Group #21444708 — subgroups

Subgroups — #21444708

Order 7808.

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

normal subgroups only

This group has 166 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	#16839889	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11758190	C₂	1	1	yes	maximal, characteristic
3	#16839839	C₂	1	1	yes	maximal, characteristic
4	#16839492	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839489	C₂	1	1	yes	maximal, characteristic
6	C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758190	C₄	1	1	yes	characteristic
9	C₂×#11758189	—	2	1	no	—
10	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	#14519709	C₄	1	1	yes	characteristic
13	#14519706	—	2	1	no	—
14	#14519702	C₄	1	1	yes	characteristic
15	#14519699	C₄	1	1	yes	characteristic
16	#14519689	—	2	1	no	—
17	#14519687	—	2	1	no	—
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	#11758198	—	2	1	no	—
32	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	#11758196	—	2	1	no	—
35	C₄×Dic₆₁	—	2	1	no	—
36	C₂×#92418	—	2	1	no	—
37	C₂×#92418	D₄	1	1	yes	characteristic
38	C₂×#92418	D₄	1	1	yes	characteristic
39	#11758190	—	2	2	no	—
40	#11758189	—	4	2	no	—
41	C₂²×D₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
45	C₂×Dic₆₁	—	2	1	no	—
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	D₂₄₄	—	2	1	no	—
49	D₂₄₄	—	2	1	no	—
50	C₄×D₆₁	—	4	1	no	—
51	C₄×D₆₁	—	4	1	no	—
52	Dic₁₂₂	—	2	1	no	—
53	Dic₁₂₂	—	2	1	no	—
54	C₈×C₆₁	—	2	1	no	—
55	C₈×C₆₁	—	4	1	no	—
56	C₈×C₆₁	—	4	1	no	—
57	C₈×C₆₁	M₁₆	1	2	yes	complemented
58	#92418	—	2	2	no	—
59	#92418	—	2	1	no	—
60	#92418	SD₁₆	1	2	yes	complemented
61	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
62	C₂×D₆₁	—	4	1	no	—
63	C₂×D₆₁	—	4	1	no	—
64	C₂×D₆₁	—	4	1	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₆₁	#131	1	1	yes	characteristic
68	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₆₁	#135	1	1	yes	characteristic
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	—	4	1	no	—
72	Dic₆₁	—	4	1	no	—
73	#1590	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	#436	1	1	yes	characteristic
75	C₂×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₆₁	#443	1	1	yes	characteristic
77	D₆₁	—	8	1	no	—
78	#446	—	61	1	no	—
79	C₄×SD₁₆	—	61	1	no	—
80	#436	—	61	1	no	—
81	#334	D₆₁	1	1	yes	characteristic, complemented
82	Q₈⋊C₈	—	61	1	no	—
83	D₄⋊C₈	—	61	1	no	—
84	C₈²	—	61	1	no	—
85	C₆₁	#1590	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	61	1	no	—
87	C₂×M₁₆	—	122	1	no	—
88	C₄×Q₈	—	61	1	no	—
89	C₄×D₄	—	61	1	no	—
90	#106	—	61	1	no	—
91	C₄⋊C₈	—	122	1	no	—
92	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
94	#103	—	61	1	no	—
95	#102	—	61	1	no	—
96	C₂²⋊C₈	—	122	1	no	—
97	#97	—	122	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	C₂×Q₈	—	61	1	no	—
102	C₂×D₄	—	61	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	SD₁₆	—	122	2	no	—
105	M₁₆	—	244	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
112	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
113	C₄⋊C₄	—	61	1	no	—
114	C₄⋊C₄	—	61	1	no	—
115	C₄⋊C₄	—	122	1	no	—
116	C₂²⋊C₄	—	122	1	no	—
117	C₄²	—	122	1	no	—
118	C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₂³	—	122	1	no	—
120	Q₈	—	122	1	no	—
121	Q₈	—	122	1	no	—
122	D₄	—	122	1	no	—
123	D₄	—	122	1	no	—
124	C₂×C₄	—	122	1	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	244	1	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄	#11758226	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	61	2	no	—
136	C₈	—	122	2	no	—
137	C₈	—	122	1	no	—
138	C₈	#11758189	1	2	yes	complemented
139	C₂²	—	244	1	no	—
140	C₂²	—	244	1	no	—
141	C₂²	—	244	1	no	—
142	C₂²	#14519739	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	244	1	no	—
145	C₄	—	244	1	no	—
146	C₄	—	244	1	no	—
147	C₄	C₂×#11758189	1	1	yes	characteristic
148	C₄	#14519785	1	1	yes	characteristic
149	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
150	C₄	#14519798	1	1	yes	characteristic
151	C₂	—	488	1	no	—
152	C₂	#16839839	1	1	yes	characteristic
153	C₂	#16839849	1	1	yes	characteristic
154	C₂	#16839878	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839889

C₂

yes

maximal, characteristic, complemented

C₄×#11758190

C₂

yes

maximal, characteristic

#16839839

C₂

yes

maximal, characteristic

#16839492

C₂

yes

maximal, characteristic, complemented

#16839489

C₂

yes

maximal, characteristic

C₈×#92418

C₂

yes

maximal, characteristic, complemented

C₆₁×#334

C₂

yes

maximal, characteristic, complemented

C₂×#11758190

C₄

yes

characteristic

C₂×#11758189

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

characteristic

#14519709

C₄

yes

characteristic

#14519706

—

#14519702

C₄

yes

characteristic

#14519699

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

#11758190

—

#11758189

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

M₁₆

yes

—

—

SD₁₆

yes

complemented

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

#135

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×SD₁₆

yes

characteristic

C₂×C₆₁

#443

yes

characteristic

D₆₁

—

#446

—

C₄×SD₁₆

—

#436

—

#334

D₆₁

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₆₁

#1590

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

122

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₆₁

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

122

—

104

SD₁₆

—

122

—

105

M₁₆

—

244

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

122

—

111

C₂×C₈

C₄×D₆₁

yes

characteristic

112

C₂×C₈

C₄×D₆₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

122

—

116

C₂²⋊C₄

—

122

—

117

C₄²

—

122

—

118

C₄²

C₂²×D₆₁

yes

characteristic

119

C₂³

—

122

—

120

Q₈

—

122

—

121

Q₈

—

122

—

122

D₄

—

122

—

123

D₄

—

122

—

124

C₂×C₄

—

122

—

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

D₄×D₆₁

yes

characteristic

131

C₂×C₄

#11758226

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

122

—

137

C₈

—

122

—

138

C₈

#11758189

yes

complemented

139

C₂²

—

244

—

140

C₂²

—

244

—

141

C₂²

—

244

—

142

C₂²

#14519739

yes

characteristic

143

C₄

—

144

C₄

—

244

—

145

C₄

—

244

—

146

C₄

—

244

—

147

C₄

C₂×#11758189

yes

characteristic

148

C₄

#14519785

yes

characteristic

149

C₄

SD₁₆×D₆₁

yes

characteristic

150

C₄

#14519798

yes

characteristic

151

C₂

—

488

—

152

C₂

#16839839

yes

characteristic

153

C₂

#16839849

yes

characteristic

154

C₂

#16839878

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#334	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#334	D₆₁
hypocenter	C₆₁	#1590
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#334

C₂

socle

C₂²×C₆₁

C₄×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16839889	C₂	semidirect
4. #16839492	C₂	semidirect
6. C₈×#92418	C₂	semidirect
7. C₆₁×#334	C₂	semidirect
57. C₈×C₆₁	M₁₆	semidirect
60. #92418	SD₁₆	semidirect
81. #334	D₆₁	semidirect
85. C₆₁	#1590	semidirect
138. C₈	#11758189	semidirect

H ≅ G/N

type

1. #16839889

C₂

semidirect

4. #16839492

C₂

semidirect

6. C₈×#92418

C₂

semidirect

7. C₆₁×#334

C₂

semidirect

57. C₈×C₆₁

M₁₆

semidirect

60. #92418

SD₁₆

semidirect

81. #334

D₆₁

semidirect

85. C₆₁

#1590

semidirect

138. C₈

#11758189

semidirect

Subgroups — #21444708

Conjugacy classes of subgroups

Complements