Group #38763334 — subgroups

Subgroups — #38763334

Order 25408.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#27163143	C₂	1	1	yes	maximal, characteristic, complemented
2	#27163142	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163129	C₂	1	3	yes	maximal, complemented
4	#27163124	C₂	1	3	yes	maximal, complemented
5	#27163118	C₂	1	3	yes	maximal, complemented
6	C₂×#20263720	C₂	1	3	yes	maximal, complemented
7	#131×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20263732	—	2	1	no	—
9	#20263731	C₂²	1	3	yes	—
10	#20263731	C₂²	1	3	yes	complemented
11	#20263730	—	2	3	no	—
12	C₂×Dic₇₉₄	C₂²	1	3	yes	complemented
13	Q₈×D₃₉₇	—	2	1	no	—
14	C₂×D₁₅₈₈	C₂²	1	3	yes	—
15	D₄×D₃₉₇	—	2	3	no	—
16	Dic₁₅₈₈	C₂²	1	3	yes	complemented
17	#20263720	C₂²	1	3	yes	complemented
18	#20263720	C₂²	1	6	yes	complemented
19	#20263720	C₂²	1	1	yes	characteristic, complemented
20	D₃₁₇₆	C₂²	1	3	yes	complemented
21	#41×C₃₉₇	C₂²	1	1	yes	characteristic
22	M₁₆×C₃₉₇	C₂²	1	3	yes	complemented
23	C₂×C₈×C₃₉₇	C₂²	1	3	yes	complemented
24	C₂²×D₃₉₇	—	2	3	no	—
25	Dic₇₉₄	—	2	3	no	—
26	Dic₇₉₄	C₂³	1	3	yes	—
27	Dic₇₉₄	C₂³	1	1	yes	characteristic
28	D₁₅₈₈	—	2	3	no	—
29	D₁₅₈₈	C₂³	1	1	yes	characteristic
30	D₁₅₈₈	C₂³	1	3	yes	—
31	C₂²⋊D₃₉₇	—	2	3	no	—
32	C₂²⋊D₃₉₇	—	2	3	no	—
33	C₂×Dic₃₉₇	—	2	3	no	—
34	C₄×D₃₉₇	—	2	3	no	—
35	C₄×D₃₉₇	—	2	3	no	—
36	Q₈×C₃₉₇	D₄	1	1	yes	characteristic
37	D₄×C₃₉₇	D₄	1	3	yes	—
38	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
39	C₈×C₃₉₇	C₂³	1	1	yes	characteristic
40	C₈×C₃₉₇	C₂³	1	3	yes	—
41	C₂²×C₃₉₇	C₂×D₄	1	3	yes	—
42	C₂×D₃₉₇	—	2	1	no	—
43	C₂×D₃₉₇	—	2	3	no	—
44	C₂×D₃₉₇	—	8	3	no	—
45	C₄×C₃₉₇	C₂×D₄	1	3	yes	—
46	C₄×C₃₉₇	C₂⁴	1	1	yes	characteristic
47	Dic₃₉₇	—	2	1	no	—
48	Dic₃₉₇	—	2	3	no	—
49	C₂×C₃₉₇	—	2	3	no	—
50	C₂×C₃₉₇	C₂²×D₄	1	1	yes	characteristic
51	D₃₉₇	—	4	1	no	—
52	D₃₉₇	—	4	3	no	—
53	C₃₉₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	397	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	397	1	no	—
56	H(5,𝔽₂)	—	397	1	no	—
57	#137	—	397	3	no	—
58	AGL(1,ℤ/8)	—	397	3	no	—
59	#135	—	397	3	no	—
60	C₂×SD₁₆	—	397	3	no	—
61	#131	D₃₉₇	1	1	yes	characteristic, complemented
62	#41	—	794	1	no	—
63	#41	—	794	3	no	—
64	#41	—	397	3	no	—
65	#41	—	397	3	no	—
66	#41	C₂×D₃₉₇	1	1	yes	characteristic
67	C₂×Q₈	—	794	1	no	—
68	C₂×Q₈	—	397	3	no	—
69	C₂×D₄	—	794	3	no	—
70	C₂×D₄	—	397	3	no	—
71	Q₁₆	—	397	3	no	—
72	SD₁₆	—	397	3	no	—
73	SD₁₆	—	397	6	no	—
74	SD₁₆	—	397	1	no	—
75	D₈	—	397	3	no	—
76	M₁₆	C₂×D₃₉₇	1	3	yes	complemented
77	C₂×C₈	C₂×D₃₉₇	1	3	yes	complemented
78	C₂³	—	794	3	no	—
79	Q₈	—	794	3	no	—
80	Q₈	—	397	3	no	—
81	Q₈	—	397	1	no	—
82	Q₈	D₁₅₈₈	1	1	yes	characteristic
83	D₄	—	794	3	no	—
84	D₄	—	794	3	no	—
85	D₄	—	794	3	no	—
86	D₄	—	397	1	no	—
87	D₄	—	397	3	no	—
88	D₄	D₁₅₈₈	1	3	yes	—
89	C₂×C₄	—	794	3	no	—
90	C₂×C₄	—	794	3	no	—
91	C₂×C₄	—	794	3	no	—
92	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
93	C₈	C₂²×D₃₉₇	1	1	yes	characteristic
94	C₈	C₂²×D₃₉₇	1	3	yes	—
95	C₂²	—	794	1	no	—
96	C₂²	—	794	3	no	—
97	C₂²	—	3176	3	no	—
98	C₂²	C₂×D₁₅₈₈	1	3	yes	—
99	C₄	—	794	1	no	—
100	C₄	—	794	3	no	—
101	C₄	C₂×D₁₅₈₈	1	3	yes	—
102	C₄	C₂³×D₃₉₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	1588	3	no	—
105	C₂	—	1588	1	no	—
106	C₂	C₂²×D₁₅₈₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27163143

C₂

yes

maximal, characteristic, complemented

#27163142

C₂

yes

maximal, characteristic, complemented

#27163129

C₂

yes

maximal, complemented

#27163124

C₂

yes

maximal, complemented

#27163118

C₂

yes

maximal, complemented

C₂×#20263720

C₂

yes

maximal, complemented

#131×C₃₉₇

C₂

yes

maximal, characteristic, complemented

#20263732

—

#20263731

C₂²

yes

—

#20263731

C₂²

yes

complemented

#20263730

—

C₂×Dic₇₉₄

C₂²

yes

complemented

Q₈×D₃₉₇

—

C₂×D₁₅₈₈

C₂²

yes

—

D₄×D₃₉₇

—

Dic₁₅₈₈

C₂²

yes

complemented

#20263720

C₂²

yes

complemented

#20263720

C₂²

yes

complemented

#20263720

C₂²

yes

characteristic, complemented

D₃₁₇₆

C₂²

yes

complemented

#41×C₃₉₇

C₂²

yes

characteristic

M₁₆×C₃₉₇

C₂²

yes

complemented

C₂×C₈×C₃₉₇

C₂²

yes

complemented

C₂²×D₃₉₇

—

Dic₇₉₄

—

Dic₇₉₄

C₂³

yes

—

Dic₇₉₄

C₂³

yes

characteristic

D₁₅₈₈

—

D₁₅₈₈

C₂³

yes

characteristic

D₁₅₈₈

C₂³

yes

—

C₂²⋊D₃₉₇

—

C₂²⋊D₃₉₇

—

C₂×Dic₃₉₇

—

C₄×D₃₉₇

—

C₄×D₃₉₇

—

Q₈×C₃₉₇

D₄

yes

characteristic

D₄×C₃₉₇

D₄

yes

—

C₂×C₄×C₃₉₇

C₂³

yes

—

C₈×C₃₉₇

C₂³

yes

characteristic

C₈×C₃₉₇

C₂³

yes

—

C₂²×C₃₉₇

C₂×D₄

yes

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

C₂×D₄

yes

—

C₄×C₃₉₇

C₂⁴

yes

characteristic

Dic₃₉₇

—

Dic₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

C₂²×D₄

yes

characteristic

D₃₉₇

—

D₃₉₇

—

C₃₉₇

#577

yes

characteristic, complemented, Sylow

#577

—

397

maximal, Sylow

2₋¹⁺⁴

—

397

—

H(5,𝔽₂)

—

397

—

#137

—

397

—

AGL(1,ℤ/8)

—

397

—

#135

—

397

—

C₂×SD₁₆

—

397

—

#131

D₃₉₇

yes

characteristic, complemented

#41

—

794

—

#41

—

794

—

#41

—

397

—

#41

—

397

—

#41

C₂×D₃₉₇

yes

characteristic

C₂×Q₈

—

794

—

C₂×Q₈

—

397

—

C₂×D₄

—

794

—

C₂×D₄

—

397

—

Q₁₆

—

397

—

SD₁₆

—

397

—

SD₁₆

—

397

—

SD₁₆

—

397

—

D₈

—

397

—

M₁₆

C₂×D₃₉₇

yes

complemented

C₂×C₈

C₂×D₃₉₇

yes

complemented

C₂³

—

794

—

Q₈

—

794

—

Q₈

—

397

—

Q₈

—

397

—

Q₈

D₁₅₈₈

yes

characteristic

D₄

—

794

—

D₄

—

794

—

D₄

—

794

—

D₄

—

397

—

D₄

—

397

—

D₄

D₁₅₈₈

yes

—

C₂×C₄

—

794

—

C₂×C₄

—

794

—

C₂×C₄

—

794

—

C₂×C₄

C₂²×D₃₉₇

yes

—

C₈

C₂²×D₃₉₇

yes

characteristic

C₈

C₂²×D₃₉₇

yes

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

3176

—

C₂²

C₂×D₁₅₈₈

yes

—

C₄

—

794

—

100

C₄

—

794

—

101

C₄

C₂×D₁₅₈₈

yes

—

102

C₄

C₂³×D₃₉₇

yes

characteristic

103

C₂

—

104

C₂

—

1588

—

105

C₂

—

1588

—

106

C₂

C₂²×D₁₅₈₈

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₅₈₈
commutator, cosocle	C₄×C₃₉₇	C₂⁴
Frattini	C₄	C₂³×D₃₉₇
Fitting	#131×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂²×D₄
hypercenter	#131	D₃₉₇
hypocenter	C₃₉₇	#577
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₁₅₈₈

commutator, cosocle

C₄×C₃₉₇

C₂⁴

Frattini

C₄

C₂³×D₃₉₇

Fitting

#131×C₃₉₇

C₂

socle

C₂×C₃₉₇

C₂²×D₄

D₃₉₇

C₃₉₇

G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #27163143	C₂	semidirect
2. #27163142	C₂	semidirect
3. #27163129	C₂	semidirect
4. #27163124	C₂	semidirect
5. #27163118	C₂	semidirect
6. C₂×#20263720	C₂	semidirect
7. #131×C₃₉₇	C₂	semidirect
10. #20263731	C₂²	semidirect
12. C₂×Dic₇₉₄	C₂²	semidirect
16. Dic₁₅₈₈	C₂²	semidirect
17. #20263720	C₂²	semidirect
18. #20263720	C₂²	semidirect
19. #20263720	C₂²	semidirect
20. D₃₁₇₆	C₂²	semidirect
22. M₁₆×C₃₉₇	C₂²	semidirect
23. C₂×C₈×C₃₉₇	C₂²	semidirect
53. C₃₉₇	#577	semidirect
61. #131	D₃₉₇	semidirect
76. M₁₆	C₂×D₃₉₇	semidirect
77. C₂×C₈	C₂×D₃₉₇	semidirect

H ≅ G/N

type

1. #27163143

C₂

semidirect

2. #27163142

C₂

semidirect

3. #27163129

C₂

semidirect

4. #27163124

C₂

semidirect

5. #27163118

C₂

semidirect

6. C₂×#20263720

C₂

semidirect

7. #131×C₃₉₇

C₂

semidirect

10. #20263731

C₂²

semidirect

12. C₂×Dic₇₉₄

C₂²

semidirect

16. Dic₁₅₈₈

C₂²

semidirect

17. #20263720

C₂²

semidirect

18. #20263720

C₂²

semidirect

19. #20263720

C₂²

semidirect

20. D₃₁₇₆

C₂²

semidirect

22. M₁₆×C₃₉₇

C₂²

semidirect

23. C₂×C₈×C₃₉₇

C₂²

semidirect

53. C₃₉₇

#577

semidirect

61. #131

D₃₉₇

semidirect

76. M₁₆

C₂×D₃₉₇

semidirect

77. C₂×C₈

C₂×D₃₉₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#27163143	2
#27163142	2
#27163129	2
#27163124	2
#27163118	2
C₂×#20263720	2
#131×C₃₉₇	2
#577	397

subgroup

index

#131×C₃₉₇

397

Subgroups — #38763334

Conjugacy classes of subgroups

Complements

Maximal subgroups