Group #21455581 — subgroups

Subgroups — C₂×#16840044 #21455581

Order 7808.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16840044	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519708	C₂	1	2	yes	maximal
4	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519700	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	2	yes	—
9	C₄×Dic₁₂₂	—	2	2	no	—
10	C₄×Dic₁₂₂	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
12	C₂×#11758196	—	2	1	no	—
13	C₂×C₄×Dic₆₁	—	2	1	no	—
14	#14519708	C₂²	1	8	yes	—
15	C₄⋊#92418	C₂²	1	4	yes	—
16	#14519700	C₂²	1	8	yes	—
17	C₂²×Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	—	2	2	no	—
19	C₄⋊Q₈×C₆₁	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
21	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	4	no	—
24	C₂×Q₈×C₆₁	—	4	4	no	—
25	C₂²×C₄×C₆₁	—	2	2	no	—
26	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₆₁	—	2	1	no	—
30	C₂²×Dic₆₁	—	2	1	no	—
31	C₂×Dic₁₂₂	—	2	1	no	—
32	C₂×Dic₁₂₂	—	2	2	no	—
33	C₂×Dic₁₂₂	Q₈	1	4	yes	complemented
34	C₂×Dic₁₂₂	C₂³	1	2	yes	—
35	C₄⋊C₄×C₆₁	—	2	4	no	—
36	C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
38	C₄²×C₆₁	D₄	1	2	yes	—
39	C₄²×C₆₁	C₂³	1	2	yes	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	C₄⋊Dic₆₁	C₂³	1	2	yes	—
42	#11758196	—	2	4	no	—
43	C₄×Dic₆₁	—	2	4	no	—
44	C₂×#92418	—	2	4	no	—
45	C₂×#92418	C₂³	1	4	yes	—
46	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₆₁	—	4	8	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	4	no	—
50	C₂×C₄×C₆₁	—	4	4	no	—
51	C₂×C₄×C₆₁	SD₁₆	1	4	yes	—
52	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₆₁	#41	1	2	yes	—
57	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	4	no	—
59	C₂×Dic₆₁	—	2	2	no	—
60	C₂×Dic₆₁	—	2	2	no	—
61	C₂×Dic₆₁	—	4	2	no	—
62	Dic₁₂₂	—	2	4	no	—
63	Dic₁₂₂	—	2	2	no	—
64	Dic₁₂₂	C₂×Q₈	1	4	yes	complemented
65	#92418	—	2	4	no	—
66	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₆₁	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₆₁	#137	1	2	yes	—
69	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	4	4	no	—
72	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
74	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
76	Dic₆₁	—	2	4	no	—
77	Dic₆₁	—	4	2	no	—
78	C₂×#475	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	#475	1	4	yes	complemented
80	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#137	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
84	C₂×C₄×Q₈	—	61	1	no	—
85	#475	—	61	8	no	—
86	C₂×#106	—	61	2	no	—
87	C₂×C₄⋊C₈	—	61	1	no	—
88	C₂×#103	—	61	2	no	—
89	C₆₁	C₂×#475	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	61	1	no	—
92	C₂²×C₈	—	61	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂×D₆₁	1	4	yes	—
95	C₄×Q₈	—	61	4	no	—
96	C₄×Q₈	—	122	2	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	61	1	no	—
99	C₂×C₄⋊C₄	—	122	1	no	—
100	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
101	C₂×C₄²	—	122	1	no	—
102	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
103	#106	—	61	8	no	—
104	C₄⋊C₈	—	61	4	no	—
105	#103	—	61	8	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	61	4	no	—
109	C₂×Q₈	—	61	2	no	—
110	C₂×Q₈	—	122	2	no	—
111	C₂×Q₈	—	122	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	122	1	no	—
114	C₂²×C₄	—	122	1	no	—
115	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
118	C₂×C₈	—	61	4	no	—
119	C₂×C₈	—	122	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	61	2	no	—
123	C₄⋊C₄	—	122	1	no	—
124	C₄⋊C₄	—	122	4	no	—
125	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
126	C₄²	—	122	4	no	—
127	C₄²	C₂²⋊D₆₁	1	2	yes	—
128	C₄²	C₂²×D₆₁	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	61	4	no	—
132	Q₈	—	122	2	no	—
133	Q₈	—	122	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	122	2	no	—
138	C₂×C₄	—	122	4	no	—
139	C₂×C₄	—	122	2	no	—
140	C₂×C₄	—	244	2	no	—
141	C₂×C₄	#11758200	1	4	yes	—
142	C₂×C₄	Q₈×D₆₁	1	2	yes	—
143	C₂×C₄	#11758226	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
147	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
148	C₈	—	122	4	no	—
149	C₂²	#14519753	1	2	yes	—
150	C₂²	C₂×#11758200	1	2	yes	—
151	C₂²	#14519808	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	4	4	no	—
155	C₄	—	122	4	no	—
156	C₄	—	244	2	no	—
157	C₄	#14519753	1	2	yes	—
158	C₄	C₂×#11758200	1	4	yes	—
159	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
160	C₄	C₂×#11758226	1	1	yes	characteristic
161	C₂	#16840044	1	4	yes	direct product factor
162	C₂	C₂×#14519753	1	1	yes	characteristic
163	C₂	C₂²×#11758200	1	1	yes	characteristic
164	C₂	C₂×#14519808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

#16840044

C₂

yes

maximal, direct product factor

C₂×#14519708

C₂

yes

maximal

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

C₂×#14519700

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

#14519708

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519700

C₂²

yes

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

Q₈

yes

complemented

C₂×Dic₁₂₂

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

C₂×Q₈

yes

complemented

#92418

—

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₂²×C₆₁

C₂×SD₁₆

yes

—

C₂²×C₆₁

#137

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×SD₁₆

yes

characteristic

C₂×C₆₁

C₂×#137

yes

characteristic

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

C₂×C₄×Q₈

—

#475

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₆₁

C₂×#475

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

100

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

101

C₂×C₄²

—

122

—

102

C₂×C₄²

C₂×D₆₁

yes

characteristic

103

#106

—

104

C₄⋊C₈

—

105

#103

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

122

—

111

C₂×Q₈

—

122

—

112

C₂²×C₄

—

113

C₂²×C₄

—

122

—

114

C₂²×C₄

—

122

—

115

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

116

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

117

C₂²×C₄

C₂²×D₆₁

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

122

—

124

C₄⋊C₄

—

122

—

125

C₄⋊C₄

C₂²×D₆₁

yes

—

126

C₄²

—

122

—

127

C₄²

C₂²⋊D₆₁

yes

—

128

C₄²

C₂²×D₆₁

yes

—

129

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

122

—

133

Q₈

—

122

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

122

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

244

—

141

C₂×C₄

#11758200

yes

—

142

C₂×C₄

Q₈×D₆₁

yes

—

143

C₂×C₄

#11758226

yes

—

144

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

145

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

147

C₂×C₄

C₂³×D₆₁

yes

characteristic

148

C₈

—

122

—

149

C₂²

#14519753

yes

—

150

C₂²

C₂×#11758200

yes

—

151

C₂²

#14519808

yes

—

152

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

122

—

156

C₄

—

244

—

157

C₄

#14519753

yes

—

158

C₄

C₂×#11758200

yes

—

159

C₄

C₂×Q₈×D₆₁

yes

characteristic

160

C₄

C₂×#11758226

yes

characteristic

161

C₂

#16840044

yes

direct product factor

162

C₂

C₂×#14519753

yes

characteristic

163

C₂

C₂²×#11758200

yes

characteristic

164

C₂

C₂×#14519808

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	C₂×#475
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

C₂×#475

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16840044	C₂	direct (N × H)
33. C₂×Dic₁₂₂	Q₈	semidirect
64. Dic₁₂₂	C₂×Q₈	semidirect
79. C₂×C₆₁	#475	semidirect
89. C₆₁	C₂×#475	semidirect
161. C₂	#16840044	direct (N × H)

H ≅ G/N

type

2. #16840044

C₂

direct (N × H)

33. C₂×Dic₁₂₂

Q₈

semidirect

64. Dic₁₂₂

C₂×Q₈

semidirect

79. C₂×C₆₁

#475

semidirect

89. C₆₁

C₂×#475

semidirect

161. C₂

#16840044

direct (N × H)

Subgroups — C₂×#16840044 #21455581

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840044 #21455581

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840044 #21455581