Group #21447106 — subgroups

Subgroups — #21447106

Order 7808.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16840562	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519705	C₂	1	2	yes	maximal, complemented
3	#16839744	C₂	1	2	yes	maximal, complemented
4	#16839520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758222	C₂²	1	2	yes	—
7	#14519810	—	2	2	no	—
8	#14519805	—	2	2	no	—
9	#14519799	—	2	2	no	—
10	C₂×#11758194	C₂²	1	2	yes	—
11	#14519705	—	2	2	no	—
12	#14519705	—	2	2	no	—
13	#14519695	—	2	2	no	—
14	#14519693	—	2	2	no	—
15	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₆₁	—	2	2	no	—
19	C₂²×C₄×C₆₁	—	2	1	no	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₆₁	—	2	2	no	—
22	#11758222	—	2	2	no	—
23	#11758222	—	2	2	no	—
24	#11758222	—	2	2	no	—
25	#11758222	—	2	2	no	—
26	C₂×D₂₄₄	D₄	1	2	yes	—
27	C₂×C₄×D₆₁	—	2	2	no	—
28	C₂×Dic₁₂₂	D₄	1	2	yes	—
29	M₁₆×C₆₁	—	2	2	no	—
30	M₁₆×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	—	2	1	no	—
32	C₂×C₈×C₆₁	D₄	1	2	yes	—
33	C₄²×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₄²×C₆₁	D₄	1	1	yes	characteristic
36	#11758194	—	2	2	no	—
37	#11758194	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	Dic₂₄₄	—	2	2	no	—
40	D₄₈₈	—	2	2	no	—
41	#11758190	—	2	4	no	—
42	C₂³×C₆₁	#41	1	1	yes	characteristic
43	C₂²×D₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#41	1	2	yes	—
51	C₂²⋊D₆₁	—	4	2	no	—
52	C₂²⋊D₆₁	—	4	2	no	—
53	C₂×Dic₆₁	—	2	2	no	—
54	D₂₄₄	—	2	2	no	—
55	D₂₄₄	—	2	2	no	—
56	C₄×D₆₁	—	4	2	no	—
57	C₄×D₆₁	—	4	2	no	—
58	Dic₁₂₂	—	2	2	no	—
59	Dic₁₂₂	—	2	2	no	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	—	4	1	no	—
62	#92418	—	4	2	no	—
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	#121	1	1	yes	characteristic
65	C₂²×C₆₁	#123	1	2	yes	—
66	C₂×D₆₁	—	4	2	no	—
67	C₂×D₆₁	—	8	2	no	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	4	1	no	—
70	C₄×C₆₁	#120	1	1	yes	characteristic
71	C₄×C₆₁	#121	1	2	yes	—
72	C₄×C₆₁	#124	1	1	yes	characteristic
73	Dic₆₁	—	4	2	no	—
74	#2050	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	—	2	2	no	—
77	C₂×C₆₁	#394	1	1	yes	characteristic
78	D₆₁	—	8	2	no	—
79	#575	—	61	1	no	—
80	#423	D₆₁	1	1	yes	characteristic, complemented
81	C₂×#104	—	61	2	no	—
82	#413	—	61	2	no	—
83	#338	—	61	1	no	—
84	C₆₁	#2050	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	61	2	no	—
86	#137	—	122	2	no	—
87	AGL(1,ℤ/8)	—	122	2	no	—
88	#135	—	122	2	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
91	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	#104	—	122	2	no	—
95	#104	—	122	2	no	—
96	#101	—	122	2	no	—
97	#100	—	122	2	no	—
98	#41	—	122	2	no	—
99	#41	—	122	2	no	—
100	#41	—	122	2	no	—
101	#41	—	122	2	no	—
102	C₂×Q₈	—	61	2	no	—
103	C₂×D₄	—	61	2	no	—
104	C₂×D₄	—	122	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	122	2	no	—
107	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
108	Q₁₆	—	122	2	no	—
109	SD₁₆	—	122	4	no	—
110	D₈	—	122	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	122	2	no	—
114	M₁₆	—	122	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	122	2	no	—
117	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	D₂₄₄	1	1	yes	characteristic
120	C₄²	D₂₄₄	1	1	yes	characteristic
121	C₂³	—	122	2	no	—
122	C₂³	#11758222	1	1	yes	characteristic
123	Q₈	—	122	2	no	—
124	Q₈	—	122	2	no	—
125	D₄	—	122	2	no	—
126	D₄	—	122	2	no	—
127	D₄	—	244	2	no	—
128	D₄	—	244	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	122	2	no	—
133	C₂×C₄	—	244	2	no	—
134	C₂×C₄	—	244	2	no	—
135	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
136	C₂×C₄	D₄×D₆₁	1	2	yes	—
137	C₂×C₄	#11758226	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	244	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	488	2	no	—
144	C₂²	—	244	2	no	—
145	C₂²	#14519751	1	1	yes	characteristic
146	C₂²	#14519762	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	244	2	no	—
150	C₄	#14519745	1	1	yes	characteristic
151	C₄	#14519752	1	2	yes	—
152	C₄	#14519766	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	488	2	no	—
156	C₂	#16839679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840562

C₂

yes

maximal, characteristic

C₂×#14519705

C₂

yes

maximal, complemented

#16839744

C₂

yes

maximal, complemented

#16839520

C₂

yes

maximal, characteristic, complemented

C₆₁×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

—

—

—

—

—

C₂×D₂₄₄

D₄

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

D₄

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

—

—

Dic₂₄₄

—

D₄₈₈

—

#11758190

—

C₂³×C₆₁

#41

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂²⋊D₆₁

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

#123

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#120

yes

characteristic

C₄×C₆₁

#121

yes

—

C₄×C₆₁

#124

yes

characteristic

Dic₆₁

—

#2050

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#394

yes

characteristic

D₆₁

—

#575

—

#423

D₆₁

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₆₁

#2050

yes

characteristic, complemented, Sylow

C₂×#41

—

#137

—

122

—

AGL(1,ℤ/8)

—

122

—

#135

—

122

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

#104

—

122

—

#104

—

122

—

#101

—

122

—

#100

—

122

—

#41

—

122

—

#41

—

122

—

100

#41

—

122

—

101

#41

—

122

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

122

—

105

C₂²×C₄

—

106

C₂²×C₄

—

122

—

107

C₂²×C₄

C₂²×D₆₁

yes

characteristic

108

Q₁₆

—

122

—

109

SD₁₆

—

122

—

110

D₈

—

122

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

122

—

114

M₁₆

—

122

—

115

C₂×C₈

—

116

C₂×C₈

—

122

—

117

C₂×C₈

C₂²⋊D₆₁

yes

—

118

C₄²

—

119

C₄²

D₂₄₄

yes

characteristic

120

C₄²

D₂₄₄

yes

characteristic

121

C₂³

—

122

—

122

C₂³

#11758222

yes

characteristic

123

Q₈

—

122

—

124

Q₈

—

122

—

125

D₄

—

122

—

126

D₄

—

122

—

127

D₄

—

244

—

128

D₄

—

244

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

244

—

134

C₂×C₄

—

244

—

135

C₂×C₄

C₂×D₂₄₄

yes

characteristic

136

C₂×C₄

D₄×D₆₁

yes

—

137

C₂×C₄

#11758226

yes

—

138

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

244

—

142

C₂²

—

143

C₂²

—

488

—

144

C₂²

—

244

—

145

C₂²

#14519751

yes

characteristic

146

C₂²

#14519762

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

244

—

150

C₄

#14519745

yes

characteristic

151

C₄

#14519752

yes

—

152

C₄

#14519766

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

488

—

156

C₂

#16839679

yes

characteristic

	subgroup	G/H
center	C₄	#14519766
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#423	C₂
socle	C₂×C₆₁	#394
hypercenter	#423	D₆₁
hypocenter	C₆₁	#2050
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#14519766

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#423

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₂×#14519705	C₂	semidirect
3. #16839744	C₂	semidirect
4. #16839520	C₂	semidirect
5. C₆₁×#423	C₂	semidirect
80. #423	D₆₁	semidirect
84. C₆₁	#2050	semidirect

H ≅ G/N

type

2. C₂×#14519705

C₂

semidirect

3. #16839744

C₂

semidirect

4. #16839520

C₂

semidirect

5. C₆₁×#423

C₂

semidirect

80. #423

D₆₁

semidirect

84. C₆₁

#2050

semidirect

Subgroups — #21447106

Conjugacy classes of subgroups

Complements