Group #21456324 — subgroups

Subgroups — #500×D₆₁ #21456324

Order 7808.

424 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₄⋊Q₈×D₆₁	C₂	1	2	yes	maximal
2	#16840167	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840166	C₂	1	4	yes	maximal, complemented
4	#16840165	C₂	1	2	yes	maximal, complemented
5	#107×D₆₁	C₂	1	4	yes	maximal
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	—	2	2	no	—
9	C₂×C₈×D₆₁	C₂²	1	2	yes	—
10	#14519777	C₂²	1	2	yes	—
11	#14519776	—	2	2	no	—
12	#14519776	C₂²	1	4	yes	—
13	#14519775	—	2	2	no	—
14	C₄⋊C₄×D₆₁	—	2	2	no	—
15	C₄⋊C₄×D₆₁	C₂²	1	4	yes	—
16	C₄²×D₆₁	C₂²	1	1	yes	characteristic
17	#14519711	C₂²	1	4	yes	—
18	#14519710	C₂²	1	8	yes	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
22	#107×C₆₁	C₂²	1	4	yes	—
23	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	—	2	2	no	—
25	Q₈×D₆₁	—	4	8	no	—
26	C₂×C₄×D₆₁	—	2	4	no	—
27	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₂	—	2	2	no	—
31	C₂×Dic₁₂₂	—	2	4	no	—
32	C₂×C₈×C₆₁	C₂³	1	2	yes	—
33	C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
35	C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₆₁	—	2	2	no	—
37	C₄⋊Dic₆₁	C₂³	1	4	yes	—
38	#11758196	—	2	4	no	—
39	#11758196	—	2	4	no	—
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	C₂×#92418	C₂³	1	2	yes	—
44	C₈×D₆₁	Q₈	1	8	yes	complemented
45	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₆₁	—	4	4	no	—
47	C₂×C₄×C₆₁	—	2	4	no	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	4	no	—
52	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄×D₆₁	—	4	8	no	—
56	C₄×D₆₁	D₈	1	4	yes	—
57	C₄×D₆₁	Q₁₆	1	4	yes	—
58	C₄×D₆₁	C₂×Q₈	1	4	yes	—
59	Dic₁₂₂	—	4	4	no	—
60	Dic₁₂₂	—	4	8	no	—
61	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
62	#92418	C₂×Q₈	1	4	yes	complemented
63	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₆₁	C₄⋊Q₈	1	2	yes	—
65	C₂×D₆₁	C₂×D₈	1	2	yes	—
66	C₂×D₆₁	C₂×Q₁₆	1	2	yes	—
67	C₄×C₆₁	—	4	4	no	—
68	C₄×C₆₁	C₂×D₈	1	2	yes	—
69	C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
70	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
71	Dic₆₁	—	4	4	no	—
72	Dic₆₁	C₄⋊Q₈	1	2	yes	—
73	Dic₆₁	C₂×D₈	1	2	yes	—
74	Dic₆₁	C₂×Q₁₆	1	2	yes	—
75	C₂×#500	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
79	D₆₁	#500	1	4	yes	direct product factor
80	C₂×C₄⋊Q₈	—	61	2	no	—
81	#500	—	61	2	no	—
82	#500	—	61	4	no	—
83	#500	—	61	1	no	—
84	#500	D₆₁	1	1	yes	characteristic, direct product factor
85	C₂×#107	—	61	4	no	—
86	C₂×C₄×C₈	—	61	1	no	—
87	C₆₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	122	2	no	—
89	C₂²×C₈	—	61	2	no	—
90	C₄⋊Q₈	—	61	4	no	—
91	C₄⋊Q₈	—	61	2	no	—
92	C₄⋊Q₈	—	122	2	no	—
93	C₄⋊Q₈	—	122	2	no	—
94	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
95	C₂×C₄⋊C₄	—	61	4	no	—
96	C₂×C₄⋊C₄	—	122	2	no	—
97	C₂×C₄²	—	61	1	no	—
98	#107	—	61	4	no	—
99	#107	—	61	8	no	—
100	#107	C₂×D₆₁	1	4	yes	—
101	C₄×C₈	—	61	2	no	—
102	C₄×C₈	—	61	1	no	—
103	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	122	4	no	—
106	C₂×Q₈	—	122	2	no	—
107	C₂×Q₈	—	244	8	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	61	1	no	—
111	C₂²×C₄	—	122	4	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	61	8	no	—
114	C₂×C₈	C₂²×D₆₁	1	2	yes	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	61	4	no	—
117	C₄⋊C₄	—	122	4	no	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	—	122	2	no	—
120	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
121	C₄²	—	61	1	no	—
122	C₄²	—	61	1	no	—
123	C₄²	—	61	1	no	—
124	C₄²	C₂²×D₆₁	1	1	yes	characteristic
125	C₂³	—	61	1	no	—
126	Q₈	—	4	4	no	—
127	Q₈	—	244	8	no	—
128	Q₈	—	244	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	61	1	no	—
131	C₂×C₄	—	61	1	no	—
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	4	no	—
134	C₂×C₄	—	61	4	no	—
135	C₂×C₄	—	61	4	no	—
136	C₂×C₄	—	122	4	no	—
137	C₂×C₄	—	244	8	no	—
138	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
139	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
141	C₈	—	61	4	no	—
142	C₈	Q₈×D₆₁	1	4	yes	complemented
143	C₂²	—	61	2	no	—
144	C₂²	—	61	2	no	—
145	C₂²	—	61	2	no	—
146	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	—	61	2	no	—
149	C₄	—	61	2	no	—
150	C₄	—	61	2	no	—
151	C₄	—	244	4	no	—
152	C₄	D₈×D₆₁	1	2	yes	—
153	C₄	Q₁₆×D₆₁	1	2	yes	—
154	C₄	C₂×Q₈×D₆₁	1	2	yes	—
155	C₂	—	61	4	no	—
156	C₂	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
157	C₂	C₂×D₈×D₆₁	1	1	yes	characteristic
158	C₂	C₂×Q₁₆×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₆₁

C₂

yes

maximal

#16840167

C₂

yes

maximal, characteristic, complemented

#16840166

C₂

yes

maximal, complemented

#16840165

C₂

yes

maximal, complemented

#107×D₆₁

C₂

yes

maximal

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₆₁×#500

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₂²

yes

—

#14519776

—

#14519776

C₂²

yes

—

#14519775

—

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

C₄×#92418

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₈×D₆₁

Q₈

yes

complemented

C₂²×D₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

D₈

yes

—

C₄×D₆₁

Q₁₆

yes

—

C₄×D₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

complemented

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

C₄⋊Q₈

yes

—

C₂×D₆₁

C₂×D₈

yes

—

C₂×D₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

Dic₆₁

C₄⋊Q₈

yes

—

Dic₆₁

C₂×D₈

yes

—

Dic₆₁

C₂×Q₁₆

yes

—

C₂×#500

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

C₂²×Q₁₆

yes

characteristic

D₆₁

#500

yes

direct product factor

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic, direct product factor

C₂×#107

—

C₂×C₄×C₈

—

C₆₁

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

#107

—

#107

—

100

#107

C₂×D₆₁

yes

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×D₆₁

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

122

—

107

C₂×Q₈

—

244

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

122

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₆₁

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

C₂²×D₆₁

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₆₁

yes

characteristic

125

C₂³

—

126

Q₈

—

127

Q₈

—

244

—

128

Q₈

—

244

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

122

—

137

C₂×C₄

—

244

—

138

C₂×C₄

D₄×D₆₁

yes

characteristic

139

C₂×C₄

D₄×D₆₁

yes

characteristic

140

C₂×C₄

C₂³×D₆₁

yes

characteristic

141

C₈

—

142

C₈

Q₈×D₆₁

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₄×D₆₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

244

—

152

C₄

D₈×D₆₁

yes

—

153

C₄

Q₁₆×D₆₁

yes

—

154

C₄

C₂×Q₈×D₆₁

yes

—

155

C₂

—

156

C₂

C₄⋊Q₈×D₆₁

yes

characteristic

157

C₂

C₂×D₈×D₆₁

yes

characteristic

158

C₂

C₂×Q₁₆×D₆₁

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#500	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#500	D₆₁
hypocenter	C₆₁	C₂×#500
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#500

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #16840167	C₂	semidirect
3. #16840166	C₂	semidirect
4. #16840165	C₂	semidirect
7. C₆₁×#500	C₂	semidirect
44. C₈×D₆₁	Q₈	semidirect
61. C₈×C₆₁	C₂×Q₈	semidirect
62. #92418	C₂×Q₈	semidirect
79. D₆₁	#500	direct (N × H)
84. #500	D₆₁	direct (N × H)
87. C₆₁	C₂×#500	semidirect
142. C₈	Q₈×D₆₁	semidirect

H ≅ G/N

type

2. #16840167

C₂

semidirect

3. #16840166

C₂

semidirect

4. #16840165

C₂

semidirect

7. C₆₁×#500

C₂

semidirect

44. C₈×D₆₁

Q₈

semidirect

61. C₈×C₆₁

C₂×Q₈

semidirect

62. #92418

C₂×Q₈

semidirect

79. D₆₁

#500

direct (N × H)

84. #500

D₆₁

direct (N × H)

87. C₆₁

C₂×#500

semidirect

142. C₈

Q₈×D₆₁

semidirect

Subgroups — #500×D₆₁ #21456324

Conjugacy classes of subgroups

Complements

Subgroups — #500×D61 #21456324

Conjugacy classes of subgroups

Complements

Subgroups — #500×D₆₁ #21456324