Group #22577646 — subgroups

Subgroups — #22577646

Order 8768.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14555493	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984064	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984059	C₂	1	1	yes	maximal, characteristic
4	#17984022	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984012	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984009	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
9	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
10	#14555493	—	2	2	no	—
11	C₂×#11764040	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
14	#14555489	—	2	1	no	—
15	C₂²⋊Dic₁₃₇	—	2	1	no	—
16	#14555487	—	2	1	no	—
17	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₃₇	—	2	2	no	—
20	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₇	D₄	1	1	yes	characteristic
22	C₂²⋊D₁₃₇	—	4	2	no	—
23	C₂×Dic₁₃₇	—	2	1	no	—
24	C₂×Dic₁₃₇	—	2	1	no	—
25	C₂×Dic₁₃₇	D₄	1	1	yes	characteristic
26	C₄×D₁₃₇	—	2	1	no	—
27	C₄×D₁₃₇	—	2	1	no	—
28	#11764040	—	2	1	no	—
29	C₂³×C₁₃₇	—	2	1	no	—
30	Q₈×C₁₃₇	—	2	1	no	—
31	Q₈×C₁₃₇	—	2	1	no	—
32	D₄×C₁₃₇	—	2	1	no	—
33	D₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄×C₁₃₇	—	2	1	no	—
35	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₃₇	D₄	1	2	yes	complemented
37	C₂²×C₁₃₇	—	4	1	no	—
38	C₂²×C₁₃₇	—	4	1	no	—
39	C₂²×C₁₃₇	—	4	1	no	—
40	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	1	no	—
42	C₂×D₁₃₇	—	2	1	no	—
43	C₂×D₁₃₇	—	2	1	no	—
44	C₄×C₁₃₇	—	4	1	no	—
45	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₃₇	#41	1	1	yes	characteristic
47	Dic₁₃₇	—	2	1	no	—
48	Dic₁₃₇	—	4	1	no	—
49	Dic₁₃₇	—	4	1	no	—
50	C₂×C₁₃₇	—	8	1	no	—
51	C₂×C₁₃₇	#121	1	1	yes	characteristic
52	C₂×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂×C₁₃₇	#137	1	1	yes	characteristic
54	D₁₃₇	—	4	1	no	—
55	C₁₃₇	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	137	1	no	maximal, Sylow
57	C₂×SD₁₆	D₁₃₇	1	1	yes	characteristic, complemented
58	C₂×M₁₆	—	137	1	no	—
59	C₂²⋊Q₈	—	137	1	no	—
60	#121	—	137	1	no	—
61	#107	—	137	1	no	—
62	#103	—	137	1	no	—
63	#102	—	137	1	no	—
64	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
65	C₂×D₄	—	274	1	no	—
66	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
67	C₂²×C₄	—	137	1	no	—
68	SD₁₆	—	2	2	no	—
69	M₁₆	—	274	2	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
72	C₄⋊C₄	—	274	1	no	—
73	C₄⋊C₄	—	137	1	no	—
74	C₄⋊C₄	—	137	1	no	—
75	C₂²⋊C₄	—	274	1	no	—
76	C₂²⋊C₄	—	274	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	137	1	no	—
79	Q₈	—	2	1	no	—
80	Q₈	—	2	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	548	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	274	1	no	—
86	C₂×C₄	—	274	1	no	—
87	C₂×C₄	—	274	1	no	—
88	C₂×C₄	—	274	1	no	—
89	C₂×C₄	—	137	1	no	—
90	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	C₈	—	274	1	no	—
92	C₈	C₂²⋊D₁₃₇	1	2	yes	complemented
93	C₂²	—	4	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	274	1	no	—
97	C₂²	—	274	1	no	—
98	C₂²	—	274	1	no	—
99	C₂²	D₄×D₁₃₇	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	274	1	no	—
102	C₄	—	548	1	no	—
103	C₄	—	548	1	no	—
104	C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
105	C₄	#14555509	1	1	yes	characteristic
106	C₂	—	8	1	no	—
107	C₂	—	548	1	no	—
108	C₂	#17984059	1	1	yes	characteristic
109	C₂	#17984115	1	1	yes	characteristic
110	C₂	#17984117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555493

C₂

yes

maximal, characteristic, complemented

#17984064

C₂

yes

maximal, characteristic, complemented

#17984059

C₂

yes

maximal, characteristic

#17984022

C₂

yes

maximal, characteristic, complemented

#17984012

C₂

yes

maximal, characteristic, complemented

#17984009

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₃₇

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

#14555493

—

C₂×#11764040

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

#14555489

—

C₂²⋊Dic₁₃₇

—

#14555487

—

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂×D₄×C₁₃₇

C₂²

yes

characteristic

SD₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

D₄

yes

characteristic

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

D₄

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₄×C₁₃₇

#41

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

C₁₃₇

#468

yes

characteristic, complemented, Sylow

#468

—

137

maximal, Sylow

C₂×SD₁₆

D₁₃₇

yes

characteristic, complemented

C₂×M₁₆

—

137

—

C₂²⋊Q₈

—

137

—

#121

—

137

—

#107

—

137

—

#103

—

137

—

#102

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

274

—

C₂×D₄

C₂×D₁₃₇

yes

characteristic

C₂²×C₄

—

137

—

SD₁₆

—

M₁₆

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₂³

—

C₂³

—

137

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₈

—

274

—

C₈

C₂²⋊D₁₃₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

D₄×D₁₃₇

yes

characteristic

100

C₄

—

101

C₄

—

274

—

102

C₄

—

548

—

103

C₄

—

548

—

104

C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

105

C₄

#14555509

yes

characteristic

106

C₂

—

107

C₂

—

548

—

108

C₂

#17984059

yes

characteristic

109

C₂

#17984115

yes

characteristic

110

C₂

#17984117

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₂×SD₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₃₇
hypocenter	C₁₃₇	#468
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₇

Fitting

C₂×SD₁₆×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₃₇

hypocenter

C₁₃₇

#468

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#14555493	C₂	semidirect
2. #17984064	C₂	semidirect
4. #17984022	C₂	semidirect
5. #17984012	C₂	semidirect
6. #17984009	C₂	semidirect
7. C₂×SD₁₆×C₁₃₇	C₂	semidirect
36. C₈×C₁₃₇	D₄	semidirect
55. C₁₃₇	#468	semidirect
57. C₂×SD₁₆	D₁₃₇	semidirect
92. C₈	C₂²⋊D₁₃₇	semidirect

H ≅ G/N

type

1. C₂×#14555493

C₂

semidirect

2. #17984064

C₂

semidirect

4. #17984022

C₂

semidirect

5. #17984012

C₂

semidirect

6. #17984009

C₂

semidirect

7. C₂×SD₁₆×C₁₃₇

C₂

semidirect

36. C₈×C₁₃₇

D₄

semidirect

55. C₁₃₇

#468

semidirect

57. C₂×SD₁₆

D₁₃₇

semidirect

92. C₈

C₂²⋊D₁₃₇

semidirect

Subgroups — #22577646

Conjugacy classes of subgroups

Complements