Group #16835665 — subgroups

Subgroups — #16835665

Order 3712.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×C₄⋊#6599	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745218	C₂	1	1	yes	maximal, characteristic
3	#14272029	C₂	1	2	yes	maximal
4	#14272029	C₂	1	2	yes	maximal
5	C₂×#11745249	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₄	1	1	yes	characteristic
7	C₂²×#6599	C₂²	1	2	yes	—
8	C₂²×#6599	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
10	C₂×#90779	C₄	1	1	yes	characteristic
11	C₂×#90779	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
15	#11745249	—	2	2	no	—
16	#11745249	—	2	2	no	—
17	#11745218	—	2	2	no	—
18	C₄⋊#6599	—	2	2	no	—
19	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	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	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	—	2	2	no	—
28	C₂×Dic₅₈	—	2	2	no	—
29	C₂×#6599	—	2	2	no	—
30	C₂×#6599	—	2	2	no	—
31	C₂×#6599	—	2	2	no	—
32	C₂×#6599	—	2	4	no	—
33	C₂×#6599	Q₈	1	4	yes	complemented
34	C₄⋊C₄×C₂₉	—	2	2	no	—
35	#90779	—	2	2	no	—
36	#90779	—	2	2	no	—
37	C₄×Dic₂₉	—	2	1	no	—
38	C₄×Dic₂₉	—	2	1	no	—
39	C₄×Dic₂₉	—	2	1	no	—
40	C₄×Dic₂₉	—	2	1	no	—
41	C₄×Dic₂₉	D₄	1	2	yes	—
42	C₄×Dic₂₉	D₄	1	2	yes	—
43	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	—	2	1	no	—
45	C₂×C₄×C₂₉	—	2	1	no	—
46	C₂×C₄×C₂₉	—	2	2	no	—
47	C₂×C₄×C₂₉	—	2	1	no	—
48	C₂×C₄×C₂₉	—	2	2	no	—
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₉	M₁₆	1	2	yes	—
51	C₂×Dic₂₉	—	2	1	no	—
52	C₂×Dic₂₉	—	2	1	no	—
53	C₂×Dic₂₉	—	2	1	no	—
54	C₂×Dic₂₉	—	2	1	no	—
55	C₂×Dic₂₉	—	2	1	no	—
56	C₂×Dic₂₉	—	2	2	no	—
57	C₂×Dic₂₉	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₂₉	M₁₆	1	2	yes	—
59	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
62	C₂×Dic₂₉	#41	1	2	yes	—
63	Dic₅₈	—	2	8	no	—
64	#6599	—	2	4	no	—
65	#6599	—	4	2	no	—
66	#1985	—	29	1	no	maximal, Sylow
67	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
69	C₂²×C₂₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₂₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂²×C₂₉	#131	1	2	yes	—
72	C₄×C₂₉	—	2	2	no	—
73	C₄×C₂₉	—	2	2	no	—
74	C₄×C₂₉	—	4	1	no	—
75	Dic₂₉	—	2	2	no	—
76	Dic₂₉	—	2	2	no	—
77	Dic₂₉	—	4	1	no	—
78	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
79	Dic₂₉	#124	1	1	yes	characteristic
80	Dic₂₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×Q₈	—	29	1	no	—
82	C₂×C₄⋊C₈	—	29	1	no	—
83	C₂×#97	—	29	1	no	—
84	#336	—	29	2	no	—
85	#336	—	29	2	no	—
86	C₂×C₂₉	#391	1	1	yes	characteristic
87	C₂×C₂₉	#407	1	1	yes	characteristic
88	C₂×C₂₉	#408	1	1	yes	characteristic
89	C₂×C₂₉	#446	1	2	yes	—
90	C₂×C₂₉	#446	1	2	yes	—
91	C₂²×Q₈	—	29	1	no	—
92	C₂²×C₈	—	29	2	no	—
93	C₂²×C₈	—	29	2	no	—
94	C₄×Q₈	—	58	2	no	—
95	C₄×Q₈	—	58	2	no	—
96	C₂×C₄⋊C₄	—	29	1	no	—
97	C₂×C₄⋊C₄	—	29	1	no	—
98	C₂×C₄⋊C₄	#1181	1	1	yes	characteristic
99	C₂×C₄²	—	29	1	no	—
100	C₂×C₄²	—	29	1	no	—
101	C₂×C₄²	—	29	1	no	—
102	C₄⋊C₈	—	58	2	no	—
103	#97	—	58	2	no	—
104	C₂₉	#1985	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	58	2	no	—
106	C₂×Q₈	—	58	2	no	—
107	C₂×Q₈	—	58	2	no	—
108	C₂²×C₄	—	29	1	no	—
109	C₂²×C₄	—	29	1	no	—
110	C₂²×C₄	—	29	1	no	—
111	C₂²×C₄	—	29	1	no	—
112	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
113	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
114	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
115	C₂×C₈	—	29	4	no	—
116	C₂×C₈	—	58	2	no	—
117	C₂×C₈	—	58	2	no	—
118	C₂×C₈	—	58	2	no	—
119	C₂×C₈	—	58	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	58	2	no	—
122	C₄⋊C₄	—	58	2	no	—
123	C₄²	—	29	2	no	—
124	C₄²	—	29	2	no	—
125	C₄²	—	58	1	no	—
126	C₄²	—	58	1	no	—
127	C₄²	—	58	1	no	—
128	C₄²	—	58	1	no	—
129	C₂³	C₂²×#1181	1	1	yes	characteristic
130	Q₈	—	58	8	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	29	2	no	—
137	C₂×C₄	—	29	2	no	—
138	C₂×C₄	—	29	1	no	—
139	C₂×C₄	—	29	1	no	—
140	C₂×C₄	—	29	2	no	—
141	C₂×C₄	—	29	2	no	—
142	C₂×C₄	—	58	1	no	—
143	C₂×C₄	—	58	1	no	—
144	C₂×C₄	—	58	1	no	—
145	C₂×C₄	—	58	1	no	—
146	C₂×C₄	—	58	2	no	—
147	C₂×C₄	—	58	1	no	—
148	C₂×C₄	#90796	1	2	yes	—
149	C₂×C₄	C₂²⋊#1181	1	2	yes	—
150	C₈	—	58	4	no	—
151	C₈	—	116	2	no	—
152	C₂²	C₂×#90796	1	1	yes	characteristic
153	C₂²	D₂₉⋊M₁₆	1	1	yes	characteristic
154	C₂²	#11745346	1	2	yes	—
155	C₂²	Q₈×#1181	1	2	yes	—
156	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	29	2	no	—
161	C₄	—	29	1	no	—
162	C₄	—	29	1	no	—
163	C₄	—	58	2	no	—
164	C₄	—	58	2	no	—
165	C₄	—	116	1	no	—
166	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
167	C₂	#14272134	1	1	yes	characteristic
168	C₂	#14272136	1	1	yes	characteristic
169	C₂	#14272213	1	2	yes	—
170	C₂	#14272215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#6599

C₂

yes

maximal, characteristic

C₂×#11745218

C₂

yes

maximal, characteristic

#14272029

C₂

yes

maximal

#14272029

C₂

yes

maximal

C₂×#11745249

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₄

yes

characteristic

C₂²×#6599

C₂²

yes

—

C₂²×#6599

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₄

yes

characteristic

C₂×#90779

C₄

yes

characteristic

C₂×#90779

C₄

yes

characteristic

C₂×C₄×Dic₂₉

C₂²

yes

characteristic

C₂×C₄×Dic₂₉

C₂²

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

—

—

—

—

—

Q₈

yes

complemented

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

M₁₆

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

—

C₂×Dic₂₉

M₁₆

yes

—

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

C₂×Q₈

yes

—

C₂×Dic₂₉

#41

yes

—

Dic₅₈

—

—

—

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×Q₈

yes

—

C₂²×C₂₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₉

#131

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

C₂²⋊Q₈

yes

—

Dic₂₉

#124

yes

characteristic

Dic₂₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

#336

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#446

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#1181

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

#97

—

104

C₂₉

#1985

yes

characteristic, complemented, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×#1181

yes

characteristic

113

C₂²×C₄

C₂×#1181

yes

characteristic

114

C₂²×C₄

C₂×#1181

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂²×#1181

yes

characteristic

130

Q₈

—

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

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#90796

yes

—

149

C₂×C₄

C₂²⋊#1181

yes

—

150

C₈

—

151

C₈

—

116

—

152

C₂²

C₂×#90796

yes

characteristic

153

C₂²

D₂₉⋊M₁₆

yes

characteristic

154

C₂²

#11745346

yes

—

155

C₂²

Q₈×#1181

yes

—

156

C₂²

C₂×C₂²⋊#1181

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

116

—

166

C₂

(C₂×#1181)⋊Q₈

yes

characteristic

167

C₂

#14272134

yes

characteristic

168

C₂

#14272136

yes

characteristic

169

C₂

#14272213

yes

—

170

C₂

#14272215

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#1181
commutator, socle, prime radical	C₂³×C₂₉	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₂₉	C₄
cosocle	C₂²×Dic₂₉	C₂³
hypercenter	C₂×C₄⋊C₄	#1181
hypocenter	C₂₉	#1985
LCS	G ▷ C₂³×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#1181

commutator, socle, prime radical

C₂³×C₂₉

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₂₉

C₄

cosocle

C₂²×Dic₂₉

C₂³

C₂×C₄⋊C₄

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
33. C₂×#6599	Q₈	semidirect
104. C₂₉	#1985	semidirect

H ≅ G/N

type

33. C₂×#6599

Q₈

semidirect

104. C₂₉

#1985

semidirect

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

subgroup	index
C₂×C₄⋊#6599	2
C₂×#11745218	2
#14272029	2
C₂×#11745249	2
#1985	29

subgroup

index

C₂×C₄⋊#6599

Subgroups — #16835665

Conjugacy classes of subgroups

Complements

Maximal subgroups