Subgroups — #22577613

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555500	C₂	1	2	yes	maximal
2	#17984088	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic
4	#17984011	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₇₄	C₂²	1	2	yes	—
7	#14555500	—	2	4	no	—
8	C₂×#11764040	C₂²	1	2	yes	—
9	C₄⋊Dic₁₃₇	—	2	1	no	—
10	C₄⋊Dic₁₃₇	—	2	1	no	—
11	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄²×C₁₃₇	—	2	1	no	—
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	Dic₂₇₄	—	2	2	no	—
17	Dic₂₇₄	—	2	2	no	—
18	C₂×Dic₁₃₇	—	2	2	no	—
19	#11764040	—	2	2	no	—
20	Q₈×C₁₃₇	—	2	1	no	—
21	Q₈×C₁₃₇	D₄	1	2	yes	—
22	C₂×C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₁₃₇	—	2	1	no	—
24	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₃₇	—	2	2	no	—
29	C₄×C₁₃₇	—	2	1	no	—
30	C₄×C₁₃₇	—	4	1	no	—
31	C₄×C₁₃₇	Q₁₆	1	2	yes	complemented
32	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₃₇	#41	1	1	yes	characteristic
34	Dic₁₃₇	—	4	2	no	—
35	C₂×C₁₃₇	#121	1	1	yes	characteristic
36	C₂×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₃₇	#137	1	1	yes	characteristic
38	C₁₃₇	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	137	1	no	maximal, Sylow
40	C₂×Q₁₆	—	137	2	no	—
41	C₄⋊Q₈	—	137	1	no	—
42	C₄×Q₈	D₁₃₇	1	1	yes	characteristic
43	C₄⋊C₈	—	137	1	no	—
44	#103	—	137	2	no	—
45	C₂×Q₈	—	137	2	no	—
46	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
47	Q₁₆	—	274	4	no	—
48	C₂×C₈	—	137	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	274	1	no	—
51	C₄⋊C₄	—	274	1	no	—
52	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	274	2	no	—
57	Q₈	—	274	2	no	—
58	Q₈	D₅₄₈	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	274	2	no	—
62	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₈	—	274	2	no	—
66	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	548	2	no	—
71	C₄	#14555500	1	2	yes	complemented
72	C₄	C₂×D₅₄₈	1	1	yes	characteristic
73	C₄	#14555510	1	1	yes	characteristic
74	C₂	#17984062	1	1	yes	characteristic
75	C₂	C₂×#14555500	1	1	yes	characteristic
76	C₂	#17984118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄×Q₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	#462
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₁₃₇	Q₁₆	semidirect
38. C₁₃₇	#462	semidirect
71. C₄	#14555500	semidirect