Group #21462601 — subgroups

Subgroups — C₂×#16840811 #21462601

Order 7808.

440 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 234 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	#16840811	C₂	1	8	yes	maximal, direct product factor
2	C₂×#14519844	C₂	1	2	yes	maximal
3	C₂×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92420	C₂	1	2	yes	maximal
5	C₂×C₄×#92420	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519742	C₂	1	1	yes	maximal, characteristic
7	C₂²×#92420	C₂²	1	2	yes	—
8	C₂²×#92420	C₂²	1	1	yes	characteristic
9	C₂²×#92420	C₂²	1	1	yes	characteristic
10	#14519844	C₂²	1	8	yes	—
11	C₄⋊#92420	C₂²	1	4	yes	—
12	C₄×#92420	C₂²	1	4	yes	—
13	C₄×#92420	C₂²	1	8	yes	—
14	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
15	#14519742	C₄	1	4	yes	—
16	#14519742	C₂²	1	4	yes	—
17	C₂×#11758196	C₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
25	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	C₈	1	8	yes	complemented
27	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
28	C₂×#92420	—	2	4	no	—
29	C₂×#92420	—	2	2	no	—
30	C₂×#92420	Q₈	1	4	yes	complemented
31	C₂×#92420	C₂³	1	4	yes	—
32	C₂×#92420	C₂³	1	2	yes	—
33	C₂×#92420	C₂³	1	2	yes	—
34	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
35	#11758196	C₂×C₄	1	8	yes	—
36	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
37	C₄×Dic₆₁	C₂×C₄	1	4	yes	—
38	C₄×Dic₆₁	C₂³	1	2	yes	—
39	C₄×Dic₆₁	C₂³	1	4	yes	—
40	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
45	C₂×Dic₆₁	—	2	2	no	—
46	C₂×Dic₆₁	C₂×C₈	1	8	yes	—
47	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
48	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
50	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
51	C₂×Dic₆₁	#41	1	2	yes	—
52	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
53	Dic₁₂₂	C₂×C₈	1	16	yes	complemented
54	#92420	—	2	4	no	—
55	#92420	—	2	2	no	—
56	#92420	C₂×Q₈	1	4	yes	complemented
57	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
58	C₂²×C₆₁	C₂²×C₈	1	2	yes	—
59	C₂²×C₆₁	#131	1	2	yes	—
60	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₆₁	—	2	4	no	—
62	C₄×C₆₁	C₂²×C₈	1	4	yes	—
63	Dic₆₁	—	2	2	no	—
64	Dic₆₁	C₄×Q₈	1	2	yes	—
65	Dic₆₁	C₂²×C₈	1	8	yes	—
66	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
67	Dic₆₁	C₂×#41	1	1	yes	characteristic
68	C₂×C₈×Q₈	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	C₈×Q₈	1	4	yes	complemented
70	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₆₁	C₂³×C₈	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
73	C₂×C₄×Q₈	—	61	1	no	—
74	C₈×Q₈	—	61	8	no	—
75	C₂×C₄⋊C₈	—	61	1	no	—
76	C₂×C₄⋊C₈	—	61	2	no	—
77	C₂×C₄×C₈	—	61	2	no	—
78	C₂×C₄×C₈	—	61	1	no	—
79	C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	61	1	no	—
81	C₂²×C₈	—	61	1	no	—
82	C₂²×C₈	—	61	2	no	—
83	C₂²×C₈	—	61	1	no	—
84	C₄×Q₈	—	61	4	no	—
85	C₄×Q₈	—	61	4	no	—
86	C₂×C₄⋊C₄	—	61	2	no	—
87	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
88	C₂×C₄²	—	61	2	no	—
89	C₂×C₄²	—	61	1	no	—
90	C₄⋊C₈	—	61	4	no	—
91	C₄⋊C₈	—	61	8	no	—
92	C₄×C₈	—	61	8	no	—
93	C₄×C₈	—	61	4	no	—
94	C₂×Q₈	—	61	8	no	—
95	C₂×Q₈	—	61	4	no	—
96	C₂²×C₄	—	61	1	no	—
97	C₂²×C₄	—	61	2	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	C₂×#6923	1	2	yes	—
100	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
101	C₂×C₈	—	61	2	no	—
102	C₂×C₈	—	61	4	no	—
103	C₂×C₈	—	61	4	no	—
104	C₂×C₈	—	61	2	no	—
105	C₂×C₈	—	122	4	no	—
106	C₂×C₈	—	122	2	no	—
107	C₄⋊C₄	—	61	8	no	—
108	C₄⋊C₄	C₂×#6923	1	4	yes	—
109	C₄²	—	61	4	no	—
110	C₄²	—	61	2	no	—
111	C₄²	—	61	4	no	—
112	C₄²	—	61	2	no	—
113	C₂³	C₂²×#6923	1	1	yes	characteristic
114	Q₈	—	61	16	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	61	2	no	—
117	C₂×C₄	—	61	2	no	—
118	C₂×C₄	—	61	8	no	—
119	C₂×C₄	—	61	1	no	—
120	C₂×C₄	—	61	4	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	2	no	—
123	C₂×C₄	—	122	2	no	—
124	C₂×C₄	D₆₁⋊C₈	1	4	yes	—
125	C₂×C₄	C₂²×#6923	1	4	yes	—
126	C₂×C₄	C₂²×#6923	1	2	yes	—
127	C₈	—	61	4	no	—
128	C₈	—	122	4	no	—
129	C₈	—	122	2	no	—
130	C₂²	Q₈×#6923	1	2	yes	—
131	C₂²	C₂×D₆₁⋊C₈	1	2	yes	—
132	C₂²	#14519864	1	2	yes	—
133	C₂²	C₂³×#6923	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	61	2	no	—
136	C₄	—	61	1	no	—
137	C₄	—	61	1	no	—
138	C₄	—	61	8	no	—
139	C₄	—	122	2	no	—
140	C₄	C₂×D₆₁⋊C₈	1	4	yes	—
141	C₂	#16840811	1	4	yes	direct product factor
142	C₂	C₂×Q₈×#6923	1	1	yes	characteristic
143	C₂	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
144	C₂	C₂×#14519864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840811

C₂

yes

maximal, direct product factor

C₂×#14519844

C₂

yes

maximal

C₂×C₄⋊#92420

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₂

yes

maximal

C₂×C₄×#92420

C₂

yes

maximal, characteristic

C₂×#14519742

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

—

C₂²×#92420

C₂²

yes

characteristic

C₂²×#92420

C₂²

yes

characteristic

#14519844

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₂²×Dic₁₂₂

C₄

yes

characteristic

#14519742

C₄

yes

—

#14519742

C₂²

yes

—

C₂×#11758196

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₈

yes

complemented

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

Q₈

yes

complemented

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×C₈

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

Dic₁₂₂

C₂×C₈

yes

—

—

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂²×C₈

yes

—

C₂²×C₆₁

#131

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²×C₈

yes

—

Dic₆₁

—

Dic₆₁

C₄×Q₈

yes

—

Dic₆₁

C₂²×C₈

yes

—

Dic₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

C₂×#41

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₂×C₆₁

C₈×Q₈

yes

complemented

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂³×C₈

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₆₁

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#6923

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

—

100

C₂²×C₄

C₂×#6923

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

122

—

106

C₂×C₈

—

122

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×#6923

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

C₂²×#6923

yes

characteristic

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

D₆₁⋊C₈

yes

—

125

C₂×C₄

C₂²×#6923

yes

—

126

C₂×C₄

C₂²×#6923

yes

—

127

C₈

—

128

C₈

—

122

—

129

C₈

—

122

—

130

C₂²

Q₈×#6923

yes

—

131

C₂²

C₂×D₆₁⋊C₈

yes

—

132

C₂²

#14519864

yes

—

133

C₂²

C₂³×#6923

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

122

—

140

C₄

C₂×D₆₁⋊C₈

yes

—

141

C₂

#16840811

yes

direct product factor

142

C₂

C₂×Q₈×#6923

yes

characteristic

143

C₂

C₂²×D₆₁⋊C₈

yes

characteristic

144

C₂

C₂×#14519864

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×#6923
commutator	C₂×C₆₁	C₂³×C₈
Frattini	C₂²	C₂³×#6923
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	C₂×C₈×Q₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×#6923

commutator

C₂×C₆₁

C₂³×C₈

Frattini

C₂²

C₂³×#6923

Fitting

C₂×C₄⋊C₄×C₆₁

C₄

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₂×C₄⋊C₄

#6923

hypocenter

C₆₁

C₂×C₈×Q₈

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #16840811	C₂	direct (N × H)
26. C₂×Dic₁₂₂	C₈	semidirect
30. C₂×#92420	Q₈	semidirect
53. Dic₁₂₂	C₂×C₈	semidirect
56. #92420	C₂×Q₈	semidirect
69. C₂×C₆₁	C₈×Q₈	semidirect
79. C₆₁	C₂×C₈×Q₈	semidirect
141. C₂	#16840811	direct (N × H)

H ≅ G/N

type

1. #16840811

C₂

direct (N × H)

26. C₂×Dic₁₂₂

C₈

semidirect

30. C₂×#92420

Q₈

semidirect

53. Dic₁₂₂

C₂×C₈

semidirect

56. #92420

C₂×Q₈

semidirect

69. C₂×C₆₁

C₈×Q₈

semidirect

79. C₆₁

C₂×C₈×Q₈

semidirect

141. C₂

#16840811

direct (N × H)

Subgroups — C₂×#16840811 #21462601

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840811 #21462601

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840811 #21462601