Group #14558198 — subgroups

Subgroups — #14558198

Order 2368.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₃₇	C₂	1	2	yes	maximal, complemented
2	#11924535	C₂	1	2	yes	maximal, complemented
3	#11924514	C₂	1	4	yes	maximal, complemented
4	#11924512	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic
7	#11924498	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×C₃₇	C₂²	1	2	yes	—
10	Q₈×D₃₇	—	2	8	no	—
11	C₂×D₁₄₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₇	C₂²	1	4	yes	—
13	C₂×C₄×D₃₇	C₂²	1	2	yes	—
14	C₂×Dic₇₄	C₂²	1	4	yes	—
15	C₂×Dic₇₄	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
18	C₄²×C₃₇	C₂²	1	1	yes	characteristic
19	#10596938	C₂²	1	4	yes	—
20	#10596938	C₂²	1	2	yes	—
21	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
23	#10596936	C₂²	1	2	yes	—
24	#10596936	C₂²	1	4	yes	—
25	C₄×Dic₃₇	C₂²	1	2	yes	—
26	C₂²×D₃₇	C₂³	1	2	yes	—
27	Q₈×C₃₇	—	2	4	no	—
28	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
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	C₂×Dic₃₇	C₂³	1	2	yes	—
33	C₂×Dic₃₇	C₂³	1	2	yes	—
34	C₂×Dic₃₇	C₂³	1	2	yes	—
35	D₁₄₈	Q₈	1	4	yes	complemented
36	C₄×D₃₇	—	2	4	no	—
37	C₄×D₃₇	—	2	8	no	—
38	Dic₇₄	—	2	8	no	—
39	Dic₇₄	D₄	1	4	yes	complemented
40	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₃₇	—	2	2	no	—
42	C₂×D₃₇	—	2	2	no	—
43	C₂×D₃₇	C₂×Q₈	1	4	yes	complemented
44	C₄×C₃₇	—	2	4	no	—
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	C₂×D₄	1	2	yes	—
47	C₄×C₃₇	C₂×Q₈	1	2	yes	complemented
48	Dic₃₇	—	2	2	no	—
49	Dic₃₇	—	2	2	no	—
50	Dic₃₇	C₂×D₄	1	4	yes	—
51	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
54	D₃₇	—	2	4	no	—
55	D₄×Q₈	—	37	1	no	maximal, Sylow
56	C₃₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	—	37	2	no	—
58	C₄⋊Q₈	—	37	2	no	—
59	C₄⋊Q₈	D₃₇	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	37	2	no	—
61	C₂²⋊Q₈	—	37	4	no	—
62	C₄×Q₈	—	37	1	no	—
63	C₄×D₄	—	37	2	no	—
64	C₄×D₄	—	37	1	no	—
65	C₂×Q₈	—	37	1	no	—
66	C₂×Q₈	—	37	4	no	—
67	C₂×Q₈	—	74	8	no	—
68	C₂×Q₈	C₂×D₃₇	1	2	yes	—
69	C₂×D₄	—	37	1	no	—
70	C₂²×C₄	—	37	4	no	—
71	C₂²×C₄	—	37	2	no	—
72	C₄⋊C₄	—	37	2	no	—
73	C₄⋊C₄	—	37	4	no	—
74	C₄⋊C₄	—	37	1	no	—
75	C₄⋊C₄	—	37	1	no	—
76	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
77	C₄⋊C₄	C₂×D₃₇	1	2	yes	—
78	C₂²⋊C₄	—	37	4	no	—
79	C₂²⋊C₄	—	37	2	no	—
80	C₄²	—	37	2	no	—
81	C₄²	C₂×D₃₇	1	1	yes	characteristic
82	C₂³	—	37	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	37	4	no	—
85	Q₈	—	74	8	no	—
86	D₄	—	37	4	no	—
87	C₂×C₄	—	37	2	no	—
88	C₂×C₄	—	37	2	no	—
89	C₂×C₄	—	37	2	no	—
90	C₂×C₄	—	74	8	no	—
91	C₂×C₄	—	74	4	no	—
92	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₃₇	1	4	yes	—
94	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	C₂²	—	37	4	no	—
97	C₂²	—	74	2	no	—
98	C₂²	—	74	2	no	—
99	C₂²	C₂³×D₃₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	37	4	no	—
103	C₄	—	74	2	no	—
104	C₄	—	74	2	no	—
105	C₄	D₄×D₃₇	1	2	yes	—
106	C₄	Q₈×D₃₇	1	2	yes	complemented
107	C₂	—	74	4	no	—
108	C₂	C₂×D₄×D₃₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₃₇	1	1	yes	characteristic
110	C₂	#11924584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₃₇

C₂

yes

maximal, complemented

#11924535

C₂

yes

maximal, complemented

#11924514

C₂

yes

maximal, complemented

#11924512

C₂

yes

maximal, complemented

C₄×Dic₇₄

C₂

yes

maximal, characteristic, complemented

C₄×D₁₄₈

C₂

yes

maximal, characteristic

#11924498

C₂

yes

maximal

C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₇

C₂²

yes

—

Q₈×D₃₇

—

C₂×D₁₄₈

C₂²

yes

characteristic

C₂×C₄×D₃₇

C₂²

yes

—

C₂×C₄×D₃₇

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

—

C₂×Dic₇₄

C₂²

yes

characteristic

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄⋊C₄×C₃₇

C₂²

yes

—

C₄²×C₃₇

C₂²

yes

characteristic

#10596938

C₂²

yes

—

#10596938

C₂²

yes

—

C₄⋊Dic₃₇

C₂²

yes

characteristic

C₄⋊Dic₃₇

C₂²

yes

characteristic

#10596936

C₂²

yes

—

#10596936

C₂²

yes

—

C₄×Dic₃₇

C₂²

yes

—

C₂²×D₃₇

C₂³

yes

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

C₂×Dic₃₇

C₂³

yes

—

D₁₄₈

Q₈

yes

complemented

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

D₄

yes

complemented

C₂²×C₃₇

C₂⁴

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×Q₈

yes

complemented

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₄

yes

—

C₄×C₃₇

C₂×Q₈

yes

complemented

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×D₄

yes

—

C₂×C₃₇

C₂²×D₄

yes

characteristic

C₂×C₃₇

C₂²×Q₈

yes

characteristic

C₂×C₃₇

2₋¹⁺⁴

yes

characteristic

D₃₇

—

D₄×Q₈

—

maximal, Sylow

C₃₇

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₃₇

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₇

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₇

yes

—

C₄⋊C₄

C₂×D₃₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₃₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

—

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₄

C₂²×D₃₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₃₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

D₄×D₃₇

yes

—

106

C₄

Q₈×D₃₇

yes

complemented

107

C₂

—

108

C₂

C₂×D₄×D₃₇

yes

characteristic

109

C₂

C₂×Q₈×D₃₇

yes

characteristic

110

C₂

#11924584

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₄⋊Q₈×C₃₇	C₂
prime radical	C₂×D₁₄₈	C₂²
hypercenter	C₄⋊Q₈	D₃₇
hypocenter	C₃₇	D₄×Q₈
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇

commutator, socle, cosocle

C₂²×C₃₇

C₂⁴

Fitting

C₄⋊Q₈×C₃₇

C₂

prime radical

C₂×D₁₄₈

C₂²

hypercenter

C₄⋊Q₈

D₃₇

hypocenter

C₃₇

D₄×Q₈

LCS

G ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₃₇	C₂	semidirect
2. #11924535	C₂	semidirect
3. #11924514	C₂	semidirect
4. #11924512	C₂	semidirect
5. C₄×Dic₇₄	C₂	semidirect
8. C₄⋊Q₈×C₃₇	C₂	semidirect
35. D₁₄₈	Q₈	semidirect
39. Dic₇₄	D₄	semidirect
43. C₂×D₃₇	C₂×Q₈	semidirect
47. C₄×C₃₇	C₂×Q₈	semidirect
56. C₃₇	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₃₇	semidirect
106. C₄	Q₈×D₃₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₃₇

C₂

semidirect

2. #11924535

C₂

semidirect

3. #11924514

C₂

semidirect

4. #11924512

C₂

semidirect

5. C₄×Dic₇₄

C₂

semidirect

8. C₄⋊Q₈×C₃₇

C₂

semidirect

35. D₁₄₈

Q₈

semidirect

39. Dic₇₄

D₄

semidirect

43. C₂×D₃₇

C₂×Q₈

semidirect

47. C₄×C₃₇

C₂×Q₈

semidirect

56. C₃₇

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₃₇

semidirect

106. C₄

Q₈×D₃₇

semidirect

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

subgroup	index
C₂×Q₈×D₃₇	2
#11924535	2
#11924514	2
#11924512	2
C₄×Dic₇₄	2
C₄×D₁₄₈	2
#11924498	2
C₄⋊Q₈×C₃₇	2
D₄×Q₈	37

subgroup

index

C₂×Q₈×D₃₇

#11924535

#11924514

#11924512

C₄×Dic₇₄

C₄×D₁₄₈

#11924498

C₄⋊Q₈×C₃₇

D₄×Q₈

Subgroups — #14558198

Conjugacy classes of subgroups

Complements

Maximal subgroups