Group #18005119 — subgroups

Subgroups — #18005119

Order 4736.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#14558086	C₂	1	2	yes	maximal
2	#14557941	C₂	1	4	yes	maximal, complemented
3	#14557940	C₂	1	4	yes	maximal, complemented
4	#14557571	C₂	1	1	yes	maximal, characteristic
5	#14557569	C₂	1	2	yes	maximal
6	C₂×#11924446	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	#11924535	C₂²	1	4	yes	—
10	#11924527	C₂²	1	4	yes	—
11	#11924513	—	2	4	no	—
12	C₄×Dic₇₄	—	2	4	no	—
13	#11924495	C₂²	1	2	yes	—
14	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
15	#11924470	C₂²	1	2	yes	—
16	#11924469	C₂²	1	4	yes	—
17	#11924468	C₂²	1	2	yes	—
18	#11924467	C₂²	1	4	yes	—
19	#11924446	C₂²	1	2	yes	—
20	#11924446	C₂²	1	2	yes	—
21	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	#117×C₃₇	C₂²	1	2	yes	—
23	#107×C₃₇	C₂²	1	2	yes	—
24	#106×C₃₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	D₄	1	2	yes	—
27	C₂×Dic₇₄	—	2	4	no	—
28	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
29	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₃₇	—	2	2	no	—
33	C₄²×C₃₇	—	2	2	no	—
34	C₂²⋊Dic₃₇	—	2	2	no	—
35	C₄⋊Dic₃₇	—	2	4	no	—
36	C₄⋊Dic₃₇	C₂³	1	4	yes	—
37	#10596936	—	2	4	no	—
38	#10596936	—	2	4	no	—
39	C₄×Dic₃₇	—	2	2	no	—
40	C₄×Dic₃₇	D₄	1	2	yes	—
41	C₄×Dic₃₇	C₂³	1	2	yes	—
42	C₂×#64436	Q₈	1	4	yes	complemented
43	C₂×#64436	C₂³	1	2	yes	—
44	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	—	2	4	no	—
46	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₇	—	2	4	no	—
51	C₂×Dic₃₇	—	4	2	no	—
52	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
53	Dic₇₄	—	4	8	no	—
54	C₈×C₃₇	C₂×Q₈	1	2	yes	—
55	C₈×C₃₇	C₂×Q₈	1	2	yes	—
56	#64436	C₂×Q₈	1	4	yes	—
57	C₂²×C₃₇	—	2	2	no	—
58	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₃₇	—	4	4	no	—
61	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
62	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
64	Dic₃₇	—	2	4	no	—
65	Dic₃₇	—	4	4	no	—
66	#3162	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	—	2	2	no	—
68	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₃₇	#575	1	2	yes	—
70	#533	—	37	2	no	—
71	#501	—	37	4	no	—
72	#501	—	37	4	no	—
73	#427	—	37	1	no	—
74	#427	—	37	2	no	—
75	#427	D₃₇	1	1	yes	characteristic
76	C₂×#97	—	37	1	no	—
77	C₃₇	#3162	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	37	1	no	—
79	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	37	4	no	—
81	#125	—	37	4	no	—
82	C₂²⋊Q₈	—	74	4	no	—
83	C₄×Q₈	—	74	4	no	—
84	#117	—	37	2	no	—
85	#117	C₂×D₃₇	1	2	yes	—
86	C₂×C₄²	—	37	1	no	—
87	#107	—	37	2	no	—
88	#107	—	37	4	no	—
89	#107	C₂×D₃₇	1	2	yes	—
90	#106	—	37	2	no	—
91	#106	—	37	4	no	—
92	#106	C₂×D₃₇	1	2	yes	—
93	#97	—	37	2	no	—
94	#97	—	37	2	no	—
95	C₂×Q₈	—	74	4	no	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
98	C₂×C₈	—	37	4	no	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	Dic₇₄	1	4	yes	complemented
101	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
103	C₄⋊C₄	—	37	4	no	—
104	C₄⋊C₄	—	74	4	no	—
105	C₄⋊C₄	—	74	4	no	—
106	C₄⋊C₄	—	74	4	no	—
107	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	74	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	37	2	no	—
112	C₄²	—	37	2	no	—
113	C₄²	—	74	2	no	—
114	C₂³	D₄×D₃₇	1	1	yes	characteristic
115	Q₈	—	148	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	37	4	no	—
118	C₂×C₄	—	74	4	no	—
119	C₂×C₄	—	148	2	no	—
120	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
121	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₃₇	1	2	yes	—
123	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
124	C₈	—	37	4	no	—
125	C₈	C₂×Dic₇₄	1	2	yes	—
126	C₈	C₂×Dic₇₄	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#11924535	1	2	yes	—
129	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	74	4	no	—
132	C₄	—	148	4	no	—
133	C₄	#11924535	1	2	yes	—
134	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
135	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#11924535	1	1	yes	characteristic
138	C₂	#14558330	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558086

C₂

yes

maximal

#14557941

C₂

yes

maximal, complemented

#14557940

C₂

yes

maximal, complemented

#14557571

C₂

yes

maximal, characteristic

#14557569

C₂

yes

maximal

C₂×#11924446

C₂

yes

maximal, characteristic

C₃₇×#427

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

#11924535

C₂²

yes

—

#11924527

C₂²

yes

—

#11924513

—

C₄×Dic₇₄

—

#11924495

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924469

C₂²

yes

—

#11924468

C₂²

yes

—

#11924467

C₂²

yes

—

#11924446

C₂²

yes

—

#11924446

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

#107×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

D₄

yes

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

C₂³

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

Q₈

yes

complemented

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#3162

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

#575

yes

—

—

—

—

—

—

D₃₇

yes

characteristic

C₂×#97

—

C₃₇

#3162

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃₇

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₃₇

yes

—

#106

—

#106

—

#106

C₂×D₃₇

yes

—

#97

—

#97

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₇₄

yes

complemented

101

C₂×C₈

C₂²×D₃₇

yes

characteristic

102

C₂×C₈

C₂²×D₃₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×D₃₇

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₃₇

yes

characteristic

115

Q₈

—

148

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

148

—

120

C₂×C₄

C₂×Dic₇₄

yes

—

121

C₂×C₄

D₄×D₃₇

yes

characteristic

122

C₂×C₄

Q₈×D₃₇

yes

—

123

C₂×C₄

C₂³×D₃₇

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₇₄

yes

—

126

C₈

C₂×Dic₇₄

yes

—

127

C₂²

—

128

C₂²

#11924535

yes

—

129

C₂²

C₂×D₄×D₃₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

148

—

133

C₄

#11924535

yes

—

134

C₄

C₂×Q₈×D₃₇

yes

characteristic

135

C₄

C₂²×Dic₇₄

yes

characteristic

136

C₂

—

137

C₂

C₂×#11924535

yes

characteristic

138

C₂

#14558330

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#427	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	#427	D₃₇
hypocenter	C₃₇	#3162
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#427

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #14557941	C₂	semidirect
3. #14557940	C₂	semidirect
28. C₂×C₈×C₃₇	Q₈	semidirect
42. C₂×#64436	Q₈	semidirect
77. C₃₇	#3162	semidirect
100. C₂×C₈	Dic₇₄	semidirect

H ≅ G/N

type

2. #14557941

C₂

semidirect

3. #14557940

C₂

semidirect

28. C₂×C₈×C₃₇

Q₈

semidirect

42. C₂×#64436

Q₈

semidirect

77. C₃₇

#3162

semidirect

100. C₂×C₈

Dic₇₄

semidirect

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

subgroup	index
#14558086	2
#14557941	2
#14557940	2
#14557571	2
#14557569	2
C₂×#11924446	2
C₃₇×#427	2
#3162	37

subgroup

index

C₃₇×#427

Subgroups — #18005119

Conjugacy classes of subgroups

Complements

Maximal subgroups