Group #22577539 — subgroups

Subgroups — #22577539

Order 8768.

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

normal subgroups only

This group has 198 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	C₂²×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₀₉₆	C₂	1	2	yes	maximal, complemented
3	#17984062	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984010	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₃₇	—	2	1	no	—
7	C₂×D₅₄₈	—	2	4	no	—
8	C₂×D₅₄₈	—	2	1	no	—
9	C₂×D₅₄₈	C₂²	1	2	yes	—
10	C₂×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
12	D₁₀₉₆	—	2	4	no	—
13	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
14	#14555487	—	2	1	no	—
15	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₃₇	C₂²	1	2	yes	complemented
17	C₂²×D₁₃₇	—	2	4	no	—
18	C₂²×D₁₃₇	—	2	2	no	—
19	C₂²×D₁₃₇	—	2	1	no	—
20	C₂²×D₁₃₇	—	4	2	no	—
21	C₂²×D₁₃₇	—	4	2	no	—
22	D₅₄₈	—	2	4	no	—
23	D₅₄₈	—	2	2	no	—
24	D₅₄₈	—	2	2	no	—
25	D₅₄₈	D₄	1	4	yes	complemented
26	C₂²⋊D₁₃₇	—	4	2	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	C₂³×C₁₃₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₈×C₁₃₇	—	2	2	no	—
33	C₂²×C₁₃₇	—	2	1	no	—
34	C₂²×C₁₃₇	—	2	1	no	—
35	C₂²×C₁₃₇	D₈	1	2	yes	complemented
36	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₃₇	—	2	4	no	—
38	C₂×D₁₃₇	—	4	8	no	—
39	C₂×D₁₃₇	—	4	2	no	—
40	C₂×D₁₃₇	—	4	2	no	—
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	2	yes	—
46	Dic₁₃₇	—	4	1	no	—
47	C₂×C₁₃₇	—	2	2	no	—
48	C₂×C₁₃₇	#120	1	1	yes	characteristic
49	C₂×C₁₃₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	D₁₃₇	—	4	4	no	—
52	D₁₃₇	—	8	1	no	—
53	C₁₃₇	#447	1	1	yes	characteristic, complemented, Sylow
54	#447	—	137	1	no	maximal, Sylow
55	C₂²×D₄	—	137	1	no	—
56	C₂×D₈	—	137	2	no	—
57	#121	—	137	1	no	—
58	#102	—	137	2	no	—
59	C₂²⋊C₈	D₁₃₇	1	1	yes	characteristic, complemented
60	C₂⁴	—	274	1	no	—
61	C₂×D₄	—	274	4	no	—
62	C₂×D₄	—	274	1	no	—
63	C₂×D₄	—	274	1	no	—
64	C₂×D₄	—	137	1	no	—
65	C₂×D₄	—	137	2	no	—
66	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
67	D₈	—	274	4	no	—
68	C₂×C₈	C₂×D₁₃₇	1	2	yes	complemented
69	C₄⋊C₄	—	137	1	no	—
70	C₂²⋊C₄	—	274	1	no	—
71	C₂³	—	274	4	no	—
72	C₂³	—	274	2	no	—
73	C₂³	—	274	1	no	—
74	C₂³	—	548	2	no	—
75	C₂³	—	548	2	no	—
76	C₂³	D₅₄₈	1	1	yes	characteristic
77	D₄	—	274	4	no	—
78	D₄	—	274	2	no	—
79	D₄	—	274	2	no	—
80	D₄	—	548	2	no	—
81	D₄	—	137	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	274	1	no	—
84	C₂×C₄	D₅₄₈	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	—	274	4	no	—
90	C₂²	—	548	8	no	—
91	C₂²	—	548	2	no	—
92	C₂²	—	548	2	no	—
93	C₂²	—	548	1	no	—
94	C₂²	—	548	1	no	—
95	C₂²	—	548	1	no	—
96	C₂²	D₁₀₉₆	1	2	yes	complemented
97	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	548	1	no	—
100	C₄	D₄×D₁₃₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	548	4	no	—
103	C₂	—	1096	1	no	—
104	C₂	#17984058	1	1	yes	characteristic
105	C₂	C₂×D₁₀₉₆	1	1	yes	characteristic
106	C₂	#17984116	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₄₈

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₉₆

C₂

yes

maximal, complemented

#17984062

C₂

yes

maximal, characteristic, complemented

#17984010

C₂

yes

maximal, complemented

C₂²⋊C₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₇

—

C₂×D₅₄₈

—

C₂×D₅₄₈

—

C₂×D₅₄₈

C₂²

yes

—

C₂×D₅₄₈

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₁₃₇

—

D₁₀₉₆

—

C₄⋊Dic₁₃₇

C₂²

yes

characteristic, complemented

#14555487

—

C₂²×C₄×C₁₃₇

C₂²

yes

characteristic

C₂×C₈×C₁₃₇

C₂²

yes

complemented

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

D₄

yes

complemented

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

D₈

yes

complemented

C₂²×C₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#120

yes

characteristic

C₂×C₁₃₇

C₂×D₈

yes

characteristic

C₂×C₁₃₇

AGL(1,ℤ/8)

yes

characteristic

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#447

yes

characteristic, complemented, Sylow

#447

—

137

maximal, Sylow

C₂²×D₄

—

137

—

C₂×D₈

—

137

—

#121

—

137

—

#102

—

137

—

C₂²⋊C₈

D₁₃₇

yes

characteristic, complemented

C₂⁴

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

D₈

—

274

—

C₂×C₈

C₂×D₁₃₇

yes

complemented

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂³

—

274

—

C₂³

—

274

—

C₂³

—

274

—

C₂³

—

548

—

C₂³

—

548

—

C₂³

D₅₄₈

yes

characteristic

D₄

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

548

—

D₄

—

137

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

D₅₄₈

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

—

274

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

—

548

—

C₂²

D₁₀₉₆

yes

complemented

C₂²

C₂×D₅₄₈

yes

characteristic

C₄

—

C₄

—

548

—

100

C₄

D₄×D₁₃₇

yes

—

101

C₂

—

102

C₂

—

548

—

103

C₂

—

1096

—

104

C₂

#17984058

yes

characteristic

105

C₂

C₂×D₁₀₉₆

yes

characteristic

106

C₂

#17984116

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	#447
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

#447

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂²×D₅₄₈	C₂	semidirect
2. C₂×D₁₀₉₆	C₂	semidirect
3. #17984062	C₂	semidirect
4. #17984010	C₂	semidirect
5. C₂²⋊C₈×C₁₃₇	C₂	semidirect
10. C₂×D₅₄₈	C₂²	semidirect
13. C₄⋊Dic₁₃₇	C₂²	semidirect
16. C₂×C₈×C₁₃₇	C₂²	semidirect
25. D₅₄₈	D₄	semidirect
35. C₂²×C₁₃₇	D₈	semidirect
53. C₁₃₇	#447	semidirect
59. C₂²⋊C₈	D₁₃₇	semidirect
68. C₂×C₈	C₂×D₁₃₇	semidirect
96. C₂²	D₁₀₉₆	semidirect

H ≅ G/N

type

1. C₂²×D₅₄₈

C₂

semidirect

2. C₂×D₁₀₉₆

C₂

semidirect

3. #17984062

C₂

semidirect

4. #17984010

C₂

semidirect

5. C₂²⋊C₈×C₁₃₇

C₂

semidirect

10. C₂×D₅₄₈

C₂²

semidirect

13. C₄⋊Dic₁₃₇

C₂²

semidirect

16. C₂×C₈×C₁₃₇

C₂²

semidirect

25. D₅₄₈

D₄

semidirect

35. C₂²×C₁₃₇

D₈

semidirect

53. C₁₃₇

#447

semidirect

59. C₂²⋊C₈

D₁₃₇

semidirect

68. C₂×C₈

C₂×D₁₃₇

semidirect

96. C₂²

D₁₀₉₆

semidirect

Subgroups — #22577539

Conjugacy classes of subgroups

Complements