Subgroups — #31780946

Order 17536.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984020	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	#17984020	—	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	SD₁₆	1	8	yes	—
25	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₁₃₇	C₄⋊C₄	1	8	yes	complemented
30	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₁₃₇	#127	1	1	yes	characteristic
32	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₂×SD₁₆	1	4	yes	—
34	C₄×C₁₃₇	#106	1	8	yes	complemented
35	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
37	Dic₁₃₇	—	8	4	no	—
38	C₂×C₁₃₇	#384	1	1	yes	characteristic
39	C₂×C₁₃₇	C₂×#106	1	2	yes	—
40	C₂×C₁₃₇	#492	1	2	yes	—
41	C₂×C₁₃₇	#498	1	2	yes	—
42	C₁₃₇	#1844	1	1	yes	characteristic, complemented, Sylow
43	#1844	—	137	1	no	maximal, Sylow
44	C₂×#106	—	137	4	no	—
45	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
46	#384	—	137	2	no	—
47	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
48	C₂×C₄⋊C₄	—	274	4	no	—
49	C₂×C₄⋊C₄	—	137	4	no	—
50	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
51	#106	—	274	8	no	—
52	C₄×C₈	Dic₁₃₇	1	4	yes	complemented
53	C₂²×C₄	—	274	4	no	—
54	C₂²×C₄	D₅₄₈	1	2	yes	—
55	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
56	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
57	C₂×C₈	D₅₄₈	1	4	yes	—
58	C₂×C₈	Dic₂₇₄	1	4	yes	—
59	C₄⋊C₄	—	274	8	no	—
60	C₄⋊C₄	—	548	8	no	—
61	C₄²	C₂×Dic₁₃₇	1	2	yes	—
62	C₄²	Dic₂₇₄	1	2	yes	—
63	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
64	C₂×C₄	—	548	4	no	—
65	C₂×C₄	—	548	8	no	—
66	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
67	C₂×C₄	#14555499	1	8	yes	—
68	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
70	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
71	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
72	C₈	C₄⋊Dic₁₃₇	1	8	yes	complemented
73	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
74	C₂²	#17984085	1	1	yes	characteristic
75	C₂²	#17984088	1	1	yes	characteristic
76	C₂²	C₂×#14555499	1	4	yes	—
77	C₄	—	1096	4	no	—
78	C₄	#17984020	1	8	yes	complemented
79	C₄	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
80	C₄	#17984088	1	2	yes	—
81	C₂	#22577273	1	1	yes	characteristic
82	C₂	C₂×#17984020	1	2	yes	—
83	C₂	#22577776	1	2	yes	—
84	C₂	#22577801	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	#1844
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
29. C₈×C₁₃₇	C₄⋊C₄	semidirect
34. C₄×C₁₃₇	#106	semidirect
42. C₁₃₇	#1844	semidirect
52. C₄×C₈	Dic₁₃₇	semidirect
72. C₈	C₄⋊Dic₁₃₇	semidirect
78. C₄	#17984020	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984020	2
#22577273	2
C₂×C₄×C₈×C₁₃₇	2
#1844	137