Subgroups — #22578277

Order 8768.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#10587739	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#10587739	—	2	2	no	—
4	C₄⋊#10587739	—	2	2	no	—
5	C₄⋊#10587739	C₄	1	4	yes	complemented
6	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×#11764040	C₄	1	1	yes	characteristic, complemented
8	C₈×D₁₃₇	—	2	2	no	—
9	C₂×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
10	C₂×#10587739	—	2	4	no	—
11	C₂×#10587739	—	4	2	no	—
12	C₂×#10587739	—	4	2	no	—
13	C₂²×D₁₃₇	D₄	1	1	yes	characteristic
14	C₂×Dic₁₃₇	Q₈	1	1	yes	characteristic
15	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
16	C₄×D₁₃₇	D₄	1	2	yes	—
17	#11764040	—	2	1	no	—
18	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₈×C₁₃₇	—	2	1	no	—
20	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₁₃₇	C₂²⋊C₄	1	2	yes	—
22	C₂×D₁₃₇	D₈	1	1	yes	characteristic
23	C₂×D₁₃₇	SD₁₆	1	2	yes	—
24	C₂×D₁₃₇	Q₁₆	1	1	yes	characteristic
25	#10587739	—	4	4	no	—
26	C₄×C₁₃₇	C₄²	1	1	yes	characteristic
27	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₁₃₇	#95	1	1	yes	characteristic
30	C₂×C₁₃₇	#106	1	1	yes	characteristic
31	C₂×C₁₃₇	#107	1	1	yes	characteristic
32	D₁₃₇	#102	1	2	yes	—
33	D₁₃₇	#103	1	2	yes	—
34	C₁₃₇	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	137	1	no	maximal, Sylow
36	C₂²×C₈	—	137	1	no	—
37	C₂×C₄⋊C₄	—	137	2	no	—
38	C₂²×C₄	—	274	2	no	—
39	C₂²×C₄	—	137	1	no	—
40	C₂×C₈	—	274	2	no	—
41	C₂×C₈	—	137	1	no	—
42	C₂×C₈	#10587739	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	274	2	no	—
44	C₄⋊C₄	—	137	4	no	—
45	C₂³	—	137	1	no	—
46	C₂×C₄	—	274	4	no	—
47	C₂×C₄	—	548	2	no	—
48	C₂×C₄	—	548	2	no	—
49	C₂×C₄	—	137	1	no	—
50	C₂×C₄	—	137	2	no	—
51	C₂×C₄	—	137	2	no	—
52	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	274	1	no	—
55	C₂²	—	137	2	no	—
56	C₂²	—	137	2	no	—
57	C₂²	—	137	1	no	—
58	C₂²	—	137	1	no	—
59	C₂²	C₄⋊#10587739	1	1	yes	characteristic
60	C₄	—	548	4	no	—
61	C₄	—	137	2	no	—
62	C₄	C₄×#10587739	1	1	yes	characteristic
63	C₄	C₂²⋊#10587739	1	1	yes	characteristic
64	C₂	—	137	2	no	—
65	C₂	—	137	2	no	—
66	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
67	C₂	#17984156	1	1	yes	characteristic
68	C₂	#17984157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#10587739
commutator	C₄×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₂×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₄⋊C₄
hypercenter	C₂×C₈	#10587739
hypocenter	C₁₃₇	#340
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#10587739	C₄	semidirect
7. C₂×#11764040	C₄	semidirect
9. C₂×C₈×C₁₃₇	C₄	semidirect
34. C₁₃₇	#340	semidirect
42. C₂×C₈	#10587739	semidirect