Group #16835359 — subgroups

Subgroups — #16835359

Order 3712.

272 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14272166	C₂	1	2	yes	maximal
2	C₂×#11745221	C₂	1	2	yes	maximal
3	#14272043	C₂	1	2	yes	maximal
4	C₂×#11745249	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
6	C₂×#90800	C₂²	1	4	yes	—
7	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
8	C₂×#90779	C₄	1	1	yes	characteristic
9	C₂×#90779	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₂₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
13	#11745249	—	2	2	no	—
14	#11745249	—	2	2	no	—
15	#11745221	—	2	4	no	—
16	#11745220	—	2	2	no	—
17	C₄⋊#6599	—	2	2	no	—
18	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	2	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	D₄	1	2	yes	—
28	C₂×Dic₅₈	Q₈	1	2	yes	—
29	#90800	—	2	4	no	—
30	#90800	—	2	4	no	—
31	C₂×#6599	—	2	4	no	—
32	C₄⋊C₄×C₂₉	—	2	2	no	—
33	#90779	—	2	2	no	—
34	#90779	—	2	2	no	—
35	C₄×Dic₂₉	—	2	1	no	—
36	C₄×Dic₂₉	—	2	1	no	—
37	C₄×Dic₂₉	—	2	1	no	—
38	C₄×Dic₂₉	—	2	1	no	—
39	C₄×Dic₂₉	D₄	1	2	yes	—
40	C₄×Dic₂₉	D₄	1	2	yes	—
41	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	2	1	no	—
43	C₂×C₄×C₂₉	—	2	1	no	—
44	C₂×C₄×C₂₉	—	2	1	no	—
45	C₂×C₄×C₂₉	—	4	1	no	—
46	C₂×C₄×C₂₉	—	4	1	no	—
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
49	C₂×Dic₂₉	—	2	1	no	—
50	C₂×Dic₂₉	—	2	1	no	—
51	C₂×Dic₂₉	—	2	2	no	—
52	C₂×Dic₂₉	—	2	1	no	—
53	C₂×Dic₂₉	—	4	1	no	—
54	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₂₉	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₂₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₂₉	#41	1	2	yes	—
61	Dic₅₈	—	2	4	no	—
62	Dic₅₈	—	2	4	no	—
63	#6599	—	4	4	no	—
64	#1801	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	—	2	1	no	—
66	C₂²×C₂₉	—	2	1	no	—
67	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₉	#117	1	1	yes	characteristic
70	C₄×C₂₉	—	2	2	no	—
71	C₄×C₂₉	—	2	2	no	—
72	C₄×C₂₉	—	4	1	no	—
73	Dic₂₉	—	2	2	no	—
74	Dic₂₉	—	2	1	no	—
75	Dic₂₉	—	4	1	no	—
76	Dic₂₉	#101	1	2	yes	complemented
77	Dic₂₉	#121	1	2	yes	—
78	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×Q₈	—	29	1	no	—
80	#423	—	29	2	no	—
81	C₂×#101	—	29	2	no	—
82	#343	—	29	2	no	—
83	C₂×C₂₉	—	2	1	no	—
84	C₂×C₂₉	—	2	1	no	—
85	C₂×C₂₉	#380	1	1	yes	characteristic
86	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
87	C₂×C₂₉	#413	1	1	yes	characteristic
88	C₂²×Q₈	—	29	1	no	—
89	C₂×M₁₆	—	29	4	no	—
90	C₄×Q₈	—	58	2	no	—
91	C₄×Q₈	—	58	2	no	—
92	C₂×C₄⋊C₄	—	29	1	no	—
93	C₂×C₄⋊C₄	—	29	1	no	—
94	C₂×C₄⋊C₄	#1181	1	1	yes	characteristic
95	C₂×C₄²	—	29	1	no	—
96	C₂×C₄²	—	29	1	no	—
97	C₂×C₄²	—	29	1	no	—
98	C₄⋊C₈	—	58	2	no	—
99	C₄⋊C₈	—	58	2	no	—
100	#101	—	58	4	no	—
101	C₂₉	#1801	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	29	2	no	—
103	C₂×Q₈	—	29	2	no	—
104	C₂×Q₈	—	58	2	no	—
105	C₂×Q₈	—	58	2	no	—
106	C₂²×C₄	—	29	1	no	—
107	C₂²×C₄	—	29	1	no	—
108	C₂²×C₄	—	29	1	no	—
109	C₂²×C₄	—	29	1	no	—
110	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
111	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
112	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
113	M₁₆	—	58	4	no	—
114	M₁₆	—	58	4	no	—
115	C₂×C₈	—	58	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	58	2	no	—
118	C₄⋊C₄	—	58	2	no	—
119	C₄²	—	29	2	no	—
120	C₄²	—	29	2	no	—
121	C₄²	—	58	1	no	—
122	C₄²	—	58	1	no	—
123	C₄²	—	58	1	no	—
124	C₄²	—	58	1	no	—
125	C₂³	C₂²×#1181	1	1	yes	characteristic
126	Q₈	—	58	4	no	—
127	Q₈	—	58	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	29	1	no	—
134	C₂×C₄	—	29	1	no	—
135	C₂×C₄	—	29	2	no	—
136	C₂×C₄	—	29	1	no	—
137	C₂×C₄	—	29	2	no	—
138	C₂×C₄	—	29	2	no	—
139	C₂×C₄	—	29	1	no	—
140	C₂×C₄	—	58	2	no	—
141	C₂×C₄	—	58	1	no	—
142	C₂×C₄	—	58	1	no	—
143	C₂×C₄	—	58	1	no	—
144	C₂×C₄	—	116	1	no	—
145	C₂×C₄	C₄⋊#1181	1	2	yes	—
146	C₂×C₄	C₂²⋊#1181	1	2	yes	—
147	C₈	—	116	4	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂×C₄⋊#1181	1	1	yes	characteristic
151	C₂²	#11745345	1	1	yes	characteristic
152	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	1	no	—
156	C₄	—	29	2	no	—
157	C₄	—	29	2	no	—
158	C₄	—	29	2	no	—
159	C₄	—	58	2	no	—
160	C₄	—	58	1	no	—
161	C₄	—	116	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#14272105	1	1	yes	characteristic
165	C₂	D₂₉⋊#101	1	1	yes	characteristic
166	C₂	#14272150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272166

C₂

yes

maximal

C₂×#11745221

C₂

yes

maximal

#14272043

C₂

yes

maximal

C₂×#11745249

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂×#90800

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

characteristic

C₂×#90779

C₄

yes

characteristic

C₂×#90779

C₂²

yes

characteristic

C₂×C₄×Dic₂₉

C₄

yes

characteristic

C₂×C₄×Dic₂₉

C₄

yes

characteristic

C₂×C₄×Dic₂₉

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

D₄

yes

—

C₂×Dic₅₈

Q₈

yes

—

#90800

—

#90800

—

C₂×#6599

—

C₄⋊C₄×C₂₉

—

#90779

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₄×Dic₂₉

D₄

yes

—

C₄×Dic₂₉

D₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₄⋊C₄

yes

—

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

—

C₂×Dic₂₉

C₂×Q₈

yes

characteristic

C₂×Dic₂₉

#41

yes

—

Dic₅₈

—

Dic₅₈

—

#6599

—

#1801

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₉

#117

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#101

yes

complemented

Dic₂₉

#121

yes

—

Dic₂₉

C₂²⋊Q₈

yes

—

C₂×C₄×Q₈

—

#423

—

C₂×#101

—

#343

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#1181

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#101

—

101

C₂₉

#1801

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×#1181

yes

characteristic

111

C₂²×C₄

C₂×#1181

yes

characteristic

112

C₂²×C₄

C₂×#1181

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₂³

C₂²×#1181

yes

characteristic

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

116

—

145

C₂×C₄

C₄⋊#1181

yes

—

146

C₂×C₄

C₂²⋊#1181

yes

—

147

C₈

—

116

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₄⋊#1181

yes

characteristic

151

C₂²

#11745345

yes

characteristic

152

C₂²

C₂×C₂²⋊#1181

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

116

—

162

C₂

—

163

C₂

—

164

C₂

#14272105

yes

characteristic

165

C₂

D₂₉⋊#101

yes

characteristic

166

C₂

#14272150

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator, prime radical	C₂³×C₂₉	C₂²×C₄
Frattini	C₂³	C₂²×#1181
Fitting	C₂×C₄⋊C₄×C₂₉	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₂₉	C₂³
hypercenter	C₂×C₄⋊C₄	#1181
hypocenter	C₂₉	#1801
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator, prime radical

C₂³×C₂₉

C₂²×C₄

Frattini

C₂³

C₂²×#1181

Fitting

C₂×C₄⋊C₄×C₂₉

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₂₉

C₂³

C₂×C₄⋊C₄

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
76. Dic₂₉	#101	semidirect
101. C₂₉	#1801	semidirect

H ≅ G/N

type

76. Dic₂₉

#101

semidirect

101. C₂₉

#1801

semidirect

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

subgroup	index
#14272166	2
C₂×#11745221	2
#14272043	2
C₂×#11745249	2
#1801	29

subgroup

index

Subgroups — #16835359

Conjugacy classes of subgroups

Complements

Maximal subgroups