Subgroups — #22577722

Order 8768.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984062	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984020	C₂	1	2	yes	maximal, complemented
4	#17984010	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
8	C₂²×Dic₁₃₇	—	2	1	no	—
9	C₄⋊Dic₁₃₇	—	2	1	no	—
10	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	#14555487	—	2	1	no	—
13	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
15	C₂²×D₁₃₇	—	2	1	no	—
16	D₅₄₈	—	2	2	no	—
17	C₂²⋊D₁₃₇	—	4	2	no	—
18	C₂×Dic₁₃₇	—	2	1	no	—
19	C₂×Dic₁₃₇	—	2	2	no	—
20	C₂×Dic₁₃₇	—	4	2	no	—
21	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	—	2	2	no	—
23	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₃₇	—	2	2	no	—
26	C₂²×C₁₃₇	—	2	1	no	—
27	C₂²×C₁₃₇	—	2	1	no	—
28	C₂²×C₁₃₇	SD₁₆	1	2	yes	—
29	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₁₃₇	—	4	1	no	—
31	C₂×D₁₃₇	—	4	1	no	—
32	C₂×D₁₃₇	—	4	1	no	—
33	C₄×C₁₃₇	—	2	1	no	—
34	C₄×C₁₃₇	#41	1	2	yes	—
35	Dic₁₃₇	—	4	1	no	—
36	Dic₁₃₇	—	4	2	no	—
37	C₂×C₁₃₇	—	2	2	no	—
38	C₂×C₁₃₇	#123	1	1	yes	characteristic
39	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₁₃₇	—	8	1	no	—
42	C₁₃₇	#481	1	1	yes	characteristic, complemented, Sylow
43	#481	—	137	1	no	maximal, Sylow
44	#121	—	137	1	no	—
45	C₂×C₄⋊C₄	—	137	1	no	—
46	#106	—	137	2	no	—
47	#102	—	137	2	no	—
48	C₂²⋊C₈	D₁₃₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	274	1	no	—
50	C₂×D₄	—	137	1	no	—
51	C₂²×C₄	—	274	1	no	—
52	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
53	C₂×C₈	C₂×D₁₃₇	1	2	yes	—
54	C₄⋊C₄	—	274	1	no	—
55	C₄⋊C₄	—	137	2	no	—
56	C₄⋊C₄	—	137	1	no	—
57	C₂²⋊C₄	—	274	1	no	—
58	C₂³	—	274	1	no	—
59	C₂³	D₅₄₈	1	1	yes	characteristic
60	D₄	—	274	2	no	—
61	D₄	—	548	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	274	1	no	—
64	C₂×C₄	—	274	2	no	—
65	C₂×C₄	—	548	2	no	—
66	C₂×C₄	D₅₄₈	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	548	1	no	—
72	C₂²	—	548	1	no	—
73	C₂²	—	548	1	no	—
74	C₂²	#14555499	1	2	yes	—
75	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	548	1	no	—
78	C₄	—	548	2	no	—
79	C₄	#14555509	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	1096	1	no	—
82	C₂	#17984072	1	1	yes	characteristic
83	C₂	C₂×#14555499	1	1	yes	characteristic
84	C₂	#17984116	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂²⋊C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₃₇
hypocenter	C₁₃₇	#481
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₃₇	C₂	semidirect
3. #17984020	C₂	semidirect
4. #17984010	C₂	semidirect
5. C₂²⋊C₈×C₁₃₇	C₂	semidirect
42. C₁₃₇	#481	semidirect
48. C₂²⋊C₈	D₁₃₇	semidirect