Subgroups — #22578285

Order 8768.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#17984167	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555493	C₂	1	1	yes	maximal, characteristic
4	#14555520	—	2	1	no	—
5	#14555519	C₄	1	2	yes	complemented
6	#14555518	—	2	1	no	—
7	C₄⋊#10587739	—	2	1	no	—
8	C₂²⋊#10587739	C₄	1	2	yes	—
9	C₄×#10587739	—	2	1	no	—
10	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
11	#14555493	—	2	1	no	—
12	#14555493	—	2	1	no	—
13	C₂×#11764040	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#10587739	—	2	2	no	—
16	#11764047	—	2	2	no	—
17	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₃₇	D₄	1	1	yes	characteristic
20	C₄×D₁₃₇	D₄	1	1	yes	characteristic
21	C₄×D₁₃₇	D₄	1	1	yes	characteristic
22	C₄×D₁₃₇	Q₈	1	1	yes	characteristic
23	#11764040	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₃₇	—	2	1	no	—
26	C₂²×C₁₃₇	C₄²	1	1	yes	characteristic
27	C₂×D₁₃₇	—	4	1	no	—
28	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
30	#10587739	—	4	2	no	—
31	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	#95	1	1	yes	characteristic
37	D₁₃₇	—	2	1	no	—
38	D₁₃₇	—	2	1	no	—
39	C₁₃₇	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	137	1	no	maximal, Sylow
41	C₂×M₁₆	—	137	1	no	—
42	C₂×M₁₆	—	137	1	no	—
43	#117	—	137	1	no	—
44	C₂²×C₄	—	137	1	no	—
45	M₁₆	—	274	1	no	—
46	M₁₆	—	274	1	no	—
47	M₁₆	—	274	1	no	—
48	M₁₆	—	137	2	no	—
49	C₂×C₈	—	274	1	no	—
50	C₂×C₈	—	137	1	no	—
51	C₂×C₈	#10587739	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	274	1	no	—
53	C₂²⋊C₄	—	137	2	no	—
54	C₄²	—	274	1	no	—
55	C₂³	—	137	1	no	—
56	C₂×C₄	—	274	2	no	—
57	C₂×C₄	—	137	1	no	—
58	C₂×C₄	—	137	1	no	—
59	C₂×C₄	—	137	1	no	—
60	C₂×C₄	—	137	1	no	—
61	C₂×C₄	—	137	1	no	—
62	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	274	2	no	—
65	C₈	—	274	1	no	—
66	C₂²	—	548	1	no	—
67	C₂²	—	137	1	no	—
68	C₂²	—	137	1	no	—
69	C₂²	C₄×#10587739	1	1	yes	characteristic
70	C₄	—	548	2	no	—
71	C₄	—	137	1	no	—
72	C₄	—	137	1	no	—
73	C₄	C₂²⋊#10587739	1	1	yes	characteristic
74	C₄	C₄⋊#10587739	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	274	1	no	—
77	C₂	—	274	1	no	—
78	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#10587739
commutator	C₂²×C₁₃₇	C₄²
Frattini	C₂×C₄	C₂×#10587739
Fitting	C₂×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#95
hypercenter	C₂×C₈	#10587739
hypocenter	C₁₃₇	#344
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #14555519	C₄	semidirect
13. C₂×#11764040	C₄	semidirect
14. C₂×C₈×C₁₃₇	C₄	semidirect
39. C₁₃₇	#344	semidirect
51. C₂×C₈	#10587739	semidirect