Group #14558107 — subgroups

Subgroups — #14558107

Order 2368.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#10596962	C₂	1	1	yes	maximal, characteristic
2	#11924534	C₂	1	2	yes	maximal, complemented
3	#11924525	C₂	1	2	yes	maximal, complemented
4	#11924519	C₂	1	4	yes	maximal, complemented
5	#11924512	C₂	1	4	yes	maximal, complemented
6	#11924495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₃₇	—	2	4	no	—
9	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
10	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
12	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₃₇	C₂²	1	2	yes	—
14	#10596962	—	2	4	no	—
15	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₇	C₂²	1	2	yes	—
17	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₃₇	C₂²	1	2	yes	—
19	#10596938	C₂²	1	8	yes	—
20	C₄⋊Dic₃₇	C₂²	1	2	yes	—
21	#10596936	C₂²	1	8	yes	—
22	C₄×Dic₃₇	C₂²	1	2	yes	—
23	C₂³×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×D₃₇	C₂³	1	2	yes	—
25	Q₈×C₃₇	—	2	4	no	—
26	Q₈×C₃₇	—	2	4	no	—
27	C₂×C₄×C₃₇	—	2	4	no	—
28	C₂×C₄×C₃₇	D₄	1	4	yes	—
29	C₂×C₄×C₃₇	C₂³	1	4	yes	—
30	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²⋊D₃₇	—	2	4	no	—
33	C₂×Dic₃₇	C₂³	1	4	yes	—
34	C₂×Dic₃₇	C₂³	1	2	yes	—
35	D₁₄₈	—	2	2	no	—
36	C₄×D₃₇	—	2	4	no	—
37	Dic₇₄	—	2	2	no	—
38	C₂²×C₃₇	—	2	2	no	—
39	C₂²×C₃₇	C₂×D₄	1	2	yes	—
40	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₃₇	—	2	2	no	—
42	C₂×D₃₇	—	2	4	no	—
43	C₄×C₃₇	—	2	4	no	—
44	C₄×C₃₇	C₂×D₄	1	2	yes	—
45	C₄×C₃₇	C₂×D₄	1	2	yes	—
46	Dic₃₇	—	2	2	no	—
47	Dic₃₇	—	2	4	no	—
48	C₂×C₃₇	—	2	2	no	—
49	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₃₇	2₋¹⁺⁴	1	2	yes	—
51	D₃₇	—	4	2	no	—
52	#536	—	37	1	no	maximal, Sylow
53	C₃₇	#536	1	1	yes	characteristic, complemented, Sylow
54	C₂×#41	—	37	1	no	—
55	C₂²×Q₈	D₃₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	37	2	no	—
57	#124	—	37	2	no	—
58	#123	—	37	4	no	—
59	C₂²⋊Q₈	—	37	4	no	—
60	#117	—	37	1	no	—
61	#41	—	74	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	37	1	no	—
64	C₂×Q₈	C₂×D₃₇	1	2	yes	—
65	C₂×Q₈	C₂×D₃₇	1	2	yes	—
66	C₂×D₄	—	37	2	no	—
67	C₂×D₄	—	37	1	no	—
68	C₂²×C₄	—	37	2	no	—
69	C₂²×C₄	C₂×D₃₇	1	2	yes	—
70	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
71	C₄⋊C₄	—	37	8	no	—
72	C₄⋊C₄	—	37	2	no	—
73	C₂²⋊C₄	—	37	2	no	—
74	C₂²⋊C₄	—	37	8	no	—
75	C₄²	—	37	2	no	—
76	C₂³	—	37	2	no	—
77	C₂³	C₂²×D₃₇	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	74	2	no	—
81	D₄	—	74	4	no	—
82	D₄	—	74	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	37	4	no	—
85	C₂×C₄	—	37	2	no	—
86	C₂×C₄	—	74	4	no	—
87	C₂×C₄	C₂²⋊D₃₇	1	4	yes	—
88	C₂×C₄	C₂²×D₃₇	1	4	yes	—
89	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	74	2	no	—
93	C₂²	—	74	4	no	—
94	C₂²	C₂×C₂²⋊D₃₇	1	2	yes	—
95	C₂²	C₂³×D₃₇	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	74	2	no	—
98	C₄	—	74	4	no	—
99	C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
100	C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	148	2	no	—
103	C₂	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
104	C₂	#11924584	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10596962

C₂

yes

maximal, characteristic

#11924534

C₂

yes

maximal, complemented

#11924525

C₂

yes

maximal, complemented

#11924519

C₂

yes

maximal, complemented

#11924512

C₂

yes

maximal, complemented

#11924495

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

C₂²

yes

—

#10596962

—

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₂²⋊Dic₃₇

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂³×C₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

D₄

yes

—

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×D₄

yes

—

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₄

yes

—

C₄×C₃₇

C₂×D₄

yes

—

Dic₃₇

—

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₄

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

—

D₃₇

—

#536

—

maximal, Sylow

C₃₇

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

D₃₇

yes

characteristic, complemented

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

—

C₂×Q₈

C₂×D₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

—

C₂²×C₄

C₂×D₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×D₃₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₃₇

yes

—

C₂×C₄

C₂²×D₃₇

yes

—

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₃₇

yes

—

C₂²

C₂³×D₃₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×C₂²⋊D₃₇

yes

—

100

C₄

C₂×C₂²⋊D₃₇

yes

—

101

C₂

—

102

C₂

—

148

—

103

C₂

C₂²×C₂²⋊D₃₇

yes

characteristic

104

C₂

#11924584

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₂²×Q₈×C₃₇	C₂
prime radical	C₂×#10596962	C₂
hypercenter	C₂²×Q₈	D₃₇
hypocenter	C₃₇	#536
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₂²×Q₈×C₃₇

C₂

prime radical

C₂×#10596962

C₂

hypercenter

C₂²×Q₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #11924534	C₂	semidirect
3. #11924525	C₂	semidirect
4. #11924519	C₂	semidirect
5. #11924512	C₂	semidirect
6. #11924495	C₂	semidirect
7. C₂²×Q₈×C₃₇	C₂	semidirect
53. C₃₇	#536	semidirect
55. C₂²×Q₈	D₃₇	semidirect

H ≅ G/N

type

2. #11924534

C₂

semidirect

3. #11924525

C₂

semidirect

4. #11924519

C₂

semidirect

5. #11924512

C₂

semidirect

6. #11924495

C₂

semidirect

7. C₂²×Q₈×C₃₇

C₂

semidirect

53. C₃₇

#536

semidirect

55. C₂²×Q₈

D₃₇

semidirect

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

subgroup	index
C₂×#10596962	2
#11924534	2
#11924525	2
#11924519	2
#11924512	2
#11924495	2
C₂²×Q₈×C₃₇	2
#536	37

subgroup

index

C₂²×Q₈×C₃₇

Subgroups — #14558107

Conjugacy classes of subgroups

Complements

Maximal subgroups