D₁₃₇	C₂²
hypercenter	D₄	#10587739
hypocenter	C₁₃₇	#102
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #14555518	C₂	semidirect
2. C₄⋊#10587739	C₂	semidirect
7. D₅₄₈	C₄	semidirect
10. D₄×C₁₃₇	C₄	semidirect
24. C₁₃₇	#102	semidirect
32. D₄	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#14555518	2
C₄⋊#10587739	2
D₄×D₁₃₇	2
#102	137