Group #21454396 — subgroups

Subgroups — C₂×#16839797 #21454396

Order 7808.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	C₂²×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519735	C₂	1	2	yes	maximal
3	#16839797	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14519710	C₂	1	2	yes	maximal, complemented
5	C₂×#14519709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₆₁	C₂²	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₆₁	C₄	1	8	yes	complemented
10	C₂×C₈×D₆₁	C₂²	1	4	yes	—
11	#14519735	—	2	4	no	—
12	#14519735	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
14	C₂×#11758198	—	2	2	no	—
15	C₂×C₄×Dic₆₁	—	2	2	no	—
16	#14519710	C₂²	1	8	yes	complemented
17	#14519709	C₂²	1	4	yes	complemented
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	#106×C₆₁	C₂²	1	4	yes	complemented
21	C₂³×D₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	—	2	2	no	—
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	2	2	no	—
25	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₂×C₄	1	4	yes	—
27	C₂×C₄×D₆₁	D₄	1	2	yes	—
28	C₂×C₄×D₆₁	Q₈	1	4	yes	—
29	C₂×C₄×D₆₁	C₂³	1	2	yes	—
30	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₆₁	C₂³	1	2	yes	—
32	C₄⋊C₄×C₆₁	—	2	2	no	—
33	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
34	#11758198	—	2	8	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	C₂³	1	4	yes	—
37	C₄×Dic₆₁	—	2	8	no	—
38	C₂×#92418	C₂×C₄	1	4	yes	—
39	C₂×#92418	C₂³	1	2	yes	—
40	C₈×D₆₁	C₂×C₄	1	16	yes	complemented
41	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₆₁	—	2	4	no	—
43	C₂²×D₆₁	C₄⋊C₄	1	4	yes	—
44	C₂²×D₆₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	—	4	4	no	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	4	no	—
52	C₂×Dic₆₁	—	4	4	no	—
53	C₂×Dic₆₁	C₄⋊C₄	1	4	yes	—
54	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
55	C₄×D₆₁	C₄⋊C₄	1	8	yes	—
56	C₄×D₆₁	C₂²×C₄	1	4	yes	—
57	C₄×D₆₁	C₂×Q₈	1	4	yes	—
58	C₈×C₆₁	C₂²×C₄	1	4	yes	—
59	#92418	C₂²×C₄	1	4	yes	—
60	C₂²×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
61	C₂²×C₆₁	#135	1	4	yes	—
62	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₆₁	—	2	4	no	—
64	C₂×D₆₁	—	2	8	no	—
65	C₂×D₆₁	C₂×C₄⋊C₄	1	4	yes	—
66	C₄×C₆₁	—	4	4	no	—
67	C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
70	Dic₆₁	—	4	4	no	—
71	Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×#427	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#427	1	4	yes	complemented
74	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₆₁	C₂×#135	1	2	yes	—
76	D₆₁	—	2	4	no	—
77	C₂³×C₈	—	61	1	no	—
78	C₂×#117	—	61	2	no	—
79	#427	—	61	8	no	—
80	C₂×#107	—	61	2	no	—
81	C₂×#106	—	61	1	no	—
82	C₂×#106	D₆₁	1	1	yes	characteristic, complemented
83	C₆₁	C₂×#427	1	1	yes	characteristic, complemented, Sylow
84	C₂³×C₄	—	61	1	no	—
85	C₂²×C₈	—	61	1	no	—
86	C₂²×C₈	—	61	8	no	—
87	C₂²×C₈	—	61	4	no	—
88	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
89	#117	—	61	8	no	—
90	#117	—	122	4	no	—
91	C₂×C₄⋊C₄	—	61	2	no	—
92	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
93	C₂×C₂²⋊C₄	—	122	2	no	—
94	C₂×C₄²	—	122	2	no	—
95	#107	—	61	8	no	—
96	#106	—	61	4	no	—
97	#106	C₂×D₆₁	1	4	yes	complemented
98	C₂⁴	—	61	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	61	1	no	—
101	C₂²×C₄	—	61	4	no	—
102	C₂²×C₄	—	61	2	no	—
103	C₂²×C₄	—	61	4	no	—
104	C₂²×C₄	—	61	2	no	—
105	C₂²×C₄	—	122	2	no	—
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	—	61	4	no	—
108	C₂×C₈	—	61	2	no	—
109	C₂×C₈	—	61	16	no	—
110	C₂×C₈	C₄×D₆₁	1	4	yes	—
111	C₂×C₈	C₂²×D₆₁	1	2	yes	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	61	4	no	—
114	C₄⋊C₄	—	122	2	no	—
115	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
116	C₂²⋊C₄	—	122	8	no	—
117	C₄²	—	122	8	no	—
118	C₂³	—	61	2	no	—
119	C₂³	—	61	4	no	—
120	C₂³	—	122	4	no	—
121	C₂³	D₄×D₆₁	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	61	2	no	—
125	C₂×C₄	—	61	4	no	—
126	C₂×C₄	—	61	4	no	—
127	C₂×C₄	—	61	4	no	—
128	C₂×C₄	—	61	8	no	—
129	C₂×C₄	—	122	4	no	—
130	C₂×C₄	—	244	4	no	—
131	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
132	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₆₁	1	2	yes	—
134	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
135	C₈	—	61	4	no	—
136	C₈	C₂×C₄×D₆₁	1	4	yes	—
137	C₂²	—	61	4	no	—
138	C₂²	—	122	4	no	—
139	C₂²	—	122	8	no	—
140	C₂²	C₄⋊C₄×D₆₁	1	2	yes	—
141	C₂²	#14519798	1	4	yes	—
142	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	61	4	no	—
145	C₄	—	244	4	no	—
146	C₄	C₄⋊C₄×D₆₁	1	2	yes	—
147	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
148	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
149	C₂	—	122	4	no	—
150	C₂	#16839797	1	4	yes	direct product factor
151	C₂	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
152	C₂	C₂×#14519798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519735

C₂

yes

maximal

#16839797

C₂

yes

maximal, direct product factor

C₂×#14519710

C₂

yes

maximal, complemented

C₂×#14519709

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₆₁

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₄

yes

complemented

C₂×C₈×D₆₁

C₂²

yes

—

#14519735

—

#14519735

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758198

—

C₂×C₄×Dic₆₁

—

#14519710

C₂²

yes

complemented

#14519709

C₂²

yes

complemented

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

complemented

C₂³×D₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

Q₈

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂³

yes

—

C₈×D₆₁

C₂×C₄

yes

complemented

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

C₄⋊C₄

yes

—

C₂²×D₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₄×D₆₁

C₄⋊C₄

yes

—

C₄×D₆₁

C₂²×C₄

yes

—

C₄×D₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂²×C₄

yes

—

#92418

C₂²×C₄

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

#135

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

C₂×C₄⋊C₄

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

D₆₁

—

C₂³×C₈

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₆₁

C₂×#427

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

#117

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₄

—

122

—

C₂×C₄²

—

122

—

#107

—

#106

—

#106

C₂×D₆₁

yes

complemented

C₂⁴

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

122

—

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄×D₆₁

yes

—

111

C₂×C₈

C₂²×D₆₁

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

C₂²×D₆₁

yes

—

116

C₂²⋊C₄

—

122

—

117

C₄²

—

122

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

122

—

121

C₂³

D₄×D₆₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

244

—

131

C₂×C₄

C₂×C₄×D₆₁

yes

—

132

C₂×C₄

D₄×D₆₁

yes

characteristic

133

C₂×C₄

Q₈×D₆₁

yes

—

134

C₂×C₄

C₂³×D₆₁

yes

characteristic

135

C₈

—

136

C₈

C₂×C₄×D₆₁

yes

—

137

C₂²

—

138

C₂²

—

122

—

139

C₂²

—

122

—

140

C₂²

C₄⋊C₄×D₆₁

yes

—

141

C₂²

#14519798

yes

—

142

C₂²

C₂×D₄×D₆₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

244

—

146

C₄

C₄⋊C₄×D₆₁

yes

—

147

C₄

C₂²×C₄×D₆₁

yes

characteristic

148

C₄

C₂×Q₈×D₆₁

yes

characteristic

149

C₂

—

122

—

150

C₂

#16839797

yes

direct product factor

151

C₂

C₂×C₄⋊C₄×D₆₁

yes

characteristic

152

C₂

C₂×#14519798

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	C₂×#427
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #16839797	C₂	direct (N × H)
4. C₂×#14519710	C₂	semidirect
5. C₂×#14519709	C₂	semidirect
6. C₂×#106×C₆₁	C₂	semidirect
9. C₂×C₈×D₆₁	C₄	semidirect
16. #14519710	C₂²	semidirect
17. #14519709	C₂²	semidirect
20. #106×C₆₁	C₂²	semidirect
40. C₈×D₆₁	C₂×C₄	semidirect
73. C₂×C₆₁	#427	semidirect
82. C₂×#106	D₆₁	semidirect
83. C₆₁	C₂×#427	semidirect
97. #106	C₂×D₆₁	semidirect
150. C₂	#16839797	direct (N × H)

H ≅ G/N

type

3. #16839797

C₂

direct (N × H)

4. C₂×#14519710

C₂

semidirect

5. C₂×#14519709

C₂

semidirect

6. C₂×#106×C₆₁

C₂

semidirect

9. C₂×C₈×D₆₁

C₄

semidirect

16. #14519710

C₂²

semidirect

17. #14519709

C₂²

semidirect

20. #106×C₆₁

C₂²

semidirect

40. C₈×D₆₁

C₂×C₄

semidirect

73. C₂×C₆₁

#427

semidirect

82. C₂×#106

D₆₁

semidirect

83. C₆₁

C₂×#427

semidirect

97. #106

C₂×D₆₁

semidirect

150. C₂

#16839797

direct (N × H)

Subgroups — C₂×#16839797 #21454396

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16839797 #21454396

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16839797 #21454396