Subgroups — #22577685

Order 8768.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic
2	#17984019	C₂	1	2	yes	maximal
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	1	yes	characteristic
7	C₂×#11764040	C₂²	1	2	yes	—
8	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#14555489	—	2	1	no	—
10	C₄×Dic₁₃₇	—	2	1	no	—
11	C₂×Q₈×C₁₃₇	—	2	1	no	—
12	C₄⋊C₄×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	Dic₂₇₄	—	2	1	no	—
16	Dic₂₇₄	Q₈	1	2	yes	complemented
17	C₂×Dic₁₃₇	—	2	1	no	—
18	C₂×Dic₁₃₇	—	2	1	no	—
19	#11764040	—	2	2	no	—
20	Q₈×C₁₃₇	—	4	2	no	—
21	C₂×C₄×C₁₃₇	—	2	2	no	—
22	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₃₇	—	2	1	no	—
27	C₄×C₁₃₇	—	4	2	no	—
28	C₄×C₁₃₇	SD₁₆	1	2	yes	—
29	C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₁₃₇	#41	1	1	yes	characteristic
31	Dic₁₃₇	—	2	2	no	—
32	Dic₁₃₇	—	4	1	no	—
33	C₂×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₃₇	#137	1	1	yes	characteristic
36	C₁₃₇	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	137	1	no	maximal, Sylow
38	C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic
39	C₄×Q₈	—	137	1	no	—
40	#106	—	137	2	no	—
41	C₄⋊C₈	—	137	1	no	—
42	#103	—	137	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	137	1	no	—
45	C₂×C₈	—	137	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	274	1	no	—
48	C₄⋊C₄	—	137	1	no	—
49	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
50	C₄²	—	274	1	no	—
51	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	274	1	no	—
54	Q₈	—	137	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	274	1	no	—
57	C₂×C₄	—	274	1	no	—
58	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
61	C₈	—	274	2	no	—
62	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	274	2	no	—
66	C₄	—	548	1	no	—
67	C₄	#14555498	1	2	yes	—
68	C₄	Q₈×D₁₃₇	1	1	yes	characteristic
69	C₄	#14555511	1	1	yes	characteristic
70	C₂	#17984064	1	1	yes	characteristic
71	C₂	C₂×#14555498	1	1	yes	characteristic
72	C₂	#17984119	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	#475
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
16. Dic₂₇₄	Q₈	semidirect
36. C₁₃₇	#475	semidirect