Subgroups — #22577741

Order 8768.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984047	C₂	1	1	yes	maximal, characteristic
2	#17984021	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984019	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984012	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984012	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#11764040	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11764040	C₂²	1	1	yes	characteristic
9	C₂×#11764040	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₁₃₇	—	2	1	no	—
13	C₄×Dic₁₃₇	—	2	1	no	—
14	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₃₇	—	2	1	no	—
17	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
19	C₂×Dic₁₃₇	—	2	1	no	—
20	C₂×Dic₁₃₇	—	2	1	no	—
21	#11764040	—	2	1	no	—
22	#11764040	—	2	1	no	—
23	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
24	Q₈×C₁₃₇	—	2	1	no	—
25	Q₈×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	—	2	1	no	—
27	C₂×C₄×C₁₃₇	—	2	1	no	—
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₁₃₇	—	2	1	no	—
33	C₂²×C₁₃₇	—	2	1	no	—
34	C₂²×C₁₃₇	—	2	1	no	—
35	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	—	4	1	no	—
38	C₄×C₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	#41	1	1	yes	characteristic
40	C₄×C₁₃₇	#41	1	1	yes	characteristic
41	Dic₁₃₇	—	4	1	no	—
42	Dic₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	—	4	1	no	—
44	C₂×C₁₃₇	#123	1	1	yes	characteristic
45	C₂×C₁₃₇	#135	1	1	yes	characteristic
46	C₂×C₁₃₇	#137	1	1	yes	characteristic
47	C₁₃₇	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	137	1	no	maximal, Sylow
49	C₂²⋊Q₈	D₁₃₇	1	1	yes	characteristic
50	#117	—	137	1	no	—
51	#107	—	137	1	no	—
52	#106	—	137	1	no	—
53	#103	—	137	1	no	—
54	#103	—	137	1	no	—
55	C₂²⋊C₈	—	137	1	no	—
56	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
58	C₂×C₈	—	137	1	no	—
59	C₂×C₈	—	137	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	137	1	no	—
62	C₄⋊C₄	—	137	1	no	—
63	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	274	1	no	—
66	C₄²	—	274	1	no	—
67	C₂³	C₂²⋊D₁₃₇	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	274	1	no	—
75	C₂×C₄	—	274	1	no	—
76	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
78	C₈	—	274	1	no	—
79	C₈	—	274	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	548	1	no	—
88	C₄	—	548	1	no	—
89	C₄	#14555509	1	1	yes	characteristic
90	C₄	#14555509	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#17984069	1	1	yes	characteristic
93	C₂	#17984108	1	1	yes	characteristic
94	C₂	#17984119	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂²⋊Q₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₂²⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	#485
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #17984021	C₂	semidirect
3. #17984019	C₂	semidirect
4. #17984012	C₂	semidirect
5. #17984012	C₂	semidirect
47. C₁₃₇	#485	semidirect