Subgroups — #22578268

Order 8768.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10587739	C₄	1	2	yes	—
5	#14555518	—	2	1	no	—
6	#14555518	—	2	1	no	—
7	C₂×#11764047	C₄	1	2	yes	complemented
8	C₄×#10587739	—	2	1	no	—
9	C₄×#10587739	—	2	1	no	—
10	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#11764040	C₄	1	1	yes	characteristic, complemented
12	C₈×D₁₃₇	—	2	1	no	—
13	C₈×D₁₃₇	—	2	1	no	—
14	C₂×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#10587739	—	2	2	no	—
16	C₂×#10587739	—	2	2	no	—
17	C₂×#10587739	C₈	1	4	yes	complemented
18	#11764047	—	2	2	no	—
19	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₁₃₇	D₄	1	1	yes	characteristic
22	C₄×D₁₃₇	D₄	1	1	yes	characteristic
23	C₄×D₁₃₇	D₄	1	1	yes	characteristic
24	C₄×D₁₃₇	Q₈	1	1	yes	characteristic
25	#11764040	—	2	1	no	—
26	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₈×C₁₃₇	—	2	1	no	—
28	C₂²×C₁₃₇	C₄²	1	1	yes	characteristic
29	C₂×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×D₁₃₇	C₂×C₈	1	2	yes	—
32	C₂×D₁₃₇	M₁₆	1	2	yes	—
33	#10587739	—	2	4	no	—
34	C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
36	Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₃₇	#95	1	1	yes	characteristic
39	C₂×C₁₃₇	C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₃₇	#97	1	1	yes	characteristic
41	D₁₃₇	C₂²⋊C₈	1	2	yes	—
42	D₁₃₇	C₄⋊C₈	1	2	yes	—
43	C₁₃₇	#336	1	1	yes	characteristic, complemented, Sylow
44	#336	—	137	1	no	maximal, Sylow
45	C₂²×C₈	—	137	1	no	—
46	C₂²×C₈	—	137	1	no	—
47	C₂×C₄²	—	137	1	no	—
48	C₂²×C₄	—	137	2	no	—
49	C₂²×C₄	—	137	1	no	—
50	C₂×C₈	—	274	1	no	—
51	C₂×C₈	—	274	1	no	—
52	C₂×C₈	—	274	1	no	—
53	C₂×C₈	—	274	1	no	—
54	C₂×C₈	—	137	2	no	—
55	C₂×C₈	—	137	1	no	—
56	C₂×C₈	#10587739	1	1	yes	characteristic, complemented
57	C₄²	—	274	1	no	—
58	C₄²	—	274	1	no	—
59	C₂³	—	137	1	no	—
60	C₂×C₄	—	274	2	no	—
61	C₂×C₄	—	274	2	no	—
62	C₂×C₄	—	137	4	no	—
63	C₂×C₄	—	137	1	no	—
64	C₂×C₄	—	137	1	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	—	137	1	no	—
67	C₂×C₄	—	137	1	no	—
68	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	274	2	no	—
71	C₈	—	274	1	no	—
72	C₂²	—	137	1	no	—
73	C₂²	—	137	1	no	—
74	C₂²	—	137	2	no	—
75	C₂²	—	137	2	no	—
76	C₂²	C₄×#10587739	1	1	yes	characteristic
77	C₄	—	274	4	no	—
78	C₄	—	137	1	no	—
79	C₄	—	137	1	no	—
80	C₄	C₂²⋊#10587739	1	1	yes	characteristic
81	C₄	C₄⋊#10587739	1	1	yes	characteristic
82	C₂	—	137	2	no	—
83	C₂	—	137	2	no	—
84	C₂	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
85	C₂	C₈×#10587739	1	1	yes	characteristic
86	C₂	#17984142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₄×C₈
Fitting	C₂×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₄²
hypercenter	C₂×C₈	#10587739
hypocenter	C₁₃₇	#336
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
7. C₂×#11764047	C₄	semidirect
11. C₂×#11764040	C₄	semidirect
14. C₂×C₈×C₁₃₇	C₄	semidirect
17. C₂×#10587739	C₈	semidirect
43. C₁₃₇	#336	semidirect
56. C₂×C₈	#10587739	semidirect