Group #21458808 — subgroups

Subgroups — #21458808

Order 7808.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16840485	C₂	1	2	yes	maximal, complemented
2	#16840146	C₂	1	2	yes	maximal
3	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839985	C₂	1	4	yes	maximal, complemented
5	#16839968	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₄₄	C₂	1	2	yes	maximal, complemented
7	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₄₄	C₂²	1	2	yes	—
10	C₂×Dic₂₄₄	C₂²	1	2	yes	—
11	C₂×#11758190	C₂²	1	4	yes	—
12	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
13	#14519777	C₂²	1	2	yes	—
14	#14519772	—	2	4	no	—
15	#14519765	C₂²	1	4	yes	—
16	#14519753	—	2	2	no	—
17	C₄×Dic₁₂₂	C₂²	1	4	yes	—
18	C₄×D₂₄₄	C₂²	1	2	yes	—
19	#14519735	—	2	2	no	—
20	#14519711	C₂²	1	1	yes	characteristic
21	#14519702	C₂²	1	4	yes	—
22	#14519699	C₂²	1	2	yes	—
23	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
26	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂²	1	2	yes	—
28	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂×D₂₄₄	C₂³	1	2	yes	—
30	C₂×C₄×D₆₁	—	2	2	no	—
31	C₂×Dic₁₂₂	C₂³	1	4	yes	—
32	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂³	1	2	yes	—
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
37	C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	C₄²×C₆₁	C₂³	1	2	yes	—
39	#11758198	—	2	4	no	—
40	#11758198	—	2	2	no	—
41	#11758198	—	2	4	no	—
42	C₄⋊Dic₆₁	—	2	2	no	—
43	C₄⋊Dic₆₁	—	2	2	no	—
44	C₄⋊Dic₆₁	C₂³	1	2	yes	—
45	#11758196	—	2	4	no	—
46	C₄×Dic₆₁	—	2	4	no	—
47	Dic₂₄₄	—	2	4	no	—
48	Dic₂₄₄	—	4	2	no	—
49	D₄₈₈	—	2	2	no	—
50	#11758190	—	4	4	no	—
51	C₂²×D₆₁	—	2	2	no	—
52	Q₈×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	—	2	4	no	—
60	D₂₄₄	—	2	2	no	—
61	D₂₄₄	—	2	2	no	—
62	C₄×D₆₁	—	4	4	no	—
63	Dic₁₂₂	—	2	4	no	—
64	Dic₁₂₂	—	2	4	no	—
65	C₈×C₆₁	—	2	2	no	—
66	C₈×C₆₁	—	2	1	no	—
67	C₈×C₆₁	#41	1	2	yes	—
68	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₆₁	—	4	2	no	—
70	C₂×D₆₁	—	4	2	no	—
71	C₂×D₆₁	—	4	2	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	2	2	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	#135	1	2	yes	—
76	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
77	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
78	Dic₆₁	—	4	4	no	—
79	Dic₆₁	—	4	2	no	—
80	#3393	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	#550	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
83	C₂×C₆₁	#578	1	1	yes	characteristic
84	D₆₁	—	8	2	no	—
85	#556	—	61	2	no	—
86	#495	—	61	2	no	—
87	#494	—	61	1	no	—
88	#464	—	61	4	no	—
89	#461	—	61	2	no	—
90	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
91	C₄×Q₁₆	—	61	2	no	—
92	C₄×D₈	—	61	1	no	—
93	C₆₁	#3393	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	61	2	no	—
95	C₂×Q₁₆	—	61	2	no	—
96	C₂×SD₁₆	—	61	4	no	—
97	C₂×D₈	—	61	1	no	—
98	C₄⋊Q₈	—	61	2	no	—
99	#126	—	122	4	no	—
100	#124	—	61	4	no	—
101	C₂²⋊Q₈	—	122	2	no	—
102	C₄×Q₈	—	61	4	no	—
103	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
104	C₄×D₄	—	61	2	no	—
105	#117	—	122	2	no	—
106	#107	—	61	1	no	—
107	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
109	#103	—	61	4	no	—
110	#102	—	61	2	no	—
111	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
112	C₄×C₈	C₂×D₆₁	1	2	yes	—
113	C₂×Q₈	—	61	4	no	—
114	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
115	C₂×D₄	—	61	2	no	—
116	C₂²×C₄	—	122	2	no	—
117	Q₁₆	—	122	4	no	—
118	Q₁₆	—	244	2	no	—
119	SD₁₆	—	244	4	no	—
120	D₈	—	122	2	no	—
121	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	C₂×C₈	C₂²×D₆₁	1	2	yes	—
123	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₄	—	61	2	no	—
125	C₄⋊C₄	—	122	4	no	—
126	C₄⋊C₄	—	122	2	no	—
127	C₄⋊C₄	—	122	2	no	—
128	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
129	C₄⋊C₄	D₂₄₄	1	2	yes	—
130	C₂²⋊C₄	—	122	4	no	—
131	C₂²⋊C₄	—	122	2	no	—
132	C₂²⋊C₄	—	122	4	no	—
133	C₄²	—	122	4	no	—
134	C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	C₄²	C₂²×D₆₁	1	2	yes	—
136	C₂³	—	122	2	no	—
137	Q₈	—	2	2	no	—
138	Q₈	—	122	4	no	—
139	Q₈	—	122	4	no	—
140	D₄	—	122	2	no	—
141	D₄	—	122	2	no	—
142	C₂×C₄	—	122	2	no	—
143	C₂×C₄	—	122	4	no	—
144	C₂×C₄	—	244	4	no	—
145	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
146	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
148	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	#11758226	1	2	yes	—
153	C₂²	—	244	2	no	—
154	C₂²	—	244	2	no	—
155	C₂²	—	244	2	no	—
156	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	244	2	no	—
161	C₄	—	244	4	no	—
162	C₄	#14519799	1	2	yes	—
163	C₄	C₂×#11758226	1	1	yes	characteristic
164	C₄	#14519823	1	1	yes	characteristic
165	C₂	—	488	2	no	—
166	C₂	#16840434	1	1	yes	characteristic
167	C₂	C₂×#14519799	1	1	yes	characteristic
168	C₂	#16840579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840485

C₂

yes

maximal, complemented

#16840146

C₂

yes

maximal

#16840142

C₂

yes

maximal, characteristic, complemented

#16839985

C₂

yes

maximal, complemented

#16839968

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, complemented

C₄×D₄₈₈

C₂

yes

maximal, characteristic

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₄₄

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×#11758190

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519777

C₂²

yes

—

#14519772

—

#14519765

C₂²

yes

—

#14519753

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519735

—

#14519711

C₂²

yes

characteristic

#14519702

C₂²

yes

—

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

#11758198

—

#11758198

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

D₄₈₈

—

#11758190

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

—

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×Q₁₆

—

C₄×D₈

—

C₆₁

#3393

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#126

—

122

—

100

#124

—

101

C₂²⋊Q₈

—

122

—

102

C₄×Q₈

—

103

C₄×Q₈

C₂×D₆₁

yes

characteristic

104

C₄×D₄

—

105

#117

—

122

—

106

#107

—

107

C₄⋊C₈

C₂×D₆₁

yes

characteristic

108

C₄⋊C₈

C₂×D₆₁

yes

—

109

#103

—

110

#102

—

111

C₄×C₈

C₂×D₆₁

yes

characteristic

112

C₄×C₈

C₂×D₆₁

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

D₂₄₄

yes

characteristic

115

C₂×D₄

—

116

C₂²×C₄

—

122

—

117

Q₁₆

—

122

—

118

Q₁₆

—

244

—

119

SD₁₆

—

244

—

120

D₈

—

122

—

121

C₂×C₈

C₂²×D₆₁

yes

characteristic

122

C₂×C₈

C₂²×D₆₁

yes

—

123

C₂×C₈

C₂²×D₆₁

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

122

—

126

C₄⋊C₄

—

122

—

127

C₄⋊C₄

—

122

—

128

C₄⋊C₄

D₂₄₄

yes

characteristic

129

C₄⋊C₄

D₂₄₄

yes

—

130

C₂²⋊C₄

—

122

—

131

C₂²⋊C₄

—

122

—

132

C₂²⋊C₄

—

122

—

133

C₄²

—

122

—

134

C₄²

C₂²×D₆₁

yes

characteristic

135

C₄²

C₂²×D₆₁

yes

—

136

C₂³

—

122

—

137

Q₈

—

138

Q₈

—

122

—

139

Q₈

—

122

—

140

D₄

—

122

—

141

D₄

—

122

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

244

—

145

C₂×C₄

C₂×D₂₄₄

yes

—

146

C₂×C₄

C₂×D₂₄₄

yes

characteristic

147

C₂×C₄

C₂×D₂₄₄

yes

—

148

C₂×C₄

C₂×D₂₄₄

yes

characteristic

149

C₂×C₄

C₂³×D₆₁

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#11758226

yes

—

153

C₂²

—

244

—

154

C₂²

—

244

—

155

C₂²

—

244

—

156

C₂²

C₂²×D₂₄₄

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

244

—

161

C₄

—

244

—

162

C₄

#14519799

yes

—

163

C₄

C₂×#11758226

yes

characteristic

164

C₄

#14519823

yes

characteristic

165

C₂

—

488

—

166

C₂

#16840434

yes

characteristic

167

C₂

C₂×#14519799

yes

characteristic

168

C₂

#16840579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#3393
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#3393

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16840485	C₂	semidirect
3. #16840142	C₂	semidirect
4. #16839985	C₂	semidirect
5. #16839968	C₂	semidirect
6. C₄×Dic₂₄₄	C₂	semidirect
8. C₈×Q₈×C₆₁	C₂	semidirect
90. C₈×Q₈	D₆₁	semidirect
93. C₆₁	#3393	semidirect

H ≅ G/N

type

1. #16840485

C₂

semidirect

3. #16840142

C₂

semidirect

4. #16839985

C₂

semidirect

5. #16839968

C₂

semidirect

6. C₄×Dic₂₄₄

C₂

semidirect

8. C₈×Q₈×C₆₁

C₂

semidirect

90. C₈×Q₈

D₆₁

semidirect

93. C₆₁

#3393

semidirect

Subgroups — #21458808

Conjugacy classes of subgroups

Complements