Group #37560163 — subgroups

Subgroups — #37560163

Order 23872.

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	#26045789	C₂	1	1	yes	maximal, characteristic, complemented
2	#26045786	C₂	1	1	yes	maximal, characteristic, complemented
3	#26045777	C₂	1	3	yes	maximal, complemented
4	#26045767	C₂	1	3	yes	maximal, complemented
5	#26045765	C₂	1	3	yes	maximal, complemented
6	#26045762	C₂	1	3	yes	maximal, complemented
7	#126×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃₇₃	C₂²	1	1	yes	characteristic
9	C₂×D₁₄₉₂	C₂²	1	3	yes	—
10	C₂×D₁₄₉₂	C₂²	1	3	yes	—
11	C₂×D₁₄₉₂	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₃₇₃	C₂²	1	3	yes	—
13	C₂×C₄×D₃₇₃	C₂²	1	3	yes	complemented
14	#20238589	C₂²	1	3	yes	complemented
15	#20238587	C₂²	1	3	yes	—
16	#20238587	C₂²	1	3	yes	—
17	#20238587	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₃₇₃	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₃₇₃	C₂²	1	3	yes	—
20	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₃₇₃	—	4	3	no	—
22	C₂²×D₃₇₃	C₂³	1	3	yes	—
23	C₂²×D₃₇₃	C₂³	1	1	yes	characteristic
24	C₂²×D₃₇₃	C₂³	1	1	yes	characteristic
25	D₁₄₉₂	—	4	3	no	—
26	D₁₄₉₂	—	4	3	no	—
27	D₁₄₉₂	—	4	3	no	—
28	C₂²⋊D₃₇₃	—	4	3	no	—
29	C₂×Dic₃₇₃	C₂³	1	3	yes	—
30	C₄×D₃₇₃	—	4	3	no	—
31	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₃₇₃	C₂³	1	3	yes	—
33	C₂×C₄×C₃₇₃	C₂³	1	3	yes	—
34	C₂²×C₃₇₃	—	2	3	no	—
35	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
36	C₂×D₃₇₃	—	2	3	no	—
37	C₂×D₃₇₃	—	2	3	no	—
38	C₂×D₃₇₃	—	2	3	no	—
39	C₂×D₃₇₃	—	2	3	no	—
40	C₂×D₃₇₃	—	2	3	no	—
41	C₂×D₃₇₃	—	4	4	no	—
42	C₄×C₃₇₃	—	2	3	no	—
43	C₄×C₃₇₃	—	2	3	no	—
44	Dic₃₇₃	—	2	3	no	—
45	C₂×C₃₇₃	—	4	1	no	—
46	C₂×C₃₇₃	H(5,𝔽₂)	1	3	yes	—
47	D₃₇₃	—	4	3	no	—
48	D₃₇₃	—	4	1	no	—
49	D₃₇₃	—	4	1	no	—
50	C₃₇₃	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	373	1	no	maximal, Sylow
52	#127	—	373	1	no	—
53	#126	—	373	1	no	—
54	#126	D₃₇₃	1	1	yes	characteristic, complemented
55	#123	—	373	3	no	—
56	#121	—	373	3	no	—
57	#121	—	373	3	no	—
58	#120	—	373	3	no	—
59	C₂⁴	—	373	1	no	—
60	C₂×D₄	—	373	3	no	—
61	C₂×D₄	—	373	3	no	—
62	C₂×D₄	—	373	3	no	—
63	C₂×D₄	—	373	3	no	—
64	C₂²×C₄	—	373	3	no	—
65	C₄⋊C₄	—	373	3	no	—
66	C₄⋊C₄	C₂×D₃₇₃	1	3	yes	complemented
67	C₂²⋊C₄	—	373	3	no	—
68	C₂²⋊C₄	—	373	3	no	—
69	C₂²⋊C₄	—	373	3	no	—
70	C₂²⋊C₄	C₂×D₃₇₃	1	3	yes	—
71	C₄²	C₂×D₃₇₃	1	1	yes	characteristic, complemented
72	C₂³	—	373	1	no	—
73	C₂³	—	373	1	no	—
74	C₂³	—	373	3	no	—
75	C₂³	—	1492	3	no	—
76	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
77	D₄	—	1492	3	no	—
78	D₄	—	1492	3	no	—
79	D₄	—	1492	3	no	—
80	D₄	—	1492	3	no	—
81	C₂×C₄	—	373	3	no	—
82	C₂×C₄	—	1492	3	no	—
83	C₂×C₄	C₂²×D₃₇₃	1	3	yes	—
84	C₂×C₄	C₂²×D₃₇₃	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	1492	4	no	—
87	C₂²	—	746	3	no	—
88	C₂²	—	746	3	no	—
89	C₂²	—	746	3	no	—
90	C₂²	—	746	3	no	—
91	C₂²	—	746	3	no	—
92	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	746	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	1492	1	no	—
98	C₂	—	1492	1	no	—
99	C₂	—	1492	3	no	—
100	C₂	#26045838	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26045789

C₂

yes

maximal, characteristic, complemented

#26045786

C₂

yes

maximal, characteristic, complemented

#26045777

C₂

yes

maximal, complemented

#26045767

C₂

yes

maximal, complemented

#26045765

C₂

yes

maximal, complemented

#26045762

C₂

yes

maximal, complemented

#126×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇₃

C₂²

yes

characteristic

C₂×D₁₄₉₂

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

complemented

C₂×C₂²⋊D₃₇₃

C₂²

yes

—

C₂×C₄×D₃₇₃

C₂²

yes

complemented

#20238589

C₂²

yes

complemented

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

complemented

C₄⋊C₄×C₃₇₃

C₂²

yes

complemented

C₂²⋊C₄×C₃₇₃

C₂²

yes

—

C₄²×C₃₇₃

C₂²

yes

characteristic, complemented

C₂²×D₃₇₃

—

C₂²×D₃₇₃

C₂³

yes

—

C₂²×D₃₇₃

C₂³

yes

characteristic

C₂²×D₃₇₃

C₂³

yes

characteristic

D₁₄₉₂

—

D₁₄₉₂

—

D₁₄₉₂

—

C₂²⋊D₃₇₃

—

C₂×Dic₃₇₃

C₂³

yes

—

C₄×D₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

H(5,𝔽₂)

yes

—

D₃₇₃

—

D₃₇₃

—

D₃₇₃

—

C₃₇₃

#560

yes

characteristic, complemented, Sylow

#560

—

373

maximal, Sylow

#127

—

373

—

#126

—

373

—

#126

D₃₇₃

yes

characteristic, complemented

#123

—

373

—

#121

—

373

—

#121

—

373

—

#120

—

373

—

C₂⁴

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂²×C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

C₂×D₃₇₃

yes

complemented

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

C₂×D₃₇₃

yes

—

C₄²

C₂×D₃₇₃

yes

characteristic, complemented

C₂³

—

373

—

C₂³

—

373

—

C₂³

—

373

—

C₂³

—

1492

—

C₂³

C₂²×D₃₇₃

yes

characteristic

D₄

—

1492

—

D₄

—

1492

—

D₄

—

1492

—

D₄

—

1492

—

C₂×C₄

—

373

—

C₂×C₄

—

1492

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂²

—

C₂²

—

1492

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

C₂³×D₃₇₃

yes

characteristic

C₄

—

C₄

—

C₄

—

746

—

C₂

—

C₂

—

1492

—

C₂

—

1492

—

C₂

—

1492

—

100

C₂

#26045838

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	#126×C₃₇₃	C₂
hypercenter	#126	D₃₇₃
hypocenter	C₃₇₃	#560
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

#126×C₃₇₃

C₂

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #26045789	C₂	semidirect
2. #26045786	C₂	semidirect
3. #26045777	C₂	semidirect
4. #26045767	C₂	semidirect
5. #26045765	C₂	semidirect
6. #26045762	C₂	semidirect
7. #126×C₃₇₃	C₂	semidirect
11. C₂×D₁₄₉₂	C₂²	semidirect
13. C₂×C₄×D₃₇₃	C₂²	semidirect
14. #20238589	C₂²	semidirect
17. #20238587	C₂²	semidirect
18. C₄⋊C₄×C₃₇₃	C₂²	semidirect
20. C₄²×C₃₇₃	C₂²	semidirect
50. C₃₇₃	#560	semidirect
54. #126	D₃₇₃	semidirect
66. C₄⋊C₄	C₂×D₃₇₃	semidirect
71. C₄²	C₂×D₃₇₃	semidirect

H ≅ G/N

type

1. #26045789

C₂

semidirect

2. #26045786

C₂

semidirect

3. #26045777

C₂

semidirect

4. #26045767

C₂

semidirect

5. #26045765

C₂

semidirect

6. #26045762

C₂

semidirect

7. #126×C₃₇₃

C₂

semidirect

11. C₂×D₁₄₉₂

C₂²

semidirect

13. C₂×C₄×D₃₇₃

C₂²

semidirect

14. #20238589

C₂²

semidirect

17. #20238587

C₂²

semidirect

18. C₄⋊C₄×C₃₇₃

C₂²

semidirect

20. C₄²×C₃₇₃

C₂²

semidirect

50. C₃₇₃

#560

semidirect

54. #126

D₃₇₃

semidirect

66. C₄⋊C₄

C₂×D₃₇₃

semidirect

71. C₄²

C₂×D₃₇₃

semidirect

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

subgroup	index
#26045789	2
#26045786	2
#26045777	2
#26045767	2
#26045765	2
#26045762	2
#126×C₃₇₃	2
#560	373

subgroup

index

#126×C₃₇₃

373

Subgroups — #37560163

Conjugacy classes of subgroups

Complements

Maximal subgroups