Group #21456331 — subgroups

Subgroups — #21456331

Order 7808.

356 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 153 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	#16840315	C₂	1	2	yes	maximal
2	#16840166	C₂	1	2	yes	maximal, complemented
3	#16840163	C₂	1	2	yes	maximal, complemented
4	#16840162	C₂	1	2	yes	maximal, complemented
5	#16840159	C₂	1	2	yes	maximal, complemented
6	#16839800	C₂	1	1	yes	maximal, characteristic
7	#16839798	C₂	1	2	yes	maximal
8	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#427	C₂	1	1	yes	maximal, characteristic
10	C₂²×#92418	C₂²	1	1	yes	characteristic
11	#14519776	C₂²	1	4	yes	—
12	#14519768	C₂²	1	4	yes	—
13	#14519754	—	2	4	no	—
14	C₄×Dic₁₂₂	—	2	4	no	—
15	#14519736	C₂²	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
17	#14519711	C₂²	1	2	yes	—
18	#14519710	C₂²	1	4	yes	—
19	#14519709	C₂²	1	2	yes	—
20	#14519708	C₂²	1	4	yes	—
21	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
24	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	#117×C₆₁	C₂²	1	2	yes	—
27	#107×C₆₁	C₂²	1	2	yes	—
28	#106×C₆₁	C₂²	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×Dic₆₁	D₄	1	2	yes	—
31	C₂×Dic₁₂₂	—	2	4	no	—
32	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₂²⋊Dic₆₁	—	2	2	no	—
39	C₄⋊Dic₆₁	—	2	4	no	—
40	C₄⋊Dic₆₁	C₂³	1	4	yes	—
41	#11758196	—	2	4	no	—
42	#11758196	—	2	4	no	—
43	C₄×Dic₆₁	—	2	2	no	—
44	C₄×Dic₆₁	D₄	1	2	yes	—
45	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
47	C₂×#92418	Q₈	1	4	yes	complemented
48	C₂×#92418	C₂³	1	1	yes	characteristic
49	C₂×#92418	C₂³	1	1	yes	characteristic
50	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	4	no	—
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	4	no	—
57	C₂×Dic₆₁	—	2	4	no	—
58	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
59	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
60	Dic₁₂₂	—	4	8	no	—
61	C₈×C₆₁	C₂×Q₈	1	2	yes	—
62	C₈×C₆₁	C₂×Q₈	1	2	yes	—
63	#92418	C₂×Q₈	1	2	yes	—
64	#92418	C₂×Q₈	1	2	yes	—
65	C₂²×C₆₁	—	2	2	no	—
66	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₆₁	—	4	4	no	—
69	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
70	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
72	Dic₆₁	—	2	2	no	—
73	Dic₆₁	—	4	4	no	—
74	Dic₆₁	#135	1	4	yes	—
75	#3160	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	2	2	no	—
77	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#135	1	2	yes	—
79	#533	—	61	2	no	—
80	#500	—	61	2	no	—
81	#499	—	61	2	no	—
82	#499	—	61	2	no	—
83	#498	—	61	2	no	—
84	#427	—	61	1	no	—
85	#427	—	61	2	no	—
86	#427	D₆₁	1	1	yes	characteristic
87	C₂×C₄×C₈	—	61	1	no	—
88	C₆₁	#3160	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	61	1	no	—
90	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
91	C₄⋊Q₈	—	61	4	no	—
92	#125	—	61	4	no	—
93	C₂²⋊Q₈	—	122	4	no	—
94	C₄×Q₈	—	122	4	no	—
95	#117	—	61	2	no	—
96	#117	C₂×D₆₁	1	2	yes	—
97	C₂×C₄²	—	61	1	no	—
98	#107	—	61	2	no	—
99	#107	—	61	4	no	—
100	#107	C₂×D₆₁	1	2	yes	—
101	#106	—	61	2	no	—
102	#106	—	61	4	no	—
103	#106	C₂×D₆₁	1	2	yes	—
104	C₄×C₈	—	61	1	no	—
105	C₄×C₈	—	61	1	no	—
106	C₄×C₈	—	61	1	no	—
107	C₄×C₈	—	61	1	no	—
108	C₂×Q₈	—	122	4	no	—
109	C₂²×C₄	—	61	2	no	—
110	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₈	—	61	4	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
115	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₄⋊C₄	—	61	4	no	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	—	122	4	no	—
120	C₄⋊C₄	—	122	4	no	—
121	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	122	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	61	2	no	—
126	C₄²	—	61	1	no	—
127	C₄²	—	61	1	no	—
128	C₄²	—	122	2	no	—
129	C₂³	D₄×D₆₁	1	1	yes	characteristic
130	Q₈	—	244	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	61	2	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	122	4	no	—
135	C₂×C₄	—	122	4	no	—
136	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
137	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	Q₈×D₆₁	1	2	yes	—
139	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
140	C₈	—	61	2	no	—
141	C₈	—	61	2	no	—
142	C₈	C₂×Dic₁₂₂	1	2	yes	—
143	C₈	C₂×Dic₁₂₂	1	2	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	#14519776	1	2	yes	—
146	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	—	61	4	no	—
149	C₄	—	122	2	no	—
150	C₄	—	244	4	no	—
151	C₄	#14519776	1	2	yes	—
152	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
153	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	C₂×#14519776	1	1	yes	characteristic
156	C₂	#135×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840315

C₂

yes

maximal

#16840166

C₂

yes

maximal, complemented

#16840163

C₂

yes

maximal, complemented

#16840162

C₂

yes

maximal, complemented

#16840159

C₂

yes

maximal, complemented

#16839800

C₂

yes

maximal, characteristic

#16839798

C₂

yes

maximal

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₆₁×#427

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

#14519776

C₂²

yes

—

#14519768

C₂²

yes

—

#14519754

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

#14519709

C₂²

yes

—

#14519708

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#117×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#135

yes

—

#3160

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

—

—

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

C₆₁

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#107

—

100

#107

C₂×D₆₁

yes

—

101

#106

—

102

#106

—

103

#106

C₂×D₆₁

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂×Q₈

—

122

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₆₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₂₂

yes

complemented

115

C₂×C₈

C₂²×D₆₁

yes

characteristic

116

C₂×C₈

C₂²×D₆₁

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

C₂²×D₆₁

yes

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

122

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

122

—

129

C₂³

D₄×D₆₁

yes

characteristic

130

Q₈

—

244

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

122

—

135

C₂×C₄

—

122

—

136

C₂×C₄

C₂×Dic₁₂₂

yes

—

137

C₂×C₄

D₄×D₆₁

yes

characteristic

138

C₂×C₄

Q₈×D₆₁

yes

—

139

C₂×C₄

C₂³×D₆₁

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×Dic₁₂₂

yes

—

143

C₈

C₂×Dic₁₂₂

yes

—

144

C₂²

—

145

C₂²

#14519776

yes

—

146

C₂²

C₂×D₄×D₆₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

122

—

150

C₄

—

244

—

151

C₄

#14519776

yes

—

152

C₄

C₂×Q₈×D₆₁

yes

characteristic

153

C₄

C₂²×Dic₁₂₂

yes

characteristic

154

C₂

—

155

C₂

C₂×#14519776

yes

characteristic

156

C₂

#135×D₆₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#427	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#427	D₆₁
hypocenter	C₆₁	#3160
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#427

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16840166	C₂	semidirect
3. #16840163	C₂	semidirect
4. #16840162	C₂	semidirect
5. #16840159	C₂	semidirect
32. C₂×C₈×C₆₁	Q₈	semidirect
47. C₂×#92418	Q₈	semidirect
88. C₆₁	#3160	semidirect
114. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16840166

C₂

semidirect

3. #16840163

C₂

semidirect

4. #16840162

C₂

semidirect

5. #16840159

C₂

semidirect

32. C₂×C₈×C₆₁

Q₈

semidirect

47. C₂×#92418

Q₈

semidirect

88. C₆₁

#3160

semidirect

114. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21456331

Conjugacy classes of subgroups

Complements