Subgroups — #22577458

Order 8768.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#17984017	C₂	1	2	yes	maximal, complemented
3	#17984000	C₂	1	2	yes	maximal
4	#17983999	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#11764040	C₂²	1	2	yes	—
10	C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
11	C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
12	C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₃₇	—	2	2	no	—
20	C₄×D₁₃₇	—	2	2	no	—
21	C₄×D₁₃₇	—	2	2	no	—
22	#11764040	—	2	2	no	—
23	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₈×C₁₃₇	—	2	2	no	—
27	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₁₃₇	—	2	1	no	—
29	C₂×D₁₃₇	—	2	1	no	—
30	C₂×D₁₃₇	M₁₆	1	2	yes	—
31	C₄×C₁₃₇	—	2	1	no	—
32	C₄×C₁₃₇	—	2	1	no	—
33	C₄×C₁₃₇	#41	1	2	yes	—
34	Dic₁₃₇	—	2	1	no	—
35	Dic₁₃₇	—	2	1	no	—
36	Dic₁₃₇	M₁₆	1	2	yes	—
37	C₂×C₁₃₇	#117	1	1	yes	characteristic
38	C₂×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
39	C₂×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
40	D₁₃₇	—	2	2	no	—
41	C₁₃₇	#432	1	1	yes	characteristic, complemented, Sylow
42	#432	—	137	1	no	maximal, Sylow
43	C₂×C₄²	—	137	1	no	—
44	C₄⋊C₈	—	137	2	no	—
45	C₂²⋊C₈	—	137	2	no	—
46	#97	—	137	1	no	—
47	#97	D₁₃₇	1	1	yes	characteristic, complemented
48	C₂²×C₄	—	137	1	no	—
49	C₂²×C₄	—	137	1	no	—
50	C₂²×C₄	—	137	1	no	—
51	C₂×C₈	—	137	2	no	—
52	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
53	C₄²	—	137	1	no	—
54	C₄²	—	137	1	no	—
55	C₄²	—	137	1	no	—
56	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
57	C₂³	—	137	1	no	—
58	C₂×C₄	—	274	2	no	—
59	C₂×C₄	—	274	2	no	—
60	C₂×C₄	—	274	2	no	—
61	C₂×C₄	—	137	1	no	—
62	C₂×C₄	—	137	1	no	—
63	C₂×C₄	—	137	1	no	—
64	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
65	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	274	2	no	—
69	C₂²	—	274	1	no	—
70	C₂²	—	274	1	no	—
71	C₂²	—	137	2	no	—
72	C₂²	C₂×C₄×D₁₃₇	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	274	1	no	—
76	C₄	—	274	1	no	—
77	C₄	—	137	2	no	—
78	C₄	#14555510	1	2	yes	—
79	C₂	—	274	2	no	—
80	C₂	#17984045	1	1	yes	characteristic
81	C₂	M₁₆×D₁₃₇	1	1	yes	characteristic
82	C₂	M₁₆×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂²×C₄
Fitting	#97×C₁₃₇	C₂
hypercenter	#97	D₁₃₇
hypocenter	C₁₃₇	#432
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #17984017	C₂	semidirect
4. #17983999	C₂	semidirect
5. #97×C₁₃₇	C₂	semidirect
41. C₁₃₇	#432	semidirect
47. #97	D₁₃₇	semidirect