Group #16820648 — subgroups

Subgroups — #16820648

Order 3712.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14271794	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271168	C₂	1	2	yes	maximal
3	#14270932	C₂	1	1	yes	maximal, characteristic, complemented
4	#14270923	C₂	1	2	yes	maximal, complemented
5	C₂₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₂₉	—	2	2	no	—
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	#11745288	C₂²	1	2	yes	—
11	#11745286	—	2	2	no	—
12	C₂×#90777	C₂²	1	2	yes	—
13	#11745252	—	2	2	no	—
14	#11745250	—	2	2	no	—
15	#11745180	—	2	2	no	—
16	#11745180	—	2	2	no	—
17	C₂⁴×C₂₉	—	2	1	no	—
18	C₂×D₄×C₂₉	—	2	2	no	—
19	C₂×D₄×C₂₉	—	2	2	no	—
20	C₂×D₄×C₂₉	—	2	1	no	—
21	C₂×D₄×C₂₉	Q₈	1	2	yes	complemented
22	C₂²×C₄×C₂₉	—	2	1	no	—
23	C₂²×C₄×C₂₉	D₄	1	2	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	2	no	—
26	C₄⋊C₄×C₂₉	—	2	2	no	—
27	C₂²⋊C₄×C₂₉	—	4	4	no	—
28	C₄²×C₂₉	—	2	2	no	—
29	C₂²⋊Dic₂₉	—	2	2	no	—
30	C₄⋊Dic₂₉	—	2	2	no	—
31	C₄⋊Dic₂₉	D₄	1	2	yes	—
32	C₄⋊Dic₂₉	D₄	1	2	yes	—
33	#90779	—	2	4	no	—
34	C₄×Dic₂₉	—	2	2	no	—
35	#90777	—	2	2	no	—
36	#90777	—	2	2	no	—
37	C₂×#6597	—	2	2	no	—
38	C₂³×C₂₉	—	2	2	no	—
39	C₂³×C₂₉	—	4	2	no	—
40	C₂³×C₂₉	—	4	2	no	—
41	C₂³×C₂₉	—	4	1	no	—
42	C₂³×C₂₉	—	4	1	no	—
43	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
44	D₄×C₂₉	—	2	4	no	—
45	D₄×C₂₉	—	4	2	no	—
46	C₂×C₄×C₂₉	—	2	2	no	—
47	C₂×C₄×C₂₉	—	2	2	no	—
48	C₂×C₄×C₂₉	—	4	1	no	—
49	C₂×C₄×C₂₉	—	4	1	no	—
50	C₂×C₄×C₂₉	—	4	2	no	—
51	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₂₉	#41	1	2	yes	—
55	C₂×Dic₂₉	—	2	2	no	—
56	C₂×Dic₂₉	—	2	2	no	—
57	Dic₅₈	—	8	2	no	—
58	#6597	—	4	2	no	—
59	#2027	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	—	2	1	no	—
61	C₂²×C₂₉	—	2	1	no	—
62	C₂²×C₂₉	—	4	4	no	—
63	C₂²×C₂₉	—	4	2	no	—
64	C₂²×C₂₉	—	8	2	no	—
65	C₂²×C₂₉	—	8	1	no	—
66	C₂²×C₂₉	—	8	1	no	—
67	C₂²×C₂₉	#120	1	1	yes	characteristic
68	C₂²×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₂₉	#124	1	1	yes	characteristic
70	C₄×C₂₉	—	4	2	no	—
71	C₄×C₂₉	—	4	2	no	—
72	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₂₉	#124	1	2	yes	—
74	Dic₂₉	—	4	2	no	—
75	Dic₂₉	—	4	2	no	—
76	#544	—	29	1	no	—
77	#418	—	29	2	no	—
78	#390	D₂₉	1	1	yes	characteristic
79	#343	—	29	1	no	—
80	#339	—	29	2	no	—
81	C₂×C₂₉	—	2	1	no	—
82	C₂×C₂₉	—	2	1	no	—
83	C₂×C₂₉	—	8	2	no	—
84	C₂×C₂₉	#393	1	1	yes	characteristic
85	C₂×C₂₉	#453	1	1	yes	characteristic
86	C₂×C₂₉	#454	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₂₉	1	1	yes	characteristic
88	C₂×M₁₆	—	29	2	no	—
89	C₄⋊Q₈	—	58	2	no	—
90	#125	—	58	2	no	—
91	C₂²⋊Q₈	—	58	2	no	—
92	#117	—	29	2	no	—
93	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
96	#102	—	58	2	no	—
97	#102	—	58	2	no	—
98	C₂₉	#2027	1	1	yes	characteristic, complemented, Sylow
99	C₂⁴	—	2	1	no	—
100	C₂×Q₈	—	58	2	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	Dic₅₈	1	2	yes	complemented
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
108	M₁₆	—	58	2	no	—
109	M₁₆	—	58	2	no	—
110	C₂×C₈	—	58	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	29	2	no	—
113	C₄⋊C₄	—	29	2	no	—
114	C₄⋊C₄	—	58	2	no	—
115	C₄⋊C₄	—	58	4	no	—
116	C₂²⋊C₄	—	4	4	no	—
117	C₂²⋊C₄	—	58	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	58	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	4	2	no	—
122	C₂³	—	4	2	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
126	Q₈	—	232	2	no	—
127	D₄	—	2	4	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	58	2	no	—
135	C₂×C₄	—	58	2	no	—
136	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
137	C₂×C₄	#90805	1	1	yes	characteristic
138	C₂×C₄	D₄×D₂₉	1	2	yes	—
139	C₂×C₄	#90807	1	2	yes	—
140	C₈	—	116	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	#11745286	1	1	yes	characteristic
149	C₂²	#11745298	1	1	yes	characteristic
150	C₂²	#11745299	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	116	2	no	—
154	C₄	—	116	2	no	—
155	C₄	#11745240	1	2	yes	—
156	C₄	#11745247	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	8	2	no	—
160	C₂	#14271075	1	1	yes	characteristic
161	C₂	#14271330	1	1	yes	characteristic
162	C₂	#14271335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271794

C₂

yes

maximal, characteristic, complemented

#14271168

C₂

yes

maximal

#14270932

C₂

yes

maximal, characteristic, complemented

#14270923

C₂

yes

maximal, complemented

C₂₉×#390

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₉

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

#11745286

—

C₂×#90777

C₂²

yes

—

—

—

—

—

C₂⁴×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

Q₈

yes

complemented

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×Dic₅₈

—

C₄⋊C₄×C₂₉

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

D₄

yes

—

C₄⋊Dic₂₉

D₄

yes

—

#90779

—

C₄×Dic₂₉

—

#90777

—

#90777

—

C₂×#6597

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

C₂×C₄×C₂₉

yes

C₂×C₄×C₂₉

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

#6597

—

#2027

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#120

yes

characteristic

C₂²×C₂₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₉

#124

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

#124

yes

—

Dic₂₉

—

Dic₂₉

—

#544

—

#418

—

#390

D₂₉

yes

characteristic

#343

—

#339

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×D₄

C₂×D₂₉

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#102

—

#102

—

C₂₉

#2027

yes

characteristic, complemented, Sylow

C₂⁴

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

Dic₅₈

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₂₉

yes

—

107

C₂²×C₄

C₂²×D₂₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

126

Q₈

—

232

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×Dic₅₈

yes

characteristic

137

C₂×C₄

#90805

yes

characteristic

138

C₂×C₄

D₄×D₂₉

yes

—

139

C₂×C₄

#90807

yes

—

140

C₈

—

116

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#11745286

yes

characteristic

149

C₂²

#11745298

yes

characteristic

150

C₂²

#11745299

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

116

—

154

C₄

—

116

—

155

C₄

#11745240

yes

—

156

C₄

#11745247

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#14271075

yes

characteristic

161

C₂

#14271330

yes

characteristic

162

C₂

#14271335

yes

characteristic

	subgroup	G/H
center	C₂²	#11745299
commutator, cosocle	C₂²×C₄×C₂₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#390	C₂
socle	C₂²×C₂₉	#120
prime radical	C₂²×D₄×C₂₉	C₂²
hypercenter	#390	D₂₉
hypocenter	C₂₉	#2027
LCS	G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

subgroup

G/H

center

C₂²

#11745299

commutator, cosocle

C₂²×C₄×C₂₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#390

C₂

socle

C₂²×C₂₉

#120

prime radical

C₂²×D₄×C₂₉

C₂²

D₂₉

C₂₉

G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

N	H ≅ G/N	type
1. #14271794	C₂	semidirect
3. #14270932	C₂	semidirect
4. #14270923	C₂	semidirect
21. C₂×D₄×C₂₉	Q₈	semidirect
98. C₂₉	#2027	semidirect
104. C₂×D₄	Dic₅₈	semidirect

H ≅ G/N

type

1. #14271794

C₂

semidirect

3. #14270932

C₂

semidirect

4. #14270923

C₂

semidirect

21. C₂×D₄×C₂₉

Q₈

semidirect

98. C₂₉

#2027

semidirect

104. C₂×D₄

Dic₅₈

semidirect

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

subgroup	index
#14271794	2
#14271168	2
#14270932	2
#14270923	2
C₂₉×#390	2
#2027	29

subgroup

index

C₂₉×#390

Subgroups — #16820648

Conjugacy classes of subgroups

Complements

Maximal subgroups