Group #16820353 — subgroups

Subgroups — #16820353

Order 3712.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745166	C₂	1	1	yes	maximal, characteristic
3	C₂×#11745164	C₂	1	2	yes	maximal
4	#14271069	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	C₂×Dic₁₁₆	—	2	2	no	—
12	C₂×Dic₁₁₆	—	2	2	no	—
13	C₂×Dic₁₁₆	C₄	1	4	yes	complemented
14	C₄×C₈×C₂₉	—	2	2	no	—
15	#11745179	—	2	4	no	—
16	#11745166	—	2	2	no	—
17	#11745164	—	2	4	no	—
18	C₂²×C₄×C₂₉	D₄	1	2	yes	—
19	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₉	—	2	2	no	—
21	C₂²×Dic₂₉	—	2	2	no	—
22	C₂×Dic₅₈	—	2	2	no	—
23	C₂×Dic₅₈	—	2	2	no	—
24	C₂×Dic₅₈	—	2	2	no	—
25	C₂×Dic₅₈	—	2	2	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₉	—	2	1	no	—
29	C₂×C₈×C₂₉	—	2	1	no	—
30	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₉	D₄	1	2	yes	—
32	C₂×C₈×C₂₉	D₄	1	2	yes	—
33	C₂×C₈×C₂₉	D₄	1	2	yes	—
34	C₄²×C₂₉	—	2	1	no	—
35	C₄²×C₂₉	—	2	1	no	—
36	C₄⋊Dic₂₉	—	2	2	no	—
37	C₄⋊Dic₂₉	—	2	2	no	—
38	Dic₁₁₆	—	2	8	no	—
39	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₉	—	2	2	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
49	C₂×Dic₂₉	—	2	4	no	—
50	C₂×Dic₂₉	—	4	2	no	—
51	C₂×Dic₂₉	—	4	2	no	—
52	C₂×Dic₂₉	—	4	2	no	—
53	C₂×Dic₂₉	—	4	4	no	—
54	Dic₅₈	—	2	4	no	—
55	Dic₅₈	—	2	4	no	—
56	Dic₅₈	—	4	2	no	—
57	Dic₅₈	—	4	2	no	—
58	C₈×C₂₉	—	2	2	no	—
59	C₈×C₂₉	C₂²⋊C₄	1	4	yes	—
60	#1969	—	29	1	no	maximal, Sylow
61	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₂₉	#121	1	1	yes	characteristic
63	C₂²×C₂₉	#127	1	1	yes	characteristic
64	C₂²×C₂₉	C₂×Q₁₆	1	2	yes	—
65	C₂²×C₂₉	#135	1	2	yes	—
66	C₄×C₂₉	—	2	4	no	—
67	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
69	C₄×C₂₉	#121	1	1	yes	characteristic
70	C₄×C₂₉	#124	1	1	yes	characteristic
71	Dic₂₉	—	4	4	no	—
72	Dic₂₉	—	8	2	no	—
73	C₂²×Q₁₆	—	29	1	no	—
74	C₂×#107	—	29	1	no	—
75	C₂×#103	—	29	2	no	—
76	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
77	#391	—	29	2	no	—
78	C₂×C₂₉	#390	1	1	yes	characteristic
79	C₂×C₂₉	C₄×Q₁₆	1	2	yes	—
80	C₂×C₂₉	#467	1	2	yes	—
81	C₂×C₂₉	#494	1	1	yes	characteristic
82	C₂×C₂₉	#495	1	1	yes	characteristic
83	C₂²×Q₈	—	29	2	no	—
84	C₂×Q₁₆	—	29	4	no	—
85	C₂×Q₁₆	—	58	2	no	—
86	C₂×Q₁₆	—	58	2	no	—
87	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	29	2	no	—
90	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
91	#107	—	58	2	no	—
92	#103	—	58	4	no	—
93	C₄×C₈	—	2	2	no	—
94	#95	—	58	4	no	—
95	C₂₉	#1969	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	29	4	no	—
97	C₂×Q₈	—	58	2	no	—
98	C₂×Q₈	—	58	2	no	—
99	C₂×Q₈	—	58	2	no	—
100	C₂×Q₈	—	58	2	no	—
101	C₂×Q₈	—	58	2	no	—
102	C₂²×C₄	—	58	2	no	—
103	C₂²×C₄	—	58	2	no	—
104	C₂²×C₄	D₁₁₆	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
106	Q₁₆	—	58	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₂₉	1	2	yes	—
110	C₂×C₈	D₁₁₆	1	2	yes	—
111	C₂×C₈	D₁₁₆	1	2	yes	—
112	C₂×C₈	C₂²⋊D₂₉	1	2	yes	—
113	C₄⋊C₄	—	58	2	no	—
114	C₄⋊C₄	—	58	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
118	Q₈	—	58	4	no	—
119	Q₈	—	58	4	no	—
120	Q₈	—	116	2	no	—
121	Q₈	—	116	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	58	4	no	—
125	C₂×C₄	—	116	2	no	—
126	C₂×C₄	—	116	2	no	—
127	C₂×C₄	—	116	2	no	—
128	C₂×C₄	—	116	4	no	—
129	C₂×C₄	Dic₁₁₆	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
131	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
133	C₂×C₄	#90805	1	1	yes	characteristic
134	C₂×C₄	#90805	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#90781	1	4	yes	—
138	C₂²	C₄×D₁₁₆	1	1	yes	characteristic
139	C₂²	#11745236	1	1	yes	characteristic
140	C₂²	#11745266	1	2	yes	—
141	C₂²	C₂×Dic₁₁₆	1	2	yes	—
142	C₂²	#11745292	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	116	4	no	—
145	C₄	—	232	2	no	—
146	C₄	C₄×D₁₁₆	1	1	yes	characteristic
147	C₄	#11745237	1	1	yes	characteristic
148	C₄	C₂×#90781	1	1	yes	characteristic
149	C₄	#11745292	1	1	yes	characteristic
150	C₂	#14271065	1	1	yes	characteristic
151	C₂	C₄×Dic₁₁₆	1	2	yes	—
152	C₂	#14271404	1	2	yes	—
153	C₂	#14271543	1	1	yes	characteristic
154	C₂	#14271547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₁₆

C₂

yes

maximal, characteristic

C₂×#11745166

C₂

yes

maximal, characteristic

C₂×#11745164

C₂

yes

maximal

#14271069

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

C₄

yes

complemented

C₄×C₈×C₂₉

—

#11745179

—

#11745166

—

#11745164

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₂×C₈×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

D₄

yes

—

C₄²×C₂₉

—

C₄²×C₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

Dic₁₁₆

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

Q₁₆

yes

complemented

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

C₂×C₄×C₂₉

yes

C₂×C₄×C₂₉

yes

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

C₂²⋊C₄

yes

—

#1969

—

maximal, Sylow

C₂²×C₂₉

C₄×D₄

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

C₂×Q₁₆

yes

—

C₂²×C₂₉

#135

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

C₄×D₄

yes

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

Dic₂₉

—

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₂₉

yes

characteristic

#391

—

C₂×C₂₉

#390

yes

characteristic

C₂×C₂₉

C₄×Q₁₆

yes

—

C₂×C₂₉

#467

yes

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#107

—

#103

—

C₄×C₈

—

#95

—

C₂₉

#1969

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₁₁₆

yes

—

105

C₂²×C₄

C₂²×D₂₉

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₂₉

yes

—

110

C₂×C₈

D₁₁₆

yes

—

111

C₂×C₈

D₁₁₆

yes

—

112

C₂×C₈

C₂²⋊D₂₉

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₁₁₆

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

116

—

121

Q₈

—

116

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

116

—

126

C₂×C₄

—

116

—

127

C₂×C₄

—

116

—

128

C₂×C₄

—

116

—

129

C₂×C₄

Dic₁₁₆

yes

complemented

130

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

131

C₂×C₄

C₂×D₁₁₆

yes

characteristic

132

C₂×C₄

C₂×D₁₁₆

yes

characteristic

133

C₂×C₄

#90805

yes

characteristic

134

C₂×C₄

#90805

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

136

C₈

—

137

C₈

#90781

yes

—

138

C₂²

C₄×D₁₁₆

yes

characteristic

139

C₂²

#11745236

yes

characteristic

140

C₂²

#11745266

yes

—

141

C₂²

C₂×Dic₁₁₆

yes

—

142

C₂²

#11745292

yes

characteristic

143

C₄

—

144

C₄

—

116

—

145

C₄

—

232

—

146

C₄

C₄×D₁₁₆

yes

characteristic

147

C₄

#11745237

yes

characteristic

148

C₄

C₂×#90781

yes

characteristic

149

C₄

#11745292

yes

characteristic

150

C₂

#14271065

yes

characteristic

151

C₂

C₄×Dic₁₁₆

yes

—

152

C₂

#14271404

yes

—

153

C₂

#14271543

yes

characteristic

154

C₂

#14271547

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₁₆
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₄×C₈×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₉
hypocenter	C₂₉	#1969
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₁₆

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₄×C₈×C₂₉

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₉

hypocenter

C₂₉

#1969

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
13. C₂×Dic₁₁₆	C₄	semidirect
42. C₂×C₄×C₂₉	Q₁₆	semidirect
95. C₂₉	#1969	semidirect
129. C₂×C₄	Dic₁₁₆	semidirect

H ≅ G/N

type

13. C₂×Dic₁₁₆

C₄

semidirect

42. C₂×C₄×C₂₉

Q₁₆

semidirect

95. C₂₉

#1969

semidirect

129. C₂×C₄

Dic₁₁₆

semidirect

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

subgroup	index
C₂²×Dic₁₁₆	2
C₂×#11745166	2
C₂×#11745164	2
#14271069	2
C₂×C₄×C₈×C₂₉	2
#1969	29

subgroup

index

C₂²×Dic₁₁₆

C₂×C₄×C₈×C₂₉

Subgroups — #16820353

Conjugacy classes of subgroups

Complements

Maximal subgroups