Group #21454907 — subgroups

Subgroups — #21454907

Order 7808.

364 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	3	yes	maximal, complemented
3	#16839889	C₂	1	3	yes	maximal, complemented
4	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839818	C₂	1	3	yes	maximal
6	#16839813	C₂	1	3	yes	maximal
7	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	C₄	1	1	yes	characteristic
9	Q₈×Dic₆₁	C₄	1	1	yes	characteristic
10	C₄×Dic₁₂₂	C₂²	1	3	yes	—
11	#14519742	C₄	1	3	yes	—
12	C₄⋊C₄×D₆₁	C₄	1	3	yes	—
13	C₄²×D₆₁	C₂²	1	3	yes	—
14	C₄⋊#92418	C₂²	1	3	yes	—
15	#14519706	C₂²	1	3	yes	—
16	#14519706	C₂²	1	3	yes	—
17	#14519689	C₂²	1	1	yes	characteristic
18	#14519689	C₂²	1	3	yes	—
19	#14519687	C₂²	1	6	yes	—
20	C₄×#92418	C₂²	1	3	yes	—
21	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
23	C₄×C₈×C₆₁	C₂²	1	3	yes	—
24	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	Q₈×D₆₁	—	2	4	no	—
26	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₂₂	C₂×C₄	1	3	yes	—
30	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂³	1	3	yes	—
32	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
33	C₄²×C₆₁	C₂³	1	3	yes	—
34	C₄⋊Dic₆₁	C₂×C₄	1	3	yes	—
35	#11758196	C₂×C₄	1	6	yes	—
36	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
37	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
38	C₄×Dic₆₁	C₂³	1	3	yes	—
39	C₂×#92418	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂³	1	3	yes	—
41	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₆₁	M₁₆	1	4	yes	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
48	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
49	C₄×D₆₁	—	2	6	no	—
50	C₄×D₆₁	—	2	6	no	—
51	C₄×D₆₁	—	2	1	no	—
52	C₄×D₆₁	—	2	1	no	—
53	Dic₁₂₂	—	2	6	no	—
54	C₈×C₆₁	—	2	3	no	—
55	C₈×C₆₁	—	2	1	no	—
56	#92418	—	2	1	no	—
57	#92418	—	2	3	no	—
58	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₆₁	—	2	1	no	—
60	C₂×D₆₁	—	2	1	no	—
61	C₂×D₆₁	#131	1	2	yes	—
62	C₄×C₆₁	—	2	3	no	—
63	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
64	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₆₁	—	2	1	no	—
67	Dic₆₁	—	2	3	no	—
68	Dic₆₁	—	2	3	no	—
69	#2984	—	61	1	no	maximal, Sylow
70	C₂×C₆₁	#519	1	1	yes	characteristic
71	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
73	D₆₁	—	2	2	no	—
74	C₂×C₄×Q₈	—	61	1	no	—
75	#446	—	61	3	no	—
76	#446	—	61	3	no	—
77	C₈×Q₈	—	61	1	no	—
78	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
79	#432	—	61	3	no	—
80	#431	—	61	3	no	—
81	C₆₁	#2984	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	61	1	no	—
83	C₄×Q₈	—	61	1	no	—
84	C₄×Q₈	—	61	3	no	—
85	C₄×Q₈	—	61	3	no	—
86	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	61	3	no	—
88	C₂×C₄²	—	61	3	no	—
89	C₄⋊C₈	—	61	3	no	—
90	C₄⋊C₈	—	61	3	no	—
91	C₄⋊C₈	—	61	3	no	—
92	C₄⋊C₈	C₂×D₆₁	1	3	yes	—
93	C₂²⋊C₈	—	61	1	no	—
94	C₂²⋊C₈	—	61	3	no	—
95	#97	—	61	6	no	—
96	C₄×C₈	—	61	3	no	—
97	C₄×C₈	C₂×D₆₁	1	3	yes	—
98	C₂×Q₈	—	61	3	no	—
99	C₂×Q₈	—	122	4	no	—
100	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
101	C₂²×C₄	—	61	3	no	—
102	C₂²×C₄	—	61	3	no	—
103	C₂²×C₄	—	61	1	no	—
104	C₂×C₈	—	61	1	no	—
105	C₂×C₈	—	61	3	no	—
106	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₆₁	1	3	yes	—
108	C₄⋊C₄	—	61	6	no	—
109	C₄⋊C₄	—	61	3	no	—
110	C₄⋊C₄	C₄×D₆₁	1	3	yes	—
111	C₄²	—	61	3	no	—
112	C₄²	—	61	3	no	—
113	C₄²	—	61	3	no	—
114	C₄²	C₂²×D₆₁	1	3	yes	—
115	C₂³	—	61	1	no	—
116	Q₈	—	122	6	no	—
117	Q₈	#11758189	1	4	yes	—
118	C₂×C₄	—	61	1	no	—
119	C₂×C₄	—	61	3	no	—
120	C₂×C₄	—	61	3	no	—
121	C₂×C₄	—	122	6	no	—
122	C₂×C₄	—	122	1	no	—
123	C₂×C₄	—	122	1	no	—
124	C₂×C₄	—	122	6	no	—
125	C₂×C₄	C₂×C₄×D₆₁	1	3	yes	—
126	C₂×C₄	C₂×C₄×D₆₁	1	3	yes	—
127	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
128	C₈	—	2	3	no	—
129	C₈	—	2	1	no	—
130	C₈	—	122	1	no	—
131	C₈	—	122	3	no	—
132	C₂²	—	61	2	no	—
133	C₂²	—	122	1	no	—
134	C₂²	—	122	1	no	—
135	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	122	1	no	—
138	C₄	—	122	3	no	—
139	C₄	—	122	3	no	—
140	C₄	C₂×#11758189	1	6	yes	—
141	C₄	#14519824	1	1	yes	characteristic
142	C₄	#14519825	1	1	yes	characteristic
143	C₂	—	122	2	no	—
144	C₂	#16840238	1	1	yes	characteristic
145	C₂	C₂²×#11758189	1	1	yes	characteristic
146	C₂	#131×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×D₆₁

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, complemented

#16839889

C₂

yes

maximal, complemented

Q₈×#92418

C₂

yes

maximal, characteristic, complemented

#16839818

C₂

yes

maximal

#16839813

C₂

yes

maximal

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

C₄

yes

characteristic

Q₈×Dic₆₁

C₄

yes

characteristic

C₄×Dic₁₂₂

C₂²

yes

—

#14519742

C₄

yes

—

C₄⋊C₄×D₆₁

C₄

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

#14519706

C₂²

yes

—

#14519689

C₂²

yes

characteristic

#14519689

C₂²

yes

—

#14519687

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

Q₈×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₂²×D₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#131

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

D₆₁

—

C₂×C₄×Q₈

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

#432

—

#431

—

C₆₁

#2984

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

122

—

100

C₂×Q₈

C₄×D₆₁

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

C₂²×D₆₁

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₄×D₆₁

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₆₁

yes

—

115

C₂³

—

116

Q₈

—

122

—

117

Q₈

#11758189

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

C₂×C₄×D₆₁

yes

—

126

C₂×C₄

C₂×C₄×D₆₁

yes

—

127

C₂×C₄

C₂³×D₆₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

122

—

131

C₈

—

122

—

132

C₂²

—

133

C₂²

—

122

—

134

C₂²

—

122

—

135

C₂²

C₂²×C₄×D₆₁

yes

characteristic

136

C₄

—

137

C₄

—

122

—

138

C₄

—

122

—

139

C₄

—

122

—

140

C₄

C₂×#11758189

yes

—

141

C₄

#14519824

yes

characteristic

142

C₄

#14519825

yes

characteristic

143

C₂

—

122

—

144

C₂

#16840238

yes

characteristic

145

C₂

C₂²×#11758189

yes

characteristic

146

C₂

#131×D₆₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₈×Q₈×C₆₁	C₂
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#2984
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₈×Q₈×C₆₁

C₂

hypercenter

C₈×Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. #16839889	C₂	semidirect
4. Q₈×#92418	C₂	semidirect
7. C₈×Q₈×C₆₁	C₂	semidirect
78. C₈×Q₈	D₆₁	semidirect
81. C₆₁	#2984	semidirect

H ≅ G/N

type

2. #16839892

C₂

semidirect

3. #16839889

C₂

semidirect

4. Q₈×#92418

C₂

semidirect

7. C₈×Q₈×C₆₁

C₂

semidirect

78. C₈×Q₈

D₆₁

semidirect

81. C₆₁

#2984

semidirect

Subgroups — #21454907

Conjugacy classes of subgroups

Complements