Group #16818160 — subgroups

Subgroups — #16818160

Order 3712.

344 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 101 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	#14271216	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271187	C₂	1	1	yes	maximal, characteristic, complemented
3	#14270907	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₉×#127	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₂₉	—	2	1	no	—
6	C₂²×D₄×C₂₉	—	2	1	no	—
7	C₂²×D₄×C₂₉	C₄	1	2	yes	—
8	C₂₉×#127	—	2	1	no	—
9	C₂₉×#127	—	2	1	no	—
10	C₂₉×#127	C₄	1	2	yes	—
11	C₂₉×#127	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90777	—	2	1	no	—
14	C₂²⋊#6597	—	2	1	no	—
15	C₄⋊#6597	—	2	1	no	—
16	C₄⋊#6597	C₂²	1	2	yes	complemented
17	C₄⋊#6597	C₂²	1	2	yes	complemented
18	C₄×#6597	—	2	1	no	—
19	C₂⁴×C₂₉	—	2	2	no	—
20	C₂×D₄×C₂₉	—	2	2	no	—
21	C₂×D₄×C₂₉	—	2	2	no	—
22	C₂×D₄×C₂₉	—	2	2	no	—
23	C₂×D₄×C₂₉	—	2	2	no	—
24	C₂×D₄×C₂₉	—	2	2	no	—
25	C₂×D₄×C₂₉	—	4	4	no	—
26	C₂×D₄×C₂₉	—	4	4	no	—
27	C₂×D₄×C₂₉	—	4	2	no	—
28	C₂×D₄×C₂₉	—	4	2	no	—
29	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
33	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₉	D₄	1	1	yes	characteristic
35	C₄²×C₂₉	D₄	1	1	yes	characteristic
36	C₄²×C₂₉	C₂³	1	1	yes	characteristic
37	#90777	—	4	2	no	—
38	C₂×#6597	—	2	2	no	—
39	C₂×#6597	—	2	2	no	—
40	C₂³×C₂₉	—	2	4	no	—
41	C₂³×C₂₉	—	4	4	no	—
42	C₂³×C₂₉	—	4	4	no	—
43	C₂³×C₂₉	—	4	4	no	—
44	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
45	D₄×C₂₉	—	4	4	no	—
46	D₄×C₂₉	—	4	4	no	—
47	D₄×C₂₉	—	4	4	no	—
48	D₄×C₂₉	—	4	4	no	—
49	D₄×C₂₉	—	4	2	no	—
50	D₄×C₂₉	—	4	2	no	—
51	D₄×C₂₉	—	4	4	no	—
52	C₂×C₄×C₂₉	—	2	2	no	—
53	C₂×C₄×C₂₉	—	2	2	no	—
54	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	D₈	1	2	yes	—
58	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
59	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
62	#6597	—	4	2	no	—
63	#6597	—	4	2	no	—
64	#1545	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	—	2	1	no	—
66	C₂²×C₂₉	—	2	1	no	—
67	C₂²×C₂₉	—	4	4	no	—
68	C₂²×C₂₉	—	4	4	no	—
69	C₂²×C₂₉	—	4	4	no	—
70	C₂²×C₂₉	—	8	8	no	—
71	C₂²×C₂₉	#102	1	2	yes	—
72	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₂₉	—	2	1	no	—
74	C₄×C₂₉	—	2	1	no	—
75	C₄×C₂₉	#100	1	2	yes	—
76	C₄×C₂₉	#102	1	2	yes	—
77	C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
78	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×#127	D₂₉	1	1	yes	characteristic
81	#431	—	29	1	no	—
82	#423	—	29	1	no	—
83	#331	—	29	4	no	—
84	C₂×C₂₉	—	2	2	no	—
85	C₂×C₂₉	—	8	4	no	—
86	C₂×C₂₉	C₂×#100	1	1	yes	characteristic
87	C₂×C₂₉	C₂×#102	1	1	yes	characteristic
88	C₂×C₂₉	#418	1	1	yes	characteristic
89	C₂²×D₄	—	2	1	no	—
90	C₂²×D₄	—	2	1	no	—
91	C₂²×D₄	Dic₂₉	1	2	yes	—
92	C₂×M₁₆	—	58	1	no	—
93	#127	—	2	1	no	—
94	#127	—	2	1	no	—
95	#127	Dic₂₉	1	2	yes	—
96	#127	C₂×D₂₉	1	2	yes	—
97	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
98	C₄⋊C₈	—	29	2	no	—
99	C₄⋊C₈	—	29	2	no	—
100	C₄⋊C₈	—	58	1	no	—
101	C₂²⋊C₈	—	58	1	no	—
102	C₄×C₈	—	58	1	no	—
103	C₂₉	#1545	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	4	4	no	—
111	C₂×D₄	—	4	4	no	—
112	C₂×D₄	—	4	2	no	—
113	C₂×D₄	—	4	2	no	—
114	C₂×D₄	C₂×Dic₂₉	1	4	yes	—
115	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
118	M₁₆	—	116	2	no	—
119	C₂×C₈	—	58	2	no	—
120	C₂×C₈	—	58	2	no	—
121	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
122	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
123	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
124	C₄²	C₂²×D₂₉	1	1	yes	characteristic
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	—	4	4	no	—
128	C₂³	—	4	4	no	—
129	C₂³	C₂²⋊Dic₂₉	1	1	yes	characteristic
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	4	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	4	no	—
136	D₄	—	4	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	#90782	1	2	yes	—
140	C₂×C₄	#90783	1	2	yes	—
141	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
147	C₈	—	116	2	no	—
148	C₈	—	116	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	4	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	8	8	no	—
155	C₂²	#11745180	1	2	yes	—
156	C₂²	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#11745180	1	2	yes	—
160	C₄	#11745181	1	2	yes	—
161	C₄	C₂×#90782	1	1	yes	characteristic
162	C₄	#11745294	1	2	yes	—
163	C₄	C₂×#90783	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	4	no	—
166	C₂	C₂×#11745181	1	1	yes	characteristic
167	C₂	C₂×#11745180	1	1	yes	characteristic
168	C₂	#14271168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271216

C₂

yes

maximal, characteristic, complemented

#14271187

C₂

yes

maximal, characteristic, complemented

#14270907

C₂

yes

maximal, complemented

C₂×C₂₉×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₉

—

C₂²×D₄×C₂₉

—

C₂²×D₄×C₂₉

C₄

yes

—

C₂₉×#127

—

C₂₉×#127

—

C₂₉×#127

C₄

yes

—

C₂₉×#127

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×#90777

—

C₂²⋊#6597

—

C₄⋊#6597

—

C₄⋊#6597

C₂²

yes

complemented

C₄⋊#6597

C₂²

yes

complemented

C₄×#6597

—

C₂⁴×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

—

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×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×C₄×C₂₉

D₈

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#102

yes

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#100

yes

—

C₄×C₂₉

#102

yes

—

C₄×C₂₉

C₂×D₈

yes

characteristic

C₄×C₂₉

C₂×SD₁₆

yes

characteristic

C₄×C₂₉

AGL(1,ℤ/8)

yes

—

C₂×#127

D₂₉

yes

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

Dic₂₉

yes

—

C₂×M₁₆

—

#127

—

#127

—

#127

Dic₂₉

yes

—

#127

C₂×D₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₂₉

#1545

yes

characteristic, complemented, Sylow

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

C₂×Dic₂₉

yes

—

115

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

116

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

117

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

118

M₁₆

—

116

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄²

C₂×Dic₂₉

yes

characteristic

122

C₄²

C₂²⋊D₂₉

yes

characteristic

123

C₄²

C₂²⋊D₂₉

yes

characteristic

124

C₄²

C₂²×D₂₉

yes

characteristic

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²⋊Dic₂₉

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#90782

yes

—

140

C₂×C₄

#90783

yes

—

141

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

144

C₂×C₄

C₂²×Dic₂₉

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

147

C₈

—

116

—

148

C₈

—

116

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#11745180

yes

—

156

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

#11745180

yes

—

160

C₄

#11745181

yes

—

161

C₄

C₂×#90782

yes

characteristic

162

C₄

#11745294

yes

—

163

C₄

C₂×#90783

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

C₂×#11745181

yes

characteristic

167

C₂

C₂×#11745180

yes

characteristic

168

C₂

#14271168

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₉
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉
Fitting, prime radical	C₂×C₂₉×#127	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
cosocle	C₄²×C₂₉	C₂³
hypercenter	C₂×#127	D₂₉
hypocenter	C₂₉	#1545
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₉

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉

Fitting, prime radical

C₂×C₂₉×#127

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

cosocle

C₄²×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #14271216	C₂	semidirect
2. #14271187	C₂	semidirect
3. #14270907	C₂	semidirect
16. C₄⋊#6597	C₂²	semidirect
17. C₄⋊#6597	C₂²	semidirect
103. C₂₉	#1545	semidirect

H ≅ G/N

type

1. #14271216

C₂

semidirect

2. #14271187

C₂

semidirect

3. #14270907

C₂

semidirect

16. C₄⋊#6597

C₂²

semidirect

17. C₄⋊#6597

C₂²

semidirect

103. C₂₉

#1545

semidirect

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

subgroup	index
#14271216	2
#14271187	2
#14270907	2
C₂×C₂₉×#127	2
#1545	29

subgroup

index

C₂×C₂₉×#127

Subgroups — #16818160

Conjugacy classes of subgroups

Complements

Maximal subgroups