Group #38763264 — subgroups

Subgroups — #38763264

Order 25408.

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

normal subgroups only

This group has 260 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	#27163085	C₂	1	2	yes	maximal, complemented
2	#27163084	C₂	1	2	yes	maximal, complemented
3	#27163082	C₂	1	4	yes	maximal, complemented
4	#27163066	C₂	1	4	yes	maximal, complemented
5	#27163064	C₂	1	2	yes	maximal, complemented
6	#124×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₉₇	C₂²	1	2	yes	complemented
8	C₂×Dic₇₉₄	C₂²	1	2	yes	complemented
9	C₂×D₁₅₈₈	C₂²	1	2	yes	—
10	C₂×D₁₅₈₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₃₉₇	C₂²	1	4	yes	—
12	C₂²⋊Dic₃₉₇	C₂²	1	2	yes	complemented
13	#20263708	C₂²	1	4	yes	complemented
14	#20263708	C₂²	1	4	yes	complemented
15	#20263708	C₂²	1	4	yes	complemented
16	C₄×Dic₃₉₇	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₃₉₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₉₇	C₂²	1	4	yes	complemented
20	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₉₇	—	4	4	no	—
22	C₂²×D₃₉₇	—	4	2	no	—
23	C₂²×D₃₉₇	C₂³	1	4	yes	—
24	Dic₇₉₄	—	4	2	no	—
25	D₁₅₈₈	—	4	2	no	—
26	D₁₅₈₈	—	4	2	no	—
27	C₂²⋊D₃₉₇	—	4	4	no	—
28	C₂×Dic₃₉₇	C₂³	1	4	yes	—
29	C₂³×C₃₉₇	C₂³	1	2	yes	—
30	Q₈×C₃₉₇	—	4	1	no	—
31	D₄×C₃₉₇	—	4	1	no	—
32	C₂×C₄×C₃₉₇	C₂³	1	2	yes	—
33	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₃₉₇	C₂³	1	2	yes	—
35	C₂²×C₃₉₇	—	2	2	no	—
36	C₂²×C₃₉₇	—	2	4	no	—
37	C₂²×C₃₉₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₃₉₇	—	2	4	no	—
39	C₂×D₃₉₇	—	2	4	no	—
40	C₂×D₃₉₇	—	2	4	no	—
41	C₂×D₃₉₇	—	4	8	no	—
42	C₄×C₃₉₇	—	2	1	no	—
43	C₄×C₃₉₇	—	2	2	no	—
44	C₄×C₃₉₇	—	2	2	no	—
45	Dic₃₉₇	—	2	4	no	—
46	C₂×C₃₉₇	—	4	2	no	—
47	C₂×C₃₉₇	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₃₉₇	H(5,𝔽₂)	1	2	yes	—
49	D₃₉₇	—	4	4	no	—
50	C₃₉₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	397	1	no	maximal, Sylow
52	#124	—	397	2	no	—
53	#124	—	397	4	no	—
54	#124	—	397	2	no	—
55	#124	D₃₉₇	1	1	yes	characteristic, complemented
56	#120	—	397	2	no	—
57	#120	—	397	4	no	—
58	C₂⁴	—	397	2	no	—
59	C₂×Q₈	—	397	2	no	—
60	C₂×Q₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
61	C₂×D₄	—	397	4	no	—
62	C₂×D₄	—	397	2	no	—
63	C₂×D₄	—	397	2	no	—
64	C₂×D₄	C₂×D₃₉₇	1	1	yes	characteristic
65	C₂²⋊C₄	—	397	2	no	—
66	C₂²⋊C₄	—	397	4	no	—
67	C₂²⋊C₄	—	397	4	no	—
68	C₂²⋊C₄	—	397	4	no	—
69	C₂²⋊C₄	C₂×D₃₉₇	1	4	yes	complemented
70	C₄²	—	397	2	no	—
71	C₄²	C₂×D₃₉₇	1	1	yes	characteristic, complemented
72	C₂³	—	1588	2	no	—
73	C₂³	—	1588	4	no	—
74	C₂³	—	397	4	no	—
75	C₂³	C₂²×D₃₉₇	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	1588	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	1588	2	no	—
80	D₄	—	1588	2	no	—
81	D₄	—	1588	4	no	—
82	C₂×C₄	—	397	4	no	—
83	C₂×C₄	C₂²×D₃₉₇	1	2	yes	—
84	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₃₉₇	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	794	4	no	—
89	C₂²	—	794	4	no	—
90	C₂²	—	794	4	no	—
91	C₂²	—	1588	8	no	—
92	C₂²	C₂³×D₃₉₇	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	794	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	1588	4	no	—
99	C₂	#27163141	1	1	yes	characteristic
100	C₂	#27163142	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27163085

C₂

yes

maximal, complemented

#27163084

C₂

yes

maximal, complemented

#27163082

C₂

yes

maximal, complemented

#27163066

C₂

yes

maximal, complemented

#27163064

C₂

yes

maximal, complemented

#124×C₃₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₉₇

C₂²

yes

complemented

C₂×Dic₇₉₄

C₂²

yes

complemented

C₂×D₁₅₈₈

C₂²

yes

—

C₂×D₁₅₈₈

C₂²

yes

—

C₂×C₂²⋊D₃₉₇

C₂²

yes

—

C₂²⋊Dic₃₉₇

C₂²

yes

complemented

#20263708

C₂²

yes

complemented

#20263708

C₂²

yes

complemented

#20263708

C₂²

yes

complemented

C₄×Dic₃₉₇

C₂²

yes

complemented

C₂×Q₈×C₃₉₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₉₇

C₂²

yes

complemented

C₄²×C₃₉₇

C₂²

yes

characteristic, complemented

C₂²×D₃₉₇

—

C₂²×D₃₉₇

—

C₂²×D₃₉₇

C₂³

yes

—

Dic₇₉₄

—

D₁₅₈₈

—

D₁₅₈₈

—

C₂²⋊D₃₉₇

—

C₂×Dic₃₉₇

C₂³

yes

—

C₂³×C₃₉₇

C₂³

yes

—

Q₈×C₃₉₇

—

D₄×C₃₉₇

—

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂×C₄×C₃₉₇

C₂³

yes

characteristic

C₂×C₄×C₃₉₇

C₂³

yes

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

—

C₂²×C₃₉₇

C₂⁴

yes

characteristic

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₂×D₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

—

C₄×C₃₉₇

—

Dic₃₉₇

—

C₂×C₃₉₇

—

C₂×C₃₉₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₉₇

H(5,𝔽₂)

yes

—

D₃₉₇

—

C₃₉₇

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

397

maximal, Sylow

#124

—

397

—

#124

—

397

—

#124

—

397

—

#124

D₃₉₇

yes

characteristic, complemented

#120

—

397

—

#120

—

397

—

C₂⁴

—

397

—

C₂×Q₈

—

397

—

C₂×Q₈

C₂×D₃₉₇

yes

characteristic, complemented

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

—

397

—

C₂×D₄

C₂×D₃₉₇

yes

characteristic

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

—

397

—

C₂²⋊C₄

C₂×D₃₉₇

yes

complemented

C₄²

—

397

—

C₄²

C₂×D₃₉₇

yes

characteristic, complemented

C₂³

—

1588

—

C₂³

—

1588

—

C₂³

—

397

—

C₂³

C₂²×D₃₉₇

yes

—

Q₈

—

Q₈

—

1588

—

D₄

—

D₄

—

1588

—

D₄

—

1588

—

D₄

—

1588

—

C₂×C₄

—

397

—

C₂×C₄

C₂²×D₃₉₇

yes

—

C₂×C₄

C₂²×D₃₉₇

yes

characteristic

C₂×C₄

C₂²×D₃₉₇

yes

—

C₂²

—

C₂²

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

794

—

C₂²

—

1588

—

C₂²

C₂³×D₃₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

794

—

C₂

—

C₂

—

1588

—

C₂

#27163141

yes

characteristic

100

C₂

#27163142

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂⁴
Fitting	#124×C₃₉₇	C₂
hypercenter	#124	D₃₉₇
hypocenter	C₃₉₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₉₇

commutator, socle, cosocle

C₂²×C₃₉₇

C₂⁴

Fitting

#124×C₃₉₇

C₂

D₃₉₇

C₃₉₇

G ▷ C₂²×C₃₉₇ ▷ C₃₉₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #27163085	C₂	semidirect
2. #27163084	C₂	semidirect
3. #27163082	C₂	semidirect
4. #27163066	C₂	semidirect
5. #27163064	C₂	semidirect
6. #124×C₃₉₇	C₂	semidirect
7. C₂³×D₃₉₇	C₂²	semidirect
8. C₂×Dic₇₉₄	C₂²	semidirect
12. C₂²⋊Dic₃₉₇	C₂²	semidirect
13. #20263708	C₂²	semidirect
14. #20263708	C₂²	semidirect
15. #20263708	C₂²	semidirect
16. C₄×Dic₃₉₇	C₂²	semidirect
17. C₂×Q₈×C₃₉₇	C₂²	semidirect
19. C₂²⋊C₄×C₃₉₇	C₂²	semidirect
20. C₄²×C₃₉₇	C₂²	semidirect
50. C₃₉₇	H(3,𝔽₄)	semidirect
55. #124	D₃₉₇	semidirect
60. C₂×Q₈	C₂×D₃₉₇	semidirect
69. C₂²⋊C₄	C₂×D₃₉₇	semidirect
71. C₄²	C₂×D₃₉₇	semidirect

H ≅ G/N

type

1. #27163085

C₂

semidirect

2. #27163084

C₂

semidirect

3. #27163082

C₂

semidirect

4. #27163066

C₂

semidirect

5. #27163064

C₂

semidirect

6. #124×C₃₉₇

C₂

semidirect

7. C₂³×D₃₉₇

C₂²

semidirect

8. C₂×Dic₇₉₄

C₂²

semidirect

12. C₂²⋊Dic₃₉₇

C₂²

semidirect

13. #20263708

C₂²

semidirect

14. #20263708

C₂²

semidirect

15. #20263708

C₂²

semidirect

16. C₄×Dic₃₉₇

C₂²

semidirect

17. C₂×Q₈×C₃₉₇

C₂²

semidirect

19. C₂²⋊C₄×C₃₉₇

C₂²

semidirect

20. C₄²×C₃₉₇

C₂²

semidirect

50. C₃₉₇

H(3,𝔽₄)

semidirect

55. #124

D₃₉₇

semidirect

60. C₂×Q₈

C₂×D₃₉₇

semidirect

69. C₂²⋊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
#27163085	2
#27163084	2
#27163082	2
#27163066	2
#27163064	2
#124×C₃₉₇	2
H(3,𝔽₄)	397

subgroup

index

#124×C₃₉₇

397

Subgroups — #38763264

Conjugacy classes of subgroups

Complements

Maximal subgroups