Group #16828211 — subgroups

Subgroups — #16828211

Order 3712.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#11745263	C₂	1	2	yes	maximal, complemented
2	#14271713	C₂	1	1	yes	maximal, characteristic
3	#14271236	C₂	1	8	yes	maximal, complemented
4	#14271187	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271129	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#90805	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745292	C₄	1	4	yes	—
12	C₂×#90777	C₂²	1	2	yes	—
13	#11745263	—	2	8	no	—
14	C₂×#90772	C₂²	1	4	yes	—
15	C₂×C₈×D₂₉	C₂²	1	4	yes	—
16	#11745237	C₄	1	2	yes	—
17	#11745236	C₄	1	1	yes	characteristic
18	C₄×D₁₁₆	C₂²	1	4	yes	—
19	#11745231	C₄	1	1	yes	characteristic
20	C₂₉×#97	C₂²	1	4	yes	—
21	#11745168	C₂²	1	8	yes	—
22	C₄⋊#6597	C₂²	1	4	yes	—
23	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₂₉	C₂×C₄	1	4	yes	—
26	#90805	—	2	4	no	—
27	#90805	—	2	4	no	—
28	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
29	C₂×D₁₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₂₉	C₂³	1	4	yes	—
31	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₉	—	2	2	no	—
33	C₂×C₈×C₂₉	D₄	1	4	yes	complemented
34	C₂×C₈×C₂₉	C₂³	1	2	yes	—
35	C₂×C₈×C₂₉	C₂³	1	2	yes	—
36	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₂₉	C₂³	1	2	yes	—
39	#90781	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
41	#90777	—	2	4	no	—
42	C₂×#6597	C₂³	1	4	yes	—
43	#90772	—	2	8	no	—
44	C₈×D₂₉	—	2	8	no	—
45	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₂₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₂₉	—	4	2	no	—
48	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₉	#41	1	2	yes	—
53	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₂₉	—	2	8	no	—
55	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
56	D₁₁₆	—	2	4	no	—
57	D₁₁₆	—	4	4	no	—
58	C₄×D₂₉	—	2	4	no	—
59	C₄×D₂₉	—	2	4	no	—
60	Dic₅₈	—	2	4	no	—
61	C₈×C₂₉	—	2	2	no	—
62	C₈×C₂₉	C₂×D₄	1	4	yes	—
63	#6597	—	2	4	no	—
64	#2932	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	—	2	2	no	—
66	C₂²×C₂₉	C₄×D₄	1	2	yes	—
67	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
68	C₂×D₂₉	—	2	4	no	—
69	C₂×D₂₉	—	2	8	no	—
70	C₄×C₂₉	—	2	2	no	—
71	C₄×C₂₉	—	2	2	no	—
72	C₄×C₂₉	C₄×D₄	1	2	yes	—
73	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
75	Dic₂₉	—	2	4	no	—
76	C₂×#131	—	29	2	no	—
77	#532	—	29	1	no	—
78	#435	—	29	8	no	—
79	#423	—	29	1	no	—
80	#408	—	29	2	no	—
81	C₂×#97	D₂₉	1	1	yes	characteristic, complemented
82	C₂×C₂₉	—	2	2	no	—
83	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
84	C₂×C₂₉	#568	1	2	yes	—
85	D₂₉	—	4	4	no	—
86	C₂×#41	—	29	2	no	—
87	#131	—	58	8	no	—
88	C₂×M₁₆	—	29	2	no	—
89	C₂×M₁₆	—	29	4	no	—
90	C₂²×C₈	—	29	4	no	—
91	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
93	C₄⋊Q₈	—	29	1	no	—
94	#127	—	29	1	no	—
95	#124	—	29	2	no	—
96	#121	—	29	4	no	—
97	C₄×D₄	—	29	4	no	—
98	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
99	C₄⋊C₈	—	29	4	no	—
100	C₂²⋊C₈	—	29	8	no	—
101	#97	C₂×D₂₉	1	4	yes	—
102	C₂₉	#2932	1	1	yes	characteristic, complemented, Sylow
103	#41	—	58	4	no	—
104	#41	—	58	4	no	—
105	C₂×Q₈	—	29	2	no	—
106	C₂×D₄	—	29	4	no	—
107	C₂×D₄	—	29	2	no	—
108	C₂×D₄	—	29	4	no	—
109	C₂²×C₄	—	29	4	no	—
110	C₂²×C₄	C₄×D₂₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
112	M₁₆	—	58	4	no	—
113	M₁₆	—	58	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	29	4	no	—
116	C₂×C₈	—	58	8	no	—
117	C₂×C₈	D₁₁₆	1	4	yes	complemented
118	C₂×C₈	C₂²×D₂₉	1	2	yes	—
119	C₂×C₈	C₂²×D₂₉	1	2	yes	—
120	C₄⋊C₄	—	29	4	no	—
121	C₂²⋊C₄	—	29	8	no	—
122	C₄²	C₄×D₂₉	1	1	yes	characteristic
123	C₄²	C₄×D₂₉	1	1	yes	characteristic
124	C₄²	C₂²×D₂₉	1	2	yes	—
125	C₂³	—	29	4	no	—
126	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
127	Q₈	—	58	4	no	—
128	D₄	—	58	8	no	—
129	D₄	—	58	4	no	—
130	D₄	—	116	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	29	4	no	—
133	C₂×C₄	—	58	4	no	—
134	C₂×C₄	—	58	4	no	—
135	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
139	C₂×C₄	#90805	1	2	yes	—
140	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	58	4	no	—
143	C₈	C₂×D₁₁₆	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	58	4	no	—
146	C₂²	—	58	8	no	—
147	C₂²	C₄×D₁₁₆	1	2	yes	—
148	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	58	4	no	—
152	C₄	C₄×D₁₁₆	1	2	yes	—
153	C₄	C₂²×D₁₁₆	1	1	yes	characteristic
154	C₄	C₂×#90805	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	116	4	no	—
157	C₂	C₂×C₄×D₁₁₆	1	1	yes	characteristic
158	C₂	#14271929	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745263

C₂

yes

maximal, complemented

#14271713

C₂

yes

maximal, characteristic

#14271236

C₂

yes

maximal, complemented

#14271187

C₂

yes

maximal, characteristic, complemented

#14271129

C₂

yes

maximal, complemented

C₂×C₂₉×#97

C₂

yes

maximal, characteristic, complemented

C₂×#90805

C₂²

yes

—

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745292

C₄

yes

—

C₂×#90777

C₂²

yes

—

#11745263

—

C₂×#90772

C₂²

yes

—

C₂×C₈×D₂₉

C₂²

yes

—

#11745237

C₄

yes

—

#11745236

C₄

yes

characteristic

C₄×D₁₁₆

C₂²

yes

—

#11745231

C₄

yes

characteristic

C₂₉×#97

C₂²

yes

—

#11745168

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂×C₂²⋊D₂₉

C₂×C₄

yes

—

#90805

—

#90805

—

C₂×D₁₁₆

C₂×C₄

yes

—

C₂×D₁₁₆

C₂×C₄

yes

—

C₂×C₄×D₂₉

C₂³

yes

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

D₄

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

—

C₂×C₈×C₂₉

C₂³

yes

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

#90781

C₂×C₄

yes

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90777

—

C₂×#6597

C₂³

yes

—

#90772

—

C₈×D₂₉

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂²×D₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂²⋊D₂₉

—

C₂×Dic₂₉

C₂²×C₄

yes

—

D₁₁₆

—

D₁₁₆

—

C₄×D₂₉

—

C₄×D₂₉

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

C₂×D₄

yes

—

#6597

—

#2932

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄×D₄

yes

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

C₂×#41

yes

characteristic

Dic₂₉

—

—

—

—

—

—

C₂×#97

D₂₉

yes

characteristic, complemented

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄×D₄

yes

characteristic

C₂×C₂₉

#568

yes

—

D₂₉

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#97

C₂×D₂₉

yes

—

102

C₂₉

#2932

yes

characteristic, complemented, Sylow

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₂₉

yes

—

111

C₂²×C₄

C₂²×D₂₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₁₁₆

yes

complemented

118

C₂×C₈

C₂²×D₂₉

yes

—

119

C₂×C₈

C₂²×D₂₉

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₂₉

yes

characteristic

123

C₄²

C₄×D₂₉

yes

characteristic

124

C₄²

C₂²×D₂₉

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₂₉

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

116

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₂₉

yes

—

136

C₂×C₄

C₂×C₄×D₂₉

yes

—

137

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

138

C₂×C₄

C₂×D₁₁₆

yes

—

139

C₂×C₄

#90805

yes

—

140

C₂×C₄

C₂³×D₂₉

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

C₂×D₁₁₆

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

C₄×D₁₁₆

yes

—

148

C₂²

C₂²×C₄×D₂₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₁₁₆

yes

—

153

C₄

C₂²×D₁₁₆

yes

characteristic

154

C₄

C₂×#90805

yes

characteristic

155

C₂

—

156

C₂

—

116

—

157

C₂

C₂×C₄×D₁₁₆

yes

characteristic

158

C₂

#14271929

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×#97	C₂
prime radical	#14271713	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#97	D₂₉
hypocenter	C₂₉	#2932
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₉

commutator, socle

C₂²×C₂₉

C₂³×C₄

Fitting

C₂×C₂₉×#97

C₂

prime radical

#14271713

C₂

cosocle

C₂×C₄×C₂₉

C₂⁴

hypercenter

C₂×#97

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#11745263	C₂	semidirect
3. #14271236	C₂	semidirect
4. #14271187	C₂	semidirect
5. #14271129	C₂	semidirect
6. C₂×C₂₉×#97	C₂	semidirect
33. C₂×C₈×C₂₉	D₄	semidirect
81. C₂×#97	D₂₉	semidirect
102. C₂₉	#2932	semidirect
117. C₂×C₈	D₁₁₆	semidirect

H ≅ G/N

type

1. C₂×#11745263

C₂

semidirect

3. #14271236

C₂

semidirect

4. #14271187

C₂

semidirect

5. #14271129

C₂

semidirect

6. C₂×C₂₉×#97

C₂

semidirect

33. C₂×C₈×C₂₉

D₄

semidirect

81. C₂×#97

D₂₉

semidirect

102. C₂₉

#2932

semidirect

117. C₂×C₈

D₁₁₆

semidirect

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

subgroup	index
C₂×#11745263	2
#14271713	2
#14271236	2
#14271187	2
#14271129	2
C₂×C₂₉×#97	2
#2932	29

subgroup

index

C₂×C₂₉×#97

Subgroups — #16828211

Conjugacy classes of subgroups

Complements

Maximal subgroups