Subgroups — #22578119

Order 8768.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984077	C₂	1	4	yes	maximal, complemented
2	#17984063	C₂	1	4	yes	maximal, complemented
3	C₄×D₅₄₈	C₂	1	2	yes	maximal, complemented
4	#17984046	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₄₈	C₂²	1	2	yes	—
7	C₂×D₅₄₈	C₂²	1	4	yes	—
8	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
9	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
10	#14555487	C₂²	1	8	yes	—
11	#14555487	C₂²	1	4	yes	—
12	C₄×Dic₁₃₇	C₂²	1	4	yes	—
13	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
15	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×D₁₃₇	C₂³	1	4	yes	—
17	D₅₄₈	—	2	4	no	—
18	D₅₄₈	—	4	4	no	—
19	C₂×Dic₁₃₇	C₂³	1	4	yes	—
20	C₄×D₁₃₇	—	2	8	no	—
21	Q₈×C₁₃₇	—	4	2	no	—
22	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
24	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
25	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₁₃₇	—	2	4	no	—
27	C₂×D₁₃₇	—	2	4	no	—
28	C₂×D₁₃₇	—	2	4	no	—
29	C₄×C₁₃₇	—	2	1	no	—
30	C₄×C₁₃₇	—	2	4	no	—
31	C₄×C₁₃₇	#41	1	4	yes	—
32	Dic₁₃₇	—	2	4	no	—
33	C₂×C₁₃₇	C₂×#41	1	2	yes	—
34	C₂×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
35	D₁₃₇	—	4	4	no	—
36	C₁₃₇	#555	1	1	yes	characteristic, complemented, Sylow
37	#555	—	137	1	no	maximal, Sylow
38	C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic, complemented
39	#124	—	137	4	no	—
40	#121	—	137	4	no	—
41	C₄×D₄	—	137	2	no	—
42	#117	—	137	4	no	—
43	C₂×Q₈	C₂×D₁₃₇	1	2	yes	—
44	C₂×D₄	—	137	2	no	—
45	C₂×D₄	—	137	4	no	—
46	C₂²×C₄	—	137	4	no	—
47	C₄⋊C₄	—	137	2	no	—
48	C₄⋊C₄	C₂×D₁₃₇	1	4	yes	—
49	C₂²⋊C₄	—	137	8	no	—
50	C₂²⋊C₄	—	137	4	no	—
51	C₄²	—	137	4	no	—
52	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
53	C₂³	—	137	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	274	4	no	—
56	D₄	—	548	4	no	—
57	C₂×C₄	—	274	8	no	—
58	C₂×C₄	—	137	4	no	—
59	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
60	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
62	C₂²	—	274	4	no	—
63	C₂²	—	274	4	no	—
64	C₂²	—	274	4	no	—
65	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	274	4	no	—
69	C₄	#14555511	1	4	yes	—
70	C₂	—	548	4	no	—
71	C₂	C₂×#14555511	1	2	yes	—
72	C₂	#17984133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₃₇	C₂
hypercenter	C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	#555
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #17984077	C₂	semidirect
2. #17984063	C₂	semidirect
3. C₄×D₅₄₈	C₂	semidirect
4. #17984046	C₂	semidirect
5. C₄⋊Q₈×C₁₃₇	C₂	semidirect
36. C₁₃₇	#555	semidirect
38. C₄⋊Q₈	D₁₃₇	semidirect