Group #21441981 — subgroups

Subgroups — C₆₁×#1940 #21441981

Order 7808.

248 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 136 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⋊C₈×C₆₁	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#336	C₂	1	2	yes	maximal
6	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
10	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₆₁	—	2	4	no	—
13	C₄⋊C₈×C₆₁	—	2	2	no	—
14	#97×C₆₁	—	2	2	no	—
15	#95×C₆₁	C₄	1	1	yes	characteristic
16	#95×C₆₁	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	—	2	1	no	—
25	C₂×C₈×C₆₁	—	2	1	no	—
26	C₂×C₈×C₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	D₄	1	2	yes	—
30	C₂×C₈×C₆₁	Q₈	1	2	yes	—
31	C₄⋊C₄×C₆₁	—	2	4	no	—
32	C₄²×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	—	2	2	no	—
34	C₄²×C₆₁	D₄	1	2	yes	—
35	C₄²×C₆₁	Q₈	1	2	yes	—
36	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₆₁	M₁₆	1	4	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₈×C₆₁	—	2	2	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	—	4	2	no	—
56	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₆₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₆₁	#131	1	2	yes	—
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	2	4	no	—
64	C₄×C₆₁	—	4	2	no	—
65	C₄×C₆₁	#121	1	1	yes	characteristic
66	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₆₁	#125	1	1	yes	characteristic
68	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
69	#1940	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₆₁	#389	1	1	yes	characteristic
71	C₂×C₆₁	#423	1	1	yes	characteristic
72	C₂×C₆₁	#424	1	1	yes	characteristic
73	C₂×C₆₁	#435	1	2	yes	—
74	C₂×C₆₁	#446	1	2	yes	—
75	C₂×C₄⋊C₈	C₂×C₆₁	1	2	yes	—
76	C₂×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
77	C₂×#97	C₂×C₆₁	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
79	#336	C₂×C₆₁	1	2	yes	—
80	C₆₁	#1940	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₈	C₂²×C₆₁	1	2	yes	—
82	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₄×C₆₁	1	2	yes	—
85	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
86	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	2	2	no	—
89	#97	—	2	2	no	—
90	#95	C₄×C₆₁	1	1	yes	characteristic
91	#95	C₄×C₆₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₆₁	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₄×C₆₁	1	2	yes	—
96	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	D₄×C₆₁	1	2	yes	—
105	C₂×C₈	Q₈×C₆₁	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	D₄×C₆₁	1	2	yes	—
110	C₄²	Q₈×C₆₁	1	2	yes	—
111	C₂³	C₂²×C₄×C₆₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
120	C₂×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
121	C₂×C₄	M₁₆×C₆₁	1	4	yes	—
122	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
124	C₂×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
126	C₂×C₄	#41×C₆₁	1	1	yes	characteristic
127	C₂×C₄	#41×C₆₁	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₂²	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
132	C₂²	C₄×D₄×C₆₁	1	1	yes	characteristic
133	C₂²	C₄×Q₈×C₆₁	1	1	yes	characteristic
134	C₂²	C₂×M₁₆×C₆₁	1	2	yes	—
135	C₂²	#131×C₆₁	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#121×C₆₁	1	1	yes	characteristic
141	C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
142	C₄	#125×C₆₁	1	1	yes	characteristic
143	C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
144	C₂	C₆₁×#389	1	1	yes	characteristic
145	C₂	C₆₁×#423	1	1	yes	characteristic
146	C₂	C₆₁×#424	1	1	yes	characteristic
147	C₂	C₆₁×#435	1	2	yes	—
148	C₂	C₆₁×#446	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal

C₂×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#97×C₆₁

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#336

C₂

yes

maximal

C₂²×C₈×C₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#97×C₆₁

—

#95×C₆₁

C₄

yes

characteristic

#95×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂²×C₆₁

C₄×Q₈

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

#131

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#121

yes

characteristic

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#125

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

#1940

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#446

yes

—

C₂×C₄⋊C₈

C₂×C₆₁

yes

—

C₂×C₄⋊C₈

C₂×C₆₁

yes

characteristic

C₂×#97

C₂×C₆₁

yes

characteristic

C₄×C₄⋊C₄

C₂×C₆₁

yes

characteristic

#336

C₂×C₆₁

yes

—

C₆₁

#1940

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₆₁

yes

—

C₂²×C₈

C₂²×C₆₁

yes

characteristic

C₂²×C₈

C₂²×C₆₁

yes

characteristic

C₂×C₄⋊C₄

C₄×C₆₁

yes

—

C₂×C₄²

C₂²×C₆₁

yes

—

C₂×C₄²

C₂²×C₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

C₄×C₆₁

yes

characteristic

#95

C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₁

yes

—

C₂²×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₁

yes

—

C₂²×C₄

C₂³×C₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×C₆₁

yes

—

105

C₂×C₈

Q₈×C₆₁

yes

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

D₄×C₆₁

yes

—

110

C₄²

Q₈×C₆₁

yes

—

111

C₂³

C₂²×C₄×C₆₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊C₄×C₆₁

yes

—

120

C₂×C₄

C₄⋊C₄×C₆₁

yes

—

121

C₂×C₄

M₁₆×C₆₁

yes

—

122

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

123

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

124

C₂×C₄

C₂×Q₈×C₆₁

yes

characteristic

125

C₂×C₄

C₂×Q₈×C₆₁

yes

characteristic

126

C₂×C₄

#41×C₆₁

yes

characteristic

127

C₂×C₄

#41×C₆₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₂×C₄⋊C₄×C₆₁

yes

characteristic

132

C₂²

C₄×D₄×C₆₁

yes

characteristic

133

C₂²

C₄×Q₈×C₆₁

yes

characteristic

134

C₂²

C₂×M₁₆×C₆₁

yes

—

135

C₂²

#131×C₆₁

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#121×C₆₁

yes

characteristic

141

C₄

C₂²⋊Q₈×C₆₁

yes

characteristic

142

C₄

#125×C₆₁

yes

characteristic

143

C₄

C₄⋊Q₈×C₆₁

yes

characteristic

144

C₂

C₆₁×#389

yes

characteristic

145

C₂

C₆₁×#423

yes

characteristic

146

C₂

C₆₁×#424

yes

characteristic

147

C₂

C₆₁×#435

yes

—

148

C₂

C₆₁×#446

yes

—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂³
commutator	C₂³	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₆₁

C₂³

commutator

C₂³

C₂²×C₄×C₆₁

Frattini, cosocle

C₂²×C₄

C₂³×C₆₁

socle, prime radical

C₂³×C₆₁

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₆₁ ◁ G

N	H ≅ G/N	type
69. #1940	C₆₁	direct (N × H)
80. C₆₁	#1940	direct (N × H)

H ≅ G/N

type

69. #1940

C₆₁

direct (N × H)

80. C₆₁

#1940

direct (N × H)

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

subgroup	index
C₂×C₄⋊C₈×C₆₁	2
C₂×#97×C₆₁	2
C₄×C₄⋊C₄×C₆₁	2
C₆₁×#336	2
#1940	61

subgroup

index

C₂×C₄⋊C₈×C₆₁

C₂×#97×C₆₁

C₄×C₄⋊C₄×C₆₁

C₆₁×#336

#1940

Subgroups — C₆₁×#1940 #21441981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1940 #21441981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1940 #21441981