Subgroups — #22578249

Order 8768.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#11764047	C₂	1	2	yes	maximal
2	C₂×#14555489	C₂	1	1	yes	maximal, characteristic
3	C₂×#11764047	—	2	2	no	—
4	C₂²×Dic₁₃₇	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₃₇	C₂²	1	1	yes	characteristic
6	#14555489	—	2	2	no	—
7	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic
8	#11764047	—	4	2	no	—
9	C₂×Dic₁₃₇	—	2	1	no	—
10	C₂×Dic₁₃₇	—	2	1	no	—
11	C₂×Dic₁₃₇	—	4	1	no	—
12	C₂×Dic₁₃₇	C₈	1	2	yes	complemented
13	C₂×Dic₁₃₇	D₄	1	2	yes	—
14	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₃₇	—	2	1	no	—
16	C₂×C₄×C₁₃₇	—	4	1	no	—
17	C₂²×C₁₃₇	—	2	1	no	—
18	C₂²×C₁₃₇	—	2	1	no	—
19	C₂²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₁₃₇	M₁₆	1	1	yes	characteristic
22	C₄×C₁₃₇	—	4	1	no	—
23	Dic₁₃₇	—	2	2	no	—
24	Dic₁₃₇	—	2	2	no	—
25	C₂×C₁₃₇	—	2	1	no	—
26	C₂×C₁₃₇	—	2	1	no	—
27	C₂×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₁₃₇	#99	1	1	yes	characteristic
29	C₂×C₁₃₇	#101	1	1	yes	characteristic
30	C₁₃₇	#324	1	1	yes	characteristic, complemented, Sylow
31	#324	—	137	1	no	maximal, Sylow
32	C₂×C₄⋊C₄	—	137	1	no	—
33	C₂²⋊C₈	—	137	2	no	—
34	C₂²×C₄	—	137	1	no	—
35	C₂²×C₄	—	137	1	no	—
36	C₂²×C₄	#10587739	1	1	yes	characteristic
37	C₂×C₈	—	274	2	no	—
38	C₄⋊C₄	—	274	2	no	—
39	C₂³	C₂×#10587739	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	—	274	1	no	—
43	C₂×C₄	—	274	1	no	—
44	C₂×C₄	—	548	1	no	—
45	C₂×C₄	—	137	2	no	—
46	C₂×C₄	—	137	2	no	—
47	C₈	—	548	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂²⋊#10587739	1	1	yes	characteristic
51	C₂²	#14555518	1	1	yes	characteristic
52	C₂²	#14555520	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	274	2	no	—
55	C₄	—	274	2	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#17984145	1	1	yes	characteristic
59	C₂	#17984146	1	1	yes	characteristic
60	C₂	#17984149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#10587739
commutator	C₂²×C₁₃₇	C₂×C₈
Frattini	C₂³	C₂×#10587739
Fitting	C₂²×C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²⋊C₄
hypercenter	C₂²×C₄	#10587739
hypocenter	C₁₃₇	#324
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
12. C₂×Dic₁₃₇	C₈	semidirect
30. C₁₃₇	#324	semidirect