Group #13056387 — subgroups

Subgroups — #101×D₂₁ #13056387

Order 1344.

292 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 77 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×Q₈×D₂₁	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₂₁	C₂	1	2	yes	maximal
3	C₃×C₇×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620675	C₂	1	1	yes	maximal, characteristic, complemented
5	#10620663	C₂	1	2	yes	maximal, complemented
6	D₇×#101	—	3	1	no	maximal, Hall
7	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
8	Q₈×D₂₁	—	2	2	no	—
9	Q₈×D₂₁	—	2	2	no	—
10	C₂×C₄×D₂₁	C₄	1	2	yes	—
11	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₂	C₄	1	2	yes	—
13	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
14	C₃×C₇×M₁₆	C₂²	1	2	yes	—
15	#66627	C₂²	1	2	yes	—
16	#66622	—	2	2	no	—
17	C₈×D₂₁	—	2	2	no	—
18	C₂×Q₈×D₇	—	3	1	no	—
19	D₇×M₁₆	—	3	2	no	—
20	C₇×#101	D₃	1	1	yes	characteristic, complemented
21	#6410	—	3	1	no	—
22	#6398	—	3	2	no	—
23	D₃×#101	—	7	1	no	maximal, Hall
24	C₂²×D₂₁	C₂×C₄	1	1	yes	characteristic
25	C₃×C₇×Q₈	—	2	2	no	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₁	C₂×C₄	1	1	yes	characteristic
30	C₄×D₂₁	—	2	2	no	—
31	C₄×D₂₁	—	2	2	no	—
32	C₄×D₂₁	D₄	1	2	yes	—
33	C₄×D₂₁	D₄	1	2	yes	—
34	Dic₄₂	—	2	2	no	—
35	Dic₄₂	—	2	4	no	—
36	C₃×C₇×C₈	—	2	2	no	—
37	C₃⋊(C₇⋊C₈)	—	2	2	no	—
38	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
39	Q₈×D₇	—	6	2	no	—
40	Q₈×D₇	—	6	2	no	—
41	C₂×C₄×D₇	—	3	1	no	—
42	C₂×C₄×D₇	—	3	2	no	—
43	C₂×Dic₁₄	—	3	2	no	—
44	C₂×Dic₁₄	—	3	1	no	—
45	C₇×M₁₆	C₂×D₃	1	2	yes	—
46	#1136	—	3	2	no	—
47	#1131	—	6	2	no	—
48	C₈×D₇	—	6	2	no	—
49	C₂×D₃×Q₈	—	7	1	no	—
50	D₃×M₁₆	—	7	2	no	—
51	C₃×#101	D₇	1	1	yes	characteristic, complemented
52	#836	—	7	1	no	—
53	#824	—	7	2	no	—
54	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×D₂₁	—	2	2	no	—
56	C₂×D₂₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₂₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₃×C₄×C₇	—	2	2	no	—
59	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
60	Dic₂₁	—	2	2	no	—
61	Dic₂₁	C₂²⋊C₄	1	2	yes	—
62	C₂×#101	—	21	1	no	Sylow
63	C₂²×D₇	—	3	1	no	—
64	C₇×Q₈	—	2	2	no	—
65	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
66	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
67	C₂×Dic₇	—	3	1	no	—
68	C₂×Dic₇	—	3	2	no	—
69	C₄×D₇	—	3	2	no	—
70	C₄×D₇	—	3	2	no	—
71	C₄×D₇	—	6	2	no	—
72	C₄×D₇	—	6	2	no	—
73	Dic₁₄	—	6	4	no	—
74	Dic₁₄	—	6	2	no	—
75	C₇×C₈	—	2	2	no	—
76	C₇⋊C₈	—	6	2	no	—
77	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
78	D₃×Q₈	—	14	2	no	—
79	D₃×Q₈	—	14	2	no	—
80	C₂×C₄×D₃	—	7	1	no	—
81	C₂×C₄×D₃	—	7	2	no	—
82	C₂×Dic₆	—	7	2	no	—
83	C₂×Dic₆	—	7	1	no	—
84	C₃×M₁₆	C₂×D₇	1	2	yes	—
85	#208	—	7	2	no	—
86	ΓL(1,𝔽₂₅)	—	14	2	no	—
87	D₃×C₈	—	14	2	no	—
88	C₂×C₃×C₇	—	2	1	no	—
89	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	D₂₁	—	2	1	no	—
91	D₂₁	#101	1	2	yes	direct product factor
92	C₂²×Q₈	—	21	1	no	—
93	C₂×M₁₆	—	21	2	no	—
94	#101	—	21	1	no	—
95	#101	—	21	2	no	—
96	#101	D₂₁	1	1	yes	characteristic, direct product factor
97	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
98	C₂×D₇	—	3	1	no	—
99	C₂×D₇	—	3	1	no	—
100	C₂×D₇	—	6	2	no	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	D₃×D₄	1	2	yes	—
103	Dic₇	—	3	2	no	—
104	Dic₇	—	6	2	no	—
105	C₂²×D₃	—	7	1	no	—
106	C₃×Q₈	—	2	2	no	—
107	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
108	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
109	C₂×Dic₃	—	7	1	no	—
110	C₂×Dic₃	—	7	2	no	—
111	C₄×D₃	—	7	2	no	—
112	C₄×D₃	—	7	2	no	—
113	C₄×D₃	—	14	2	no	—
114	C₄×D₃	—	14	2	no	—
115	Dic₆	—	14	4	no	—
116	Dic₆	—	14	2	no	—
117	C₃×C₈	—	2	2	no	—
118	C₃⋊C₈	—	14	2	no	—
119	C₃×C₇	C₂×#101	1	1	yes	characteristic, complemented, Hall
120	C₂×Q₈	—	21	2	no	—
121	C₂×Q₈	—	21	1	no	—
122	C₂×Q₈	—	42	2	no	—
123	C₂×Q₈	—	42	2	no	—
124	C₂×Q₈	C₂×D₂₁	1	1	yes	characteristic
125	C₂²×C₄	—	21	1	no	—
126	C₂²×C₄	—	21	2	no	—
127	M₁₆	—	21	2	no	—
128	M₁₆	—	42	2	no	—
129	M₁₆	C₂×D₂₁	1	2	yes	—
130	C₂×C₈	—	42	2	no	—
131	C₂×C₇	—	2	1	no	—
132	C₂×C₇	D₃×C₂²⋊C₄	1	1	yes	characteristic
133	D₇	—	3	2	no	—
134	D₇	—	6	1	no	—
135	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
136	C₂×D₃	—	7	1	no	—
137	C₂×D₃	—	7	1	no	—
138	C₂×D₃	—	14	2	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	D₄×D₇	1	2	yes	—
141	Dic₃	—	7	2	no	—
142	Dic₃	—	14	2	no	—
143	C₂³	—	21	1	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	42	4	no	—
146	Q₈	—	42	2	no	—
147	C₂×C₄	—	21	1	no	—
148	C₂×C₄	—	21	2	no	—
149	C₂×C₄	—	21	2	no	—
150	C₂×C₄	—	21	2	no	—
151	C₂×C₄	—	42	2	no	—
152	C₂×C₄	—	42	2	no	—
153	C₂×C₄	C₄×D₂₁	1	2	yes	—
154	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	42	2	no	—
157	C₇	D₃×#101	1	1	yes	characteristic, complemented, Sylow
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	D₇×C₂²⋊C₄	1	1	yes	characteristic
160	D₃	—	7	2	no	—
161	D₃	—	14	1	no	—
162	C₂²	—	21	1	no	—
163	C₂²	—	21	1	no	—
164	C₂²	—	42	2	no	—
165	C₂²	C₂×C₄×D₂₁	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	21	2	no	—
168	C₄	—	42	2	no	—
169	C₄	D₄×D₂₁	1	2	yes	—
170	C₃	D₇×#101	1	1	yes	characteristic, complemented, Sylow
171	C₂	—	2	1	no	—
172	C₂	—	21	2	no	—
173	C₂	—	42	1	no	—
174	C₂	C₂²⋊C₄×D₂₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₂₁

C₂

yes

maximal, characteristic

M₁₆×D₂₁

C₂

yes

maximal

C₃×C₇×#101

C₂

yes

maximal, characteristic, complemented

#10620675

C₂

yes

maximal, characteristic, complemented

#10620663

C₂

yes

maximal, complemented

D₇×#101

—

maximal, Hall

C₂×C₃×C₇×Q₈

C₂²

yes

characteristic

Q₈×D₂₁

—

Q₈×D₂₁

—

C₂×C₄×D₂₁

C₄

yes

—

C₂×C₄×D₂₁

C₂²

yes

characteristic

C₂×Dic₄₂

C₄

yes

—

C₂×Dic₄₂

C₂²

yes

characteristic

C₃×C₇×M₁₆

C₂²

yes

—

#66627

C₂²

yes

—

#66622

—

C₈×D₂₁

—

C₂×Q₈×D₇

—

D₇×M₁₆

—

C₇×#101

D₃

yes

characteristic, complemented

—

—

—

C₂²×D₂₁

C₂×C₄

yes

characteristic

C₃×C₇×Q₈

—

C₂×C₃×C₄×C₇

C₂×C₄

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₂₁

C₂×C₄

yes

—

C₂×Dic₂₁

C₂×C₄

yes

characteristic

C₄×D₂₁

—

C₄×D₂₁

—

C₄×D₂₁

D₄

yes

—

C₄×D₂₁

D₄

yes

—

Dic₄₂

—

Dic₄₂

—

C₃×C₇×C₈

—

C₃⋊(C₇⋊C₈)

—

C₂×C₇×Q₈

C₂×D₃

yes

characteristic

Q₈×D₇

—

Q₈×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₇×M₁₆

C₂×D₃

yes

—

#1136

—

#1131

—

C₈×D₇

—

C₂×D₃×Q₈

—

D₃×M₁₆

—

C₃×#101

D₇

yes

characteristic, complemented

#836

—

#824

—

C₂²×C₃×C₇

C₂²×C₄

yes

characteristic

C₂×D₂₁

—

C₂×D₂₁

C₂²⋊C₄

yes

characteristic

C₂×D₂₁

C₂²⋊C₄

yes

characteristic

C₃×C₄×C₇

—

C₃×C₄×C₇

C₂×D₄

yes

—

Dic₂₁

—

Dic₂₁

C₂²⋊C₄

yes

—

C₂×#101

—

Sylow

C₂²×D₇

—

C₇×Q₈

—

C₂×C₄×C₇

C₄×D₃

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂×C₃×Q₈

C₂×D₇

yes

characteristic

D₃×Q₈

—

D₃×Q₈

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

C₂×D₇

yes

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂×C₃×C₇

—

C₂×C₃×C₇

C₂×C₂²⋊C₄

yes

characteristic

D₂₁

—

D₂₁

#101

yes

direct product factor

C₂²×Q₈

—

C₂×M₁₆

—

#101

—

#101

—

#101

D₂₁

yes

characteristic, direct product factor

C₂²×C₇

C₂×C₄×D₃

yes

characteristic

C₂×D₇

—

C₂×D₇

—

100

C₂×D₇

—

101

C₄×C₇

—

102

C₄×C₇

D₃×D₄

yes

—

103

Dic₇

—

104

Dic₇

—

105

C₂²×D₃

—

106

C₃×Q₈

—

107

C₂×C₃×C₄

C₄×D₇

yes

—

108

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

C₄×D₃

—

112

C₄×D₃

—

113

C₄×D₃

—

114

C₄×D₃

—

115

Dic₆

—

116

Dic₆

—

117

C₃×C₈

—

118

C₃⋊C₈

—

119

C₃×C₇

C₂×#101

yes

characteristic, complemented, Hall

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

C₂×D₂₁

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

127

M₁₆

—

128

M₁₆

—

129

M₁₆

C₂×D₂₁

yes

—

130

C₂×C₈

—

131

C₂×C₇

—

132

C₂×C₇

D₃×C₂²⋊C₄

yes

characteristic

133

D₇

—

134

D₇

—

135

C₂²×C₃

C₂×C₄×D₇

yes

characteristic

136

C₂×D₃

—

137

C₂×D₃

—

138

C₂×D₃

—

139

C₃×C₄

—

140

C₃×C₄

D₄×D₇

yes

—

141

Dic₃

—

142

Dic₃

—

143

C₂³

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₄×D₂₁

yes

—

154

C₂×C₄

C₂²×D₂₁

yes

characteristic

155

C₈

—

156

C₈

—

157

C₇

D₃×#101

yes

characteristic, complemented, Sylow

158

C₂×C₃

—

159

C₂×C₃

D₇×C₂²⋊C₄

yes

characteristic

160

D₃

—

161

D₃

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×C₄×D₂₁

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

D₄×D₂₁

yes

—

170

C₃

D₇×#101

yes

characteristic, complemented, Sylow

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

C₂²⋊C₄×D₂₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₂₁
commutator	C₂²×C₃×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₇×#101	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₁	C₂×C₄
cosocle	C₂×C₃×C₄×C₇	C₂³
hypercenter	#101	D₂₁
hypocenter	C₃×C₇	C₂×#101
LCS	G ▷ C₂²×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

C₂²⋊C₄×D₂₁

commutator

C₂²×C₃×C₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₂₁

Fitting

C₃×C₇×#101

C₂

socle

C₂×C₃×C₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₁

C₂×C₄

cosocle

C₂×C₃×C₄×C₇

C₂³

D₂₁

C₃×C₇

G ▷ C₂²×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
3. C₃×C₇×#101	C₂	semidirect
4. #10620675	C₂	semidirect
5. #10620663	C₂	semidirect
20. C₇×#101	D₃	semidirect
51. C₃×#101	D₇	semidirect
91. D₂₁	#101	direct (N × H)
96. #101	D₂₁	direct (N × H)
119. C₃×C₇	C₂×#101	semidirect
157. C₇	D₃×#101	semidirect
170. C₃	D₇×#101	semidirect

H ≅ G/N

type

3. C₃×C₇×#101

C₂

semidirect

4. #10620675

C₂

semidirect

5. #10620663

C₂

semidirect

20. C₇×#101

D₃

semidirect

51. C₃×#101

D₇

semidirect

91. D₂₁

#101

direct (N × H)

96. #101

D₂₁

direct (N × H)

119. C₃×C₇

C₂×#101

semidirect

157. C₇

D₃×#101

semidirect

170. C₃

D₇×#101

semidirect

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

subgroup	index
C₂×Q₈×D₂₁	2
M₁₆×D₂₁	2
C₃×C₇×#101	2
#10620675	2
#10620663	2
D₇×#101	3
D₃×#101	7

subgroup

index

C₂×Q₈×D₂₁

M₁₆×D₂₁

C₃×C₇×#101

Subgroups — #101×D₂₁ #13056387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×D21 #13056387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×D₂₁ #13056387