Group #16819323 — subgroups

Subgroups — #16819323

Order 3712.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₅₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745164	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₂₉	C₂	1	1	yes	maximal, characteristic
4	#14270926	C₂	1	2	yes	maximal
5	C₂×C₂₉×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
11	C₂×#90779	—	2	1	no	—
12	C₂×C₄×Dic₂₉	—	2	1	no	—
13	C₂×C₄×Dic₂₉	—	2	1	no	—
14	C₄×Dic₅₈	—	2	2	no	—
15	C₄×Dic₅₈	C₄	1	4	yes	complemented
16	C₂₉×#97	—	2	2	no	—
17	#11745179	—	2	1	no	—
18	#11745164	C₄	1	8	yes	complemented
19	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₉	—	2	1	no	—
23	C₂²×Dic₂₉	—	2	1	no	—
24	C₂²×Dic₂₉	—	2	2	no	—
25	C₂×Dic₅₈	—	2	1	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
28	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₉	—	2	2	no	—
30	C₂×C₈×C₂₉	—	2	2	no	—
31	C₂×C₈×C₂₉	C₂×C₄	1	4	yes	—
32	C₄²×C₂₉	C₂×C₄	1	2	yes	—
33	C₄²×C₂₉	D₄	1	2	yes	—
34	C₄⋊Dic₂₉	—	2	1	no	—
35	C₄⋊Dic₂₉	—	2	2	no	—
36	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
38	#90779	—	2	4	no	—
39	C₄×Dic₂₉	—	2	4	no	—
40	C₄×Dic₂₉	—	4	2	no	—
41	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	2	1	no	—
43	C₂×C₄×C₂₉	—	2	1	no	—
44	C₂×C₄×C₂₉	—	2	2	no	—
45	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	#41	1	2	yes	—
51	C₂×Dic₂₉	—	2	2	no	—
52	C₂×Dic₂₉	—	2	4	no	—
53	C₂×Dic₂₉	—	2	2	no	—
54	C₂×Dic₂₉	—	2	4	no	—
55	C₂×Dic₂₉	—	4	2	no	—
56	C₂×Dic₂₉	—	4	1	no	—
57	C₂×Dic₂₉	—	4	1	no	—
58	C₂×Dic₂₉	—	4	2	no	—
59	Dic₅₈	—	2	2	no	—
60	Dic₅₈	—	2	4	no	—
61	Dic₅₈	C₄²	1	4	yes	complemented
62	C₈×C₂₉	—	2	4	no	—
63	#1763	—	29	1	no	maximal, Sylow
64	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₂₉	C₄×D₄	1	2	yes	—
66	C₂²×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₂₉	#137	1	1	yes	characteristic
68	C₂²×C₂₉	#137	1	2	yes	—
69	C₄×C₂₉	—	2	2	no	—
70	C₄×C₂₉	—	2	2	no	—
71	C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₂₉	#117	1	1	yes	characteristic
73	C₄×C₂₉	C₄×D₄	1	2	yes	—
74	Dic₂₉	—	2	4	no	—
75	Dic₂₉	—	4	2	no	—
76	Dic₂₉	—	4	4	no	—
77	C₂×C₄×Q₈	—	29	1	no	—
78	C₂×#103	—	29	2	no	—
79	C₂×#97	D₂₉	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	29	1	no	—
81	#340	—	29	2	no	—
82	C₂×C₂₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₂₉	#417	1	1	yes	characteristic
84	C₂×C₂₉	#419	1	1	yes	characteristic
85	C₂×C₂₉	#440	1	2	yes	—
86	C₂×C₂₉	#441	1	2	yes	—
87	C₂²×Q₈	—	29	1	no	—
88	C₂²×C₈	C₂×D₂₉	1	2	yes	—
89	C₄×Q₈	—	29	4	no	—
90	C₄×Q₈	—	58	2	no	—
91	C₂×C₄⋊C₄	—	29	1	no	—
92	C₂×C₄⋊C₄	—	29	2	no	—
93	C₂×C₄⋊C₄	—	58	1	no	—
94	C₂×C₄²	—	58	1	no	—
95	C₂×C₄²	—	58	1	no	—
96	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
97	#103	—	29	8	no	—
98	#97	—	2	2	no	—
99	#95	—	58	1	no	—
100	C₂₉	#1763	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	29	4	no	—
102	C₂×Q₈	—	29	2	no	—
103	C₂×Q₈	—	58	2	no	—
104	C₂×Q₈	—	58	1	no	—
105	C₂²×C₄	—	58	1	no	—
106	C₂²×C₄	—	58	2	no	—
107	C₂²×C₄	—	58	1	no	—
108	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
109	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₂₉	1	4	yes	—
114	C₄⋊C₄	—	29	4	no	—
115	C₄⋊C₄	—	29	2	no	—
116	C₄⋊C₄	—	58	4	no	—
117	C₄⋊C₄	—	58	2	no	—
118	C₄⋊C₄	—	58	1	no	—
119	C₄²	—	58	4	no	—
120	C₄²	—	116	2	no	—
121	C₄²	C₄×D₂₉	1	2	yes	—
122	C₄²	C₂²⋊D₂₉	1	2	yes	—
123	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
124	Q₈	—	29	4	no	—
125	Q₈	—	58	4	no	—
126	Q₈	—	58	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	58	2	no	—
131	C₂×C₄	—	58	4	no	—
132	C₂×C₄	—	58	2	no	—
133	C₂×C₄	—	58	4	no	—
134	C₂×C₄	—	116	2	no	—
135	C₂×C₄	—	116	1	no	—
136	C₂×C₄	—	116	2	no	—
137	C₂×C₄	—	116	1	no	—
138	C₂×C₄	#90781	1	2	yes	—
139	C₂×C₄	#90781	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
141	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
142	C₂×C₄	#90805	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	C₄×D₁₁₆	1	2	yes	—
146	C₂²	#11745270	1	1	yes	characteristic
147	C₂²	#11745271	1	2	yes	—
148	C₂²	#11745271	1	1	yes	characteristic
149	C₂²	C₂×#90781	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	58	4	no	—
153	C₄	—	116	4	no	—
154	C₄	—	116	2	no	—
155	C₄	C₄²×D₂₉	1	1	yes	characteristic
156	C₄	#11745234	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₂₉	1	2	yes	—
158	C₂	C₄×#90781	1	1	yes	characteristic
159	C₂	#14271165	1	1	yes	characteristic
160	C₂	#14271173	1	1	yes	characteristic
161	C₂	#14271265	1	2	yes	—
162	C₂	#14271271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₈

C₂

yes

maximal, characteristic

C₂×#11745164

C₂

yes

maximal

C₄×C₄⋊Dic₂₉

C₂

yes

maximal, characteristic

#14270926

C₂

yes

maximal

C₂×C₂₉×#97

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

characteristic

C₂×#90779

—

C₂×C₄×Dic₂₉

—

C₂×C₄×Dic₂₉

—

C₄×Dic₅₈

—

C₄×Dic₅₈

C₄

yes

complemented

C₂₉×#97

—

#11745179

—

#11745164

C₄

yes

complemented

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

C₄²

yes

complemented

C₈×C₂₉

—

#1763

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×D₄

yes

—

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×C₄²

yes

characteristic

C₄×C₂₉

#117

yes

characteristic

C₄×C₂₉

C₄×D₄

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

D₂₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂×C₂₉

C₄×C₂²⋊C₄

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#441

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

—

—

—

100

C₂₉

#1763

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₁₁₆

yes

characteristic

109

C₂²×C₄

D₁₁₆

yes

characteristic

110

C₂²×C₄

C₂²×D₂₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₂₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

116

—

121

C₄²

C₄×D₂₉

yes

—

122

C₄²

C₂²⋊D₂₉

yes

—

123

C₂³

C₂×D₁₁₆

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

116

—

135

C₂×C₄

—

116

—

136

C₂×C₄

—

116

—

137

C₂×C₄

—

116

—

138

C₂×C₄

#90781

yes

—

139

C₂×C₄

#90781

yes

—

140

C₂×C₄

C₂×C₄×D₂₉

yes

—

141

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

142

C₂×C₄

#90805

yes

—

143

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

144

C₈

—

145

C₂²

C₄×D₁₁₆

yes

—

146

C₂²

#11745270

yes

characteristic

147

C₂²

#11745271

yes

—

148

C₂²

#11745271

yes

characteristic

149

C₂²

C₂×#90781

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

116

—

154

C₄

—

116

—

155

C₄

C₄²×D₂₉

yes

characteristic

156

C₄

#11745234

yes

characteristic

157

C₄

C₄×C₂²⋊D₂₉

yes

—

158

C₂

C₄×#90781

yes

characteristic

159

C₂

#14271165

yes

characteristic

160

C₂

#14271173

yes

characteristic

161

C₂

#14271265

yes

—

162

C₂

#14271271

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₆
commutator	C₄×C₂₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×#97	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×#97	D₂₉
hypocenter	C₂₉	#1763
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₆

commutator

C₄×C₂₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×#97

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×#97

D₂₉

hypocenter

C₂₉

#1763

LCS

G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
15. C₄×Dic₅₈	C₄	semidirect
18. #11745164	C₄	semidirect
61. Dic₅₈	C₄²	semidirect
100. C₂₉	#1763	semidirect

H ≅ G/N

type

15. C₄×Dic₅₈

C₄

semidirect

18. #11745164

C₄

semidirect

61. Dic₅₈

C₄²

semidirect

100. C₂₉

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₅₈	2
C₂×#11745164	2
C₄×C₄⋊Dic₂₉	2
#14270926	2
C₂×C₂₉×#97	2
#1763	29

subgroup

index

C₂×C₄×Dic₅₈

C₂×#11745164

C₄×C₄⋊Dic₂₉

#14270926

C₂×C₂₉×#97

#1763

Subgroups — #16819323

Conjugacy classes of subgroups

Complements

Maximal subgroups