Group #22577632 — subgroups

Subgroups — #22577632

Order 8768.

124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 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×#14555499	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984067	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984062	C₂	1	1	yes	maximal, characteristic
4	#17984020	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984013	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984010	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
9	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
11	#14555499	—	2	2	no	—
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	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	—	2	2	no	—
19	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₇	—	2	1	no	—
22	Dic₂₇₄	—	2	1	no	—
23	Dic₂₇₄	—	2	1	no	—
24	D₅₄₈	—	2	1	no	—
25	D₅₄₈	—	2	1	no	—
26	C₂²⋊D₁₃₇	—	4	2	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂×Dic₁₃₇	—	2	1	no	—
29	C₂×Dic₁₃₇	—	2	1	no	—
30	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄×C₁₃₇	—	2	1	no	—
33	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₃₇	—	2	1	no	—
36	C₈×C₁₃₇	D₄	1	2	yes	complemented
37	C₂²×C₁₃₇	—	2	1	no	—
38	C₂²×C₁₃₇	—	2	1	no	—
39	C₂²×C₁₃₇	SD₁₆	1	2	yes	complemented
40	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₃₇	—	4	1	no	—
42	C₂×D₁₃₇	—	4	1	no	—
43	C₂×D₁₃₇	—	4	1	no	—
44	C₄×C₁₃₇	—	2	1	no	—
45	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₃₇	#41	1	1	yes	characteristic
47	Dic₁₃₇	—	4	1	no	—
48	Dic₁₃₇	—	4	1	no	—
49	Dic₁₃₇	—	4	1	no	—
50	C₂×C₁₃₇	—	2	2	no	—
51	C₂×C₁₃₇	#121	1	1	yes	characteristic
52	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂×C₁₃₇	#135	1	1	yes	characteristic
54	D₁₃₇	—	8	1	no	—
55	C₁₃₇	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	137	1	no	maximal, Sylow
57	C₂×SD₁₆	—	137	1	no	—
58	C₂²×C₈	D₁₃₇	1	1	yes	characteristic, complemented
59	C₂²⋊Q₈	—	137	1	no	—
60	#121	—	137	1	no	—
61	#106	—	137	1	no	—
62	#103	—	137	1	no	—
63	#102	—	137	1	no	—
64	C₂×Q₈	—	137	1	no	—
65	C₂×D₄	—	274	1	no	—
66	C₂×D₄	—	137	1	no	—
67	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
68	SD₁₆	—	274	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
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	—	274	1	no	—
78	C₂³	D₅₄₈	1	1	yes	characteristic
79	Q₈	—	274	1	no	—
80	Q₈	—	274	1	no	—
81	D₄	—	274	1	no	—
82	D₄	—	274	1	no	—
83	D₄	—	548	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	274	1	no	—
87	C₂×C₄	—	274	1	no	—
88	C₂×C₄	—	274	1	no	—
89	C₂×C₄	D₅₄₈	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	C₂²⋊D₁₃₇	1	2	yes	complemented
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	548	1	no	—
96	C₂²	—	548	1	no	—
97	C₂²	—	548	1	no	—
98	C₂²	#14555499	1	2	yes	complemented
99	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	548	1	no	—
102	C₄	—	548	1	no	—
103	C₄	—	548	1	no	—
104	C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
105	C₄	#14555510	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	1096	1	no	—
108	C₂	#17984062	1	1	yes	characteristic
109	C₂	C₂×#14555499	1	1	yes	characteristic
110	C₂	#17984110	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555499

C₂

yes

maximal, characteristic, complemented

#17984067

C₂

yes

maximal, characteristic, complemented

#17984062

C₂

yes

maximal, characteristic

#17984020

C₂

yes

maximal, characteristic, complemented

#17984013

C₂

yes

maximal, characteristic, complemented

#17984010

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

characteristic

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₇

—

#14555499

—

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

#14555489

—

C₂²⋊Dic₁₃₇

—

#14555487

—

C₂²×C₄×C₁₃₇

C₂²

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

—

Dic₂₇₄

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

D₄

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

SD₁₆

yes

complemented

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

#121

yes

characteristic

C₂×C₁₃₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₇

#135

yes

characteristic

D₁₃₇

—

C₁₃₇

#465

yes

characteristic, complemented, Sylow

#465

—

137

maximal, Sylow

C₂×SD₁₆

—

137

—

C₂²×C₈

D₁₃₇

yes

characteristic, complemented

C₂²⋊Q₈

—

137

—

#121

—

137

—

#106

—

137

—

#103

—

137

—

#102

—

137

—

C₂×Q₈

—

137

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

SD₁₆

—

274

—

C₂×C₈

—

C₂×C₈

C₂×D₁₃₇

yes

characteristic

C₂×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₂³

—

274

—

C₂³

D₅₄₈

yes

characteristic

Q₈

—

274

—

Q₈

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

548

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

D₅₄₈

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₈

—

C₈

C₂²⋊D₁₃₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

#14555499

yes

complemented

C₂²

C₂×D₅₄₈

yes

characteristic

100

C₄

—

101

C₄

—

548

—

102

C₄

—

548

—

103

C₄

—

548

—

104

C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

105

C₄

#14555510

yes

characteristic

106

C₂

—

107

C₂

—

1096

—

108

C₂

#17984062

yes

characteristic

109

C₂

C₂×#14555499

yes

characteristic

110

C₂

#17984110

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	#465
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

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

#465

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#14555499	C₂	semidirect
2. #17984067	C₂	semidirect
4. #17984020	C₂	semidirect
5. #17984013	C₂	semidirect
6. #17984010	C₂	semidirect
7. C₂²×C₈×C₁₃₇	C₂	semidirect
36. C₈×C₁₃₇	D₄	semidirect
39. C₂²×C₁₃₇	SD₁₆	semidirect
55. C₁₃₇	#465	semidirect
58. C₂²×C₈	D₁₃₇	semidirect
92. C₈	C₂²⋊D₁₃₇	semidirect
98. C₂²	#14555499	semidirect

H ≅ G/N

type

1. C₂×#14555499

C₂

semidirect

2. #17984067

C₂

semidirect

4. #17984020

C₂

semidirect

5. #17984013

C₂

semidirect

6. #17984010

C₂

semidirect

7. C₂²×C₈×C₁₃₇

C₂

semidirect

36. C₈×C₁₃₇

D₄

semidirect

39. C₂²×C₁₃₇

SD₁₆

semidirect

55. C₁₃₇

#465

semidirect

58. C₂²×C₈

D₁₃₇

semidirect

92. C₈

C₂²⋊D₁₃₇

semidirect

98. C₂²

#14555499

semidirect

Subgroups — #22577632

Conjugacy classes of subgroups

Complements