Group #21454746 — subgroups

Subgroups — C₂×#16839892 #21454746

Order 7808.

400 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 221 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×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519706	C₂	1	2	yes	maximal
5	C₂×#14519687	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	2	yes	—
9	C₄×Dic₁₂₂	C₄	1	4	yes	—
10	C₄×Dic₁₂₂	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758196	C₄	1	2	yes	—
13	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
14	C₄⋊#92418	C₂²	1	4	yes	—
15	#14519706	C₂²	1	8	yes	—
16	#14519687	C₂²	1	8	yes	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₆₁	C₂²	1	4	yes	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
25	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₂₂	—	2	4	no	—
27	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
30	C₂×C₈×C₆₁	C₂³	1	2	yes	—
31	C₂×C₈×C₆₁	C₂³	1	2	yes	—
32	C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₄²×C₆₁	C₂³	1	2	yes	—
34	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
35	#11758196	C₂×C₄	1	8	yes	—
36	C₄×Dic₆₁	C₂×C₄	1	4	yes	—
37	C₄×Dic₆₁	C₂³	1	4	yes	—
38	C₂×#92418	—	2	4	no	—
39	C₂×#92418	C₂³	1	4	yes	—
40	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	M₁₆	1	4	yes	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₆₁	#41	1	2	yes	—
48	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₆₁	—	2	4	no	—
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
52	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
53	Dic₁₂₂	—	2	8	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
56	#92418	—	2	4	no	—
57	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
58	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
59	C₂²×C₆₁	#131	1	2	yes	—
60	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	C₄×Q₈	1	2	yes	—
63	C₄×C₆₁	C₂×M₁₆	1	4	yes	—
64	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
66	Dic₆₁	—	2	4	no	—
67	Dic₆₁	—	2	4	no	—
68	C₂×#446	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	#446	1	4	yes	complemented
70	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
73	C₂×C₄×Q₈	—	61	1	no	—
74	#446	—	61	8	no	—
75	C₂×C₄⋊C₈	—	61	2	no	—
76	C₂×C₄⋊C₈	—	61	1	no	—
77	C₂×#97	—	61	2	no	—
78	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
79	C₆₁	C₂×#446	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	61	1	no	—
81	C₂²×C₈	—	61	2	no	—
82	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
84	C₄×Q₈	—	61	4	no	—
85	C₄×Q₈	—	61	4	no	—
86	C₂×C₄⋊C₄	—	61	2	no	—
87	C₂×C₄⋊C₄	—	61	1	no	—
88	C₂×C₄²	—	61	2	no	—
89	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
90	C₄⋊C₈	—	61	8	no	—
91	C₄⋊C₈	—	61	4	no	—
92	#97	—	61	8	no	—
93	C₄×C₈	C₂×D₆₁	1	4	yes	—
94	C₂×Q₈	—	61	4	no	—
95	C₂×Q₈	—	122	4	no	—
96	C₂²×C₄	—	61	2	no	—
97	C₂²×C₄	—	61	2	no	—
98	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	61	4	no	—
103	C₂×C₈	—	122	4	no	—
104	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
105	C₂×C₈	C₂²×D₆₁	1	2	yes	—
106	C₂×C₈	C₂²×D₆₁	1	2	yes	—
107	C₄⋊C₄	—	61	8	no	—
108	C₄⋊C₄	—	61	4	no	—
109	C₄²	—	61	4	no	—
110	C₄²	—	61	4	no	—
111	C₄²	C₄×D₆₁	1	2	yes	—
112	C₄²	C₂²×D₆₁	1	2	yes	—
113	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
114	Q₈	—	122	8	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	61	4	no	—
117	C₂×C₄	—	61	4	no	—
118	C₂×C₄	—	122	4	no	—
119	C₂×C₄	—	122	4	no	—
120	C₂×C₄	#11758189	1	4	yes	—
121	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
125	C₂×C₄	#11758222	1	2	yes	—
126	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	122	4	no	—
129	C₈	C₂×Dic₁₂₂	1	4	yes	complemented
130	C₂²	C₄×Dic₁₂₂	1	2	yes	—
131	C₂²	C₂×#11758189	1	2	yes	—
132	C₂²	#14519783	1	2	yes	—
133	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	122	4	no	—
136	C₄	—	122	4	no	—
137	C₄	C₄×Dic₁₂₂	1	2	yes	—
138	C₄	C₂×#11758189	1	4	yes	—
139	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
140	C₄	C₂×#11758222	1	1	yes	characteristic
141	C₂	#16839892	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
143	C₂	C₂²×#11758189	1	1	yes	characteristic
144	C₂	C₂×#14519783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, direct product factor

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

C₂×#14519706

C₂

yes

maximal

C₂×#14519687

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₄

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₄×Dic₁₂₂

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

#14519687

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

complemented

#92418

—

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

#131

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

C₂×C₄×Q₈

—

#446

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

D₆₁

yes

characteristic

C₆₁

C₂×#446

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

100

C₂²×C₄

C₂²×D₆₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

Dic₁₂₂

yes

complemented

105

C₂×C₈

C₂²×D₆₁

yes

—

106

C₂×C₈

C₂²×D₆₁

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₆₁

yes

—

112

C₄²

C₂²×D₆₁

yes

—

113

C₂³

C₂×C₄×D₆₁

yes

characteristic

114

Q₈

—

122

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

122

—

120

C₂×C₄

#11758189

yes

—

121

C₂×C₄

C₂×Dic₁₂₂

yes

—

122

C₂×C₄

C₂×C₄×D₆₁

yes

—

123

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

124

C₂×C₄

C₂×C₄×D₆₁

yes

—

125

C₂×C₄

#11758222

yes

—

126

C₂×C₄

C₂³×D₆₁

yes

characteristic

127

C₈

—

128

C₈

—

122

—

129

C₈

C₂×Dic₁₂₂

yes

complemented

130

C₂²

C₄×Dic₁₂₂

yes

—

131

C₂²

C₂×#11758189

yes

—

132

C₂²

#14519783

yes

—

133

C₂²

C₂²×C₄×D₆₁

yes

characteristic

134

C₄

—

135

C₄

—

122

—

136

C₄

—

122

—

137

C₄

C₄×Dic₁₂₂

yes

—

138

C₄

C₂×#11758189

yes

—

139

C₄

C₂²×Dic₁₂₂

yes

characteristic

140

C₄

C₂×#11758222

yes

characteristic

141

C₂

#16839892

yes

direct product factor

142

C₂

C₂×C₄×Dic₁₂₂

yes

characteristic

143

C₂

C₂²×#11758189

yes

characteristic

144

C₂

C₂×#14519783

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×#446
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16839892	C₂	direct (N × H)
29. C₂×C₈×C₆₁	Q₈	semidirect
55. C₈×C₆₁	C₂×Q₈	semidirect
69. C₂×C₆₁	#446	semidirect
79. C₆₁	C₂×#446	semidirect
104. C₂×C₈	Dic₁₂₂	semidirect
129. C₈	C₂×Dic₁₂₂	semidirect
141. C₂	#16839892	direct (N × H)

H ≅ G/N

type

2. #16839892

C₂

direct (N × H)

29. C₂×C₈×C₆₁

Q₈

semidirect

55. C₈×C₆₁

C₂×Q₈

semidirect

69. C₂×C₆₁

#446

semidirect

79. C₆₁

C₂×#446

semidirect

104. C₂×C₈

Dic₁₂₂

semidirect

129. C₈

C₂×Dic₁₂₂

semidirect

141. C₂

#16839892

direct (N × H)

Subgroups — C₂×#16839892 #21454746

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16839892 #21454746

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16839892 #21454746