Subgroups — #31791313

Order 17536.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577960	C₂	1	2	yes	maximal
2	#22577812	C₂	1	8	yes	maximal, complemented
3	#22577441	C₂	1	4	yes	maximal
4	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984088	C₂²	1	4	yes	—
6	#17984080	C₂²	1	4	yes	—
7	#17984067	—	2	4	no	—
8	C₄×Dic₂₇₄	—	2	4	no	—
9	#17984047	C₂²	1	4	yes	—
10	#17984022	C₂²	1	8	yes	—
11	#17984020	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	#97×C₁₃₇	C₂²	1	4	yes	—
15	C₂×Dic₂₇₄	—	2	4	no	—
16	C₄⋊Dic₁₃₇	—	2	4	no	—
17	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
18	#14555489	—	2	8	no	—
19	C₂²⋊Dic₁₃₇	—	2	4	no	—
20	C₄×Dic₁₃₇	—	2	4	no	—
21	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	Q₈	1	8	yes	complemented
24	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
25	C₄²×C₁₃₇	D₄	1	2	yes	—
26	C₄²×C₁₃₇	C₂³	1	2	yes	—
27	Dic₂₇₄	—	4	8	no	—
28	C₂×Dic₁₃₇	—	2	8	no	—
29	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	4	2	no	—
31	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
35	C₈×C₁₃₇	C₂×Q₈	1	8	yes	—
36	C₂²×C₁₃₇	—	2	2	no	—
37	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₁₃₇	—	2	4	no	—
40	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
41	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
42	Dic₁₃₇	—	4	8	no	—
43	C₂×C₁₃₇	—	2	2	no	—
44	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₃₇	#575	1	2	yes	—
46	C₁₃₇	#3162	1	1	yes	characteristic, complemented, Sylow
47	#3162	—	137	1	no	maximal, Sylow
48	#533	—	137	2	no	—
49	#501	—	137	8	no	—
50	#427	—	137	4	no	—
51	C₂×#97	D₁₃₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
53	C₄⋊Q₈	—	137	4	no	—
54	#125	—	137	4	no	—
55	C₂²⋊Q₈	—	274	4	no	—
56	C₄×Q₈	—	274	4	no	—
57	#117	—	137	4	no	—
58	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
59	#107	—	137	8	no	—
60	#106	—	137	8	no	—
61	#97	C₂×D₁₃₇	1	4	yes	—
62	C₂×Q₈	—	274	4	no	—
63	C₂²×C₄	D₅₄₈	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₂×C₈	Dic₂₇₄	1	8	yes	complemented
66	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
67	C₄⋊C₄	—	274	8	no	—
68	C₄⋊C₄	—	274	4	no	—
69	C₄⋊C₄	—	137	8	no	—
70	C₂²⋊C₄	—	274	4	no	—
71	C₄²	—	274	4	no	—
72	C₄²	D₅₄₈	1	2	yes	—
73	C₄²	C₂²×D₁₃₇	1	2	yes	—
74	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
75	Q₈	—	548	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	274	8	no	—
78	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
79	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
81	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
82	C₈	C₂×Dic₂₇₄	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#17984088	1	2	yes	—
85	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	548	8	no	—
88	C₄	#17984088	1	2	yes	—
89	C₄	C₂²×Dic₂₇₄	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×#17984088	1	1	yes	characteristic
92	C₂	#22578203	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#3162
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22577812	C₂	semidirect
23. C₂×C₈×C₁₃₇	Q₈	semidirect
46. C₁₃₇	#3162	semidirect
65. C₂×C₈	Dic₂₇₄	semidirect

Maximal subgroups

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

subgroup	index
#22577960	2
#22577812	2
#22577441	2
C₂×#97×C₁₃₇	2
#3162	137