Subgroups — #22578471

Order 8768.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10587739	C₂	1	8	yes	maximal, complemented
2	#17984167	C₂	1	6	yes	maximal
3	C₂×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#10587739	C₂²	1	12	yes	—
5	C₂²⋊#10587739	C₂²	1	4	yes	—
6	C₄×#10587739	C₂²	1	12	yes	—
7	C₂×Dic₂₇₄	C₄	1	3	yes	complemented
8	Q₈×D₁₃₇	C₄	1	4	yes	complemented
9	Q₈×D₁₃₇	C₂²	1	4	yes	—
10	C₂×C₄×D₁₃₇	C₂²	1	3	yes	—
11	C₂×Q₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂×#10587739	C₂³	1	8	yes	—
13	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
14	Dic₂₇₄	C₂×C₄	1	12	yes	—
15	C₂×Dic₁₃₇	C₂×C₄	1	3	yes	—
16	C₄×D₁₃₇	C₂×C₄	1	6	yes	—
17	C₄×D₁₃₇	C₂³	1	6	yes	—
18	Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
20	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₁₃₇	—	2	2	no	—
22	C₂×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×D₁₃₇	C₂⁴	1	1	yes	characteristic
24	#10587739	—	2	8	no	—
25	C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
26	Dic₁₃₇	C₂²×C₄	1	6	yes	—
27	C₂×C₁₃₇	—	2	1	no	—
28	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
29	D₁₃₇	—	2	1	no	—
30	D₁₃₇	2₋¹⁺⁴	1	2	yes	—
31	C₁₃₇	#519	1	1	yes	characteristic, complemented, Sylow
32	#519	—	137	1	no	maximal, Sylow
33	C₂²×Q₈	—	137	1	no	—
34	C₄×Q₈	—	137	8	no	—
35	#117	—	137	6	no	—
36	C₂×Q₈	—	137	4	no	—
37	C₂×Q₈	—	137	4	no	—
38	C₂×Q₈	—	137	3	no	—
39	C₂×Q₈	#10587739	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	137	3	no	—
41	C₄⋊C₄	—	137	12	no	—
42	C₂²⋊C₄	—	137	4	no	—
43	C₄²	—	137	12	no	—
44	C₂³	—	137	1	no	—
45	Q₈	—	137	12	no	—
46	Q₈	C₂×#10587739	1	4	yes	—
47	C₂×C₄	—	137	8	no	—
48	C₂×C₄	—	137	3	no	—
49	C₂×C₄	—	137	6	no	—
50	C₂×C₄	—	137	6	no	—
51	C₂×C₄	C₂×#10587739	1	3	yes	—
52	C₂²	—	274	2	no	—
53	C₂²	—	137	1	no	—
54	C₂²	—	137	1	no	—
55	C₂²	C₂²×#10587739	1	1	yes	characteristic
56	C₄	—	274	8	no	—
57	C₄	—	137	6	no	—
58	C₄	C₂²×#10587739	1	6	yes	—
59	C₂	—	2	1	no	—
60	C₂	—	274	1	no	—
61	C₂	—	137	2	no	—
62	C₂	C₂³×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#10587739
commutator, socle	C₂×C₁₃₇	C₂³×C₄
Fitting	C₂×Q₈×C₁₃₇	C₄
hypercenter	C₂×Q₈	#10587739
hypocenter	C₁₃₇	#519
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#10587739	C₂	semidirect
7. C₂×Dic₂₇₄	C₄	semidirect
8. Q₈×D₁₃₇	C₄	semidirect
11. C₂×Q₈×C₁₃₇	C₄	semidirect
31. C₁₃₇	#519	semidirect
39. C₂×Q₈	#10587739	semidirect