Group #38763325 — subgroups

Subgroups — #38763325

Order 25408.

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

normal subgroups only

This group has 268 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	#27163142	C₂	1	1	yes	maximal, characteristic, complemented
2	#27163122	C₂	1	3	yes	maximal, complemented
3	#27163121	C₂	1	3	yes	maximal, complemented
4	#27163116	C₂	1	3	yes	maximal, complemented
5	C₂×#20263716	C₂	1	3	yes	maximal, complemented
6	#27163103	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20263732	—	2	1	no	—
9	#20263731	C₂²	1	3	yes	complemented
10	C₂×D₁₅₈₈	C₂²	1	3	yes	complemented
11	D₄×D₃₉₇	—	2	3	no	—
12	#20263721	C₂²	1	1	yes	characteristic, complemented
13	#20263719	C₂²	1	3	yes	complemented
14	#20263718	C₂²	1	3	yes	complemented
15	#20263716	C₂²	1	3	yes	complemented
16	#20263716	C₂²	1	6	yes	complemented
17	#20263715	C₂²	1	3	yes	complemented
18	C₂×#15691787	C₂²	1	3	yes	complemented
19	#41×C₃₉₇	—	2	1	no	—
20	#41×C₃₉₇	C₂²	1	1	yes	characteristic
21	#41×C₃₉₇	C₂²	1	3	yes	complemented
22	C₂×D₄×C₃₉₇	—	2	3	no	—
23	C₂×D₄×C₃₉₇	C₂²	1	3	yes	complemented
24	C₂²×D₃₉₇	—	2	3	no	—
25	Dic₇₉₄	C₂³	1	1	yes	characteristic
26	D₁₅₈₈	—	2	3	no	—
27	D₁₅₈₈	C₂³	1	3	yes	—
28	C₂²⋊D₃₉₇	—	2	3	no	—
29	C₄×D₃₉₇	—	2	3	no	—
30	#15691787	C₂³	1	1	yes	characteristic
31	#15691787	C₂³	1	3	yes	—
32	C₂³×C₃₉₇	—	2	3	no	—
33	Q₈×C₃₉₇	D₄	1	1	yes	characteristic
34	Q₈×C₃₉₇	C₂³	1	1	yes	characteristic
35	D₄×C₃₉₇	—	2	3	no	—
36	D₄×C₃₉₇	—	2	3	no	—
37	D₄×C₃₉₇	D₄	1	3	yes	—
38	D₄×C₃₉₇	C₂³	1	3	yes	—
39	C₂×C₄×C₃₉₇	—	2	3	no	—
40	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
41	C₂²×C₃₉₇	—	2	3	no	—
42	C₂²×C₃₉₇	—	8	3	no	—
43	C₂²×C₃₉₇	C₂×D₄	1	3	yes	—
44	C₂×D₃₉₇	—	2	3	no	—
45	C₂×D₃₉₇	—	8	3	no	—
46	C₄×C₃₉₇	—	2	1	no	—
47	C₄×C₃₉₇	C₂×D₄	1	3	yes	—
48	C₄×C₃₉₇	C₂⁴	1	1	yes	characteristic
49	Dic₃₉₇	—	2	1	no	—
50	C₂×C₃₉₇	—	2	3	no	—
51	C₂×C₃₉₇	—	4	3	no	—
52	C₂×C₃₉₇	C₂²×D₄	1	1	yes	characteristic
53	D₃₉₇	—	4	3	no	—
54	C₃₉₇	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	397	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	397	1	no	—
57	H(5,𝔽₂)	D₃₉₇	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	397	3	no	—
59	AGL(1,ℤ/8)	—	397	3	no	—
60	#135	—	397	3	no	—
61	C₂×D₈	—	397	3	no	—
62	#131	—	397	1	no	—
63	#41	—	2	1	no	—
64	#41	—	794	1	no	—
65	#41	—	397	3	no	—
66	#41	C₂×D₃₉₇	1	1	yes	characteristic
67	#41	C₂×D₃₉₇	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	794	3	no	—
70	C₂×D₄	—	397	3	no	—
71	C₂×D₄	C₂×D₃₉₇	1	3	yes	complemented
72	Q₁₆	—	397	1	no	—
73	SD₁₆	—	397	3	no	—
74	SD₁₆	—	397	3	no	—
75	D₈	—	397	3	no	—
76	D₈	—	397	6	no	—
77	M₁₆	—	397	3	no	—
78	C₂×C₈	—	397	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	794	3	no	—
81	Q₈	—	397	1	no	—
82	Q₈	C₂²⋊D₃₉₇	1	1	yes	characteristic
83	Q₈	C₂²×D₃₉₇	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	794	3	no	—
87	D₄	—	794	3	no	—
88	D₄	—	397	3	no	—
89	D₄	C₂²⋊D₃₉₇	1	3	yes	—
90	D₄	C₂²×D₃₉₇	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	794	3	no	—
93	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
94	C₈	—	397	1	no	—
95	C₈	—	397	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	794	3	no	—
99	C₂²	—	3176	3	no	—
100	C₂²	C₂×C₂²⋊D₃₉₇	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	794	1	no	—
103	C₄	C₂×C₂²⋊D₃₉₇	1	3	yes	—
104	C₄	C₂³×D₃₉₇	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	1588	3	no	—
108	C₂	C₂²×C₂²⋊D₃₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27163142

C₂

yes

maximal, characteristic, complemented

#27163122

C₂

yes

maximal, complemented

#27163121

C₂

yes

maximal, complemented

#27163116

C₂

yes

maximal, complemented

C₂×#20263716

C₂

yes

maximal, complemented

#27163103

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₃₉₇

C₂

yes

maximal, characteristic, complemented

#20263732

—

#20263731

C₂²

yes

complemented

C₂×D₁₅₈₈

C₂²

yes

complemented

D₄×D₃₉₇

—

#20263721

C₂²

yes

characteristic, complemented

#20263719

C₂²

yes

complemented

#20263718

C₂²

yes

complemented

#20263716

C₂²

yes

complemented

#20263716

C₂²

yes

complemented

#20263715

C₂²

yes

complemented

C₂×#15691787

C₂²

yes

complemented

#41×C₃₉₇

—

#41×C₃₉₇

C₂²

yes

characteristic

#41×C₃₉₇

C₂²

yes

complemented

C₂×D₄×C₃₉₇

—

C₂×D₄×C₃₉₇

C₂²

yes

complemented

C₂²×D₃₉₇

—

Dic₇₉₄

C₂³

yes

characteristic

D₁₅₈₈

—

D₁₅₈₈

C₂³

yes

—

C₂²⋊D₃₉₇

—

C₄×D₃₉₇

—

#15691787

C₂³

yes

characteristic

#15691787

C₂³

yes

—

C₂³×C₃₉₇

—

Q₈×C₃₉₇

D₄

yes

characteristic

Q₈×C₃₉₇

C₂³

yes

characteristic

D₄×C₃₉₇

—

D₄×C₃₉₇

—

D₄×C₃₉₇

D₄

yes

—

D₄×C₃₉₇

C₂³

yes

—

C₂×C₄×C₃₉₇

—

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

C₂×D₄

yes

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

C₂×D₄

yes

—

C₄×C₃₉₇

C₂⁴

yes

characteristic

Dic₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

C₂²×D₄

yes

characteristic

D₃₉₇

—

C₃₉₇

#576

yes

characteristic, complemented, Sylow

#576

—

397

maximal, Sylow

H(5,𝔽₂)

—

397

—

H(5,𝔽₂)

D₃₉₇

yes

characteristic, complemented

AGL(1,ℤ/8)

—

397

—

AGL(1,ℤ/8)

—

397

—

#135

—

397

—

C₂×D₈

—

397

—

#131

—

397

—

#41

—

#41

—

794

—

#41

—

397

—

#41

C₂×D₃₉₇

yes

characteristic

#41

C₂×D₃₉₇

yes

complemented

C₂×D₄

—

C₂×D₄

—

794

—

C₂×D₄

—

397

—

C₂×D₄

C₂×D₃₉₇

yes

complemented

Q₁₆

—

397

—

SD₁₆

—

397

—

SD₁₆

—

397

—

D₈

—

397

—

D₈

—

397

—

M₁₆

—

397

—

C₂×C₈

—

397

—

C₂³

—

C₂³

—

794

—

Q₈

—

397

—

Q₈

C₂²⋊D₃₉₇

yes

characteristic

Q₈

C₂²×D₃₉₇

yes

characteristic

D₄

—

D₄

—

D₄

—

794

—

D₄

—

794

—

D₄

—

397

—

D₄

C₂²⋊D₃₉₇

yes

—

D₄

C₂²×D₃₉₇

yes

—

C₂×C₄

—

C₂×C₄

—

794

—

C₂×C₄

C₂²×D₃₉₇

yes

—

C₈

—

397

—

C₈

—

397

—

C₂²

—

C₂²

—

C₂²

—

794

—

C₂²

—

3176

—

100

C₂²

C₂×C₂²⋊D₃₉₇

yes

—

101

C₄

—

102

C₄

—

794

—

103

C₄

C₂×C₂²⋊D₃₉₇

yes

—

104

C₄

C₂³×D₃₉₇

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

1588

—

108

C₂

C₂²×C₂²⋊D₃₉₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₃₉₇
commutator, cosocle	C₄×C₃₉₇	C₂⁴
Frattini	C₄	C₂³×D₃₉₇
Fitting	H(5,𝔽₂)×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₃₉₇
hypocenter	C₃₉₇	#576
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₃₉₇

commutator, cosocle

C₄×C₃₉₇

C₂⁴

Frattini

C₄

C₂³×D₃₉₇

Fitting

H(5,𝔽₂)×C₃₉₇

C₂

socle

C₂×C₃₉₇

C₂²×D₄

D₃₉₇

C₃₉₇

G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #27163142	C₂	semidirect
2. #27163122	C₂	semidirect
3. #27163121	C₂	semidirect
4. #27163116	C₂	semidirect
5. C₂×#20263716	C₂	semidirect
6. #27163103	C₂	semidirect
7. H(5,𝔽₂)×C₃₉₇	C₂	semidirect
9. #20263731	C₂²	semidirect
10. C₂×D₁₅₈₈	C₂²	semidirect
12. #20263721	C₂²	semidirect
13. #20263719	C₂²	semidirect
14. #20263718	C₂²	semidirect
15. #20263716	C₂²	semidirect
16. #20263716	C₂²	semidirect
17. #20263715	C₂²	semidirect
18. C₂×#15691787	C₂²	semidirect
21. #41×C₃₉₇	C₂²	semidirect
23. C₂×D₄×C₃₉₇	C₂²	semidirect
54. C₃₉₇	#576	semidirect
57. H(5,𝔽₂)	D₃₉₇	semidirect
67. #41	C₂×D₃₉₇	semidirect
71. C₂×D₄	C₂×D₃₉₇	semidirect

H ≅ G/N

type

1. #27163142

C₂

semidirect

2. #27163122

C₂

semidirect

3. #27163121

C₂

semidirect

4. #27163116

C₂

semidirect

5. C₂×#20263716

C₂

semidirect

6. #27163103

C₂

semidirect

7. H(5,𝔽₂)×C₃₉₇

C₂

semidirect

9. #20263731

C₂²

semidirect

10. C₂×D₁₅₈₈

C₂²

semidirect

12. #20263721

C₂²

semidirect

13. #20263719

C₂²

semidirect

14. #20263718

C₂²

semidirect

15. #20263716

C₂²

semidirect

16. #20263716

C₂²

semidirect

17. #20263715

C₂²

semidirect

18. C₂×#15691787

C₂²

semidirect

21. #41×C₃₉₇

C₂²

semidirect

23. C₂×D₄×C₃₉₇

C₂²

semidirect

54. C₃₉₇

#576

semidirect

57. H(5,𝔽₂)

D₃₉₇

semidirect

67. #41

C₂×D₃₉₇

semidirect

71. C₂×D₄

C₂×D₃₉₇

semidirect

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

subgroup	index
#27163142	2
#27163122	2
#27163121	2
#27163116	2
C₂×#20263716	2
#27163103	2
H(5,𝔽₂)×C₃₉₇	2
#576	397

subgroup

index

H(5,𝔽₂)×C₃₉₇

397

Subgroups — #38763325

Conjugacy classes of subgroups

Complements

Maximal subgroups