Group #14558275 — subgroups

Subgroups — #14558275

Order 2368.

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	#11924525	C₂	1	2	yes	maximal, complemented
2	#11924524	C₂	1	2	yes	maximal, complemented
3	#11924522	C₂	1	4	yes	maximal, complemented
4	#11924506	C₂	1	4	yes	maximal, complemented
5	#11924504	C₂	1	2	yes	maximal, complemented
6	#124×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₇	C₂²	1	2	yes	complemented
8	C₂×Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₃₇	C₂²	1	4	yes	—
11	C₂×D₁₄₈	C₂²	1	2	yes	—
12	C₂×D₁₄₈	C₂²	1	2	yes	—
13	C₂×Dic₇₄	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₃₇	C₂²	1	4	yes	complemented
15	C₄²×C₃₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊Dic₃₇	C₂²	1	2	yes	complemented
17	#10596938	C₂²	1	4	yes	complemented
18	#10596938	C₂²	1	4	yes	complemented
19	#10596938	C₂²	1	4	yes	complemented
20	C₄×Dic₃₇	C₂²	1	2	yes	complemented
21	C₂³×C₃₇	C₂³	1	2	yes	—
22	C₂²×D₃₇	—	4	2	no	—
23	C₂²×D₃₇	—	4	4	no	—
24	C₂²×D₃₇	C₂³	1	4	yes	—
25	Q₈×C₃₇	—	4	1	no	—
26	D₄×C₃₇	—	4	1	no	—
27	C₂×C₄×C₃₇	C₂³	1	2	yes	—
28	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂³	1	2	yes	—
30	C₂²⋊D₃₇	—	4	4	no	—
31	C₂×Dic₃₇	C₂³	1	4	yes	—
32	D₁₄₈	—	4	2	no	—
33	D₁₄₈	—	4	2	no	—
34	Dic₇₄	—	4	2	no	—
35	C₂²×C₃₇	—	2	4	no	—
36	C₂²×C₃₇	—	2	2	no	—
37	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₃₇	—	2	4	no	—
39	C₂×D₃₇	—	2	4	no	—
40	C₂×D₃₇	—	2	4	no	—
41	C₂×D₃₇	—	4	8	no	—
42	C₄×C₃₇	—	2	2	no	—
43	C₄×C₃₇	—	2	2	no	—
44	C₄×C₃₇	—	2	1	no	—
45	Dic₃₇	—	2	4	no	—
46	C₂×C₃₇	—	4	2	no	—
47	C₂×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₃₇	H(5,𝔽₂)	1	2	yes	—
49	D₃₇	—	4	4	no	—
50	H(3,𝔽₄)	—	37	1	no	maximal, Sylow
51	C₃₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
52	#124	—	37	2	no	—
53	#124	—	37	4	no	—
54	#124	—	37	2	no	—
55	#124	D₃₇	1	1	yes	characteristic, complemented
56	#120	—	37	2	no	—
57	#120	—	37	4	no	—
58	C₂⁴	—	37	2	no	—
59	C₂×Q₈	—	37	2	no	—
60	C₂×Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
61	C₂×D₄	—	37	4	no	—
62	C₂×D₄	—	37	2	no	—
63	C₂×D₄	—	37	2	no	—
64	C₂×D₄	C₂×D₃₇	1	1	yes	characteristic
65	C₂²⋊C₄	—	37	2	no	—
66	C₂²⋊C₄	—	37	4	no	—
67	C₂²⋊C₄	—	37	4	no	—
68	C₂²⋊C₄	—	37	4	no	—
69	C₂²⋊C₄	C₂×D₃₇	1	4	yes	complemented
70	C₄²	—	37	2	no	—
71	C₄²	C₂×D₃₇	1	1	yes	characteristic, complemented
72	C₂³	—	37	4	no	—
73	C₂³	—	148	2	no	—
74	C₂³	—	148	4	no	—
75	C₂³	C₂²×D₃₇	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	148	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	148	2	no	—
80	D₄	—	148	2	no	—
81	D₄	—	148	4	no	—
82	C₂×C₄	—	37	4	no	—
83	C₂×C₄	C₂²×D₃₇	1	2	yes	—
84	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₃₇	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	74	4	no	—
89	C₂²	—	74	4	no	—
90	C₂²	—	74	4	no	—
91	C₂²	—	148	8	no	—
92	C₂²	C₂³×D₃₇	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	74	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	148	4	no	—
99	C₂	#11924581	1	1	yes	characteristic
100	C₂	#11924582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11924525

C₂

yes

maximal, complemented

#11924524

C₂

yes

maximal, complemented

#11924522

C₂

yes

maximal, complemented

#11924506

C₂

yes

maximal, complemented

#11924504

C₂

yes

maximal, complemented

#124×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇

C₂²

yes

complemented

C₂×Q₈×C₃₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

—

C₂×D₁₄₈

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

complemented

C₂²⋊C₄×C₃₇

C₂²

yes

complemented

C₄²×C₃₇

C₂²

yes

characteristic, complemented

C₂²⋊Dic₃₇

C₂²

yes

complemented

#10596938

C₂²

yes

complemented

#10596938

C₂²

yes

complemented

#10596938

C₂²

yes

complemented

C₄×Dic₃₇

C₂²

yes

complemented

C₂³×C₃₇

C₂³

yes

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇

H(5,𝔽₂)

yes

—

D₃₇

—

H(3,𝔽₄)

—

maximal, Sylow

C₃₇

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

—

—

—

D₃₇

yes

characteristic, complemented

#120

—

#120

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₇

yes

complemented

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic, complemented

C₂³

—

C₂³

—

148

—

C₂³

—

148

—

C₂³

C₂²×D₃₇

yes

—

Q₈

—

Q₈

—

148

—

D₄

—

D₄

—

148

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

C₂²×D₃₇

yes

—

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

148

—

C₂²

C₂³×D₃₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

148

—

C₂

#11924581

yes

characteristic

100

C₂

#11924582

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	#124×C₃₇	C₂
hypercenter	#124	D₃₇
hypocenter	C₃₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

#124×C₃₇

C₂

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #11924525	C₂	semidirect
2. #11924524	C₂	semidirect
3. #11924522	C₂	semidirect
4. #11924506	C₂	semidirect
5. #11924504	C₂	semidirect
6. #124×C₃₇	C₂	semidirect
7. C₂³×D₃₇	C₂²	semidirect
8. C₂×Q₈×C₃₇	C₂²	semidirect
13. C₂×Dic₇₄	C₂²	semidirect
14. C₂²⋊C₄×C₃₇	C₂²	semidirect
15. C₄²×C₃₇	C₂²	semidirect
16. C₂²⋊Dic₃₇	C₂²	semidirect
17. #10596938	C₂²	semidirect
18. #10596938	C₂²	semidirect
19. #10596938	C₂²	semidirect
20. C₄×Dic₃₇	C₂²	semidirect
51. C₃₇	H(3,𝔽₄)	semidirect
55. #124	D₃₇	semidirect
60. C₂×Q₈	C₂×D₃₇	semidirect
69. C₂²⋊C₄	C₂×D₃₇	semidirect
71. C₄²	C₂×D₃₇	semidirect

H ≅ G/N

type

1. #11924525

C₂

semidirect

2. #11924524

C₂

semidirect

3. #11924522

C₂

semidirect

4. #11924506

C₂

semidirect

5. #11924504

C₂

semidirect

6. #124×C₃₇

C₂

semidirect

7. C₂³×D₃₇

C₂²

semidirect

8. C₂×Q₈×C₃₇

C₂²

semidirect

13. C₂×Dic₇₄

C₂²

semidirect

14. C₂²⋊C₄×C₃₇

C₂²

semidirect

15. C₄²×C₃₇

C₂²

semidirect

16. C₂²⋊Dic₃₇

C₂²

semidirect

17. #10596938

C₂²

semidirect

18. #10596938

C₂²

semidirect

19. #10596938

C₂²

semidirect

20. C₄×Dic₃₇

C₂²

semidirect

51. C₃₇

H(3,𝔽₄)

semidirect

55. #124

D₃₇

semidirect

60. C₂×Q₈

C₂×D₃₇

semidirect

69. C₂²⋊C₄

C₂×D₃₇

semidirect

71. C₄²

C₂×D₃₇

semidirect

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

subgroup	index
#11924525	2
#11924524	2
#11924522	2
#11924506	2
#11924504	2
#124×C₃₇	2
H(3,𝔽₄)	37

subgroup

index

#124×C₃₇

Subgroups — #14558275

Conjugacy classes of subgroups

Complements

Maximal subgroups