Group #18010130 — subgroups

Subgroups — #18010130

Order 4736.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#14558556	C₂	1	2	yes	maximal
2	#14558393	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
4	#11924603	C₂²	1	4	yes	—
5	#11924593	—	2	2	no	—
6	#11924591	—	2	2	no	—
7	C₂×Q₈×D₃₇	C₄	1	2	yes	—
8	#11924536	C₄	1	1	yes	characteristic
9	#11924535	—	2	2	no	—
10	#11924534	C₂²	1	2	yes	—
11	C₄⋊C₄×D₃₇	—	2	2	no	—
12	C₄²×D₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₃₇	C₂×C₄	1	2	yes	—
15	Q₈×D₃₇	—	4	4	no	—
16	C₂×C₄×D₃₇	—	2	2	no	—
17	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₃₇	D₄	1	2	yes	—
19	C₂×Dic₇₄	—	2	2	no	—
20	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
21	C₂×#64438	—	2	4	no	—
22	C₄⋊C₄×C₃₇	—	2	2	no	—
23	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₃₇	—	2	2	no	—
25	#10596936	—	2	4	no	—
26	C₄×Dic₃₇	D₄	1	2	yes	—
27	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
28	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₃₇	—	4	2	no	—
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	—	2	2	no	—
34	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
36	C₄×D₃₇	—	2	2	no	—
37	C₄×D₃₇	—	2	4	no	—
38	C₄×D₃₇	—	4	4	no	—
39	Dic₇₄	—	4	4	no	—
40	Dic₇₄	—	4	2	no	—
41	#64438	—	4	4	no	—
42	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₃₇	—	2	2	no	—
44	C₂×D₃₇	#100	1	2	yes	—
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	—	2	2	no	—
47	C₄×C₃₇	—	4	2	no	—
48	Dic₃₇	—	2	1	no	—
49	Dic₃₇	—	4	2	no	—
50	Dic₃₇	#137	1	4	yes	—
51	#1558	—	37	1	no	maximal, Sylow
52	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
53	C₂×C₃₇	#419	1	2	yes	—
54	D₃₇	—	2	2	no	—
55	C₂×C₄⋊Q₈	—	37	1	no	—
56	#432	—	37	2	no	—
57	#333	—	37	4	no	—
58	C₃₇	#1558	1	1	yes	characteristic, complemented, Sylow
59	C₂²×Q₈	—	37	2	no	—
60	C₄⋊Q₈	—	37	2	no	—
61	C₄⋊Q₈	—	37	1	no	—
62	C₄⋊Q₈	—	74	2	no	—
63	C₄⋊Q₈	#3866	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	74	2	no	—
65	C₂×C₄²	—	37	1	no	—
66	C₄⋊C₈	—	37	4	no	—
67	C₂²⋊C₈	—	74	2	no	—
68	#97	—	74	2	no	—
69	C₂×Q₈	—	37	2	no	—
70	C₂×Q₈	—	74	2	no	—
71	C₂×Q₈	—	148	4	no	—
72	C₂×Q₈	C₂×#3866	1	2	yes	—
73	C₂²×C₄	—	37	2	no	—
74	C₂²×C₄	—	37	1	no	—
75	C₂²×C₄	—	74	2	no	—
76	C₂×C₈	—	74	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	74	4	no	—
79	C₄⋊C₄	—	74	2	no	—
80	C₄²	—	37	2	no	—
81	C₄²	—	37	1	no	—
82	C₄²	C₂×#3866	1	1	yes	characteristic
83	C₂³	—	37	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	148	2	no	—
86	Q₈	—	148	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	37	2	no	—
89	C₂×C₄	—	37	1	no	—
90	C₂×C₄	—	74	4	no	—
91	C₂×C₄	—	74	2	no	—
92	C₂×C₄	—	74	2	no	—
93	C₂×C₄	—	148	4	no	—
94	C₂×C₄	C₂²⋊#3866	1	2	yes	—
95	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
96	C₈	—	148	4	no	—
97	C₂²	—	37	2	no	—
98	C₂²	—	74	2	no	—
99	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	37	4	no	—
104	C₄	—	74	1	no	—
105	C₄	—	148	2	no	—
106	C₂	—	74	2	no	—
107	C₂	D₃₇⋊#100	1	1	yes	characteristic
108	C₂	#14558526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558556

C₂

yes

maximal

#14558393

C₂

yes

maximal, complemented

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

#11924603

C₂²

yes

—

#11924593

—

#11924591

—

C₂×Q₈×D₃₇

C₄

yes

—

#11924536

C₄

yes

characteristic

#11924535

—

#11924534

C₂²

yes

—

C₄⋊C₄×D₃₇

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂²×D₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#100

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#137

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₃₇

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#3866

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

C₂×#3866

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

148

—

Q₈

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

C₂²⋊#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

148

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#3866

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

148

—

106

C₂

—

107

C₂

D₃₇⋊#100

yes

characteristic

108

C₂

#14558526

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1558
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄⋊Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₄⋊Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #14558393	C₂	semidirect
58. C₃₇	#1558	semidirect

H ≅ G/N

type

2. #14558393

C₂

semidirect

58. C₃₇

#1558

semidirect

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

subgroup	index
#14558556	2
#14558393	2
C₄⋊Q₈×D₃₇	2
#1558	37

subgroup

index

#14558556

#14558393

C₄⋊Q₈×D₃₇

#1558

Subgroups — #18010130

Conjugacy classes of subgroups

Complements

Maximal subgroups