Group #37589092 — subgroups

Subgroups — #37589092

Order 24512.

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	#27133838	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133835	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133826	C₂	1	3	yes	maximal, complemented
4	#27133816	C₂	1	3	yes	maximal, complemented
5	#27133814	C₂	1	3	yes	maximal, complemented
6	#27133811	C₂	1	3	yes	maximal, complemented
7	#126×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃₈₃	C₂²	1	1	yes	characteristic
9	C₂×D₁₅₃₂	C₂²	1	3	yes	—
10	C₂×D₁₅₃₂	C₂²	1	3	yes	—
11	C₂×D₁₅₃₂	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₃₈₃	C₂²	1	3	yes	—
13	C₂×C₄×D₃₈₃	C₂²	1	3	yes	complemented
14	Dic₃₈₃⋊C₄	C₂²	1	3	yes	complemented
15	(C₂×D₃₈₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₃₈₃)⋊C₄	C₂²	1	3	yes	—
17	(C₂×D₃₈₃)⋊C₄	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₃₈₃	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₃₈₃	C₂²	1	3	yes	—
20	C₄²×C₃₈₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₈₃	—	4	3	no	—
22	C₂²×D₃₈₃	C₂³	1	1	yes	characteristic
23	C₂²×D₃₈₃	C₂³	1	1	yes	characteristic
24	C₂²×D₃₈₃	C₂³	1	3	yes	—
25	D₁₅₃₂	—	4	3	no	—
26	D₁₅₃₂	—	4	3	no	—
27	D₁₅₃₂	—	4	3	no	—
28	C₂²⋊D₃₈₃	—	4	3	no	—
29	C₂×Dic₃₈₃	C₂³	1	3	yes	—
30	C₄×D₃₈₃	—	4	3	no	—
31	C₂³×C₃₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₃₈₃	C₂³	1	3	yes	—
33	C₂×C₄×C₃₈₃	C₂³	1	3	yes	—
34	C₂²×C₃₈₃	—	2	3	no	—
35	C₂²×C₃₈₃	C₂⁴	1	1	yes	characteristic
36	C₂×D₃₈₃	—	2	3	no	—
37	C₂×D₃₈₃	—	2	3	no	—
38	C₂×D₃₈₃	—	2	3	no	—
39	C₂×D₃₈₃	—	2	3	no	—
40	C₂×D₃₈₃	—	2	3	no	—
41	C₂×D₃₈₃	—	4	4	no	—
42	C₄×C₃₈₃	—	2	3	no	—
43	C₄×C₃₈₃	—	2	3	no	—
44	Dic₃₈₃	—	2	3	no	—
45	C₂×C₃₈₃	—	4	1	no	—
46	C₂×C₃₈₃	H(5,𝔽₂)	1	3	yes	—
47	D₃₈₃	—	4	3	no	—
48	D₃₈₃	—	4	1	no	—
49	D₃₈₃	—	4	1	no	—
50	C₃₈₃	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	383	1	no	maximal, Sylow
52	#127	—	383	1	no	—
53	#126	—	383	1	no	—
54	#126	D₃₈₃	1	1	yes	characteristic, complemented
55	#123	—	383	3	no	—
56	#121	—	383	3	no	—
57	#121	—	383	3	no	—
58	#120	—	383	3	no	—
59	C₂⁴	—	383	1	no	—
60	C₂×D₄	—	383	3	no	—
61	C₂×D₄	—	383	3	no	—
62	C₂×D₄	—	383	3	no	—
63	C₂×D₄	—	383	3	no	—
64	C₂²×C₄	—	383	3	no	—
65	C₄⋊C₄	—	383	3	no	—
66	C₄⋊C₄	C₂×D₃₈₃	1	3	yes	complemented
67	C₂²⋊C₄	—	383	3	no	—
68	C₂²⋊C₄	—	383	3	no	—
69	C₂²⋊C₄	—	383	3	no	—
70	C₂²⋊C₄	C₂×D₃₈₃	1	3	yes	—
71	C₄²	C₂×D₃₈₃	1	1	yes	characteristic, complemented
72	C₂³	—	383	1	no	—
73	C₂³	—	383	1	no	—
74	C₂³	—	383	3	no	—
75	C₂³	—	1532	3	no	—
76	C₂³	C₂²×D₃₈₃	1	1	yes	characteristic
77	D₄	—	1532	3	no	—
78	D₄	—	1532	3	no	—
79	D₄	—	1532	3	no	—
80	D₄	—	1532	3	no	—
81	C₂×C₄	—	383	3	no	—
82	C₂×C₄	—	1532	3	no	—
83	C₂×C₄	C₂²×D₃₈₃	1	3	yes	—
84	C₂×C₄	C₂²×D₃₈₃	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	1532	4	no	—
87	C₂²	—	766	3	no	—
88	C₂²	—	766	3	no	—
89	C₂²	—	766	3	no	—
90	C₂²	—	766	3	no	—
91	C₂²	—	766	3	no	—
92	C₂²	C₂³×D₃₈₃	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	766	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	1532	1	no	—
98	C₂	—	1532	1	no	—
99	C₂	—	1532	3	no	—
100	C₂	#27133887	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133838

C₂

yes

maximal, characteristic, complemented

#27133835

C₂

yes

maximal, characteristic, complemented

#27133826

C₂

yes

maximal, complemented

#27133816

C₂

yes

maximal, complemented

#27133814

C₂

yes

maximal, complemented

#27133811

C₂

yes

maximal, complemented

#126×C₃₈₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₈₃

C₂²

yes

characteristic

C₂×D₁₅₃₂

C₂²

yes

—

C₂×D₁₅₃₂

C₂²

yes

—

C₂×D₁₅₃₂

C₂²

yes

complemented

C₂×C₂²⋊D₃₈₃

C₂²

yes

—

C₂×C₄×D₃₈₃

C₂²

yes

complemented

Dic₃₈₃⋊C₄

C₂²

yes

complemented

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

—

(C₂×D₃₈₃)⋊C₄

C₂²

yes

complemented

C₄⋊C₄×C₃₈₃

C₂²

yes

complemented

C₂²⋊C₄×C₃₈₃

C₂²

yes

—

C₄²×C₃₈₃

C₂²

yes

characteristic, complemented

C₂²×D₃₈₃

—

C₂²×D₃₈₃

C₂³

yes

characteristic

C₂²×D₃₈₃

C₂³

yes

characteristic

C₂²×D₃₈₃

C₂³

yes

—

D₁₅₃₂

—

D₁₅₃₂

—

D₁₅₃₂

—

C₂²⋊D₃₈₃

—

C₂×Dic₃₈₃

C₂³

yes

—

C₄×D₃₈₃

—

C₂³×C₃₈₃

C₂³

yes

characteristic

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂×C₄×C₃₈₃

C₂³

yes

—

C₂²×C₃₈₃

—

C₂²×C₃₈₃

C₂⁴

yes

characteristic

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₂×D₃₈₃

—

C₄×C₃₈₃

—

C₄×C₃₈₃

—

Dic₃₈₃

—

C₂×C₃₈₃

—

C₂×C₃₈₃

H(5,𝔽₂)

yes

—

D₃₈₃

—

D₃₈₃

—

D₃₈₃

—

C₃₈₃

#560

yes

characteristic, complemented, Sylow

#560

—

383

maximal, Sylow

#127

—

383

—

#126

—

383

—

#126

D₃₈₃

yes

characteristic, complemented

#123

—

383

—

#121

—

383

—

#121

—

383

—

#120

—

383

—

C₂⁴

—

383

—

C₂×D₄

—

383

—

C₂×D₄

—

383

—

C₂×D₄

—

383

—

C₂×D₄

—

383

—

C₂²×C₄

—

383

—

C₄⋊C₄

—

383

—

C₄⋊C₄

C₂×D₃₈₃

yes

complemented

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

—

383

—

C₂²⋊C₄

C₂×D₃₈₃

yes

—

C₄²

C₂×D₃₈₃

yes

characteristic, complemented

C₂³

—

383

—

C₂³

—

383

—

C₂³

—

383

—

C₂³

—

1532

—

C₂³

C₂²×D₃₈₃

yes

characteristic

D₄

—

1532

—

D₄

—

1532

—

D₄

—

1532

—

D₄

—

1532

—

C₂×C₄

—

383

—

C₂×C₄

—

1532

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₂×C₄

C₂²×D₃₈₃

yes

—

C₂²

—

C₂²

—

1532

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

—

766

—

C₂²

C₂³×D₃₈₃

yes

characteristic

C₄

—

C₄

—

C₄

—

766

—

C₂

—

C₂

—

1532

—

C₂

—

1532

—

C₂

—

1532

—

100

C₂

#27133887

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₃
commutator, socle, cosocle	C₂²×C₃₈₃	C₂⁴
Fitting	#126×C₃₈₃	C₂
hypercenter	#126	D₃₈₃
hypocenter	C₃₈₃	#560
LCS	G ▷ C₂²×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₈₃

commutator, socle, cosocle

C₂²×C₃₈₃

C₂⁴

Fitting

#126×C₃₈₃

C₂

D₃₈₃

C₃₈₃

G ▷ C₂²×C₃₈₃ ▷ C₃₈₃

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #27133838	C₂	semidirect
2. #27133835	C₂	semidirect
3. #27133826	C₂	semidirect
4. #27133816	C₂	semidirect
5. #27133814	C₂	semidirect
6. #27133811	C₂	semidirect
7. #126×C₃₈₃	C₂	semidirect
11. C₂×D₁₅₃₂	C₂²	semidirect
13. C₂×C₄×D₃₈₃	C₂²	semidirect
14. Dic₃₈₃⋊C₄	C₂²	semidirect
17. (C₂×D₃₈₃)⋊C₄	C₂²	semidirect
18. C₄⋊C₄×C₃₈₃	C₂²	semidirect
20. C₄²×C₃₈₃	C₂²	semidirect
50. C₃₈₃	#560	semidirect
54. #126	D₃₈₃	semidirect
66. C₄⋊C₄	C₂×D₃₈₃	semidirect
71. C₄²	C₂×D₃₈₃	semidirect

H ≅ G/N

type

1. #27133838

C₂

semidirect

2. #27133835

C₂

semidirect

3. #27133826

C₂

semidirect

4. #27133816

C₂

semidirect

5. #27133814

C₂

semidirect

6. #27133811

C₂

semidirect

7. #126×C₃₈₃

C₂

semidirect

11. C₂×D₁₅₃₂

C₂²

semidirect

13. C₂×C₄×D₃₈₃

C₂²

semidirect

14. Dic₃₈₃⋊C₄

C₂²

semidirect

17. (C₂×D₃₈₃)⋊C₄

C₂²

semidirect

18. C₄⋊C₄×C₃₈₃

C₂²

semidirect

20. C₄²×C₃₈₃

C₂²

semidirect

50. C₃₈₃

#560

semidirect

54. #126

D₃₈₃

semidirect

66. C₄⋊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
#27133838	2
#27133835	2
#27133826	2
#27133816	2
#27133814	2
#27133811	2
#126×C₃₈₃	2
#560	383

subgroup

index

#126×C₃₈₃

383

Subgroups — #37589092

Conjugacy classes of subgroups

Complements

Maximal subgroups