Group #18005148 — subgroups

Subgroups — #18005148

Order 4736.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14558090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924528	C₂	1	1	yes	maximal, characteristic
3	#14557942	C₂	1	4	yes	maximal, complemented
4	#14557935	C₂	1	4	yes	maximal, complemented
5	#14557575	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
7	#11924536	C₂²	1	2	yes	—
8	#11924528	—	2	2	no	—
9	#11924528	C₂²	1	2	yes	—
10	#11924528	C₂²	1	2	yes	—
11	#11924528	C₂²	1	2	yes	—
12	#11924515	—	2	2	no	—
13	C₄×Dic₇₄	—	2	2	no	—
14	#11924495	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
16	C₂×#10596936	—	2	2	no	—
17	#11924470	C₂²	1	8	yes	—
18	#11924468	C₂²	1	8	yes	—
19	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	#97×C₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂²×C₄×C₃₇	D₄	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₇	—	2	2	no	—
26	C₂×Dic₇₄	—	2	2	no	—
27	M₁₆×C₃₇	Q₈	1	8	yes	complemented
28	C₂×C₈×C₃₇	C₂³	1	4	yes	—
29	C₄²×C₃₇	D₄	1	2	yes	—
30	C₄²×C₃₇	C₂³	1	1	yes	characteristic
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₃₇	—	2	2	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	2	no	—
35	C₄⋊Dic₃₇	C₂³	1	4	yes	—
36	C₄⋊Dic₃₇	C₂³	1	4	yes	—
37	#10596936	—	2	4	no	—
38	#10596936	—	2	4	no	—
39	#10596936	—	2	4	no	—
40	C₄×Dic₃₇	—	2	2	no	—
41	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	—	4	2	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₇	—	2	4	no	—
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	—	4	4	no	—
51	Dic₇₄	—	4	4	no	—
52	C₈×C₃₇	C₂×Q₈	1	8	yes	—
53	C₂²×C₃₇	—	2	2	no	—
54	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₃₇	—	2	2	no	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
59	C₄×C₃₇	C₂²×Q₈	1	2	yes	—
60	Dic₃₇	—	4	4	no	—
61	Dic₃₇	—	4	4	no	—
62	#3166	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	—	2	2	no	—
64	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₃₇	#575	1	2	yes	—
66	#533	—	37	1	no	—
67	C₂×#125	—	37	1	no	—
68	#501	—	37	4	no	—
69	#499	—	37	4	no	—
70	#428	—	37	4	no	—
71	C₄×M₁₆	D₃₇	1	1	yes	characteristic
72	C₃₇	#3166	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
74	C₄⋊Q₈	—	37	2	no	—
75	#125	—	37	2	no	—
76	#125	—	37	2	no	—
77	#125	—	37	2	no	—
78	#125	—	74	2	no	—
79	C₂²⋊Q₈	—	74	2	no	—
80	C₄×Q₈	—	74	2	no	—
81	#117	—	37	2	no	—
82	C₂×C₄⋊C₄	—	37	2	no	—
83	C₂×C₄⋊C₄	—	74	2	no	—
84	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
85	#107	—	37	8	no	—
86	#106	—	37	8	no	—
87	#97	C₂×D₃₇	1	2	yes	—
88	C₄×C₈	C₂×D₃₇	1	2	yes	—
89	C₂×Q₈	—	74	2	no	—
90	C₂²×C₄	—	74	2	no	—
91	C₂²×C₄	D₁₄₈	1	2	yes	—
92	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
93	M₁₆	Dic₇₄	1	8	yes	complemented
94	C₂×C₈	C₂²×D₃₇	1	4	yes	—
95	C₄⋊C₄	—	37	4	no	—
96	C₄⋊C₄	—	37	4	no	—
97	C₄⋊C₄	—	74	4	no	—
98	C₄⋊C₄	—	74	4	no	—
99	C₄⋊C₄	—	74	4	no	—
100	C₄⋊C₄	—	74	2	no	—
101	C₄⋊C₄	—	74	2	no	—
102	C₂²⋊C₄	—	74	2	no	—
103	C₄²	—	74	2	no	—
104	C₄²	D₁₄₈	1	2	yes	—
105	C₄²	C₂²×D₃₇	1	1	yes	characteristic
106	C₄²	C₂²×D₃₇	1	1	yes	characteristic
107	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
108	Q₈	—	148	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	74	4	no	—
111	C₂×C₄	—	74	4	no	—
112	C₂×C₄	—	148	4	no	—
113	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
114	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
115	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
116	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
118	C₈	C₂×Dic₇₄	1	8	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	#11924536	1	2	yes	—
121	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	148	4	no	—
125	C₄	—	148	4	no	—
126	C₄	#11924536	1	2	yes	—
127	C₄	C₂²×Dic₇₄	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#11924536	1	1	yes	characteristic
130	C₂	#14558333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558090

C₂

yes

maximal, characteristic

C₂×#11924528

C₂

yes

maximal, characteristic

#14557942

C₂

yes

maximal, complemented

#14557935

C₂

yes

maximal, complemented

#14557575

C₂

yes

maximal

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic

#11924536

C₂²

yes

—

#11924528

—

#11924528

C₂²

yes

—

#11924528

C₂²

yes

—

#11924528

C₂²

yes

—

#11924515

—

C₄×Dic₇₄

—

#11924495

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

M₁₆×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

#3166

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

#575

yes

—

—

—

—

—

—

C₄×M₁₆

D₃₇

yes

characteristic

C₃₇

#3166

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₃₇

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

—

—

C₂×D₃₇

yes

—

C₄×C₈

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

Dic₇₄

yes

complemented

C₂×C₈

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

D₁₄₈

yes

—

105

C₄²

C₂²×D₃₇

yes

characteristic

106

C₄²

C₂²×D₃₇

yes

characteristic

107

C₂³

C₂×D₁₄₈

yes

characteristic

108

Q₈

—

148

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

148

—

113

C₂×C₄

C₂×Dic₇₄

yes

—

114

C₂×C₄

C₂×D₁₄₈

yes

—

115

C₂×C₄

C₂×D₁₄₈

yes

—

116

C₂×C₄

C₂×D₁₄₈

yes

characteristic

117

C₂×C₄

C₂³×D₃₇

yes

characteristic

118

C₈

C₂×Dic₇₄

yes

—

119

C₂²

—

120

C₂²

#11924536

yes

—

121

C₂²

C₂²×D₁₄₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

148

—

125

C₄

—

148

—

126

C₄

#11924536

yes

—

127

C₄

C₂²×Dic₇₄

yes

—

128

C₂

—

129

C₂

C₂×#11924536

yes

characteristic

130

C₂

#14558333

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×M₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#3166
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₄×M₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂³×C₃₇

C₂×D₄

hypercenter

C₄×M₁₆

D₃₇

hypocenter

C₃₇

#3166

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14557942	C₂	semidirect
4. #14557935	C₂	semidirect
27. M₁₆×C₃₇	Q₈	semidirect
72. C₃₇	#3166	semidirect
93. M₁₆	Dic₇₄	semidirect

H ≅ G/N

type

3. #14557942

C₂

semidirect

4. #14557935

C₂

semidirect

27. M₁₆×C₃₇

Q₈

semidirect

72. C₃₇

#3166

semidirect

93. M₁₆

Dic₇₄

semidirect

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

subgroup	index
#14558090	2
C₂×#11924528	2
#14557942	2
#14557935	2
#14557575	2
C₄×M₁₆×C₃₇	2
#3166	37

subgroup

index

C₄×M₁₆×C₃₇

Subgroups — #18005148

Conjugacy classes of subgroups

Complements

Maximal subgroups