Group #21446732 — subgroups

Subgroups — #21446732

Order 7808.

276 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 89 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×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#16839772	C₂	1	2	yes	maximal
3	C₂×#14519695	C₂	1	2	yes	maximal
4	#16839717	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	C₂×Dic₂₄₄	—	2	1	no	—
8	C₂×Dic₂₄₄	—	2	1	no	—
9	C₂×Dic₂₄₄	—	2	2	no	—
10	C₂×Dic₂₄₄	C₄	1	4	yes	complemented
11	C₂×#11758194	C₂²	1	1	yes	characteristic
12	#14519736	C₂²	1	1	yes	characteristic
13	#14519702	—	2	4	no	—
14	#14519695	—	2	4	no	—
15	#14519687	—	2	1	no	—
16	C₈×Dic₆₁	—	2	1	no	—
17	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	#108×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₆₁	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	—	2	2	no	—
24	C₂×Dic₁₂₂	—	2	2	no	—
25	C₂×Dic₁₂₂	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
28	M₁₆×C₆₁	—	2	2	no	—
29	M₁₆×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	D₄	1	2	yes	—
35	C₂²⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	D₄	1	2	yes	—
37	C₄×Dic₆₁	—	2	1	no	—
38	#11758194	—	2	1	no	—
39	#11758194	D₄	1	2	yes	—
40	C₂×#92418	—	2	1	no	—
41	Dic₂₄₄	—	2	4	no	—
42	Dic₂₄₄	—	2	4	no	—
43	C₂³×C₆₁	#41	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	2	2	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	C₂×Dic₆₁	—	4	2	no	—
54	Dic₁₂₂	—	2	4	no	—
55	Dic₁₂₂	—	2	4	no	—
56	Dic₁₂₂	—	4	2	no	—
57	Dic₁₂₂	—	4	2	no	—
58	C₈×C₆₁	—	4	2	no	—
59	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
60	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
61	#92418	—	2	2	no	—
62	C₂²×C₆₁	—	2	2	no	—
63	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₆₁	#121	1	1	yes	characteristic
65	C₂²×C₆₁	#124	1	1	yes	characteristic
66	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₄×C₆₁	#121	1	1	yes	characteristic
69	C₄×C₆₁	#127	1	1	yes	characteristic
70	Dic₆₁	—	4	4	no	—
71	Dic₆₁	—	4	2	no	—
72	#1979	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	2	1	no	—
74	C₂×C₆₁	—	2	1	no	—
75	C₂×C₆₁	#390	1	1	yes	characteristic
76	C₂×C₆₁	#473	1	2	yes	—
77	C₂²×Q₁₆	—	61	1	no	—
78	C₂×#108	D₆₁	1	1	yes	characteristic
79	#419	—	61	2	no	—
80	C₂×#101	—	61	2	no	—
81	#405	—	61	1	no	—
82	C₆₁	#1979	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	61	2	no	—
84	C₂×Q₁₆	—	61	4	no	—
85	C₂×Q₁₆	—	122	1	no	—
86	C₂×Q₁₆	—	122	1	no	—
87	C₂×Q₁₆	—	122	2	no	—
88	C₂×M₁₆	—	61	1	no	—
89	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
90	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
91	#117	—	61	1	no	—
92	#108	—	2	2	no	—
93	#103	—	122	4	no	—
94	#101	—	122	4	no	—
95	#97	—	122	1	no	—
96	C₄×C₈	—	122	1	no	—
97	C₂×Q₈	—	61	4	no	—
98	C₂×Q₈	—	122	2	no	—
99	C₂×Q₈	—	122	2	no	—
100	C₂×Q₈	—	122	2	no	—
101	C₂×Q₈	—	122	2	no	—
102	C₂×Q₈	—	122	2	no	—
103	C₂²×C₄	—	122	2	no	—
104	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
105	Q₁₆	—	122	4	no	—
106	Q₁₆	—	122	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	61	2	no	—
110	M₁₆	—	122	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	122	1	no	—
113	C₂×C₈	C₄×D₆₁	1	2	yes	—
114	C₂×C₈	D₂₄₄	1	1	yes	characteristic
115	C₂×C₈	D₂₄₄	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
117	C₄⋊C₄	—	61	2	no	—
118	C₂²⋊C₄	—	122	1	no	—
119	C₄²	—	122	1	no	—
120	C₂³	#11758226	1	1	yes	characteristic
121	Q₈	—	122	4	no	—
122	Q₈	—	122	4	no	—
123	Q₈	—	244	2	no	—
124	Q₈	—	244	2	no	—
125	C₂×C₄	—	122	4	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	—	244	2	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
131	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	#11758226	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	122	2	no	—
137	C₈	#11758198	1	2	yes	—
138	C₈	#11758198	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#14519739	1	1	yes	characteristic
141	C₂²	#14519752	1	1	yes	characteristic
142	C₂²	#14519766	1	1	yes	characteristic
143	C₄	—	244	2	no	—
144	C₄	—	244	4	no	—
145	C₄	C₂×#11758198	1	1	yes	characteristic
146	C₄	#14519739	1	1	yes	characteristic
147	C₄	#14519752	1	1	yes	characteristic
148	C₄	#14519773	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#16839663	1	1	yes	characteristic
152	C₂	#16840038	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₄₄

C₂

yes

maximal, characteristic

#16839772

C₂

yes

maximal

C₂×#14519695

C₂

yes

maximal

#16839717

C₂

yes

maximal, characteristic

C₂×#108×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₄

yes

complemented

C₂×#11758194

C₂²

yes

characteristic

#14519736

C₂²

yes

—

—

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄×Dic₆₁

—

#11758194

—

#11758194

D₄

yes

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

C₂³×C₆₁

#41

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

—

C₈×C₆₁

C₂²⋊C₄

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

#1979

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

#473

yes

—

C₂²×Q₁₆

—

C₂×#108

D₆₁

yes

—

—

—

C₆₁

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₂²×C₈

C₂×D₆₁

yes

—

—

—

122

—

#101

—

122

—

#97

—

122

—

C₄×C₈

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

100

C₂×Q₈

—

122

—

101

C₂×Q₈

—

122

—

102

C₂×Q₈

—

122

—

103

C₂²×C₄

—

122

—

104

C₂²×C₄

C₂²×D₆₁

yes

characteristic

105

Q₁₆

—

122

—

106

Q₁₆

—

122

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

122

—

111

C₂×C₈

—

112

C₂×C₈

—

122

—

113

C₂×C₈

C₄×D₆₁

yes

—

114

C₂×C₈

D₂₄₄

yes

characteristic

115

C₂×C₈

D₂₄₄

yes

characteristic

116

C₂×C₈

C₂²⋊D₆₁

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

122

—

119

C₄²

—

122

—

120

C₂³

#11758226

yes

characteristic

121

Q₈

—

122

—

122

Q₈

—

122

—

123

Q₈

—

244

—

124

Q₈

—

244

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

244

—

128

C₂×C₄

—

244

—

129

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

130

C₂×C₄

C₂×D₂₄₄

yes

characteristic

131

C₂×C₄

D₄×D₆₁

yes

characteristic

132

C₂×C₄

D₄×D₆₁

yes

characteristic

133

C₂×C₄

#11758226

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

135

C₈

—

136

C₈

—

122

—

137

C₈

#11758198

yes

—

138

C₈

#11758198

yes

—

139

C₂²

—

140

C₂²

#14519739

yes

characteristic

141

C₂²

#14519752

yes

characteristic

142

C₂²

#14519766

yes

characteristic

143

C₄

—

244

—

144

C₄

—

244

—

145

C₄

C₂×#11758198

yes

characteristic

146

C₄

#14519739

yes

characteristic

147

C₄

#14519752

yes

characteristic

148

C₄

#14519773

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#16839663

yes

characteristic

152

C₂

#16840038

yes

—

	subgroup	G/H
center	C₂²	#14519752
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#108×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₂×#108	D₆₁
hypocenter	C₆₁	#1979
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14519752

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#108×C₆₁

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
10. C₂×Dic₂₄₄	C₄	semidirect
82. C₆₁	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₂₄₄

C₄

semidirect

82. C₆₁

#1979

semidirect

Subgroups — #21446732

Conjugacy classes of subgroups

Complements