Subgroups — #22577756

Order 8768.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984079	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984075	C₂	1	1	yes	maximal, characteristic
3	#17984013	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984011	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984009	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
9	C₂×#11764040	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₇	—	2	1	no	—
11	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	#14555489	—	2	1	no	—
13	C₂²⋊Dic₁₃₇	—	2	1	no	—
14	C₂²⋊Dic₁₃₇	—	2	1	no	—
15	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	Dic₂₇₄	—	2	1	no	—
20	Dic₂₇₄	—	2	1	no	—
21	C₂×Dic₁₃₇	—	2	1	no	—
22	C₂×Dic₁₃₇	—	2	1	no	—
23	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
24	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
25	#11764040	—	2	1	no	—
26	C₂³×C₁₃₇	—	2	1	no	—
27	D₄×C₁₃₇	—	2	1	no	—
28	D₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₈×C₁₃₇	—	2	1	no	—
32	C₂²×C₁₃₇	—	4	1	no	—
33	C₂²×C₁₃₇	—	4	1	no	—
34	C₂²×C₁₃₇	—	4	1	no	—
35	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₃₇	—	4	1	no	—
37	C₄×C₁₃₇	#41	1	1	yes	characteristic
38	C₄×C₁₃₇	#41	1	1	yes	characteristic
39	Dic₁₃₇	—	2	1	no	—
40	Dic₁₃₇	—	2	1	no	—
41	Dic₁₃₇	—	4	1	no	—
42	Dic₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	—	8	1	no	—
44	C₂×C₁₃₇	#124	1	1	yes	characteristic
45	C₂×C₁₃₇	#135	1	1	yes	characteristic
46	C₂×C₁₃₇	#135	1	1	yes	characteristic
47	C₁₃₇	#488	1	1	yes	characteristic, complemented, Sylow
48	#488	—	137	1	no	maximal, Sylow
49	#125	—	137	1	no	—
50	#124	—	137	1	no	—
51	#103	—	137	1	no	—
52	#103	—	137	1	no	—
53	#102	—	137	1	no	—
54	#102	D₁₃₇	1	1	yes	characteristic
55	C₄×C₈	—	137	1	no	—
56	C₂×Q₈	—	137	1	no	—
57	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
58	C₂×C₈	—	137	1	no	—
59	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
60	C₄⋊C₄	—	274	1	no	—
61	C₄⋊C₄	—	274	1	no	—
62	C₄⋊C₄	—	137	1	no	—
63	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
64	C₂²⋊C₄	—	274	1	no	—
65	C₂²⋊C₄	—	274	1	no	—
66	C₄²	—	137	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	274	1	no	—
69	Q₈	—	274	1	no	—
70	D₄	—	2	1	no	—
71	D₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	274	1	no	—
74	C₂×C₄	—	274	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	274	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	274	1	no	—
86	C₄	—	274	1	no	—
87	C₄	—	548	1	no	—
88	C₄	—	548	1	no	—
89	C₄	#14555509	1	1	yes	characteristic
90	C₄	#14555510	1	1	yes	characteristic
91	C₂	—	8	1	no	—
92	C₂	#17984074	1	1	yes	characteristic
93	C₂	#17984107	1	1	yes	characteristic
94	C₂	#17984109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#102×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	#102	D₁₃₇
hypocenter	C₁₃₇	#488
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #17984079	C₂	semidirect
3. #17984013	C₂	semidirect
4. #17984011	C₂	semidirect
6. C₈×Dic₁₃₇	C₂	semidirect
47. C₁₃₇	#488	semidirect