Group #16835635 — subgroups

Subgroups — #16835635

Order 3712.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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	1	yes	maximal, characteristic
2	C₂×#11745221	C₂	1	4	yes	maximal
3	C₄×#90800	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745250	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₂²	1	2	yes	—
6	C₂×#90800	C₂²	1	2	yes	—
7	C₂×#90800	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
10	C₂×#90779	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
12	#11745250	—	2	1	no	—
13	#11745250	—	2	1	no	—
14	#11745250	—	2	1	no	—
15	#11745250	—	2	1	no	—
16	#11745221	—	2	8	no	—
17	#11745220	—	2	1	no	—
18	#11745218	—	2	1	no	—
19	C₄⋊#6599	—	2	1	no	—
20	C₄×#6599	—	2	1	no	—
21	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₅₈	—	2	4	no	—
27	C₂×Dic₅₈	—	2	2	no	—
28	C₂×Dic₅₈	—	2	2	no	—
29	C₂×Dic₅₈	—	2	4	no	—
30	#90800	—	2	2	no	—
31	#90800	—	2	4	no	—
32	#90800	D₄	1	4	yes	—
33	C₂×#6599	—	2	2	no	—
34	C₂×#6599	—	2	2	no	—
35	C₄⋊C₄×C₂₉	—	2	1	no	—
36	C₄⋊C₄×C₂₉	—	2	1	no	—
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	—	2	1	no	—
39	#90779	—	2	2	no	—
40	#90779	—	2	2	no	—
41	C₄×Dic₂₉	D₄	1	1	yes	characteristic
42	C₄×Dic₂₉	D₄	1	1	yes	characteristic
43	C₄×Dic₂₉	D₄	1	1	yes	characteristic
44	C₄×Dic₂₉	D₄	1	1	yes	characteristic
45	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	—	2	2	no	—
47	C₂×C₄×C₂₉	—	2	2	no	—
48	C₂×C₄×C₂₉	—	2	4	no	—
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₂₉	—	2	2	no	—
52	C₂×Dic₂₉	—	2	2	no	—
53	C₂×Dic₂₉	—	2	4	no	—
54	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
58	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₂₉	#41	1	2	yes	—
60	Dic₅₈	—	4	4	no	—
61	Dic₅₈	—	4	2	no	—
62	Dic₅₈	—	4	2	no	—
63	#6599	—	2	4	no	—
64	#6599	—	4	2	no	—
65	#1966	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	—	2	1	no	—
67	C₂²×C₂₉	—	2	1	no	—
68	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₉	C₄×D₄	1	2	yes	—
70	C₄×C₂₉	—	2	1	no	—
71	C₄×C₂₉	—	4	2	no	—
72	C₄×C₂₉	#101	1	2	yes	complemented
73	Dic₂₉	—	2	1	no	—
74	Dic₂₉	—	4	2	no	—
75	Dic₂₉	#101	1	2	yes	complemented
76	Dic₂₉	#121	1	2	yes	—
77	Dic₂₉	#124	1	1	yes	characteristic
78	Dic₂₉	#127	1	1	yes	characteristic
79	C₂×C₄⋊Q₈	—	29	1	no	—
80	#423	—	29	1	no	—
81	C₂×#101	—	29	4	no	—
82	C₄×M₁₆	—	29	1	no	—
83	C₂×C₂₉	—	2	2	no	—
84	C₂×C₂₉	#390	1	1	yes	characteristic
85	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
86	C₂×C₂₉	C₂×#101	1	1	yes	characteristic
87	C₂²×Q₈	—	29	2	no	—
88	C₂×M₁₆	—	29	2	no	—
89	C₂×M₁₆	—	29	2	no	—
90	C₄⋊Q₈	—	58	1	no	—
91	C₄⋊Q₈	—	58	1	no	—
92	C₄⋊Q₈	—	58	1	no	—
93	C₄⋊Q₈	—	58	1	no	—
94	C₂×C₄⋊C₄	—	29	2	no	—
95	C₂×C₄⋊C₄	—	29	1	no	—
96	C₂×C₄⋊C₄	#1181	1	1	yes	characteristic
97	C₂×C₄²	—	29	1	no	—
98	C₄⋊C₈	—	58	1	no	—
99	C₄⋊C₈	—	58	1	no	—
100	#101	—	58	8	no	—
101	#97	—	58	1	no	—
102	C₄×C₈	—	58	1	no	—
103	C₂₉	#1966	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	58	4	no	—
105	C₂×Q₈	—	58	2	no	—
106	C₂×Q₈	—	58	2	no	—
107	C₂×Q₈	—	58	4	no	—
108	C₂²×C₄	—	29	1	no	—
109	C₂²×C₄	—	29	1	no	—
110	C₂²×C₄	—	29	2	no	—
111	C₂²×C₄	C₂×#1181	1	2	yes	—
112	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
113	M₁₆	—	29	4	no	—
114	M₁₆	—	58	2	no	—
115	M₁₆	—	58	4	no	—
116	C₂×C₈	—	58	2	no	—
117	C₂×C₈	—	58	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	58	2	no	—
121	C₄⋊C₄	—	58	2	no	—
122	C₄⋊C₄	—	58	1	no	—
123	C₄⋊C₄	—	58	1	no	—
124	C₄²	—	29	1	no	—
125	C₄²	—	29	1	no	—
126	C₄²	—	29	1	no	—
127	C₄²	—	29	1	no	—
128	C₂³	C₂²×#1181	1	1	yes	characteristic
129	Q₈	—	116	4	no	—
130	Q₈	—	116	2	no	—
131	Q₈	—	116	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	29	2	no	—
136	C₂×C₄	—	29	2	no	—
137	C₂×C₄	—	29	1	no	—
138	C₂×C₄	—	29	1	no	—
139	C₂×C₄	—	29	1	no	—
140	C₂×C₄	—	29	1	no	—
141	C₂×C₄	—	58	2	no	—
142	C₂×C₄	—	58	4	no	—
143	C₂×C₄	—	58	2	no	—
144	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
146	C₈	—	58	4	no	—
147	C₈	—	116	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	D₄×#1181	1	2	yes	—
151	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	29	2	no	—
155	C₄	—	29	1	no	—
156	C₄	—	29	1	no	—
157	C₄	—	29	2	no	—
158	C₄	—	58	1	no	—
159	C₄	—	116	2	no	—
160	C₄	#11745221	1	2	yes	complemented
161	C₂	—	2	2	no	—
162	C₂	#14272115	1	1	yes	characteristic
163	C₂	C₂×#11745221	1	1	yes	characteristic
164	C₂	D₂₉⋊#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14272166

C₂

yes

maximal, characteristic

C₂×#11745221

C₂

yes

maximal

C₄×#90800

C₂

yes

maximal, characteristic

C₂×#11745250

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂×#90800

C₂²

yes

—

C₂×#90800

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

characteristic

C₂×C₄⋊Dic₂₉

C₄

yes

characteristic

C₂×#90779

C₄

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂²×Dic₂₉

C₂×C₄

yes

characteristic

C₂²×Dic₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

#90800

—

#90800

—

#90800

D₄

yes

—

C₂×#6599

—

C₂×#6599

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

#90779

—

#90779

—

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

—

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

#41

yes

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#101

yes

complemented

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#101

yes

complemented

Dic₂₉

#121

yes

—

Dic₂₉

yes

Dic₂₉

yes

C₂×C₄⋊Q₈

—

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#1181

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#101

—

101

#97

—

102

C₄×C₈

—

103

C₂₉

#1966

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×#1181

yes

—

112

C₂²×C₄

C₂×#1181

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂³

C₂²×#1181

yes

characteristic

129

Q₈

—

116

—

130

Q₈

—

116

—

131

Q₈

—

116

—

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

C₂²⋊#1181

yes

characteristic

145

C₂×C₄

C₂²⋊#1181

yes

characteristic

146

C₈

—

147

C₈

—

116

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄×#1181

yes

—

151

C₂²

C₂×C₂²⋊#1181

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

116

—

160

C₄

#11745221

yes

complemented

161

C₂

—

162

C₂

#14272115

yes

characteristic

163

C₂

C₂×#11745221

yes

characteristic

164

C₂

D₂₉⋊#101

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	#1966
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
72. C₄×C₂₉	#101	semidirect
75. Dic₂₉	#101	semidirect
103. C₂₉	#1966	semidirect
160. C₄	#11745221	semidirect

H ≅ G/N

type

72. C₄×C₂₉

#101

semidirect

75. Dic₂₉

#101

semidirect

103. C₂₉

#1966

semidirect

160. C₄

#11745221

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
C₄×#90800	2
C₂×#11745250	2
#1966	29

subgroup

index

Subgroups — #16835635

Conjugacy classes of subgroups

Complements

Maximal subgroups