Subgroups — #22578309

Order 8768.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984147	C₂	1	2	yes	maximal
2	#17984077	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#10587739	—	2	2	no	—
4	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
5	#14555487	—	2	2	no	—
6	C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
7	C₂×Q₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×#10587739	—	4	2	no	—
9	C₂²×D₁₃₇	D₄	1	2	yes	—
10	D₅₄₈	—	4	1	no	—
11	C₂×Dic₁₃₇	—	2	1	no	—
12	Q₈×C₁₃₇	—	4	1	no	—
13	C₂×C₄×C₁₃₇	—	2	1	no	—
14	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
16	C₂×D₁₃₇	—	2	2	no	—
17	C₂×D₁₃₇	—	4	2	no	—
18	#10587739	—	8	2	no	—
19	C₄×C₁₃₇	—	2	1	no	—
20	C₄×C₁₃₇	—	4	1	no	—
21	Dic₁₃₇	—	4	1	no	—
22	C₂×C₁₃₇	—	2	1	no	—
23	C₂×C₁₃₇	#99	1	1	yes	characteristic
24	D₁₃₇	—	4	2	no	—
25	C₁₃₇	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	137	1	no	maximal, Sylow
27	#124	—	137	1	no	—
28	#99	—	137	2	no	—
29	C₂×Q₈	#10587739	1	1	yes	characteristic, complemented
30	C₂×D₄	—	137	1	no	—
31	C₂²⋊C₄	—	274	2	no	—
32	C₂²⋊C₄	—	274	2	no	—
33	C₄²	—	137	1	no	—
34	C₂³	—	137	2	no	—
35	Q₈	—	4	1	no	—
36	D₄	—	548	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	274	1	no	—
39	C₂×C₄	—	548	2	no	—
40	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
41	C₂²	—	274	2	no	—
42	C₂²	—	548	2	no	—
43	C₂²	C₂²⋊#10587739	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	548	1	no	—
47	C₄	—	1096	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	548	2	no	—
50	C₂	#17984146	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17984146
commutator	C₂×C₄×C₁₃₇	C₂×C₄
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₂×Q₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#99
hypercenter	C₂×Q₈	#10587739
hypocenter	C₁₃₇	#354
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×Dic₁₃₇	C₄	semidirect
7. C₂×Q₈×C₁₃₇	C₄	semidirect
25. C₁₃₇	#354	semidirect
29. C₂×Q₈	#10587739	semidirect