Group #21445288 — subgroups

Subgroups — #21445288

Order 7808.

168 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840044	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839959	C₂	1	1	yes	maximal, characteristic
3	#16839892	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839508	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839506	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#492	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758200	—	2	1	no	—
9	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	#14519708	—	2	1	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519706	—	2	1	no	—
15	#14519700	—	2	1	no	—
16	#14519698	—	2	1	no	—
17	#14519687	—	2	1	no	—
18	C₂×SD₁₆×C₆₁	—	2	1	no	—
19	C₂×SD₁₆×C₆₁	—	2	1	no	—
20	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	#127×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₄×C₆₁	—	2	1	no	—
27	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
28	SD₁₆×C₆₁	—	4	2	no	—
29	SD₁₆×C₆₁	—	4	2	no	—
30	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758200	—	4	2	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	#11758196	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	2	1	no	—
41	C₂×#92418	—	2	1	no	—
42	C₂³×C₆₁	—	4	1	no	—
43	Q₈×C₆₁	—	4	1	no	—
44	Q₈×C₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	4	1	no	—
49	D₄×C₆₁	—	8	1	no	—
50	C₂×C₄×C₆₁	—	4	1	no	—
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂×Dic₆₁	—	2	1	no	—
55	C₂×Dic₆₁	—	2	1	no	—
56	Dic₁₂₂	—	2	1	no	—
57	Dic₁₂₂	—	2	1	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	SD₁₆	1	2	yes	complemented
60	#92418	—	4	1	no	—
61	#92418	—	4	1	no	—
62	#92418	—	4	1	no	—
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	#121	1	1	yes	characteristic
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	8	1	no	—
69	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₆₁	#137	1	1	yes	characteristic
73	Dic₆₁	—	4	1	no	—
74	Dic₆₁	—	4	1	no	—
75	#1686	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	16	1	no	—
77	C₂×C₆₁	#460	1	1	yes	characteristic
78	C₂×C₆₁	#468	1	1	yes	characteristic
79	C₂×C₆₁	#471	1	1	yes	characteristic
80	#492	D₆₁	1	1	yes	characteristic
81	#475	—	61	1	no	—
82	#460	—	61	1	no	—
83	#446	—	61	1	no	—
84	#335	—	61	1	no	—
85	#332	—	61	1	no	—
86	#331	—	61	1	no	—
87	C₆₁	#1686	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	2	1	no	—
89	C₂×SD₁₆	—	2	1	no	—
90	C₂×SD₁₆	—	122	1	no	—
91	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
92	#127	C₂×D₆₁	1	1	yes	characteristic
93	C₄×Q₈	—	61	1	no	—
94	#106	—	122	1	no	—
95	C₄⋊C₈	—	61	1	no	—
96	C₄⋊C₈	—	61	1	no	—
97	C₄⋊C₈	—	61	1	no	—
98	C₄⋊C₈	—	122	1	no	—
99	#103	—	122	1	no	—
100	#102	—	122	1	no	—
101	#97	—	122	1	no	—
102	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	61	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	SD₁₆	—	4	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	244	2	no	—
111	C₂×C₈	—	122	1	no	—
112	C₂×C₈	—	122	1	no	—
113	C₂×C₈	—	122	1	no	—
114	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	61	1	no	—
119	C₄⋊C₄	—	122	1	no	—
120	C₄²	—	122	1	no	—
121	C₄²	C₂²×D₆₁	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	122	1	no	—
126	Q₈	—	122	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	122	1	no	—
134	C₂×C₄	—	122	1	no	—
135	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
136	C₂×C₄	#11758224	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	244	1	no	—
140	C₈	—	244	1	no	—
141	C₈	—	244	1	no	—
142	C₈	#11758200	1	2	yes	complemented
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	#14519748	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	244	1	no	—
150	C₄	—	244	1	no	—
151	C₄	C₂×#11758200	1	1	yes	characteristic
152	C₄	#14519803	1	1	yes	characteristic
153	C₄	#14519804	1	1	yes	characteristic
154	C₄	#14519806	1	1	yes	characteristic
155	C₂	—	16	1	no	—
156	C₂	#16839959	1	1	yes	characteristic
157	C₂	#16840005	1	1	yes	characteristic
158	C₂	#16840022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840044

C₂

yes

maximal, characteristic, complemented

#16839959

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, characteristic, complemented

#16839515

C₂

yes

maximal, characteristic, complemented

#16839508

C₂

yes

maximal, characteristic, complemented

#16839506

C₂

yes

maximal, characteristic

C₆₁×#492

C₂

yes

maximal, characteristic

C₂×#11758200

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519708

—

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

—

C₂×SD₁₆×C₆₁

—

C₂×SD₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×Dic₁₂₂

D₄

yes

characteristic

SD₁₆×C₆₁

—

SD₁₆×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758200

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×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

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

SD₁₆

yes

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

#1686

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

—

—

—

—

—

C₆₁

#1686

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

—

122

—

100

#102

—

122

—

101

#97

—

122

—

102

C₄×C₈

C₂×D₆₁

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

109

SD₁₆

—

110

SD₁₆

—

244

—

111

C₂×C₈

—

122

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

115

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

122

—

120

C₄²

—

122

—

121

C₄²

C₂²×D₆₁

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

122

—

126

Q₈

—

122

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

122

—

135

C₂×C₄

D₄×D₆₁

yes

characteristic

136

C₂×C₄

#11758224

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

C₈

—

139

C₈

—

244

—

140

C₈

—

244

—

141

C₈

—

244

—

142

C₈

#11758200

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#14519748

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

244

—

150

C₄

—

244

—

151

C₄

C₂×#11758200

yes

characteristic

152

C₄

#14519803

yes

characteristic

153

C₄

#14519804

yes

characteristic

154

C₄

#14519806

yes

characteristic

155

C₂

—

156

C₂

#16839959

yes

characteristic

157

C₂

#16840005

yes

characteristic

158

C₂

#16840022

yes

characteristic

	subgroup	G/H
center	C₂²	#14519748
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#492	C₂
socle	C₂²×C₆₁	#121
hypercenter	#492	D₆₁
hypocenter	C₆₁	#1686
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14519748

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#492

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16840044	C₂	semidirect
3. #16839892	C₂	semidirect
4. #16839515	C₂	semidirect
5. #16839508	C₂	semidirect
59. C₈×C₆₁	SD₁₆	semidirect
87. C₆₁	#1686	semidirect
142. C₈	#11758200	semidirect

H ≅ G/N

type

1. #16840044

C₂

semidirect

3. #16839892

C₂

semidirect

4. #16839515

C₂

semidirect

5. #16839508

C₂

semidirect

59. C₈×C₆₁

SD₁₆

semidirect

87. C₆₁

#1686

semidirect

142. C₈

#11758200

semidirect

Subgroups — #21445288

Conjugacy classes of subgroups

Complements