Group #16820044 — subgroups

Subgroups — #16820044

Order 3712.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14271734	C₂	1	1	yes	maximal, characteristic
2	#14271147	C₂	1	1	yes	maximal, characteristic
3	C₂×#11745172	C₂	1	2	yes	maximal, complemented
4	#14270935	C₂	1	2	yes	maximal, complemented
5	C₂₉×#568	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
8	C₂₉×#131	—	2	4	no	—
9	C₂×M₁₆×C₂₉	—	2	2	no	—
10	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
11	#11745306	—	2	1	no	—
12	#11745298	—	2	1	no	—
13	#11745297	C₄	1	4	yes	—
14	#11745288	C₂²	1	1	yes	characteristic
15	#11745286	—	2	2	no	—
16	C₂×#90777	C₂²	1	2	yes	—
17	#11745184	—	2	1	no	—
18	#11745181	—	2	1	no	—
19	#11745172	—	2	4	no	—
20	#41×C₂₉	—	2	2	no	—
21	#41×C₂₉	—	2	2	no	—
22	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
23	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	D₄	1	2	yes	—
26	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
28	C₂×Dic₅₈	—	2	1	no	—
29	C₂×Dic₅₈	—	2	1	no	—
30	C₂×Dic₅₈	—	2	2	no	—
31	C₂×Dic₅₈	—	2	2	no	—
32	M₁₆×C₂₉	—	2	4	no	—
33	M₁₆×C₂₉	—	2	2	no	—
34	M₁₆×C₂₉	D₄	1	4	yes	—
35	C₂×C₈×C₂₉	—	2	4	no	—
36	C₂×C₈×C₂₉	—	2	2	no	—
37	C₂²⋊Dic₂₉	—	2	4	no	—
38	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₂₉	—	2	1	no	—
40	#90779	—	2	4	no	—
41	C₄×Dic₂₉	—	2	1	no	—
42	#90777	—	2	2	no	—
43	#90777	—	2	2	no	—
44	C₂×#6597	—	2	2	no	—
45	C₂³×C₂₉	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₉	—	2	2	no	—
48	D₄×C₂₉	—	2	4	no	—
49	D₄×C₂₉	—	2	2	no	—
50	C₂×C₄×C₂₉	—	2	2	no	—
51	C₂×C₄×C₂₉	—	2	2	no	—
52	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉	#41	1	2	yes	—
57	C₂×Dic₂₉	—	2	2	no	—
58	C₂×Dic₂₉	—	2	2	no	—
59	C₂×Dic₂₉	—	2	4	no	—
60	Dic₅₈	—	4	1	no	—
61	Dic₅₈	—	4	2	no	—
62	Dic₅₈	—	4	1	no	—
63	C₈×C₂₉	—	2	4	no	—
64	#6597	—	4	2	no	—
65	#1915	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	—	2	2	no	—
67	C₂²×C₂₉	—	4	1	no	—
68	C₂²×C₂₉	—	4	2	no	—
69	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₉	C₄×D₄	1	2	yes	—
71	C₄×C₂₉	—	2	2	no	—
72	C₄×C₂₉	#120	1	1	yes	characteristic
73	C₄×C₂₉	#121	1	2	yes	—
74	C₄×C₂₉	#123	1	1	yes	characteristic
75	Dic₂₉	—	4	2	no	—
76	Dic₂₉	—	4	2	no	—
77	#568	D₂₉	1	1	yes	characteristic
78	#536	—	29	1	no	—
79	#413	—	29	1	no	—
80	C₂×#101	—	29	2	no	—
81	#344	—	29	2	no	—
82	C₂×C₂₉	—	2	1	no	—
83	C₂×C₂₉	—	2	2	no	—
84	C₂×C₂₉	—	4	2	no	—
85	C₂×C₂₉	#386	1	1	yes	characteristic
86	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
87	C₂²×Q₈	—	29	1	no	—
88	#131	—	2	4	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	29	2	no	—
91	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
92	C₄⋊Q₈	—	58	1	no	—
93	#124	—	58	1	no	—
94	#123	—	29	4	no	—
95	C₂²⋊Q₈	—	58	2	no	—
96	#117	—	29	1	no	—
97	#101	—	58	1	no	—
98	#101	—	58	4	no	—
99	#100	—	58	1	no	—
100	C₂₉	#1915	1	1	yes	characteristic, complemented, Sylow
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	58	1	no	—
104	C₂×Q₈	—	58	2	no	—
105	C₂×Q₈	—	58	2	no	—
106	C₂×Q₈	—	58	1	no	—
107	C₂×Q₈	D₁₁₆	1	1	yes	characteristic
108	C₂×D₄	C₄×D₂₉	1	2	yes	—
109	C₂×D₄	D₁₁₆	1	1	yes	characteristic
110	C₂²×C₄	—	29	2	no	—
111	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	58	2	no	—
116	M₁₆	—	58	2	no	—
117	M₁₆	C₂²⋊D₂₉	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	58	2	no	—
121	C₄⋊C₄	—	58	4	no	—
122	C₄⋊C₄	—	58	1	no	—
123	C₂²⋊C₄	—	29	2	no	—
124	C₂²⋊C₄	—	58	4	no	—
125	C₄²	—	58	1	no	—
126	C₂³	#90781	1	2	yes	—
127	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	116	1	no	—
130	Q₈	—	116	2	no	—
131	Q₈	—	116	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	58	2	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	—	58	4	no	—
139	C₂×C₄	#90781	1	2	yes	—
140	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
141	C₂×C₄	#90805	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	116	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#90781	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₂₉	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	116	2	no	—
153	C₄	—	116	2	no	—
154	C₄	#11745291	1	1	yes	characteristic
155	C₄	#11745292	1	2	yes	—
156	C₄	#11745315	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#14271042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271734

C₂

yes

maximal, characteristic

#14271147

C₂

yes

maximal, characteristic

C₂×#11745172

C₂

yes

maximal, complemented

#14270935

C₂

yes

maximal, complemented

C₂₉×#568

C₂

yes

maximal, characteristic

C₂×#41×C₂₉

C₂²

yes

characteristic

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂₉×#131

—

C₂×M₁₆×C₂₉

—

C₂×M₁₆×C₂₉

C₂²

yes

—

#11745306

—

#11745298

—

#11745297

C₄

yes

—

#11745288

C₂²

yes

characteristic

#11745286

—

C₂×#90777

C₂²

yes

—

#11745184

—

#11745181

—

#11745172

—

#41×C₂₉

—

#41×C₂₉

—

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂×D₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

M₁₆×C₂₉

—

M₁₆×C₂₉

—

M₁₆×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂²⋊Dic₂₉

—

C₂²⋊Dic₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

—

#90779

—

C₄×Dic₂₉

—

#90777

—

#90777

—

C₂×#6597

—

C₂³×C₂₉

C₂²⋊C₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

#6597

—

#1915

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

—

C₄×C₂₉

#120

yes

characteristic

C₄×C₂₉

#121

yes

—

C₄×C₂₉

#123

yes

characteristic

Dic₂₉

—

Dic₂₉

—

#568

D₂₉

yes

—

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

#386

yes

characteristic

C₂×#41

C₂×D₂₉

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₉

yes

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

100

C₂₉

#1915

yes

characteristic, complemented, Sylow

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

D₁₁₆

yes

characteristic

108

C₂×D₄

C₄×D₂₉

yes

—

109

C₂×D₄

D₁₁₆

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₂₉

yes

—

112

C₂²×C₄

C₂²×D₂₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

C₂²⋊D₂₉

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₂³

#90781

yes

—

127

C₂³

C₂×C₄×D₂₉

yes

characteristic

128

Q₈

—

129

Q₈

—

116

—

130

Q₈

—

116

—

131

Q₈

—

116

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#90781

yes

—

140

C₂×C₄

C₂×D₁₁₆

yes

characteristic

141

C₂×C₄

#90805

yes

—

142

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

144

C₈

—

145

C₈

—

116

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#90781

yes

characteristic

150

C₂²

C₄×C₂²⋊D₂₉

yes

—

151

C₄

—

152

C₄

—

116

—

153

C₄

—

116

—

154

C₄

#11745291

yes

characteristic

155

C₄

#11745292

yes

—

156

C₄

#11745315

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#14271042

yes

characteristic

	subgroup	G/H
center	C₂	#14271042
commutator	C₂³×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#568	C₂
socle	C₂×C₂₉	#386
prime radical	C₂×#41×C₂₉	C₂²
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	#568	D₂₉
hypocenter	C₂₉	#1915
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#14271042

commutator

C₂³×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#568

C₂

socle

C₂×C₂₉

#386

prime radical

C₂×#41×C₂₉

C₂²

cosocle

C₂²×C₄×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
3. C₂×#11745172	C₂	semidirect
4. #14270935	C₂	semidirect
100. C₂₉	#1915	semidirect

H ≅ G/N

type

3. C₂×#11745172

C₂

semidirect

4. #14270935

C₂

semidirect

100. C₂₉

#1915

semidirect

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

subgroup	index
#14271734	2
#14271147	2
C₂×#11745172	2
#14270935	2
C₂₉×#568	2
#1915	29

subgroup

index

C₂₉×#568

Subgroups — #16820044

Conjugacy classes of subgroups

Complements

Maximal subgroups