Subgroups — #31780949

Order 17536.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984022	C₂	1	4	yes	maximal
2	#22577273	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₃₇	—	2	4	no	—
5	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
6	#17984022	—	2	8	no	—
7	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₃₇	C₄	1	4	yes	complemented
10	C₂²×Dic₁₃₇	—	2	4	no	—
11	C₄⋊Dic₁₃₇	—	2	8	no	—
12	C₄⋊Dic₁₃₇	—	4	8	no	—
13	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
14	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
17	C₂×C₈×C₁₃₇	Q₈	1	4	yes	—
18	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₃₇	Q₈	1	2	yes	—
20	C₂×Dic₁₃₇	—	4	4	no	—
21	C₂×Dic₁₃₇	—	4	8	no	—
22	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
24	C₂×C₄×C₁₃₇	D₈	1	4	yes	—
25	C₂×C₄×C₁₃₇	Q₁₆	1	4	yes	—
26	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
29	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₁₃₇	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₁₃₇	#127	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₁₃₇	C₂×D₈	1	2	yes	—
35	C₂²×C₁₃₇	C₂×Q₁₆	1	2	yes	—
36	C₄×C₁₃₇	#107	1	8	yes	complemented
37	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
39	Dic₁₃₇	—	8	4	no	—
40	C₂×C₁₃₇	#384	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂×#107	1	2	yes	—
42	C₂×C₁₃₇	#493	1	1	yes	characteristic
43	C₂×C₁₃₇	#494	1	1	yes	characteristic
44	C₂×C₁₃₇	#500	1	2	yes	—
45	C₁₃₇	#1845	1	1	yes	characteristic, complemented, Sylow
46	#1845	—	137	1	no	maximal, Sylow
47	C₂×#107	—	137	4	no	—
48	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
49	#384	—	137	2	no	—
50	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
51	C₂×C₄⋊C₄	—	274	4	no	—
52	C₂×C₄⋊C₄	—	137	4	no	—
53	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
54	#107	—	274	8	no	—
55	C₄×C₈	Dic₁₃₇	1	4	yes	complemented
56	C₂²×C₄	—	274	4	no	—
57	C₂²×C₄	D₅₄₈	1	2	yes	—
58	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
59	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
60	C₂×C₈	D₅₄₈	1	4	yes	—
61	C₂×C₈	Dic₂₇₄	1	4	yes	—
62	C₄⋊C₄	—	274	8	no	—
63	C₄⋊C₄	—	548	8	no	—
64	C₄²	C₂×Dic₁₃₇	1	2	yes	—
65	C₄²	Dic₂₇₄	1	2	yes	—
66	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
67	C₂×C₄	—	548	4	no	—
68	C₂×C₄	—	548	8	no	—
69	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
70	C₂×C₄	D₁₀₉₆	1	4	yes	—
71	C₂×C₄	Dic₅₄₈	1	4	yes	—
72	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
73	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
74	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
75	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
76	C₈	C₄⋊Dic₁₃₇	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
78	C₂²	#17984085	1	1	yes	characteristic
79	C₂²	#17984088	1	1	yes	characteristic
80	C₂²	C₂×D₁₀₉₆	1	2	yes	—
81	C₂²	C₂×Dic₅₄₈	1	2	yes	—
82	C₄	—	1096	4	no	—
83	C₄	#17984022	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
85	C₄	#17984088	1	2	yes	—
86	C₂	#22577273	1	1	yes	characteristic
87	C₂	C₂×#17984022	1	2	yes	—
88	C₂	#22577780	1	1	yes	characteristic
89	C₂	#22577783	1	1	yes	characteristic
90	C₂	#22577808	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#1845
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₁₃₇	C₄	semidirect
30. C₈×C₁₃₇	C₄⋊C₄	semidirect
36. C₄×C₁₃₇	#107	semidirect
45. C₁₃₇	#1845	semidirect
55. C₄×C₈	Dic₁₃₇	semidirect
76. C₈	C₄⋊Dic₁₃₇	semidirect
83. C₄	#17984022	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984022	2
#22577273	2
C₂×C₄×C₈×C₁₃₇	2
#1845	137