Subgroups — #22577366

Order 8768.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#14555494	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984000	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₃₇	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₃₇	—	2	2	no	—
8	C₂×C₄×D₁₃₇	—	2	2	no	—
9	C₂×C₄×D₁₃₇	C₄	1	2	yes	—
10	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
11	#14555494	—	2	2	no	—
12	C₂×#11764040	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₃₇	—	2	2	no	—
15	C₂×C₈×C₁₃₇	C₂²	1	2	yes	complemented
16	C₂²×D₁₃₇	—	2	2	no	—
17	C₂²×D₁₃₇	—	2	4	no	—
18	C₂²×D₁₃₇	C₂×C₄	1	2	yes	—
19	C₂×Dic₁₃₇	—	2	2	no	—
20	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
21	C₄×D₁₃₇	—	2	4	no	—
22	C₄×D₁₃₇	—	2	2	no	—
23	C₄×D₁₃₇	—	2	2	no	—
24	#11764040	—	2	2	no	—
25	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
28	C₂×C₄×C₁₃₇	D₄	1	2	yes	—
29	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₁₃₇	—	2	2	no	—
32	C₂²×C₁₃₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₃₇	—	2	8	no	—
35	C₂×D₁₃₇	—	2	2	no	—
36	C₂×D₁₃₇	—	2	2	no	—
37	C₂×D₁₃₇	M₁₆	1	4	yes	complemented
38	C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
39	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	Dic₁₃₇	—	2	2	no	—
42	C₂×C₁₃₇	—	2	2	no	—
43	C₂×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₃₇	C₂×M₁₆	1	2	yes	—
45	D₁₃₇	—	2	4	no	—
46	C₁₃₇	#407	1	1	yes	characteristic, complemented, Sylow
47	#407	—	137	1	no	maximal, Sylow
48	C₂³×C₄	—	137	1	no	—
49	C₂×M₁₆	—	137	1	no	—
50	C₂×M₁₆	D₁₃₇	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	137	4	no	—
52	C₂⁴	—	137	1	no	—
53	C₂²×C₄	—	274	2	no	—
54	C₂²×C₄	—	274	2	no	—
55	C₂²×C₄	—	137	1	no	—
56	C₂²×C₄	—	137	2	no	—
57	C₂²×C₄	—	137	2	no	—
58	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	274	2	no	—
61	C₂×C₈	—	137	2	no	—
62	C₂×C₈	C₂×D₁₃₇	1	2	yes	complemented
63	C₂³	—	274	4	no	—
64	C₂³	—	274	2	no	—
65	C₂³	—	137	2	no	—
66	C₂³	C₄×D₁₃₇	1	1	yes	characteristic
67	C₂×C₄	—	274	2	no	—
68	C₂×C₄	—	274	4	no	—
69	C₂×C₄	—	274	2	no	—
70	C₂×C₄	—	274	2	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₁₃₇	1	2	yes	—
74	C₂×C₄	D₅₄₈	1	2	yes	—
75	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	274	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	274	8	no	—
80	C₂²	—	274	2	no	—
81	C₂²	—	274	2	no	—
82	C₂²	—	137	4	no	—
83	C₂²	#14555487	1	2	yes	—
84	C₂²	C₂×C₄×D₁₃₇	1	1	yes	characteristic
85	C₄	—	274	2	no	—
86	C₄	#14555487	1	2	yes	—
87	C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
88	C₄	C₂×D₅₄₈	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	274	4	no	—
91	C₂	C₂×#14555487	1	1	yes	characteristic
92	C₂	M₁₆×D₁₃₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂²×C₄
Fitting	C₂×M₁₆×C₁₃₇	C₂
hypercenter	C₂×M₁₆	D₁₃₇
hypocenter	C₁₃₇	#407
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#14555494	C₂	semidirect
3. #17984000	C₂	semidirect
4. C₂×M₁₆×C₁₃₇	C₂	semidirect
12. C₂×#11764040	C₂²	semidirect
15. C₂×C₈×C₁₃₇	C₂²	semidirect
37. C₂×D₁₃₇	M₁₆	semidirect
46. C₁₃₇	#407	semidirect
50. C₂×M₁₆	D₁₃₇	semidirect
62. C₂×C₈	C₂×D₁₃₇	semidirect