Group #16835670 — subgroups

Subgroups — #16835670

Order 3712.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	D₂₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1181	C₂	1	1	yes	maximal, characteristic
3	#14272036	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₂₉	—	2	1	no	—
6	C₂×C₄⋊#1181	C₂²	1	2	yes	—
7	C₂×C₄×#1181	—	2	1	no	—
8	C₂×D₂₉⋊C₈	C₂²	1	2	yes	—
9	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	#11745306	—	2	1	no	—
11	C₄⋊C₄×D₂₉	—	2	1	no	—
12	C₄⋊C₄×D₂₉	C₂²	1	2	yes	—
13	#11745250	—	2	1	no	—
14	#11745250	C₄	1	2	yes	complemented
15	C₄²×D₂₉	C₂²	1	1	yes	characteristic
16	#11745231	C₄	1	1	yes	characteristic, complemented
17	(C₂×#1181)⋊C₄	—	2	1	no	—
18	#11745220	—	2	1	no	—
19	C₄⋊#6599	—	2	1	no	—
20	C₂²×#1181	—	2	2	no	—
21	C₂×Q₈×C₂₉	—	2	1	no	—
22	Q₈×D₂₉	—	4	4	no	—
23	C₂×C₄×D₂₉	—	2	2	no	—
24	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₅₈	—	2	1	no	—
28	C₂×Dic₅₈	—	2	2	no	—
29	C₂×#6599	—	2	2	no	—
30	C₄⋊#1181	—	2	2	no	—
31	C₄⋊#1181	Q₈	1	4	yes	complemented
32	C₄×#1181	—	4	2	no	—
33	D₂₉⋊C₈	—	2	4	no	—
34	C₄⋊C₄×C₂₉	—	2	1	no	—
35	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
36	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
39	#90779	—	2	2	no	—
40	#90779	—	2	2	no	—
41	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₂₉	D₄	1	1	yes	characteristic
43	C₄×Dic₂₉	D₄	1	1	yes	characteristic
44	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
45	C₂×#1181	—	2	4	no	—
46	C₂×#1181	—	4	2	no	—
47	C₂×#1181	—	4	2	no	—
48	Q₈×C₂₉	—	4	2	no	—
49	C₂×C₄×C₂₉	—	2	2	no	—
50	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₂₉	—	2	2	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	C₄×D₂₉	—	2	2	no	—
58	C₄×D₂₉	—	4	4	no	—
59	C₄×D₂₉	SD₁₆	1	2	yes	—
60	C₄×D₂₉	Q₁₆	1	2	yes	—
61	C₄×D₂₉	C₂×Q₈	1	2	yes	—
62	C₄×D₂₉	#41	1	2	yes	—
63	Dic₅₈	—	4	4	no	—
64	Dic₅₈	—	4	2	no	—
65	#6599	—	4	2	no	—
66	#1987	—	29	1	no	maximal, Sylow
67	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₂₉	C₂²⋊Q₈	1	2	yes	—
69	C₂×D₂₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×D₂₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×D₂₉	AGL(1,ℤ/8)	1	2	yes	—
72	#1181	—	4	4	no	—
73	C₄×C₂₉	—	2	1	no	—
74	C₄×C₂₉	—	4	2	no	—
75	C₄×C₂₉	#103	1	2	yes	—
76	C₄×C₂₉	C₄×Q₈	1	2	yes	—
77	Dic₂₉	—	2	1	no	—
78	Dic₂₉	—	4	2	no	—
79	Dic₂₉	#103	1	2	yes	—
80	Dic₂₉	#124	1	1	yes	characteristic
81	Dic₂₉	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄⋊Q₈	—	29	1	no	—
83	C₂×C₄⋊C₈	—	29	1	no	—
84	C₄×C₄⋊C₄	—	29	1	no	—
85	#340	—	29	4	no	—
86	C₂×C₂₉	#391	1	1	yes	characteristic
87	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
88	C₂×C₂₉	#418	1	1	yes	characteristic
89	D₂₉	#476	1	2	yes	—
90	D₂₉	#477	1	2	yes	—
91	C₂²×Q₈	—	58	1	no	—
92	C₂²×C₈	—	29	2	no	—
93	C₄⋊Q₈	—	29	2	no	—
94	C₄⋊Q₈	—	29	1	no	—
95	C₄⋊Q₈	—	58	1	no	—
96	C₄⋊Q₈	—	58	1	no	—
97	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
98	C₂×C₄⋊C₄	—	29	2	no	—
99	C₂×C₄⋊C₄	—	29	2	no	—
100	C₂×C₄⋊C₄	—	58	1	no	—
101	C₂×C₄²	—	29	1	no	—
102	C₂×C₄²	—	58	1	no	—
103	C₄⋊C₈	—	58	1	no	—
104	C₄⋊C₈	—	58	1	no	—
105	#95	—	58	1	no	—
106	C₂₉	#1987	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	58	2	no	—
109	C₂×Q₈	—	58	1	no	—
110	C₂×Q₈	—	116	4	no	—
111	C₂²×C₄	—	29	1	no	—
112	C₂²×C₄	—	29	1	no	—
113	C₂²×C₄	—	29	1	no	—
114	C₂²×C₄	—	58	2	no	—
115	C₂²×C₄	—	58	2	no	—
116	C₂×C₈	—	58	2	no	—
117	C₂×C₈	—	58	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	29	4	no	—
120	C₄⋊C₄	—	29	2	no	—
121	C₄⋊C₄	—	58	2	no	—
122	C₄⋊C₄	—	58	2	no	—
123	C₄⋊C₄	—	58	2	no	—
124	C₄⋊C₄	—	58	1	no	—
125	C₄⋊C₄	C₂×#1181	1	2	yes	—
126	C₄²	—	29	1	no	—
127	C₄²	—	29	1	no	—
128	C₄²	—	29	1	no	—
129	C₄²	—	116	2	no	—
130	C₄²	C₂×#1181	1	1	yes	characteristic
131	C₂³	—	29	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	116	2	no	—
134	Q₈	—	116	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	29	1	no	—
137	C₂×C₄	—	29	1	no	—
138	C₂×C₄	—	29	2	no	—
139	C₂×C₄	—	29	2	no	—
140	C₂×C₄	—	29	2	no	—
141	C₂×C₄	—	29	2	no	—
142	C₂×C₄	—	29	1	no	—
143	C₂×C₄	—	58	2	no	—
144	C₂×C₄	—	58	4	no	—
145	C₂×C₄	—	58	2	no	—
146	C₂×C₄	—	116	4	no	—
147	C₂×C₄	—	116	2	no	—
148	C₂×C₄	—	116	2	no	—
149	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
151	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
152	C₈	—	116	2	no	—
153	C₂²	—	29	2	no	—
154	C₂²	—	29	1	no	—
155	C₂²	—	29	1	no	—
156	C₂²	—	29	2	no	—
157	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	29	1	no	—
161	C₄	—	29	1	no	—
162	C₄	—	29	2	no	—
163	C₄	—	58	1	no	—
164	C₄	—	116	4	no	—
165	C₄	—	116	2	no	—
166	C₄	#11745225	1	2	yes	—
167	C₄	Q₈×#1181	1	2	yes	—
168	C₂	—	29	2	no	—
169	C₂	—	29	2	no	—
170	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
171	C₂	C₂×#11745225	1	1	yes	characteristic
172	C₂	#14272155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₂₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊#1181

C₂

yes

maximal, characteristic

#14272036

C₂

yes

maximal

C₄⋊Q₈×D₂₉

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₉

—

C₂×C₄⋊#1181

C₂²

yes

—

C₂×C₄×#1181

—

C₂×D₂₉⋊C₈

C₂²

yes

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic, complemented

#11745306

—

C₄⋊C₄×D₂₉

—

C₄⋊C₄×D₂₉

C₂²

yes

—

#11745250

—

#11745250

C₄

yes

complemented

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic, complemented

(C₂×#1181)⋊C₄

—

#11745220

—

C₄⋊#6599

—

C₂²×#1181

—

C₂×Q₈×C₂₉

—

Q₈×D₂₉

—

C₂×C₄×D₂₉

—

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

characteristic

C₂×C₄×D₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×#6599

—

C₄⋊#1181

—

C₄⋊#1181

Q₈

yes

complemented

C₄×#1181

—

D₂₉⋊C₈

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

#90779

—

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₄×Dic₂₉

D₄

yes

characteristic

C₂²×D₂₉

C₂×D₄

yes

—

—

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

characteristic

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

SD₁₆

yes

—

C₄×D₂₉

Q₁₆

yes

—

C₄×D₂₉

C₂×Q₈

yes

—

C₄×D₂₉

#41

yes

—

Dic₅₈

—

Dic₅₈

—

#6599

—

#1987

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

C₂²⋊Q₈

yes

—

C₂×D₂₉

C₂×SD₁₆

yes

characteristic

C₂×D₂₉

C₂×Q₁₆

yes

characteristic

C₂×D₂₉

AGL(1,ℤ/8)

yes

—

#1181

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#103

yes

—

C₄×C₂₉

C₄×Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#103

yes

—

Dic₂₉

#124

yes

characteristic

Dic₂₉

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

D₂₉

yes

—

D₂₉

#477

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#95

—

106

C₂₉

#1987

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

116

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

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

C₄⋊C₄

—

125

C₄⋊C₄

C₂×#1181

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

116

—

130

C₄²

C₂×#1181

yes

characteristic

131

C₂³

—

132

Q₈

—

133

Q₈

—

116

—

134

Q₈

—

116

—

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

—

116

—

147

C₂×C₄

—

116

—

148

C₂×C₄

—

116

—

149

C₂×C₄

C₂²⋊#1181

yes

characteristic

150

C₂×C₄

C₂²⋊#1181

yes

characteristic

151

C₂×C₄

C₂²×#1181

yes

characteristic

152

C₈

—

116

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#1181

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

116

—

165

C₄

—

116

—

166

C₄

#11745225

yes

—

167

C₄

Q₈×#1181

yes

—

168

C₂

—

169

C₂

—

170

C₂

(C₂×#1181)⋊Q₈

yes

characteristic

171

C₂

C₂×#11745225

yes

characteristic

172

C₂

#14272155

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉	C₂×D₄
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1987
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂₉×C₄⋊Q₈

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉

C₂×D₄

cosocle

C₂×C₄×D₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
9. C₂₉×C₄⋊Q₈	C₄	semidirect
14. #11745250	C₄	semidirect
16. #11745231	C₄	semidirect
31. C₄⋊#1181	Q₈	semidirect
97. C₄⋊Q₈	#1181	semidirect
106. C₂₉	#1987	semidirect

H ≅ G/N

type

9. C₂₉×C₄⋊Q₈

C₄

semidirect

14. #11745250

C₄

semidirect

16. #11745231

C₄

semidirect

31. C₄⋊#1181

Q₈

semidirect

97. C₄⋊Q₈

#1181

semidirect

106. C₂₉

#1987

semidirect

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

subgroup	index
D₂₉⋊(C₄⋊C₈)	2
C₄×C₄⋊#1181	2
#14272036	2
C₄⋊Q₈×D₂₉	2
#1987	29

subgroup

index

D₂₉⋊(C₄⋊C₈)

C₄×C₄⋊#1181

#14272036

C₄⋊Q₈×D₂₉

#1987

Subgroups — #16835670

Conjugacy classes of subgroups

Complements

Maximal subgroups