Group #21461109 — subgroups

Subgroups — C₂×#16840612 #21461109

Order 7808.

280 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 106 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⋊#92420	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#92420	C₂	1	1	yes	maximal, characteristic
3	#16840612	C₂	1	4	yes	maximal, direct product factor
4	C₂×#14519742	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92420	—	2	1	no	—
6	C₂²×#92420	—	2	1	no	—
7	C₄⋊#92420	—	2	1	no	—
8	C₄⋊#92420	C₂²	1	2	yes	—
9	C₄×#92420	—	2	1	no	—
10	C₄×#92420	C₂²	1	2	yes	—
11	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
12	#14519742	—	2	1	no	—
13	#14519742	—	2	1	no	—
14	#14519742	C₄	1	2	yes	—
15	#14519742	C₂²	1	2	yes	—
16	C₂×#11758196	—	2	1	no	—
17	C₂×C₄×Dic₆₁	—	2	1	no	—
18	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	—	2	1	no	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	1	no	—
23	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₈	1	4	yes	complemented
28	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₂×#92420	—	2	4	no	—
30	C₂×#92420	—	2	2	no	—
31	C₂×#92420	—	2	2	no	—
32	C₂×#92420	—	4	2	no	—
33	C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₄×Dic₆₁	—	2	2	no	—
39	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₆₁	D₄	1	1	yes	characteristic
41	C₄×Dic₆₁	D₄	1	1	yes	characteristic
42	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
43	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	2	2	no	—
52	C₂×Dic₆₁	—	2	2	no	—
53	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	SD₁₆	1	2	yes	—
56	C₂×Dic₆₁	Q₁₆	1	2	yes	—
57	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	Dic₁₂₂	—	2	4	no	—
60	Dic₁₂₂	—	2	2	no	—
61	Dic₁₂₂	C₂×C₈	1	4	yes	complemented
62	#92420	—	2	4	no	—
63	#92420	—	4	2	no	—
64	C₂²×C₆₁	C₂²⋊C₈	1	2	yes	—
65	C₂²×C₆₁	#103	1	2	yes	—
66	C₂²×C₆₁	#104	1	2	yes	—
67	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₆₁	—	4	2	no	—
69	C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
70	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
71	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	Dic₆₁	—	2	4	no	—
73	Dic₆₁	—	2	1	no	—
74	Dic₆₁	—	2	1	no	—
75	Dic₆₁	#103	1	2	yes	—
76	Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Q₈⋊C₈	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	Q₈⋊C₈	1	4	yes	complemented
80	C₂×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#104	1	1	yes	characteristic
83	C₂×C₄×Q₈	—	61	1	no	—
84	C₂×C₄⋊C₈	—	61	1	no	—
85	C₂×C₄×C₈	—	61	1	no	—
86	Q₈⋊C₈	—	61	4	no	—
87	C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	61	1	no	—
89	C₂²×C₈	—	122	1	no	—
90	C₂²×C₈	—	122	1	no	—
91	C₄×Q₈	—	61	2	no	—
92	C₄×Q₈	—	61	2	no	—
93	C₄×Q₈	—	122	1	no	—
94	C₄×Q₈	—	122	1	no	—
95	C₂×C₄⋊C₄	—	122	1	no	—
96	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
97	C₂×C₄²	—	61	1	no	—
98	C₂×C₄²	—	122	1	no	—
99	C₄⋊C₈	—	61	2	no	—
100	C₄⋊C₈	—	122	1	no	—
101	C₄×C₈	—	61	2	no	—
102	C₄×C₈	—	122	1	no	—
103	C₂×Q₈	—	61	4	no	—
104	C₂×Q₈	—	61	2	no	—
105	C₂×Q₈	—	122	2	no	—
106	C₂×Q₈	—	122	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	122	1	no	—
111	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
112	C₂×C₈	—	122	2	no	—
113	C₂×C₈	—	122	4	no	—
114	C₂×C₈	—	122	2	no	—
115	C₂×C₈	—	244	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	122	2	no	—
118	C₄⋊C₄	—	122	2	no	—
119	C₄⋊C₄	C₂×#6923	1	2	yes	—
120	C₄²	—	61	1	no	—
121	C₄²	—	61	1	no	—
122	C₄²	—	61	1	no	—
123	C₄²	—	61	1	no	—
124	C₄²	—	122	2	no	—
125	C₄²	—	122	2	no	—
126	C₂³	C₂²⋊#6923	1	1	yes	characteristic
127	Q₈	—	61	4	no	—
128	Q₈	—	122	4	no	—
129	Q₈	—	122	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	61	2	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	122	4	no	—
139	C₂×C₄	—	122	2	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	D₆₁⋊C₈	1	2	yes	—
142	C₂×C₄	#11758213	1	2	yes	—
143	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
144	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
145	C₈	—	122	4	no	—
146	C₈	—	244	2	no	—
147	C₂²	#14519834	1	2	yes	—
148	C₂²	#14519840	1	2	yes	—
149	C₂²	#14519841	1	2	yes	—
150	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	61	2	no	—
153	C₄	—	61	1	no	—
154	C₄	—	61	1	no	—
155	C₄	—	122	4	no	—
156	C₄	—	122	1	no	—
157	C₄	—	122	1	no	—
158	C₄	#14519834	1	2	yes	—
159	C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
160	C₄	C₂×#11758213	1	1	yes	characteristic
161	C₂	#16840612	1	4	yes	direct product factor
162	C₂	C₂×#14519834	1	1	yes	characteristic
163	C₂	C₂×#14519840	1	1	yes	characteristic
164	C₂	C₂×#14519841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92420

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₂

yes

maximal, characteristic

#16840612

C₂

yes

maximal, direct product factor

C₂×#14519742

C₂

yes

maximal, characteristic

C₂²×#92420

—

C₂²×#92420

—

C₄⋊#92420

—

C₄⋊#92420

C₂²

yes

—

C₄×#92420

—

C₄×#92420

C₂²

yes

—

C₂²×Dic₁₂₂

C₄

yes

—

—

C₄

yes

—

#14519742

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₈

yes

complemented

C₂×Dic₁₂₂

C₂×C₄

yes

—

—

—

—

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

SD₁₆

yes

—

C₂×Dic₆₁

Q₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

C₂×C₈

yes

complemented

#92420

—

#92420

—

C₂²×C₆₁

C₂²⋊C₈

yes

—

C₂²×C₆₁

#103

yes

—

C₂²×C₆₁

#104

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊C₈

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#103

yes

—

Dic₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

C₂×Q₁₆

yes

characteristic

C₂×Q₈⋊C₈

—

maximal, Sylow

C₂×C₆₁

Q₈⋊C₈

yes

complemented

C₂×C₆₁

C₂×C₂²⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₆₁

C₂×Q₈⋊C₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

122

—

C₂²×C₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

#6923

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

122

—

C₄⋊C₈

—

100

C₄⋊C₈

—

122

—

101

C₄×C₈

—

102

C₄×C₈

—

122

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

122

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

122

—

111

C₂²×C₄

C₂×#6923

yes

characteristic

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

122

—

115

C₂×C₈

—

244

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

C₂×#6923

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

122

—

125

C₄²

—

122

—

126

C₂³

C₂²⋊#6923

yes

characteristic

127

Q₈

—

128

Q₈

—

122

—

129

Q₈

—

122

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

D₆₁⋊C₈

yes

—

142

C₂×C₄

#11758213

yes

—

143

C₂×C₄

C₂²⋊#6923

yes

characteristic

144

C₂×C₄

C₂²×#6923

yes

characteristic

145

C₈

—

122

—

146

C₈

—

244

—

147

C₂²

#14519834

yes

—

148

C₂²

#14519840

yes

—

149

C₂²

#14519841

yes

—

150

C₂²

C₂×C₂²⋊#6923

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

122

—

156

C₄

—

122

—

157

C₄

—

122

—

158

C₄

#14519834

yes

—

159

C₄

C₂×D₆₁⋊C₈

yes

characteristic

160

C₄

C₂×#11758213

yes

characteristic

161

C₂

#16840612

yes

direct product factor

162

C₂

C₂×#14519834

yes

characteristic

163

C₂

C₂×#14519840

yes

characteristic

164

C₂

C₂×#14519841

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
socle	C₂³×C₆₁	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	C₂×Q₈⋊C₈
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂×C₄⋊C₄×C₆₁

C₄

socle

C₂³×C₆₁

C₂²⋊C₄

hypercenter

C₂×C₄⋊C₄

#6923

hypocenter

C₆₁

C₂×Q₈⋊C₈

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #16840612	C₂	direct (N × H)
27. C₂×Dic₁₂₂	C₈	semidirect
61. Dic₁₂₂	C₂×C₈	semidirect
79. C₂×C₆₁	Q₈⋊C₈	semidirect
87. C₆₁	C₂×Q₈⋊C₈	semidirect
161. C₂	#16840612	direct (N × H)

H ≅ G/N

type

3. #16840612

C₂

direct (N × H)

27. C₂×Dic₁₂₂

C₈

semidirect

61. Dic₁₂₂

C₂×C₈

semidirect

79. C₂×C₆₁

Q₈⋊C₈

semidirect

87. C₆₁

C₂×Q₈⋊C₈

semidirect

161. C₂

#16840612

direct (N × H)

Subgroups — C₂×#16840612 #21461109

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840612 #21461109

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840612 #21461109