Subgroups — #31783036

Order 17536.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577783	C₂	1	1	yes	maximal, characteristic
2	#22577206	C₂	1	4	yes	maximal
3	#22577179	C₂	1	1	yes	maximal, characteristic
4	#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₄₈	—	2	1	no	—
6	C₂×Dic₅₄₈	C₂²	1	2	yes	—
7	#17984088	—	2	1	no	—
8	C₂×#14555485	C₂²	1	2	yes	—
9	#125×C₁₃₇	—	2	1	no	—
10	#107×C₁₃₇	C₂²	1	2	yes	—
11	#106×C₁₃₇	—	2	1	no	—
12	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₇₄	—	2	2	no	—
14	Dic₅₄₈	—	2	4	no	—
15	Dic₅₄₈	—	4	2	no	—
16	C₄⋊Dic₁₃₇	—	4	2	no	—
17	#14555485	—	2	2	no	—
18	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	—	2	2	no	—
21	C₄⋊C₄×C₁₃₇	—	4	2	no	—
22	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
23	Dic₂₇₄	—	4	2	no	—
24	Dic₂₇₄	—	4	2	no	—
25	C₂×Dic₁₃₇	—	4	2	no	—
26	C₂×C₄×C₁₃₇	—	4	2	no	—
27	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₃₇	SD₁₆	1	2	yes	—
30	C₈×C₁₃₇	#41	1	2	yes	—
31	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₃₇	—	2	2	no	—
33	C₄×C₁₃₇	—	8	2	no	—
34	C₄×C₁₃₇	#124	1	1	yes	characteristic
35	C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₁₃₇	—	8	2	no	—
37	C₂×C₁₃₇	#486	1	1	yes	characteristic
38	C₂×C₁₃₇	#510	1	2	yes	—
39	C₁₃₇	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	137	1	no	maximal, Sylow
41	#499	D₁₃₇	1	1	yes	characteristic
42	#494	—	137	1	no	—
43	#358	—	137	4	no	—
44	#346	—	137	1	no	—
45	C₂×Q₁₆	—	274	1	no	—
46	C₂×Q₁₆	—	137	2	no	—
47	C₄⋊Q₈	—	274	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	—	137	2	no	—
50	#107	C₂×D₁₃₇	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
53	C₂×Q₈	—	274	2	no	—
54	Q₁₆	—	274	4	no	—
55	Q₁₆	—	548	2	no	—
56	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	548	2	no	—
61	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
62	C₁₆	—	274	2	no	—
63	Q₈	—	548	2	no	—
64	Q₈	—	548	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	548	2	no	—
67	C₂×C₄	#14555495	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
69	C₈	#14555497	1	2	yes	—
70	C₈	#14555511	1	2	yes	—
71	C₂²	C₂×#14555495	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	1096	2	no	—
75	C₄	#17984077	1	1	yes	characteristic
76	C₄	C₂×#14555497	1	1	yes	characteristic
77	C₂	#22577751	1	1	yes	characteristic
78	C₂	#22577843	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14555495
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#499×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
hypercenter	#499	D₁₃₇
hypocenter	C₁₃₇	#2245
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
39. C₁₃₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#22577783	2
#22577206	2
#22577179	2
#499×C₁₃₇	2
#2245	137