Subgroups — #22578253

Order 8768.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11764047	C₂	1	1	yes	maximal, characteristic
2	C₄×#11764047	C₂	1	1	yes	maximal, characteristic
3	#17984053	C₂	1	1	yes	maximal, characteristic
4	C₂×#11764047	—	2	1	no	—
5	C₂×#11764047	—	2	1	no	—
6	C₂×Dic₂₇₄	C₄	1	1	yes	characteristic
7	#14555489	—	2	1	no	—
8	C₄×Dic₁₃₇	—	2	1	no	—
9	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
11	#11764047	—	2	2	no	—
12	#11764047	—	4	1	no	—
13	Dic₂₇₄	—	2	1	no	—
14	Dic₂₇₄	C₈	1	2	yes	complemented
15	C₂×Dic₁₃₇	—	2	1	no	—
16	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
17	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₇	—	2	1	no	—
19	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₃₇	—	4	1	no	—
22	C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
24	Dic₁₃₇	—	2	2	no	—
25	Dic₁₃₇	—	2	1	no	—
26	Dic₁₃₇	SD₁₆	1	1	yes	characteristic
27	Dic₁₃₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₃₇	#103	1	1	yes	characteristic
30	C₂×C₁₃₇	#104	1	1	yes	characteristic
31	C₁₃₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	137	1	no	maximal, Sylow
33	C₄×Q₈	—	137	1	no	—
34	C₄⋊C₈	—	137	1	no	—
35	C₄×C₈	—	137	1	no	—
36	C₂×Q₈	—	137	1	no	—
37	C₂×C₈	—	274	1	no	—
38	C₂×C₈	—	274	1	no	—
39	C₄⋊C₄	—	274	1	no	—
40	C₄⋊C₄	#10587739	1	1	yes	characteristic
41	C₄²	—	274	1	no	—
42	C₄²	—	137	1	no	—
43	Q₈	—	274	1	no	—
44	Q₈	—	137	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	274	1	no	—
47	C₂×C₄	—	137	1	no	—
48	C₂×C₄	—	137	1	no	—
49	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
50	C₈	—	274	2	no	—
51	C₈	—	548	1	no	—
52	C₂²	C₂²⋊#10587739	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	274	2	no	—
55	C₄	—	274	1	no	—
56	C₄	—	137	1	no	—
57	C₄	—	137	1	no	—
58	C₄	#14555518	1	1	yes	characteristic
59	C₄	#14555520	1	1	yes	characteristic
60	C₂	#17984145	1	1	yes	characteristic
61	C₂	#17984151	1	1	yes	characteristic
62	C₂	#17984152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#10587739
commutator	C₄×C₁₃₇	C₂×C₈
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₄⋊C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²⋊C₄
hypercenter	C₄⋊C₄	#10587739
hypocenter	C₁₃₇	Q₈⋊C₈
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Dic₂₇₄	C₈	semidirect
31. C₁₃₇	Q₈⋊C₈	semidirect