Group #16829881 — subgroups

Subgroups — #16829881

Order 3712.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11745237	C₂	1	1	yes	maximal, characteristic
2	#14271525	C₂	1	4	yes	maximal, complemented
3	#14271520	C₂	1	4	yes	maximal, complemented
4	#14271171	C₂	1	2	yes	maximal, complemented
5	#14271167	C₂	1	2	yes	maximal
6	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₁₆	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
10	C₂₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
11	C₂₉×#125	C₂²	1	2	yes	complemented
12	C₂₉×C₂²⋊Q₈	—	2	2	no	—
13	C₄×Q₈×C₂₉	—	2	2	no	—
14	C₂₉×#117	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
16	C₂×#90781	—	2	2	no	—
17	C₂×#90777	C₂²	1	2	yes	—
18	#11745237	—	2	2	no	—
19	#11745237	C₂²	1	2	yes	—
20	#11745237	C₂²	1	2	yes	—
21	#11745158	C₂²	1	8	yes	complemented
22	#11745157	C₂²	1	8	yes	—
23	#11745151	C₂²	1	2	yes	complemented
24	C₄×#6597	C₂²	1	2	yes	complemented
25	C₂³×D₂₉	—	2	1	no	—
26	C₂×Q₈×C₂₉	—	2	2	no	—
27	C₂²×C₄×C₂₉	D₄	1	2	yes	—
28	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₉	—	2	1	no	—
30	C₂×D₁₁₆	—	2	4	no	—
31	C₂×D₁₁₆	—	2	1	no	—
32	C₂×D₁₁₆	C₂³	1	2	yes	—
33	C₂×Dic₅₈	—	2	4	no	—
34	C₂×Dic₅₈	—	2	1	no	—
35	C₂×Dic₅₈	C₂³	1	2	yes	—
36	C₄⋊C₄×C₂₉	—	2	4	no	—
37	C₄⋊C₄×C₂₉	—	2	2	no	—
38	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₂₉	—	2	2	no	—
40	C₄²×C₂₉	—	2	2	no	—
41	C₄²×C₂₉	D₄	1	2	yes	—
42	C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₄²×C₂₉	C₂³	1	1	yes	characteristic
44	#90781	—	2	4	no	—
45	#90781	—	2	4	no	—
46	#90777	—	2	4	no	—
47	C₂×#6597	C₂³	1	4	yes	—
48	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₂₉	—	2	2	no	—
50	C₂²×D₂₉	—	4	4	no	—
51	C₂²×D₂₉	—	4	2	no	—
52	Q₈×C₂₉	—	4	4	no	—
53	C₂×C₄×C₂₉	—	2	4	no	—
54	C₂×C₄×C₂₉	—	4	2	no	—
55	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉	#41	1	4	yes	—
59	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₂₉	—	2	2	no	—
61	C₂×Dic₂₉	—	4	2	no	—
62	D₁₁₆	—	2	4	no	—
63	D₁₁₆	—	4	2	no	—
64	Dic₅₈	—	2	4	no	—
65	Dic₅₈	—	4	2	no	—
66	#6597	—	2	4	no	—
67	#3140	—	29	1	no	maximal, Sylow
68	C₂²×C₂₉	—	2	2	no	—
69	C₂²×C₂₉	#124	1	2	yes	—
70	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
71	C₂×D₂₉	—	4	2	no	—
72	C₂×D₂₉	—	4	4	no	—
73	C₂×D₂₉	—	8	4	no	—
74	C₄×C₂₉	—	2	2	no	—
75	C₄×C₂₉	—	2	2	no	—
76	C₄×C₂₉	—	4	4	no	—
77	C₄×C₂₉	#124	1	2	yes	—
78	C₄×C₂₉	C₂×#41	1	2	yes	—
79	Dic₂₉	—	4	2	no	—
80	#533	D₂₉	1	1	yes	characteristic, complemented
81	C₂×#124	—	29	1	no	—
82	#489	—	29	4	no	—
83	#488	—	29	4	no	—
84	#419	—	29	2	no	—
85	#418	—	29	2	no	—
86	C₄×M₁₆	—	29	1	no	—
87	C₂×C₂₉	—	2	2	no	—
88	C₂×C₂₉	C₂×#124	1	1	yes	characteristic
89	C₂×C₂₉	#575	1	2	yes	—
90	D₂₉	—	8	2	no	—
91	C₂²×Q₈	—	29	1	no	—
92	C₂²×D₄	—	29	1	no	—
93	C₂×M₁₆	—	29	2	no	—
94	C₄⋊Q₈	C₂×D₂₉	1	2	yes	complemented
95	#125	C₂×D₂₉	1	2	yes	complemented
96	#124	—	29	2	no	—
97	#124	—	29	2	no	—
98	#124	—	58	2	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	C₂×D₂₉	1	2	yes	—
102	C₂×C₂²⋊C₄	—	58	2	no	—
103	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
104	#103	—	29	8	no	—
105	#102	—	29	8	no	—
106	#97	—	29	2	no	—
107	C₄×C₈	—	29	2	no	—
108	C₂₉	#3140	1	1	yes	characteristic, complemented, Sylow
109	C₂⁴	—	58	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	29	2	no	—
112	C₂×Q₈	—	58	1	no	—
113	C₂×Q₈	—	58	4	no	—
114	C₂×D₄	—	29	2	no	—
115	C₂×D₄	—	58	4	no	—
116	C₂×D₄	—	58	1	no	—
117	C₂²×C₄	—	58	1	no	—
118	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
119	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
120	M₁₆	—	58	4	no	—
121	C₂×C₈	—	29	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₂₉	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	58	4	no	—
127	C₂²⋊C₄	—	58	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₂₉	1	2	yes	—
130	C₄²	C₂²×D₂₉	1	1	yes	characteristic
131	C₄²	C₂²×D₂₉	1	1	yes	characteristic
132	C₂³	—	58	2	no	—
133	C₂³	—	116	4	no	—
134	C₂³	—	116	2	no	—
135	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	58	4	no	—
138	Q₈	—	116	2	no	—
139	D₄	—	58	4	no	—
140	D₄	—	116	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	58	2	no	—
144	C₂×C₄	—	116	2	no	—
145	C₂×C₄	#90809	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
150	C₈	—	58	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	116	2	no	—
153	C₂²	—	116	4	no	—
154	C₂²	—	232	4	no	—
155	C₂²	#11745309	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	116	2	no	—
161	C₄	#11745309	1	2	yes	—
162	C₄	C₂×#90809	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	232	2	no	—
165	C₂	C₂×#11745309	1	1	yes	characteristic
166	C₂	#14271961	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745237

C₂

yes

maximal, characteristic

#14271525

C₂

yes

maximal, complemented

#14271520

C₂

yes

maximal, complemented

#14271171

C₂

yes

maximal, complemented

#14271167

C₂

yes

maximal

C₄×#90777

C₂

yes

maximal, characteristic, complemented

C₂₉×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₁₆

C₂²

yes

characteristic

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂₉×C₄⋊Q₈

C₂²

yes

complemented

C₂₉×#125

C₂²

yes

complemented

C₂₉×C₂²⋊Q₈

—

C₄×Q₈×C₂₉

—

C₂₉×#117

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×#90781

—

C₂×#90777

C₂²

yes

—

#11745237

—

#11745237

C₂²

yes

—

#11745237

C₂²

yes

—

#11745158

C₂²

yes

complemented

#11745157

C₂²

yes

—

#11745151

C₂²

yes

complemented

C₄×#6597

C₂²

yes

complemented

C₂³×D₂₉

—

C₂×Q₈×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₂×D₁₁₆

C₂³

yes

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂³

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂³

yes

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

—

—

C₂³

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂²×D₂₉

—

C₂²×D₂₉

—

C₂²×D₂₉

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

D₁₁₆

—

D₁₁₆

—

Dic₅₈

—

Dic₅₈

—

#6597

—

#3140

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

#124

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#124

yes

—

C₄×C₂₉

C₂×#41

yes

—

Dic₂₉

—

#533

D₂₉

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#124

yes

characteristic

C₂×C₂₉

#575

yes

—

D₂₉

—

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₉

yes

complemented

#125

C₂×D₂₉

yes

—

—

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

C₂×D₂₉

yes

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂×D₂₉

yes

characteristic

104

#103

—

105

#102

—

106

#97

—

107

C₄×C₈

—

108

C₂₉

#3140

yes

characteristic, complemented, Sylow

109

C₂⁴

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²⋊D₂₉

yes

—

119

C₂²×C₄

C₂²×D₂₉

yes

characteristic

120

M₁₆

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₂₉

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²⋊D₂₉

yes

—

130

C₄²

C₂²×D₂₉

yes

characteristic

131

C₄²

C₂²×D₂₉

yes

characteristic

132

C₂³

—

133

C₂³

—

116

—

134

C₂³

—

116

—

135

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

136

Q₈

—

137

Q₈

—

138

Q₈

—

116

—

139

D₄

—

140

D₄

—

116

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

116

—

145

C₂×C₄

#90809

yes

—

146

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

148

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

149

C₂×C₄

C₂³×D₂₉

yes

characteristic

150

C₈

—

151

C₂²

—

152

C₂²

—

116

—

153

C₂²

—

116

—

154

C₂²

—

232

—

155

C₂²

#11745309

yes

—

156

C₂²

C₂²×C₂²⋊D₂₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

116

—

161

C₄

#11745309

yes

—

162

C₄

C₂×#90809

yes

—

163

C₂

—

164

C₂

—

232

—

165

C₂

C₂×#11745309

yes

characteristic

166

C₂

#14271961

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#533	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂²×D₁₁₆	C₂²
hypercenter	#533	D₂₉
hypocenter	C₂₉	#3140
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₉

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂₉×#533

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂²×D₁₁₆

C₂²

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14271525	C₂	semidirect
3. #14271520	C₂	semidirect
4. #14271171	C₂	semidirect
6. C₄×#90777	C₂	semidirect
7. C₂₉×#533	C₂	semidirect
10. C₂₉×C₄⋊Q₈	C₂²	semidirect
11. C₂₉×#125	C₂²	semidirect
21. #11745158	C₂²	semidirect
23. #11745151	C₂²	semidirect
24. C₄×#6597	C₂²	semidirect
80. #533	D₂₉	semidirect
94. C₄⋊Q₈	C₂×D₂₉	semidirect
95. #125	C₂×D₂₉	semidirect
108. C₂₉	#3140	semidirect

H ≅ G/N

type

2. #14271525

C₂

semidirect

3. #14271520

C₂

semidirect

4. #14271171

C₂

semidirect

6. C₄×#90777

C₂

semidirect

7. C₂₉×#533

C₂

semidirect

10. C₂₉×C₄⋊Q₈

C₂²

semidirect

11. C₂₉×#125

C₂²

semidirect

21. #11745158

C₂²

semidirect

23. #11745151

C₂²

semidirect

24. C₄×#6597

C₂²

semidirect

80. #533

D₂₉

semidirect

94. C₄⋊Q₈

C₂×D₂₉

semidirect

95. #125

C₂×D₂₉

semidirect

108. C₂₉

#3140

semidirect

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

subgroup	index
C₂×#11745237	2
#14271525	2
#14271520	2
#14271171	2
#14271167	2
C₄×#90777	2
C₂₉×#533	2
#3140	29

subgroup

index

C₂₉×#533

Subgroups — #16829881

Conjugacy classes of subgroups

Complements

Maximal subgroups