Group #18010398 — subgroups

Subgroups — #18010398

Order 4736.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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	D₃₇⋊#108	C₂	1	1	yes	maximal, characteristic
2	#14558543	C₂	1	1	yes	maximal, characteristic
3	#14558496	C₂	1	2	yes	maximal
4	#14558404	C₂	1	2	yes	maximal
5	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
6	D₃₇⋊M₁₆	C₂²	1	2	yes	—
7	#11924615	C₂²	1	2	yes	—
8	#11924607	C₄	1	2	yes	complemented
9	#11924606	C₄	1	2	yes	complemented
10	#11924605	C₄	1	2	yes	complemented
11	#11924604	C₄	1	2	yes	complemented
12	#11924590	—	2	2	no	—
13	C₈×#3866	—	2	2	no	—
14	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
16	C₄²×D₃₇	C₂²	1	1	yes	characteristic
17	C₄×#64436	C₄	1	1	yes	characteristic, complemented
18	C₈×Dic₃₇	C₄	1	1	yes	characteristic, complemented
19	C₈×Dic₃₇	C₄	1	1	yes	characteristic, complemented
20	C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×D₃₇	—	2	1	no	—
22	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
23	C₂²⋊#3866	—	2	2	no	—
24	#10596957	—	2	2	no	—
25	C₄⋊#3866	C₂×C₄	1	4	yes	—
26	C₄×#3866	—	2	2	no	—
27	#10596953	C₂×C₄	1	4	yes	—
28	D₃₇⋊C₈	—	2	2	no	—
29	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₃₇	—	2	1	no	—
33	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#64436	C₂×C₄	1	1	yes	characteristic
35	C₂×#64436	C₂×C₄	1	1	yes	characteristic
36	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
37	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
38	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
39	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
40	C₈×D₃₇	D₄	1	1	yes	characteristic
41	C₈×D₃₇	D₄	1	1	yes	characteristic
42	C₈×D₃₇	Q₈	1	1	yes	characteristic
43	C₈×D₃₇	Q₈	1	1	yes	characteristic
44	C₂²×D₃₇	#41	1	1	yes	characteristic
45	C₂×#3866	—	2	4	no	—
46	C₂×C₄×C₃₇	—	2	1	no	—
47	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₃₇	—	2	1	no	—
49	C₂×Dic₃₇	#41	1	1	yes	characteristic
50	C₄×D₃₇	—	2	4	no	—
51	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
54	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
55	#64438	—	2	4	no	—
56	C₈×C₃₇	C₄²	1	1	yes	characteristic
57	C₈×C₃₇	C₄²	1	1	yes	characteristic
58	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
60	#64436	C₄²	1	1	yes	characteristic
61	#64436	C₄²	1	1	yes	characteristic
62	#64436	C₄⋊C₄	1	1	yes	characteristic
63	#64436	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃₇	#117	1	1	yes	characteristic
65	C₂×D₃₇	—	2	2	no	—
66	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
67	C₂×D₃₇	C₄×Q₈	1	1	yes	characteristic
68	#3866	—	4	4	no	—
69	C₄×C₃₇	—	2	2	no	—
70	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	Dic₃₇	—	2	2	no	—
73	Dic₃₇	C₄×D₄	1	1	yes	characteristic
74	Dic₃₇	C₄×Q₈	1	1	yes	characteristic
75	#1772	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	1	no	—
77	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	D₃₇	—	2	1	no	—
79	D₃₇	#443	1	2	yes	—
80	C₂×#108	—	37	1	no	—
81	#427	—	37	1	no	—
82	#405	—	37	2	no	—
83	C₂×C₄×C₈	—	37	1	no	—
84	#339	—	37	2	no	—
85	C₃₇	#1772	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	37	2	no	—
87	C₂²×C₈	—	37	1	no	—
88	C₂²×C₈	—	37	1	no	—
89	#117	—	37	2	no	—
90	C₂×C₄²	—	37	1	no	—
91	#108	—	37	2	no	—
92	#108	—	37	2	no	—
93	#107	—	37	2	no	—
94	#106	—	37	2	no	—
95	#97	—	74	2	no	—
96	C₄×C₈	—	37	1	no	—
97	C₄×C₈	—	37	1	no	—
98	C₄×C₈	—	37	1	no	—
99	C₄×C₈	—	74	2	no	—
100	C₄×C₈	#3866	1	1	yes	characteristic, complemented
101	C₂²×C₄	—	37	1	no	—
102	C₂²×C₄	—	74	1	no	—
103	M₁₆	—	37	4	no	—
104	M₁₆	—	74	2	no	—
105	C₂×C₈	—	37	1	no	—
106	C₂×C₈	—	37	1	no	—
107	C₂×C₈	—	37	1	no	—
108	C₂×C₈	—	37	1	no	—
109	C₂×C₈	—	37	1	no	—
110	C₂×C₈	—	37	1	no	—
111	C₂×C₈	—	37	1	no	—
112	C₂×C₈	—	37	1	no	—
113	C₂×C₈	—	37	1	no	—
114	C₂×C₈	—	37	1	no	—
115	C₂×C₈	—	74	2	no	—
116	C₂×C₈	C₂×#3866	1	1	yes	characteristic
117	C₂×C₈	C₂×#3866	1	1	yes	characteristic
118	C₄⋊C₄	—	37	4	no	—
119	C₂²⋊C₄	—	74	2	no	—
120	C₄²	—	37	1	no	—
121	C₄²	—	74	1	no	—
122	C₄²	—	74	2	no	—
123	C₄²	C₂×#3866	1	1	yes	characteristic
124	C₂³	—	37	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	37	1	no	—
127	C₂×C₄	—	37	1	no	—
128	C₂×C₄	—	37	1	no	—
129	C₂×C₄	—	37	1	no	—
130	C₂×C₄	—	37	1	no	—
131	C₂×C₄	—	74	4	no	—
132	C₂×C₄	—	74	1	no	—
133	C₂×C₄	—	74	4	no	—
134	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
135	C₈	—	37	1	no	—
136	C₈	—	37	1	no	—
137	C₈	—	37	1	no	—
138	C₈	—	37	1	no	—
139	C₈	—	74	4	no	—
140	C₈	C₄×#3866	1	1	yes	characteristic
141	C₈	C₄×#3866	1	1	yes	characteristic
142	C₈	C₄⋊#3866	1	1	yes	characteristic
143	C₈	C₄⋊#3866	1	1	yes	characteristic
144	C₂²	—	37	1	no	—
145	C₂²	—	37	1	no	—
146	C₂²	—	74	2	no	—
147	C₂²	#11924615	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	37	1	no	—
150	C₄	—	37	1	no	—
151	C₄	—	74	2	no	—
152	C₄	—	148	4	no	—
153	C₄	C₂×C₄×#3866	1	1	yes	characteristic
154	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	37	2	no	—
157	C₂	—	74	1	no	—
158	C₂	C₄×C₄⋊#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₃₇⋊#108

C₂

yes

maximal, characteristic

#14558543

C₂

yes

maximal, characteristic

#14558496

C₂

yes

maximal

#14558404

C₂

yes

maximal

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

D₃₇⋊M₁₆

C₂²

yes

—

#11924615

C₂²

yes

—

#11924607

C₄

yes

complemented

#11924606

C₄

yes

complemented

#11924605

C₄

yes

complemented

#11924604

C₄

yes

complemented

#11924590

—

C₈×#3866

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₄²×D₃₇

C₂²

yes

characteristic

C₄×#64436

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

characteristic, complemented

C₈×Dic₃₇

C₄

yes

characteristic, complemented

C₄×C₈×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

—

#10596957

—

C₄⋊#3866

C₂×C₄

yes

—

C₄×#3866

—

#10596953

C₂×C₄

yes

—

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

Q₈

yes

characteristic

C₈×D₃₇

Q₈

yes

C₂²×D₃₇

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

#41

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

characteristic

#64438

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

#64436

C₄²

yes

characteristic

#64436

C₄²

yes

characteristic

#64436

C₄⋊C₄

yes

characteristic

#64436

C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#117

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×D₄

yes

characteristic

C₂×D₃₇

C₄×Q₈

yes

characteristic

#3866

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

Dic₃₇

—

Dic₃₇

C₄×D₄

yes

characteristic

Dic₃₇

C₄×Q₈

yes

characteristic

#1772

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

D₃₇

—

D₃₇

#443

yes

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

#339

—

C₃₇

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

#3866

yes

characteristic, complemented

101

C₂²×C₄

—

102

C₂²×C₄

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂×#3866

yes

characteristic

117

C₂×C₈

C₂×#3866

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂×#3866

yes

characteristic

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂²×#3866

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

C₄×#3866

yes

characteristic

141

C₈

C₄×#3866

yes

characteristic

142

C₈

C₄⋊#3866

yes

characteristic

143

C₈

C₄⋊#3866

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11924615

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

148

—

153

C₄

C₂×C₄×#3866

yes

characteristic

154

C₄

C₂×C₄⋊#3866

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

C₄×C₄⋊#3866

yes

characteristic

	subgroup	G/H
center	C₈	C₄⋊#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄×C₈×C₃₇	C₄
socle	C₂×C₃₇	C₄×C₄⋊C₄
prime radical	C₂²×D₃₇	#41
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄×C₈	#3866
hypocenter	C₃₇	#1772
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄⋊#3866

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄×C₈×C₃₇

C₄

socle

C₂×C₃₇

C₄×C₄⋊C₄

prime radical

C₂²×D₃₇

#41

cosocle

C₂×C₄×D₃₇

C₂³

C₄×C₈

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. #11924607	C₄	semidirect
9. #11924606	C₄	semidirect
10. #11924605	C₄	semidirect
11. #11924604	C₄	semidirect
17. C₄×#64436	C₄	semidirect
18. C₈×Dic₃₇	C₄	semidirect
19. C₈×Dic₃₇	C₄	semidirect
20. C₄×C₈×C₃₇	C₄	semidirect
85. C₃₇	#1772	semidirect
100. C₄×C₈	#3866	semidirect

H ≅ G/N

type

8. #11924607

C₄

semidirect

9. #11924606

C₄

semidirect

10. #11924605

C₄

semidirect

11. #11924604

C₄

semidirect

17. C₄×#64436

C₄

semidirect

18. C₈×Dic₃₇

C₄

semidirect

19. C₈×Dic₃₇

C₄

semidirect

20. C₄×C₈×C₃₇

C₄

semidirect

85. C₃₇

#1772

semidirect

100. C₄×C₈

#3866

semidirect

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

subgroup	index
D₃₇⋊#108	2
#14558543	2
#14558496	2
#14558404	2
C₄×C₈×D₃₇	2
#1772	37

subgroup

index

D₃₇⋊#108

C₄×C₈×D₃₇

Subgroups — #18010398

Conjugacy classes of subgroups

Complements

Maximal subgroups