Subgroups — #359×C₁₃₇ #22576894

Order 8768.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#41×C₁₃₇	—	2	1	no	—
5	Q₁₆×C₁₃₇	C₄	1	1	yes	characteristic
6	SD₁₆×C₁₃₇	—	2	1	no	—
7	D₈×C₁₃₇	C₄	1	1	yes	characteristic
8	M₁₆×C₁₃₇	—	2	1	no	—
9	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₃₇	—	2	1	no	—
11	Q₈×C₁₃₇	—	2	1	no	—
12	D₄×C₁₃₇	—	2	1	no	—
13	D₄×C₁₃₇	—	4	1	no	—
14	C₂×C₄×C₁₃₇	—	4	1	no	—
15	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
16	C₈×C₁₃₇	—	4	1	no	—
17	C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₃₇	—	4	1	no	—
20	C₂²×C₁₃₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₃₇	—	4	1	no	—
22	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₃₇	D₈	1	1	yes	characteristic
24	C₂×C₁₃₇	—	2	1	no	—
25	C₂×C₁₃₇	—	8	1	no	—
26	C₂×C₁₃₇	#102	1	1	yes	characteristic
27	C₁₃₇	#359	1	1	yes	characteristic, direct product factor, Sylow
28	#359	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#135	C₂×C₁₃₇	1	1	yes	characteristic
30	C₂×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
31	#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
32	#41	—	2	1	no	—
33	Q₁₆	C₄×C₁₃₇	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	C₄×C₁₃₇	1	1	yes	characteristic
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	D₄×C₁₃₇	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
46	C₈	D₄×C₁₃₇	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	SD₁₆×C₁₃₇	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
51	C₄	D₈×C₁₃₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#102×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	D₈
commutator	C₈	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₈	C₂²×C₁₃₇
socle	C₂×C₁₃₇	#102
prime radical	#135×C₁₃₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×C₁₃₇	C₂	semidirect
3. #108×C₁₃₇	C₂	semidirect
27. C₁₃₇	#359	direct (N × H)
28. #359	C₁₃₇	direct (N × H)
30. C₂×C₁₆	C₂×C₁₃₇	semidirect
31. #108	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#135×C₁₃₇	2
C₂×C₁₆×C₁₃₇	2
#108×C₁₃₇	2
#359	137

Subgroups — #359×C137 #22576894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #359×C₁₃₇ #22576894