Group #21445331 — subgroups

Subgroups — #21445331

Order 7808.

166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16840165	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839968	C₂	1	1	yes	maximal, characteristic
3	#16839964	C₂	1	1	yes	maximal, characteristic
4	#16839839	C₂	1	1	yes	maximal, characteristic
5	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839507	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758190	—	2	1	no	—
9	C₂×#11758201	—	2	1	no	—
10	C₂×#11758189	—	2	1	no	—
11	#14519777	C₂²	1	1	yes	characteristic
12	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
13	#14519710	—	2	1	no	—
14	#14519710	—	2	1	no	—
15	C₄⋊#92418	C₂²	1	1	yes	characteristic
16	#14519702	—	2	1	no	—
17	#14519701	—	2	1	no	—
18	#14519689	—	2	1	no	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₂₄₄	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758201	—	4	2	no	—
33	#11758198	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	D₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	#11758190	—	4	2	no	—
41	#11758189	—	4	2	no	—
42	C₂²×D₆₁	—	2	1	no	—
43	Q₈×C₆₁	—	4	1	no	—
44	Q₈×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂×Dic₆₁	—	2	1	no	—
50	C₂×Dic₆₁	—	4	1	no	—
51	D₂₄₄	—	2	1	no	—
52	D₂₄₄	—	2	1	no	—
53	C₄×D₆₁	—	4	1	no	—
54	C₄×D₆₁	—	4	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	Dic₁₂₂	—	4	1	no	—
57	C₈×C₆₁	—	4	1	no	—
58	C₈×C₆₁	—	4	1	no	—
59	#92418	—	2	1	no	—
60	#92418	—	4	1	no	—
61	#92418	SD₁₆	1	2	yes	complemented
62	C₂²×C₆₁	#121	1	1	yes	characteristic
63	C₂×D₆₁	—	4	1	no	—
64	C₂×D₆₁	—	4	1	no	—
65	C₂×D₆₁	—	4	1	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	8	1	no	—
68	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₆₁	#137	1	1	yes	characteristic
72	Dic₆₁	—	4	1	no	—
73	Dic₆₁	—	8	1	no	—
74	#1693	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	#461	1	1	yes	characteristic
76	C₂×C₆₁	#469	1	1	yes	characteristic
77	C₂×C₆₁	#473	1	1	yes	characteristic
78	D₆₁	—	8	1	no	—
79	#500	—	61	1	no	—
80	#461	—	61	1	no	—
81	#461	—	61	1	no	—
82	#436	—	61	1	no	—
83	#334	—	61	1	no	—
84	#332	—	61	1	no	—
85	#332	D₆₁	1	1	yes	characteristic, complemented
86	C₆₁	#1693	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₁₆	—	122	1	no	—
88	C₂×SD₁₆	—	122	1	no	—
89	C₂×M₁₆	—	122	1	no	—
90	C₄⋊Q₈	—	61	1	no	—
91	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄×D₄	—	61	1	no	—
93	#107	—	122	1	no	—
94	#107	—	122	1	no	—
95	C₄⋊C₈	—	61	1	no	—
96	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
97	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
98	#103	—	122	1	no	—
99	#103	—	122	1	no	—
100	C₂²⋊C₈	—	122	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	122	1	no	—
104	C₂×D₄	—	61	1	no	—
105	C₂²×C₄	—	122	1	no	—
106	SD₁₆	—	244	2	no	—
107	SD₁₆	—	244	2	no	—
108	M₁₆	—	244	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	122	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	61	1	no	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄⋊C₄	—	122	1	no	—
119	C₂²⋊C₄	—	122	1	no	—
120	C₄²	C₂²×D₆₁	1	1	yes	characteristic
121	C₂³	—	122	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	244	1	no	—
125	Q₈	—	244	1	no	—
126	D₄	—	122	1	no	—
127	D₄	—	122	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	244	1	no	—
131	C₂×C₄	—	244	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	#11758222	1	1	yes	characteristic
134	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
135	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	61	2	no	—
139	C₈	—	122	1	no	—
140	C₈	—	244	1	no	—
141	C₂²	—	244	1	no	—
142	C₂²	—	244	1	no	—
143	C₂²	—	244	1	no	—
144	C₂²	#14519750	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	488	1	no	—
148	C₄	—	244	1	no	—
149	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
150	C₄	#14519801	1	1	yes	characteristic
151	C₄	#14519804	1	1	yes	characteristic
152	C₄	#14519809	1	1	yes	characteristic
153	C₂	—	488	1	no	—
154	C₂	#16839966	1	1	yes	characteristic
155	C₂	#16840014	1	1	yes	characteristic
156	C₂	#16840036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840165

C₂

yes

maximal, characteristic, complemented

#16839968

C₂

yes

maximal, characteristic

#16839964

C₂

yes

maximal, characteristic

#16839839

C₂

yes

maximal, characteristic

#16839512

C₂

yes

maximal, characteristic, complemented

#16839507

C₂

yes

maximal, characteristic, complemented

C₆₁×#332

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758201

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

#11758190

—

#11758189

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

SD₁₆

yes

complemented

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₆₁

#1693

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₂×M₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#107

—

122

—

#107

—

122

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#103

—

122

—

100

C₂²⋊C₈

—

122

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

122

—

104

C₂×D₄

—

105

C₂²×C₄

—

122

—

106

SD₁₆

—

244

—

107

SD₁₆

—

244

—

108

M₁₆

—

244

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

122

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

122

—

118

C₄⋊C₄

—

122

—

119

C₂²⋊C₄

—

122

—

120

C₄²

C₂²×D₆₁

yes

characteristic

121

C₂³

—

122

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

244

—

125

Q₈

—

244

—

126

D₄

—

122

—

127

D₄

—

122

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

244

—

131

C₂×C₄

—

244

—

132

C₂×C₄

—

244

—

133

C₂×C₄

#11758222

yes

characteristic

134

C₂×C₄

D₄×D₆₁

yes

characteristic

135

C₂×C₄

D₄×D₆₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

122

—

140

C₈

—

244

—

141

C₂²

—

244

—

142

C₂²

—

244

—

143

C₂²

—

244

—

144

C₂²

#14519750

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

488

—

148

C₄

—

244

—

149

C₄

SD₁₆×D₆₁

yes

characteristic

150

C₄

#14519801

yes

characteristic

151

C₄

#14519804

yes

characteristic

152

C₄

#14519809

yes

characteristic

153

C₂

—

488

—

154

C₂

#16839966

yes

characteristic

155

C₂

#16840014

yes

characteristic

156

C₂

#16840036

yes

characteristic

	subgroup	G/H
center	C₂²	#14519750
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#332	C₂
socle	C₂²×C₆₁	#121
hypercenter	#332	D₆₁
hypocenter	C₆₁	#1693
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14519750

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#332

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #16840165	C₂	semidirect
5. #16839512	C₂	semidirect
6. #16839507	C₂	semidirect
7. C₆₁×#332	C₂	semidirect
61. #92418	SD₁₆	semidirect
85. #332	D₆₁	semidirect
86. C₆₁	#1693	semidirect

H ≅ G/N

type

1. #16840165

C₂

semidirect

5. #16839512

C₂

semidirect

6. #16839507

C₂

semidirect

7. C₆₁×#332

C₂

semidirect

61. #92418

SD₁₆

semidirect

85. #332

D₆₁

semidirect

86. C₆₁

#1693

semidirect

Subgroups — #21445331

Conjugacy classes of subgroups

Complements