Group #21456326 — subgroups

Subgroups — C₂×#16840166 #21456326

Order 7808.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14519776	C₂	1	2	yes	maximal
2	#16840166	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519711	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519710	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	—	2	2	no	—
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	#14519776	—	2	4	no	—
10	#14519776	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×#11758196	—	2	2	no	—
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519711	C₂²	1	4	yes	—
15	#14519710	C₂²	1	8	yes	—
16	C₈×Dic₆₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	#107×C₆₁	C₂²	1	4	yes	—
20	C₂²×C₄×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	2	8	no	—
26	C₂×Dic₁₂₂	—	4	8	no	—
27	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
28	C₂×C₈×C₆₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	C₂³	1	4	yes	—
33	#11758196	—	2	8	no	—
34	C₄×Dic₆₁	D₄	1	2	yes	—
35	C₄×Dic₆₁	C₂³	1	2	yes	—
36	C₂×#92418	Q₈	1	4	yes	complemented
37	C₂×#92418	C₂³	1	2	yes	—
38	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	4	4	no	—
41	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₆₁	—	2	4	no	—
46	C₂×Dic₆₁	—	4	4	no	—
47	C₂×Dic₆₁	D₈	1	4	yes	—
48	C₂×Dic₆₁	Q₁₆	1	4	yes	—
49	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
50	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
51	Dic₁₂₂	—	4	16	no	—
52	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
53	#92418	C₂×Q₈	1	4	yes	complemented
54	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₆₁	C₂×D₈	1	2	yes	—
56	C₂²×C₆₁	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
62	Dic₆₁	—	4	4	no	—
63	Dic₆₁	C₂×D₈	1	4	yes	—
64	Dic₆₁	C₂×Q₁₆	1	4	yes	—
65	C₂×#500	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	#500	1	4	yes	complemented
67	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
69	C₂×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	61	2	no	—
71	#500	—	61	8	no	—
72	C₂×#107	—	61	1	no	—
73	C₂×#107	—	61	2	no	—
74	C₂×#107	D₆₁	1	1	yes	characteristic
75	C₂×C₄×C₈	—	61	1	no	—
76	C₆₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
77	C₂²×Q₈	—	122	2	no	—
78	C₂²×C₈	—	61	1	no	—
79	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	61	8	no	—
81	C₄⋊Q₈	—	122	4	no	—
82	C₂×C₄⋊C₄	—	61	2	no	—
83	C₂×C₄⋊C₄	—	122	2	no	—
84	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
85	C₂×C₄²	—	61	1	no	—
86	#107	—	61	4	no	—
87	#107	—	61	8	no	—
88	#107	C₂×D₆₁	1	4	yes	—
89	C₄×C₈	—	61	4	no	—
90	C₂×Q₈	—	122	8	no	—
91	C₂×Q₈	—	244	8	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	61	1	no	—
94	C₂²×C₄	—	61	1	no	—
95	C₂²×C₄	—	122	2	no	—
96	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
97	C₂×C₈	—	61	4	no	—
98	C₂×C₈	—	61	2	no	—
99	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
100	C₂×C₈	C₂²×D₆₁	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	61	4	no	—
103	C₄⋊C₄	—	122	8	no	—
104	C₄⋊C₄	—	122	2	no	—
105	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
106	C₄²	—	61	2	no	—
107	C₄²	—	61	2	no	—
108	C₂³	D₄×D₆₁	1	1	yes	characteristic
109	Q₈	—	244	16	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	61	2	no	—
113	C₂×C₄	—	61	2	no	—
114	C₂×C₄	—	61	4	no	—
115	C₂×C₄	—	61	4	no	—
116	C₂×C₄	—	122	4	no	—
117	C₂×C₄	—	244	4	no	—
118	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
119	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₆₁	1	2	yes	—
121	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
122	C₈	—	61	4	no	—
123	C₈	C₂×Dic₁₂₂	1	4	yes	complemented
124	C₂²	#14519776	1	2	yes	—
125	C₂²	D₈×D₆₁	1	2	yes	—
126	C₂²	Q₁₆×D₆₁	1	2	yes	—
127	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	61	4	no	—
130	C₄	—	61	4	no	—
131	C₄	—	244	4	no	—
132	C₄	#14519776	1	2	yes	—
133	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
134	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
135	C₂	#16840166	1	4	yes	direct product factor
136	C₂	C₂×#14519776	1	1	yes	characteristic
137	C₂	C₂×D₈×D₆₁	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519776

C₂

yes

maximal

#16840166

C₂

yes

maximal, direct product factor

C₂×#14519711

C₂

yes

maximal, characteristic

C₂×#14519710

C₂

yes

maximal

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×#107×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

—

C₂²×#92418

C₂²

yes

characteristic

#14519776

—

#14519776

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

—

C₂×Dic₆₁

Q₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

complemented

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂×D₈

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

C₂×D₈

yes

—

Dic₆₁

C₂×Q₁₆

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

C₂²×Q₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

C₆₁

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₆₁

yes

—

C₄×C₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

complemented

100

C₂×C₈

C₂²×D₆₁

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

C₂²×D₆₁

yes

—

106

C₄²

—

107

C₄²

—

108

C₂³

D₄×D₆₁

yes

characteristic

109

Q₈

—

244

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

244

—

118

C₂×C₄

C₂×Dic₁₂₂

yes

—

119

C₂×C₄

D₄×D₆₁

yes

characteristic

120

C₂×C₄

Q₈×D₆₁

yes

—

121

C₂×C₄

C₂³×D₆₁

yes

characteristic

122

C₈

—

123

C₈

C₂×Dic₁₂₂

yes

complemented

124

C₂²

#14519776

yes

—

125

C₂²

D₈×D₆₁

yes

—

126

C₂²

Q₁₆×D₆₁

yes

—

127

C₂²

C₂×D₄×D₆₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

244

—

132

C₄

#14519776

yes

—

133

C₄

C₂×Q₈×D₆₁

yes

characteristic

134

C₄

C₂²×Dic₁₂₂

yes

characteristic

135

C₂

#16840166

yes

direct product factor

136

C₂

C₂×#14519776

yes

characteristic

137

C₂

C₂×D₈×D₆₁

yes

characteristic

138

C₂

C₂×Q₁₆×D₆₁

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	C₂×#500
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #16840166	C₂	direct (N × H)
27. C₂×C₈×C₆₁	Q₈	semidirect
36. C₂×#92418	Q₈	semidirect
52. C₈×C₆₁	C₂×Q₈	semidirect
53. #92418	C₂×Q₈	semidirect
66. C₂×C₆₁	#500	semidirect
76. C₆₁	C₂×#500	semidirect
99. C₂×C₈	Dic₁₂₂	semidirect
123. C₈	C₂×Dic₁₂₂	semidirect
135. C₂	#16840166	direct (N × H)

H ≅ G/N

type

2. #16840166

C₂

direct (N × H)

27. C₂×C₈×C₆₁

Q₈

semidirect

36. C₂×#92418

Q₈

semidirect

52. C₈×C₆₁

C₂×Q₈

semidirect

53. #92418

C₂×Q₈

semidirect

66. C₂×C₆₁

#500

semidirect

76. C₆₁

C₂×#500

semidirect

99. C₂×C₈

Dic₁₂₂

semidirect

123. C₈

C₂×Dic₁₂₂

semidirect

135. C₂

#16840166

direct (N × H)

Subgroups — C₂×#16840166 #21456326

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840166 #21456326

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840166 #21456326