Subgroups — #22577451

Order 8768.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555494	C₂	1	1	yes	maximal, characteristic
2	C₂×#14555494	C₂	1	1	yes	maximal, characteristic
3	#17984023	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#14555494	—	2	1	no	—
6	#14555494	C₄	1	2	yes	—
7	#14555494	C₄	1	2	yes	—
8	#14555494	C₂²	1	2	yes	—
9	C₂×#11764040	—	2	1	no	—
10	C₂×#11764040	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	M₁₆×C₁₃₇	—	2	1	no	—
13	M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₈×C₁₃₇	—	2	1	no	—
15	#11764040	—	2	2	no	—
16	#11764040	C₂×C₄	1	4	yes	—
17	C₂³×C₁₃₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₁₃₇	—	4	1	no	—
26	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	—	2	1	no	—
35	C₂×C₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₁₃₇	#430	1	1	yes	characteristic, complemented, Sylow
38	#430	—	137	1	no	maximal, Sylow
39	C₂×M₁₆	—	137	1	no	—
40	C₂×M₁₆	—	137	1	no	—
41	C₂×M₁₆	D₁₃₇	1	1	yes	characteristic
42	#108	—	137	4	no	—
43	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
44	M₁₆	—	2	1	no	—
45	M₁₆	—	274	1	no	—
46	M₁₆	—	137	2	no	—
47	M₁₆	—	137	2	no	—
48	M₁₆	—	137	2	no	—
49	M₁₆	C₂×D₁₃₇	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	274	1	no	—
52	C₂×C₈	—	137	2	no	—
53	C₂³	Dic₂₇₄	1	1	yes	characteristic
54	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
55	C₂×C₄	C₄×D₁₃₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
58	C₂×C₄	Dic₂₇₄	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	274	2	no	—
62	C₈	—	137	4	no	—
63	C₂²	—	4	1	no	—
64	C₂²	#14555489	1	1	yes	characteristic
65	C₂²	#14555489	1	1	yes	characteristic
66	C₂²	C₂×Dic₂₇₄	1	1	yes	characteristic
67	C₄	#14555489	1	1	yes	characteristic
68	C₄	#14555489	1	1	yes	characteristic
69	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
70	C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#14555489	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂×M₁₆×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	D₁₃₇
hypocenter	C₁₃₇	#430
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #17984023	C₂	semidirect
37. C₁₃₇	#430	semidirect