Group #18010086 — subgroups

Subgroups — #18010086

Order 4736.

238 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 56 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, complemented
2	#14558389	C₂	1	4	yes	maximal, complemented
3	#14558084	C₂	1	1	yes	maximal, characteristic
4	#11924603	—	2	2	no	—
5	#11924593	—	2	2	no	—
6	#11924591	C₂²	1	4	yes	—
7	C₂×#10596966	C₄	1	2	yes	—
8	#11924533	C₄	1	1	yes	characteristic
9	#11924525	C₂²	1	2	yes	—
10	#11924511	—	2	2	no	—
11	#11924498	—	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	#10596966	—	4	4	no	—
16	C₂×D₁₄₈	—	2	2	no	—
17	C₂×D₁₄₈	—	2	2	no	—
18	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×D₃₇	—	2	2	no	—
20	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₃₇	D₄	1	2	yes	—
22	C₂×#64438	—	2	4	no	—
23	C₄⋊C₄×C₃₇	—	2	2	no	—
24	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
25	#10596938	—	2	4	no	—
26	C₄×Dic₃₇	D₄	1	2	yes	—
27	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
28	C₂²×D₃₇	—	2	2	no	—
29	C₂²×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₃₇	—	4	2	no	—
31	C₂×C₄×C₃₇	—	2	2	no	—
32	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
36	D₁₄₈	—	4	2	no	—
37	D₁₄₈	—	4	4	no	—
38	D₁₄₈	—	4	4	no	—
39	C₄×D₃₇	—	2	2	no	—
40	C₄×D₃₇	—	4	2	no	—
41	C₄×D₃₇	—	4	4	no	—
42	#64438	—	4	4	no	—
43	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×D₃₇	—	2	1	no	—
45	C₂×D₃₇	—	2	2	no	—
46	C₂×D₃₇	—	4	2	no	—
47	C₂×D₃₇	—	4	4	no	—
48	C₄×C₃₇	—	2	1	no	—
49	C₄×C₃₇	—	2	2	no	—
50	C₄×C₃₇	—	4	2	no	—
51	Dic₃₇	—	2	2	no	—
52	Dic₃₇	#100	1	2	yes	—
53	#1536	—	37	1	no	maximal, Sylow
54	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
55	C₂×C₃₇	#421	1	2	yes	—
56	D₃₇	—	4	1	no	—
57	D₃₇	—	8	2	no	—
58	#532	—	37	1	no	—
59	#432	—	37	2	no	—
60	#329	—	37	4	no	—
61	C₃₇	#1536	1	1	yes	characteristic, complemented, Sylow
62	C₂×#41	—	37	2	no	—
63	C₄⋊Q₈	#3866	1	1	yes	characteristic
64	#127	—	37	1	no	—
65	#124	—	37	2	no	—
66	#121	—	74	2	no	—
67	C₄×D₄	—	74	2	no	—
68	C₂×C₄²	—	37	1	no	—
69	C₄⋊C₈	—	74	2	no	—
70	C₂²⋊C₈	—	74	2	no	—
71	#97	—	37	4	no	—
72	#41	—	148	4	no	—
73	C₂×Q₈	C₂×#3866	1	2	yes	—
74	C₂×D₄	—	37	2	no	—
75	C₂×D₄	—	74	2	no	—
76	C₂×D₄	—	74	2	no	—
77	C₂²×C₄	—	37	1	no	—
78	C₂²×C₄	—	37	2	no	—
79	C₂²×C₄	—	74	2	no	—
80	C₂×C₈	—	74	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	74	4	no	—
83	C₄²	—	37	2	no	—
84	C₄²	—	37	1	no	—
85	C₄²	C₂×#3866	1	1	yes	characteristic
86	C₂³	—	37	1	no	—
87	C₂³	—	74	2	no	—
88	Q₈	—	4	2	no	—
89	D₄	—	148	2	no	—
90	D₄	—	148	4	no	—
91	D₄	—	148	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	37	1	no	—
94	C₂×C₄	—	37	2	no	—
95	C₂×C₄	—	74	2	no	—
96	C₂×C₄	—	148	2	no	—
97	C₂×C₄	—	148	4	no	—
98	C₂×C₄	C₂²⋊#3866	1	2	yes	—
99	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
100	C₈	—	148	4	no	—
101	C₂²	—	74	1	no	—
102	C₂²	—	74	2	no	—
103	C₂²	—	148	2	no	—
104	C₂²	—	148	4	no	—
105	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	37	2	no	—
110	C₄	—	74	2	no	—
111	C₂	—	296	2	no	—
112	C₂	—	148	1	no	—
113	C₂	D₃₇⋊#100	1	1	yes	characteristic
114	C₂	#14558532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558556

C₂

yes

maximal, complemented

#14558389

C₂

yes

maximal, complemented

#14558084

C₂

yes

maximal, characteristic

#11924603

—

#11924593

—

#11924591

C₂²

yes

—

C₂×#10596966

C₄

yes

—

#11924533

C₄

yes

characteristic

#11924525

C₂²

yes

—

#11924511

—

#11924498

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic

C₂×Q₈×C₃₇

C₂×C₄

yes

—

#10596966

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×#64438

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂²×D₃₇

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

#100

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

D₃₇

—

#532

—

#432

—

#329

—

C₃₇

#1536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#41

—

148

—

C₂×Q₈

C₂×#3866

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

148

—

D₄

—

148

—

D₄

—

148

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₂²⋊#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

100

C₈

—

148

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

148

—

104

C₂²

—

148

—

105

C₂²

C₂×C₂²⋊#3866

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

296

—

112

C₂

—

148

—

113

C₂

D₃₇⋊#100

yes

characteristic

114

C₂

#14558532

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	#14558084	C₂
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1536
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

#14558084

C₂

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
1. #14558556	C₂	semidirect
2. #14558389	C₂	semidirect
61. C₃₇	#1536	semidirect

H ≅ G/N

type

1. #14558556

C₂

semidirect

2. #14558389

C₂

semidirect

61. C₃₇

#1536

semidirect

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

subgroup	index
#14558556	2
#14558389	2
#14558084	2
#1536	37

subgroup

index

Subgroups — #18010086

Conjugacy classes of subgroups

Complements

Maximal subgroups