Group #38763263 — subgroups

Subgroups — #38763263

Order 25408.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#27163093	C₂	1	1	yes	maximal, characteristic, complemented
2	#27163090	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163081	C₂	1	3	yes	maximal, complemented
4	#27163071	C₂	1	3	yes	maximal, complemented
5	#27163069	C₂	1	3	yes	maximal, complemented
6	#27163066	C₂	1	3	yes	maximal, complemented
7	#126×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃₉₇	C₂²	1	1	yes	characteristic
9	C₂×D₁₅₈₈	C₂²	1	3	yes	—
10	C₂×D₁₅₈₈	C₂²	1	3	yes	—
11	C₂×D₁₅₈₈	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₃₉₇	C₂²	1	3	yes	—
13	C₂×C₄×D₃₉₇	C₂²	1	3	yes	complemented
14	#20263710	C₂²	1	3	yes	complemented
15	#20263708	C₂²	1	3	yes	—
16	#20263708	C₂²	1	3	yes	—
17	#20263708	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₃₉₇	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₃₉₇	C₂²	1	3	yes	—
20	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₉₇	—	4	3	no	—
22	C₂²×D₃₉₇	C₂³	1	1	yes	characteristic
23	C₂²×D₃₉₇	C₂³	1	1	yes	characteristic
24	C₂²×D₃₉₇	C₂³	1	3	yes	—
25	D₁₅₈₈	—	4	3	no	—
26	D₁₅₈₈	—	4	3	no	—
27	D₁₅₈₈	—	4	3	no	—
28	C₂²⋊D₃₉₇	—	4	3	no	—
29	C₂×Dic₃₉₇	C₂³	1	3	yes	—
30	C₄×D₃₉₇	—	4	3	no	—
31	C₂³×C₃₉₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
33	C₂×C₄×C₃₉₇	C₂³	1	3	yes	—
34	C₂²×C₃₉₇	—	2	3	no	—
35	C₂²×C₃₉₇	C₂⁴	1	1	yes	characteristic
36	C₂×D₃₉₇	—	2	3	no	—
37	C₂×D₃₉₇	—	2	3	no	—
38	C₂×D₃₉₇	—	2	3	no	—
39	C₂×D₃₉₇	—	2	3	no	—
40	C₂×D₃₉₇	—	2	3	no	—
41	C₂×D₃₉₇	—	4	4	no	—
42	C₄×C₃₉₇	—	2	3	no	—
43	C₄×C₃₉₇	—	2	3	no	—
44	Dic₃₉₇	—	2	3	no	—
45	C₂×C₃₉₇	—	4	1	no	—
46	C₂×C₃₉₇	H(5,𝔽₂)	1	3	yes	—
47	D₃₉₇	—	4	3	no	—
48	D₃₉₇	—	4	1	no	—
49	D₃₉₇	—	4	1	no	—
50	C₃₉₇	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	397	1	no	maximal, Sylow
52	#127	—	397	1	no	—
53	#126	—	397	1	no	—
54	#126	D₃₉₇	1	1	yes	characteristic, complemented
55	#123	—	397	3	no	—
56	#121	—	397	3	no	—
57	#121	—	397	3	no	—
58	#120	—	397	3	no	—
59	C₂⁴	—	397	1	no	—
60	C₂×D₄	—	397	3	no	—
61	C₂×D₄	—	397	3	no	—
62	C₂×D₄	—	397	3	no	—
63	C₂×D₄	—	397	3	no	—
64	C₂²×C₄	—	397	3	no	—
65	C₄⋊C₄	—	397	3	no	—
66	C₄⋊C₄	C₂×D₃₉₇	1	3	yes	complemented
67	C₂²⋊C₄	—	397	3	no	—
68	C₂²⋊C₄	—	397	3	no	—
69	C₂²⋊C₄	—	397	3	no	—
70	C₂²⋊C₄	C₂×D₃₉₇	1	3	yes	—
71	C₄²	C₂×D₃₉₇	1	1	yes	characteristic, complemented
72	C₂³	—	1588	3	no	—
73	C₂³	—	397	1	no	—
74	C₂³	—	397	1	no	—
75	C₂³	—	397	3	no	—
76	C₂³	C₂²×D₃₉₇	1	1	yes	characteristic
77	D₄	—	1588	3	no	—
78	D₄	—	1588	3	no	—
79	D₄	—	1588	3	no	—
80	D₄	—	1588	3	no	—
81	C₂×C₄	—	1588	3	no	—
82	C₂×C₄	—	397	3	no	—
83	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
84	C₂×C₄	C₂²×D₃₉₇	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	794	3	no	—
87	C₂²	—	794	3	no	—
88	C₂²	—	794	3	no	—
89	C₂²	—	794	3	no	—
90	C₂²	—	794	3	no	—
91	C₂²	—	1588	4	no	—
92	C₂²	C₂³×D₃₉₇	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	794	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	1588	1	no	—
98	C₂	—	1588	1	no	—
99	C₂	—	1588	3	no	—
100	C₂	#27163142	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27163093

C₂

yes

maximal, characteristic, complemented

#27163090

C₂

yes

maximal, characteristic, complemented

#27163081

C₂

yes

maximal, complemented

#27163071

C₂

yes

maximal, complemented

#27163069

C₂

yes

maximal, complemented

#27163066

C₂

yes

maximal, complemented

#126×C₃₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₉₇

C₂²

yes

characteristic

C₂×D₁₅₈₈

C₂²

yes

—

C₂×D₁₅₈₈

C₂²

yes

—

C₂×D₁₅₈₈

C₂²

yes

complemented

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂×C₄×D₃₉₇

C₂²

yes

complemented

#20263710

C₂²

yes

complemented

#20263708

C₂²

yes

—

#20263708

C₂²

yes

—

#20263708

C₂²

yes

complemented

C₄⋊C₄×C₃₉₇

C₂²

yes

complemented

C₂²⋊C₄×C₃₉₇

C₂²

yes

—

C₄²×C₃₉₇

C₂²

yes

characteristic, complemented

C₂²×D₃₉₇

—

C₂²×D₃₉₇

C₂³

yes

characteristic

C₂²×D₃₉₇

C₂³

yes

characteristic

C₂²×D₃₉₇

C₂³

yes

—

D₁₅₈₈

—

D₁₅₈₈

—

D₁₅₈₈

—

C₂²⋊D₃₉₇

—

C₂×Dic₃₉₇

C₂³

yes

—

C₄×D₃₉₇

—

C₂³×C₃₉₇

C₂³

yes

characteristic

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

C₂⁴

yes

characteristic

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

—

Dic₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

H(5,𝔽₂)

yes

—

D₃₉₇

—

D₃₉₇

—

D₃₉₇

—

C₃₉₇

#560

yes

characteristic, complemented, Sylow

#560

—

397

maximal, Sylow

#127

—

397

—

#126

—

397

—

#126

D₃₉₇

yes

characteristic, complemented

#123

—

397

—

#121

—

397

—

#121

—

397

—

#120

—

397

—

C₂⁴

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂²×C₄

—

397

—

C₄⋊C₄

—

397

—

C₄⋊C₄

C₂×D₃₉₇

yes

complemented

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

C₂×D₃₉₇

yes

—

C₄²

C₂×D₃₉₇

yes

characteristic, complemented

C₂³

—

1588

—

C₂³

—

397

—

C₂³

—

397

—

C₂³

—

397

—

C₂³

C₂²×D₃₉₇

yes

characteristic

D₄

—

1588

—

D₄

—

1588

—

D₄

—

1588

—

D₄

—

1588

—

C₂×C₄

—

1588

—

C₂×C₄

—

397

—

C₂×C₄

C₂²×D₃₉₇

yes

—

C₂×C₄

C₂²×D₃₉₇

yes

—

C₂²

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

1588

—

C₂²

C₂³×D₃₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

794

—

C₂

—

C₂

—

1588

—

C₂

—

1588

—

C₂

—

1588

—

100

C₂

#27163142

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂⁴
Fitting	#126×C₃₉₇	C₂
hypercenter	#126	D₃₉₇
hypocenter	C₃₉₇	#560
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₉₇

commutator, socle, cosocle

C₂²×C₃₉₇

C₂⁴

Fitting

#126×C₃₉₇

C₂

D₃₉₇

C₃₉₇

G ▷ C₂²×C₃₉₇ ▷ C₃₉₇

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #27163093	C₂	semidirect
2. #27163090	C₂	semidirect
3. #27163081	C₂	semidirect
4. #27163071	C₂	semidirect
5. #27163069	C₂	semidirect
6. #27163066	C₂	semidirect
7. #126×C₃₉₇	C₂	semidirect
11. C₂×D₁₅₈₈	C₂²	semidirect
13. C₂×C₄×D₃₉₇	C₂²	semidirect
14. #20263710	C₂²	semidirect
17. #20263708	C₂²	semidirect
18. C₄⋊C₄×C₃₉₇	C₂²	semidirect
20. C₄²×C₃₉₇	C₂²	semidirect
50. C₃₉₇	#560	semidirect
54. #126	D₃₉₇	semidirect
66. C₄⋊C₄	C₂×D₃₉₇	semidirect
71. C₄²	C₂×D₃₉₇	semidirect

H ≅ G/N

type

1. #27163093

C₂

semidirect

2. #27163090

C₂

semidirect

3. #27163081

C₂

semidirect

4. #27163071

C₂

semidirect

5. #27163069

C₂

semidirect

6. #27163066

C₂

semidirect

7. #126×C₃₉₇

C₂

semidirect

11. C₂×D₁₅₈₈

C₂²

semidirect

13. C₂×C₄×D₃₉₇

C₂²

semidirect

14. #20263710

C₂²

semidirect

17. #20263708

C₂²

semidirect

18. C₄⋊C₄×C₃₉₇

C₂²

semidirect

20. C₄²×C₃₉₇

C₂²

semidirect

50. C₃₉₇

#560

semidirect

54. #126

D₃₉₇

semidirect

66. C₄⋊C₄

C₂×D₃₉₇

semidirect

71. C₄²

C₂×D₃₉₇

semidirect

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

subgroup	index
#27163093	2
#27163090	2
#27163081	2
#27163071	2
#27163069	2
#27163066	2
#126×C₃₉₇	2
#560	397

subgroup

index

#126×C₃₉₇

397

Subgroups — #38763263

Conjugacy classes of subgroups

Complements

Maximal subgroups