Group #17995696 — subgroups

Subgroups — #17995696

Order 4736.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#10596942	C₂	1	2	yes	maximal
2	C₂×#11924460	C₂	1	2	yes	maximal
3	#14557439	C₂	1	1	yes	maximal, characteristic
4	#14557288	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
7	C₂×#10596942	—	2	4	no	—
8	C₂×#10596942	—	2	4	no	—
9	C₂×#10596942	—	2	4	no	—
10	C₂²×#64436	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
12	#11924460	—	2	4	no	—
13	#11924443	—	2	2	no	—
14	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₃₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₃₇	—	2	2	no	—
19	C₂×Q₈×C₃₇	—	2	2	no	—
20	C₂×Q₈×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	—	2	2	no	—
22	C₂×Q₈×C₃₇	—	2	2	no	—
23	C₂×Q₈×C₃₇	D₄	1	4	yes	—
24	C₂²×C₄×C₃₇	—	2	2	no	—
25	C₂²×C₄×C₃₇	D₄	1	2	yes	—
26	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃₇	—	2	1	no	—
28	C₂²×Dic₃₇	—	2	1	no	—
29	C₂×Dic₇₄	—	2	2	no	—
30	C₂×Dic₇₄	—	2	2	no	—
31	C₂×Dic₇₄	—	2	1	no	—
32	C₂×Dic₇₄	D₄	1	2	yes	—
33	C₄⋊C₄×C₃₇	—	2	8	no	—
34	C₄²×C₃₇	—	2	2	no	—
35	#10596942	—	2	16	no	—
36	C₄⋊Dic₃₇	—	2	2	no	—
37	C₂×#64436	—	2	2	no	—
38	C₂×#64436	—	2	2	no	—
39	C₂×#64436	D₄	1	4	yes	—
40	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₃₇	—	2	4	no	—
42	Q₈×C₃₇	—	2	4	no	—
43	Q₈×C₃₇	—	4	2	no	—
44	Q₈×C₃₇	—	4	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	2	4	no	—
48	C₂×C₄×C₃₇	—	4	2	no	—
49	C₂×C₄×C₃₇	—	4	2	no	—
50	C₂×C₄×C₃₇	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
55	C₂×Dic₃₇	—	2	2	no	—
56	C₂×Dic₃₇	—	4	1	no	—
57	C₂×Dic₃₇	—	4	1	no	—
58	C₂×Dic₃₇	—	4	1	no	—
59	C₂×Dic₃₇	—	4	2	no	—
60	Dic₇₄	—	2	4	no	—
61	Dic₇₄	—	4	2	no	—
62	#64436	—	2	4	no	—
63	C₂²×C₃₇	#120	1	1	yes	characteristic
64	C₂²×C₃₇	#121	1	1	yes	characteristic
65	C₂²×C₃₇	#127	1	1	yes	characteristic
66	C₂²×C₃₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₃₇	#137	1	2	yes	—
68	C₄×C₃₇	—	2	4	no	—
69	C₄×C₃₇	—	4	4	no	—
70	C₄×C₃₇	#120	1	2	yes	—
71	C₄×C₃₇	#121	1	2	yes	—
72	Dic₃₇	—	4	2	no	—
73	Dic₃₇	—	8	1	no	—
74	#2016	—	37	1	no	maximal, Sylow
75	C₂×C₃₇	#392	1	1	yes	characteristic
76	C₂×C₃₇	#451	1	2	yes	—
77	C₂×C₃₇	#462	1	2	yes	—
78	C₂×C₃₇	#494	1	1	yes	characteristic
79	C₂×C₃₇	#497	1	1	yes	characteristic
80	C₂²×Q₁₆	—	37	2	no	—
81	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic
82	C₂×#103	—	37	2	no	—
83	#391	—	37	1	no	—
84	#336	—	37	1	no	—
85	C₃₇	#2016	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	37	1	no	—
87	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
88	C₂×Q₁₆	—	74	4	no	—
89	C₂×Q₁₆	—	74	4	no	—
90	C₂×Q₁₆	—	74	4	no	—
91	C₂²×C₈	—	37	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	37	1	no	—
95	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
96	#103	—	74	4	no	—
97	#95	—	74	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	37	2	no	—
104	C₂×Q₈	—	74	1	no	—
105	C₂×Q₈	—	74	2	no	—
106	C₂×Q₈	—	74	2	no	—
107	C₂×Q₈	C₂²⋊D₃₇	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	74	1	no	—
110	C₂²×C₄	—	74	1	no	—
111	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
113	Q₁₆	—	74	16	no	—
114	C₂×C₈	—	37	4	no	—
115	C₂×C₈	—	74	2	no	—
116	C₂×C₈	—	74	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	74	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	74	4	no	—
126	Q₈	—	148	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	74	2	no	—
133	C₂×C₄	—	148	1	no	—
134	C₂×C₄	—	148	2	no	—
135	C₂×C₄	—	148	1	no	—
136	C₂×C₄	—	148	1	no	—
137	C₂×C₄	#10596942	1	4	yes	complemented
138	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₃₇	1	2	yes	—
140	C₂×C₄	#10596964	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
142	C₈	—	74	4	no	—
143	C₂²	#11924505	1	1	yes	characteristic
144	C₂²	#11924507	1	1	yes	characteristic
145	C₂²	#11924532	1	1	yes	characteristic
146	C₂²	C₂×#10596942	1	2	yes	—
147	C₂²	#11924567	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	296	1	no	—
151	C₄	—	148	2	no	—
152	C₄	#11924504	1	2	yes	—
153	C₄	#11924508	1	2	yes	—
154	C₂	#14557441	1	1	yes	characteristic
155	C₂	#14557689	1	2	yes	—
156	C₂	#14557740	1	2	yes	—
157	C₂	#14557912	1	1	yes	characteristic
158	C₂	#14557925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10596942

C₂

yes

maximal

C₂×#11924460

C₂

yes

maximal

#14557439

C₂

yes

maximal, characteristic

#14557288

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

—

—

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

#11924460

—

#11924443

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

—

#10596942

—

C₄⋊Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

Q₁₆

yes

complemented

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

#64436

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×Q₁₆

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

—

C₄×C₃₇

#121

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#462

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

D₃₇

yes

—

—

—

C₃₇

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₃₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₃₇

yes

—

112

C₂²×C₄

C₂²×D₃₇

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

148

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

—

148

—

135

C₂×C₄

—

148

—

136

C₂×C₄

—

148

—

137

C₂×C₄

#10596942

yes

complemented

138

C₂×C₄

D₄×D₃₇

yes

characteristic

139

C₂×C₄

D₄×D₃₇

yes

—

140

C₂×C₄

#10596964

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

142

C₈

—

143

C₂²

#11924505

yes

characteristic

144

C₂²

#11924507

yes

characteristic

145

C₂²

#11924532

yes

characteristic

146

C₂²

C₂×#10596942

yes

—

147

C₂²

#11924567

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

296

—

151

C₄

—

148

—

152

C₄

#11924504

yes

—

153

C₄

#11924508

yes

—

154

C₂

#14557441

yes

characteristic

155

C₂

#14557689

yes

—

156

C₂

#14557740

yes

—

157

C₂

#14557912

yes

characteristic

158

C₂

#14557925

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄⋊Q₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	#2016
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×C₄⋊Q₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₃₇

hypocenter

C₃₇

#2016

LCS

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
50. C₂×C₄×C₃₇	Q₁₆	semidirect
85. C₃₇	#2016	semidirect
137. C₂×C₄	#10596942	semidirect

H ≅ G/N

type

50. C₂×C₄×C₃₇

Q₁₆

semidirect

85. C₃₇

#2016

semidirect

137. C₂×C₄

#10596942

semidirect

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

subgroup	index
C₂²×#10596942	2
C₂×#11924460	2
#14557439	2
#14557288	2
C₂×C₄⋊Q₈×C₃₇	2
#2016	37

subgroup

index

C₂×C₄⋊Q₈×C₃₇

Subgroups — #17995696

Conjugacy classes of subgroups

Complements

Maximal subgroups