Group #16820453 — subgroups

Subgroups — #16820453

Order 3712.

300 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11745231	C₂	1	1	yes	maximal, characteristic
2	#14271033	C₂	1	1	yes	maximal, characteristic
3	#14270926	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	—	2	1	no	—
6	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₉	—	2	1	no	—
9	C₂×C₄⋊Dic₂₉	—	2	1	no	—
10	C₂×C₄⋊Dic₂₉	—	2	1	no	—
11	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
13	#11745231	—	2	2	no	—
14	#11745231	C₄	1	4	yes	complemented
15	C₄×C₈×C₂₉	—	2	2	no	—
16	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₉	—	2	2	no	—
20	C₂²×Dic₂₉	—	2	2	no	—
21	C₂×Dic₅₈	—	2	4	no	—
22	C₂×Dic₅₈	—	4	2	no	—
23	C₂×Dic₅₈	—	4	2	no	—
24	C₂×C₈×C₂₉	—	2	2	no	—
25	C₂×C₈×C₂₉	—	2	2	no	—
26	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
27	C₄²×C₂₉	C₂×C₄	1	2	yes	—
28	C₄²×C₂₉	D₄	1	2	yes	—
29	C₄⋊Dic₂₉	—	2	4	no	—
30	C₄⋊Dic₂₉	—	2	2	no	—
31	C₄⋊Dic₂₉	—	2	4	no	—
32	C₄⋊Dic₂₉	—	4	2	no	—
33	C₄⋊Dic₂₉	—	4	2	no	—
34	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
35	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₉	D₈	1	2	yes	—
39	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
40	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
42	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₉	#41	1	2	yes	—
46	C₂×Dic₂₉	—	2	4	no	—
47	C₂×Dic₂₉	—	4	2	no	—
48	C₂×Dic₂₉	—	4	2	no	—
49	C₂×Dic₂₉	—	4	2	no	—
50	C₂×Dic₂₉	—	4	2	no	—
51	C₂×Dic₂₉	—	4	2	no	—
52	Dic₅₈	—	4	8	no	—
53	C₈×C₂₉	—	2	4	no	—
54	#1990	—	29	1	no	maximal, Sylow
55	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₉	#124	1	1	yes	characteristic
57	C₂²×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₂₉	#102	1	4	yes	—
63	C₄×C₂₉	#103	1	4	yes	—
64	C₄×C₂₉	C₄×Q₈	1	2	yes	—
65	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
66	Dic₂₉	—	4	4	no	—
67	Dic₂₉	—	8	2	no	—
68	C₂×C₄⋊Q₈	—	29	1	no	—
69	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
70	#384	—	29	1	no	—
71	#340	—	29	4	no	—
72	C₂×C₂₉	#391	1	1	yes	characteristic
73	C₂×C₂₉	C₂×#102	1	1	yes	characteristic
74	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
75	C₂×C₂₉	#486	1	1	yes	characteristic
76	C₂×C₂₉	#487	1	1	yes	characteristic
77	C₂×C₂₉	#498	1	1	yes	characteristic
78	C₂×C₂₉	#500	1	1	yes	characteristic
79	C₂²×Q₈	—	58	1	no	—
80	C₂²×C₈	C₂×D₂₉	1	2	yes	—
81	C₄⋊Q₈	—	29	4	no	—
82	C₄⋊Q₈	—	58	2	no	—
83	C₂×C₄⋊C₄	—	29	2	no	—
84	C₂×C₄⋊C₄	—	29	2	no	—
85	C₂×C₄⋊C₄	—	58	1	no	—
86	C₂×C₄⋊C₄	—	58	1	no	—
87	C₂×C₄⋊C₄	—	58	1	no	—
88	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
89	C₄×C₈	—	2	2	no	—
90	C₂₉	#1990	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	58	4	no	—
92	C₂×Q₈	—	116	2	no	—
93	C₂×Q₈	—	116	2	no	—
94	C₂²×C₄	—	58	2	no	—
95	C₂²×C₄	—	58	2	no	—
96	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
97	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₅₈	1	4	yes	complemented
102	C₄⋊C₄	—	29	4	no	—
103	C₄⋊C₄	—	58	4	no	—
104	C₄⋊C₄	—	58	2	no	—
105	C₄⋊C₄	—	58	4	no	—
106	C₄⋊C₄	—	116	2	no	—
107	C₄⋊C₄	—	116	2	no	—
108	C₄²	C₄×D₂₉	1	2	yes	—
109	C₄²	C₂²⋊D₂₉	1	2	yes	—
110	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
111	Q₈	—	116	8	no	—
112	C₂×C₄	—	58	4	no	—
113	C₂×C₄	—	116	2	no	—
114	C₂×C₄	—	116	2	no	—
115	C₂×C₄	—	116	2	no	—
116	C₂×C₄	—	116	2	no	—
117	C₂×C₄	—	116	2	no	—
118	C₂×C₄	#90773	1	2	yes	—
119	C₂×C₄	#90773	1	2	yes	—
120	C₂×C₄	D₂₃₂	1	2	yes	—
121	C₂×C₄	Dic₁₁₆	1	2	yes	—
122	C₂×C₄	#90781	1	2	yes	—
123	C₂×C₄	#90781	1	2	yes	—
124	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
125	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
126	C₂×C₄	#90805	1	2	yes	—
127	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
128	C₈	—	2	4	no	—
129	C₂²	#11745231	1	1	yes	characteristic
130	C₂²	#11745237	1	1	yes	characteristic
131	C₂²	C₂×#90773	1	1	yes	characteristic
132	C₂²	C₂×#90773	1	1	yes	characteristic
133	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
134	C₂²	C₂×Dic₁₁₆	1	1	yes	characteristic
135	C₂²	C₂×#90781	1	1	yes	characteristic
136	C₄	—	116	4	no	—
137	C₄	—	232	2	no	—
138	C₄	#11745164	1	4	yes	—
139	C₄	#11745169	1	4	yes	—
140	C₄	C₄×Dic₅₈	1	2	yes	—
141	C₄	#11745286	1	2	yes	—
142	C₂	#14271069	1	1	yes	characteristic
143	C₂	C₂×#11745169	1	1	yes	characteristic
144	C₂	C₂×#11745164	1	1	yes	characteristic
145	C₂	#14271510	1	1	yes	characteristic
146	C₂	#14271514	1	1	yes	characteristic
147	C₂	#14271561	1	1	yes	characteristic
148	C₂	#14271568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

#14271033

C₂

yes

maximal, characteristic

#14270926

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

—

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

#11745231

—

#11745231

C₄

yes

complemented

C₄×C₈×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

D₈

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

C₈×C₂₉

—

#1990

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

#124

yes

characteristic

C₂²×C₂₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₉

C₂×D₈

yes

characteristic

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×Q₁₆

yes

characteristic

C₄×C₂₉

#102

yes

—

C₄×C₂₉

#103

yes

—

C₄×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

D₂₉

yes

characteristic

#384

—

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄×C₈

—

C₂₉

#1990

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂×Q₈

—

116

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₅₈

yes

complemented

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

116

—

107

C₄⋊C₄

—

116

—

108

C₄²

C₄×D₂₉

yes

—

109

C₄²

C₂²⋊D₂₉

yes

—

110

C₂³

C₂×D₁₁₆

yes

characteristic

111

Q₈

—

116

—

112

C₂×C₄

—

113

C₂×C₄

—

116

—

114

C₂×C₄

—

116

—

115

C₂×C₄

—

116

—

116

C₂×C₄

—

116

—

117

C₂×C₄

—

116

—

118

C₂×C₄

#90773

yes

—

119

C₂×C₄

#90773

yes

—

120

C₂×C₄

D₂₃₂

yes

—

121

C₂×C₄

Dic₁₁₆

yes

—

122

C₂×C₄

#90781

yes

—

123

C₂×C₄

#90781

yes

—

124

C₂×C₄

C₂×Dic₅₈

yes

—

125

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

126

C₂×C₄

#90805

yes

—

127

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

128

C₈

—

129

C₂²

#11745231

yes

characteristic

130

C₂²

#11745237

yes

characteristic

131

C₂²

C₂×#90773

yes

characteristic

132

C₂²

C₂×#90773

yes

characteristic

133

C₂²

C₂×D₂₃₂

yes

characteristic

134

C₂²

C₂×Dic₁₁₆

yes

characteristic

135

C₂²

C₂×#90781

yes

characteristic

136

C₄

—

116

—

137

C₄

—

232

—

138

C₄

#11745164

yes

—

139

C₄

#11745169

yes

—

140

C₄

C₄×Dic₅₈

yes

—

141

C₄

#11745286

yes

—

142

C₂

#14271069

yes

characteristic

143

C₂

C₂×#11745169

yes

characteristic

144

C₂

C₂×#11745164

yes

characteristic

145

C₂

#14271510

yes

characteristic

146

C₂

#14271514

yes

characteristic

147

C₂

#14271561

yes

characteristic

148

C₂

#14271568

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	#1990
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

#1990

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
14. #11745231	C₄	semidirect
26. C₂×C₈×C₂₉	Q₈	semidirect
90. C₂₉	#1990	semidirect
101. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

14. #11745231

C₄

semidirect

26. C₂×C₈×C₂₉

Q₈

semidirect

90. C₂₉

#1990

semidirect

101. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
C₂×#11745231	2
#14271033	2
#14270926	2
C₂×C₄×C₈×C₂₉	2
#1990	29

subgroup

index

C₂×C₄×C₈×C₂₉

Subgroups — #16820453

Conjugacy classes of subgroups

Complements

Maximal subgroups