Group #37589093 — subgroups

Subgroups — #37589093

Order 24512.

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	#27133830	C₂	1	2	yes	maximal, complemented
2	#27133829	C₂	1	2	yes	maximal, complemented
3	#27133827	C₂	1	4	yes	maximal, complemented
4	#27133811	C₂	1	4	yes	maximal, complemented
5	#27133809	C₂	1	2	yes	maximal, complemented
6	#124×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₈₃	C₂²	1	2	yes	complemented
8	C₂×Dic₇₆₆	C₂²	1	2	yes	complemented
9	C₂×D₁₅₃₂	C₂²	1	2	yes	—
10	C₂×D₁₅₃₂	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₃₈₃	C₂²	1	4	yes	—
12	C₂²⋊Dic₃₈₃	C₂²	1	2	yes	complemented
13	(C₂×D₃₈₃)⋊C₄	C₂²	1	4	yes	complemented
14	(C₂×D₃₈₃)⋊C₄	C₂²	1	4	yes	complemented
15	(C₂×D₃₈₃)⋊C₄	C₂²	1	4	yes	complemented
16	C₄×Dic₃₈₃	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₃₈₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₃₈₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₈₃	C₂²	1	4	yes	complemented
20	C₄²×C₃₈₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₈₃	—	4	4	no	—
22	C₂²×D₃₈₃	—	4	2	no	—
23	C₂²×D₃₈₃	C₂³	1	4	yes	—
24	Dic₇₆₆	—	4	2	no	—
25	D₁₅₃₂	—	4	2	no	—
26	D₁₅₃₂	—	4	2	no	—
27	C₂²⋊D₃₈₃	—	4	4	no	—
28	C₂×Dic₃₈₃	C₂³	1	4	yes	—
29	C₂³×C₃₈₃	C₂³	1	2	yes	—
30	Q₈×C₃₈₃	—	4	1	no	—
31	D₄×C₃₈₃	—	4	1	no	—
32	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
33	C₂×C₄×C₃₈₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₃₈₃	C₂³	1	2	yes	—
35	C₂²×C₃₈₃	—	2	2	no	—
36	C₂²×C₃₈₃	—	2	4	no	—
37	C₂²×C₃₈₃	C₂⁴	1	1	yes	characteristic
38	C₂×D₃₈₃	—	2	4	no	—
39	C₂×D₃₈₃	—	2	4	no	—
40	C₂×D₃₈₃	—	2	4	no	—
41	C₂×D₃₈₃	—	4	8	no	—
42	C₄×C₃₈₃	—	2	1	no	—
43	C₄×C₃₈₃	—	2	2	no	—
44	C₄×C₃₈₃	—	2	2	no	—
45	Dic₃₈₃	—	2	4	no	—
46	C₂×C₃₈₃	—	4	2	no	—
47	C₂×C₃₈₃	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₃₈₃	H(5,𝔽₂)	1	2	yes	—
49	D₃₈₃	—	4	4	no	—
50	C₃₈₃	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	383	1	no	maximal, Sylow
52	#124	—	383	2	no	—
53	#124	—	383	4	no	—
54	#124	—	383	2	no	—
55	#124	D₃₈₃	1	1	yes	characteristic, complemented
56	#120	—	383	2	no	—
57	#120	—	383	4	no	—
58	C₂⁴	—	383	2	no	—
59	C₂×Q₈	—	383	2	no	—
60	C₂×Q₈	C₂×D₃₈₃	1	1	yes	characteristic, complemented
61	C₂×D₄	—	383	4	no	—
62	C₂×D₄	—	383	2	no	—
63	C₂×D₄	—	383	2	no	—
64	C₂×D₄	C₂×D₃₈₃	1	1	yes	characteristic
65	C₂²⋊C₄	—	383	2	no	—
66	C₂²⋊C₄	—	383	4	no	—
67	C₂²⋊C₄	—	383	4	no	—
68	C₂²⋊C₄	—	383	4	no	—
69	C₂²⋊C₄	C₂×D₃₈₃	1	4	yes	complemented
70	C₄²	—	383	2	no	—
71	C₄²	C₂×D₃₈₃	1	1	yes	characteristic, complemented
72	C₂³	—	383	4	no	—
73	C₂³	—	1532	2	no	—
74	C₂³	—	1532	4	no	—
75	C₂³	C₂²×D₃₈₃	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	1532	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	1532	2	no	—
80	D₄	—	1532	2	no	—
81	D₄	—	1532	4	no	—
82	C₂×C₄	—	383	4	no	—
83	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
84	C₂×C₄	C₂²×D₃₈₃	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₃₈₃	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	1532	8	no	—
89	C₂²	—	766	4	no	—
90	C₂²	—	766	4	no	—
91	C₂²	—	766	4	no	—
92	C₂²	C₂³×D₃₈₃	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	766	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	1532	4	no	—
99	C₂	#27133886	1	1	yes	characteristic
100	C₂	#27133887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133830

C₂

yes

maximal, complemented

#27133829

C₂

yes

maximal, complemented

#27133827

C₂

yes

maximal, complemented

#27133811

C₂

yes

maximal, complemented

#27133809

C₂

yes

maximal, complemented

#124×C₃₈₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₈₃

C₂²

yes

complemented

C₂×Dic₇₆₆

C₂²

yes

complemented

C₂×D₁₅₃₂

C₂²

yes

—

C₂×D₁₅₃₂

C₂²

yes

—

C₂×C₂²⋊D₃₈₃

C₂²

yes

—

C₂²⋊Dic₃₈₃

C₂²

yes

complemented

(C₂×D₃₈₃)⋊C₄

C₂²

yes

complemented

(C₂×D₃₈₃)⋊C₄

C₂²

yes

complemented

(C₂×D₃₈₃)⋊C₄

C₂²

yes

complemented

C₄×Dic₃₈₃

C₂²

yes

complemented

C₂×Q₈×C₃₈₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₈₃

C₂²

yes

characteristic

C₂²⋊C₄×C₃₈₃

C₂²

yes

complemented

C₄²×C₃₈₃

C₂²

yes

characteristic, complemented

C₂²×D₃₈₃

—

C₂²×D₃₈₃

—

C₂²×D₃₈₃

C₂³

yes

—

Dic₇₆₆

—

D₁₅₃₂

—

D₁₅₃₂

—

C₂²⋊D₃₈₃

—

C₂×Dic₃₈₃

C₂³

yes

—

C₂³×C₃₈₃

C₂³

yes

—

Q₈×C₃₈₃

—

D₄×C₃₈₃

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂×C₄×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂²×C₃₈₃

—

C₂²×C₃₈₃

—

C₂²×C₃₈₃

C₂⁴

yes

characteristic

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

Dic₃₈₃

—

C₂×C₃₈₃

—

C₂×C₃₈₃

H(5,𝔽₂)

yes

characteristic

C₂×C₃₈₃

H(5,𝔽₂)

yes

—

D₃₈₃

—

C₃₈₃

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

383

maximal, Sylow

#124

—

383

—

#124

—

383

—

#124

—

383

—

#124

D₃₈₃

yes

characteristic, complemented

#120

—

383

—

#120

—

383

—

C₂⁴

—

383

—

C₂×Q₈

—

383

—

C₂×Q₈

C₂×D₃₈₃

yes

characteristic, complemented

C₂×D₄

—

383

—

C₂×D₄

—

383

—

C₂×D₄

—

383

—

C₂×D₄

C₂×D₃₈₃

yes

characteristic

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

C₂×D₃₈₃

yes

complemented

C₄²

—

383

—

C₄²

C₂×D₃₈₃

yes

characteristic, complemented

C₂³

—

383

—

C₂³

—

1532

—

C₂³

—

1532

—

C₂³

C₂²×D₃₈₃

yes

—

Q₈

—

Q₈

—

1532

—

D₄

—

D₄

—

1532

—

D₄

—

1532

—

D₄

—

1532

—

C₂×C₄

—

383

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₂×C₄

C₂²×D₃₈₃

yes

characteristic

C₂×C₄

C₂²×D₃₈₃

yes

—

C₂²

—

C₂²

—

C₂²

—

1532

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

C₂³×D₃₈₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

766

—

C₂

—

C₂

—

1532

—

C₂

#27133886

yes

characteristic

100

C₂

#27133887

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₃
commutator, socle, cosocle	C₂²×C₃₈₃	C₂⁴
Fitting	#124×C₃₈₃	C₂
hypercenter	#124	D₃₈₃
hypocenter	C₃₈₃	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₈₃

commutator, socle, cosocle

C₂²×C₃₈₃

C₂⁴

Fitting

#124×C₃₈₃

C₂

D₃₈₃

C₃₈₃

G ▷ C₂²×C₃₈₃ ▷ C₃₈₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #27133830	C₂	semidirect
2. #27133829	C₂	semidirect
3. #27133827	C₂	semidirect
4. #27133811	C₂	semidirect
5. #27133809	C₂	semidirect
6. #124×C₃₈₃	C₂	semidirect
7. C₂³×D₃₈₃	C₂²	semidirect
8. C₂×Dic₇₆₆	C₂²	semidirect
12. C₂²⋊Dic₃₈₃	C₂²	semidirect
13. (C₂×D₃₈₃)⋊C₄	C₂²	semidirect
14. (C₂×D₃₈₃)⋊C₄	C₂²	semidirect
15. (C₂×D₃₈₃)⋊C₄	C₂²	semidirect
16. C₄×Dic₃₈₃	C₂²	semidirect
17. C₂×Q₈×C₃₈₃	C₂²	semidirect
19. C₂²⋊C₄×C₃₈₃	C₂²	semidirect
20. C₄²×C₃₈₃	C₂²	semidirect
50. C₃₈₃	H(3,𝔽₄)	semidirect
55. #124	D₃₈₃	semidirect
60. C₂×Q₈	C₂×D₃₈₃	semidirect
69. C₂²⋊C₄	C₂×D₃₈₃	semidirect
71. C₄²	C₂×D₃₈₃	semidirect

H ≅ G/N

type

1. #27133830

C₂

semidirect

2. #27133829

C₂

semidirect

3. #27133827

C₂

semidirect

4. #27133811

C₂

semidirect

5. #27133809

C₂

semidirect

6. #124×C₃₈₃

C₂

semidirect

7. C₂³×D₃₈₃

C₂²

semidirect

8. C₂×Dic₇₆₆

C₂²

semidirect

12. C₂²⋊Dic₃₈₃

C₂²

semidirect

13. (C₂×D₃₈₃)⋊C₄

C₂²

semidirect

14. (C₂×D₃₈₃)⋊C₄

C₂²

semidirect

15. (C₂×D₃₈₃)⋊C₄

C₂²

semidirect

16. C₄×Dic₃₈₃

C₂²

semidirect

17. C₂×Q₈×C₃₈₃

C₂²

semidirect

19. C₂²⋊C₄×C₃₈₃

C₂²

semidirect

20. C₄²×C₃₈₃

C₂²

semidirect

50. C₃₈₃

H(3,𝔽₄)

semidirect

55. #124

D₃₈₃

semidirect

60. C₂×Q₈

C₂×D₃₈₃

semidirect

69. C₂²⋊C₄

C₂×D₃₈₃

semidirect

71. C₄²

C₂×D₃₈₃

semidirect

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

subgroup	index
#27133830	2
#27133829	2
#27133827	2
#27133811	2
#27133809	2
#124×C₃₈₃	2
H(3,𝔽₄)	383

subgroup

index

#124×C₃₈₃

383

Subgroups — #37589093

Conjugacy classes of subgroups

Complements

Maximal subgroups