Group #16828572 — subgroups

Subgroups — #16828572

Order 3712.

396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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	C₈×Dic₅₈	C₂	1	4	yes	maximal, complemented
3	#14271287	C₂	1	4	yes	maximal, complemented
4	#14271216	C₂	1	1	yes	maximal, characteristic
5	#14271213	C₂	1	4	yes	maximal
6	C₂₉×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₈	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
10	C₂×#90779	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
12	C₄×Dic₅₈	C₄	1	2	yes	—
13	C₄×Dic₅₈	C₄	1	2	yes	—
14	C₄×Dic₅₈	C₂²	1	2	yes	—
15	C₄×Dic₅₈	C₂²	1	2	yes	—
16	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₄×C₈×C₂₉	C₂²	1	2	yes	—
19	C₂²⋊#6597	C₂²	1	2	yes	—
20	#11745162	C₂²	1	8	yes	—
21	C₈×Dic₂₉	C₂²	1	8	yes	—
22	C₄⋊#6597	C₂²	1	2	yes	—
23	C₄×#6597	C₂²	1	2	yes	—
24	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
28	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₅₈	C₈	1	8	yes	complemented
30	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
31	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₉	C₂³	1	4	yes	—
33	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₂₉	C₂³	1	1	yes	characteristic
36	C₄²×C₂₉	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
39	#90779	C₂×C₄	1	4	yes	—
40	#90779	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₉	C₂×C₄	1	4	yes	—
42	C₄×Dic₂₉	C₂³	1	4	yes	—
43	C₂×#6597	C₂³	1	4	yes	—
44	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	—	2	2	no	—
46	C₂×C₄×C₂₉	—	4	1	no	—
47	C₂×C₄×C₂₉	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₂₉	—	4	2	no	—
55	C₂×Dic₂₉	C₂×C₈	1	8	yes	—
56	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
57	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
58	Dic₅₈	C₂×C₈	1	8	yes	—
59	Dic₅₈	C₂×C₈	1	8	yes	—
60	C₈×C₂₉	—	2	4	no	—
61	#6597	—	2	4	no	—
62	#2982	—	29	1	no	maximal, Sylow
63	C₂²×C₂₉	—	2	1	no	—
64	C₂²×C₂₉	—	2	1	no	—
65	C₂²×C₂₉	C₂²×C₈	1	2	yes	—
66	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₂₉	—	2	1	no	—
68	C₄×C₂₉	—	2	1	no	—
69	C₄×C₂₉	—	2	1	no	—
70	C₄×C₂₉	C₂²×C₈	1	2	yes	—
71	C₄×C₂₉	C₂²×C₈	1	2	yes	—
72	C₄×C₂₉	2₋¹⁺⁴	1	2	yes	—
73	Dic₂₉	—	2	4	no	—
74	Dic₂₉	C₂²×C₈	1	8	yes	—
75	C₂×C₄×Q₈	—	29	1	no	—
76	C₈×Q₈	—	29	4	no	—
77	C₈×Q₈	—	29	4	no	—
78	#431	—	29	1	no	—
79	#431	—	29	4	no	—
80	#431	D₂₉	1	1	yes	characteristic
81	C₂×C₂₉	—	2	2	no	—
82	C₂×C₂₉	#519	1	1	yes	characteristic
83	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
84	C₂×C₂₉	#568	1	1	yes	characteristic
85	C₂²×Q₈	—	29	1	no	—
86	C₄×Q₈	—	29	2	no	—
87	C₄×Q₈	—	29	2	no	—
88	C₄×Q₈	—	29	2	no	—
89	C₄×Q₈	—	29	2	no	—
90	C₂×C₄⋊C₄	—	29	2	no	—
91	C₂×C₄⋊C₄	—	29	1	no	—
92	C₂×C₄²	—	29	2	no	—
93	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
94	C₄⋊C₈	—	29	8	no	—
95	C₄⋊C₈	—	29	2	no	—
96	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
97	C₂²⋊C₈	—	29	2	no	—
98	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
99	C₄×C₈	—	29	8	no	—
100	C₄×C₈	—	29	2	no	—
101	C₄×C₈	C₂×D₂₉	1	2	yes	—
102	C₂₉	#2982	1	1	yes	characteristic, complemented, Sylow
103	C₂×Q₈	—	29	8	no	—
104	C₂×Q₈	—	29	2	no	—
105	C₂×Q₈	—	29	2	no	—
106	C₂²×C₄	—	29	2	no	—
107	C₂²×C₄	—	29	2	no	—
108	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
111	C₂×C₈	—	29	4	no	—
112	C₂×C₈	C₂²×D₂₉	1	4	yes	—
113	C₄⋊C₄	—	29	4	no	—
114	C₄⋊C₄	—	29	4	no	—
115	C₄⋊C₄	—	29	2	no	—
116	C₄⋊C₄	—	29	2	no	—
117	C₄²	—	29	4	no	—
118	C₄²	—	29	4	no	—
119	C₄²	C₄×D₂₉	1	1	yes	characteristic
120	C₄²	C₄×D₂₉	1	1	yes	characteristic
121	C₄²	C₂²×D₂₉	1	1	yes	characteristic
122	C₄²	C₂²×D₂₉	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
124	Q₈	—	29	8	no	—
125	Q₈	—	29	8	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	29	4	no	—
129	C₂×C₄	—	29	8	no	—
130	C₂×C₄	—	29	4	no	—
131	C₂×C₄	—	116	2	no	—
132	C₂×C₄	C₈×D₂₉	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	58	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×C₈×D₂₉	1	2	yes	—
144	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	29	8	no	—
149	C₄	—	58	4	no	—
150	C₄	C₂×C₈×D₂₉	1	2	yes	—
151	C₄	C₂×C₈×D₂₉	1	2	yes	—
152	C₄	#11745365	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#14271640	1	1	yes	characteristic
155	C₂	C₂²×C₈×D₂₉	1	1	yes	characteristic
156	C₂	#14271929	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₈

C₂

yes

maximal, characteristic

C₈×Dic₅₈

C₂

yes

maximal, complemented

#14271287

C₂

yes

maximal, complemented

#14271216

C₂

yes

maximal, characteristic

#14271213

C₂

yes

maximal

C₂₉×#431

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₄

yes

characteristic

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₄

yes

characteristic

C₂×#90779

C₄

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

C₄×Dic₅₈

C₄

yes

—

C₄×Dic₅₈

C₄

yes

—

C₄×Dic₅₈

C₂²

yes

—

C₄×Dic₅₈

C₂²

yes

—

C₂₉×C₄⋊C₈

C₂²

yes

—

C₂₉×C₂²⋊C₈

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

—

C₂²⋊#6597

C₂²

yes

—

#11745162

C₂²

yes

—

C₈×Dic₂₉

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₄×#6597

C₂²

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂×Dic₅₈

C₈

yes

complemented

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

C₂³

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

C₂×C₄

yes

—

#90779

C₂×C₄

yes

—

C₄×Dic₂₉

C₂×C₄

yes

—

C₄×Dic₂₉

C₂³

yes

—

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×C₈

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂×C₈

yes

—

C₂×Dic₂₉

C₂²×C₄

yes

—

C₂×Dic₂₉

C₂²×C₄

yes

—

Dic₅₈

C₂×C₈

yes

—

Dic₅₈

C₂×C₈

yes

—

C₈×C₂₉

—

#6597

—

#2982

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂²×C₈

yes

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂²×C₈

yes

—

C₄×C₂₉

C₂²×C₈

yes

—

C₄×C₂₉

2₋¹⁺⁴

yes

—

Dic₂₉

—

Dic₂₉

C₂²×C₈

yes

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

#431

—

#431

—

#431

D₂₉

yes

characteristic

C₂×C₂₉

—

C₂×C₂₉

#519

yes

characteristic

C₂×C₂₉

C₂³×C₈

yes

characteristic

C₂×C₂₉

#568

yes

characteristic

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₂₉

yes

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₂₉

yes

—

102

C₂₉

#2982

yes

characteristic, complemented, Sylow

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₂₉

yes

characteristic

109

C₂²×C₄

C₄×D₂₉

yes

characteristic

110

C₂²×C₄

C₂²×D₂₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₂₉

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₂₉

yes

characteristic

120

C₄²

C₄×D₂₉

yes

characteristic

121

C₄²

C₂²×D₂₉

yes

characteristic

122

C₄²

C₂²×D₂₉

yes

characteristic

123

C₂³

C₂×C₄×D₂₉

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

116

—

132

C₂×C₄

C₈×D₂₉

yes

—

133

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

134

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

135

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

136

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

137

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

138

C₂×C₄

C₂³×D₂₉

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₈×D₂₉

yes

—

144

C₂²

C₂²×C₄×D₂₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×C₈×D₂₉

yes

—

151

C₄

C₂×C₈×D₂₉

yes

—

152

C₄

#11745365

yes

—

153

C₂

—

154

C₂

#14271640

yes

characteristic

155

C₂

C₂²×C₈×D₂₉

yes

characteristic

156

C₂

#14271929

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator	C₂×C₂₉	C₂³×C₈
Fitting	C₂₉×#431	C₂
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#431	D₂₉
hypocenter	C₂₉	#2982
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₉

commutator

C₂×C₂₉

C₂³×C₈

Fitting

C₂₉×#431

C₂

socle

C₂²×C₂₉

C₂³×C₄

prime radical

C₂³×C₂₉

C₂²×C₄

cosocle

C₂×C₄×C₂₉

C₂⁴

D₂₉

C₂₉

G ▷ C₂×C₂₉ ▷ C₂₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. C₈×Dic₅₈	C₂	semidirect
3. #14271287	C₂	semidirect
29. C₂×Dic₅₈	C₈	semidirect
102. C₂₉	#2982	semidirect

H ≅ G/N

type

2. C₈×Dic₅₈

C₂

semidirect

3. #14271287

C₂

semidirect

29. C₂×Dic₅₈

C₈

semidirect

102. C₂₉

#2982

semidirect

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

subgroup	index
C₂×C₄×Dic₅₈	2
C₈×Dic₅₈	2
#14271287	2
#14271216	2
#14271213	2
C₂₉×#431	2
#2982	29

subgroup

index

C₂×C₄×Dic₅₈

C₈×Dic₅₈

C₂₉×#431

Subgroups — #16828572

Conjugacy classes of subgroups

Complements

Maximal subgroups