Group #21445162 — subgroups

Subgroups — #21445162

Order 7808.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840136	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839957	C₂	1	1	yes	maximal, characteristic
3	D₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839508	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758201	—	2	1	no	—
9	C₂×#11758201	—	2	1	no	—
10	C₂×#11758199	—	2	1	no	—
11	C₂²×#92418	—	2	1	no	—
12	#14519774	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	#14519697	—	2	1	no	—
16	C₂²⋊#92418	—	2	1	no	—
17	C₄×#92418	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
20	#107×C₆₁	—	2	1	no	—
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	#102×C₆₁	—	2	1	no	—
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×D₂₄₄	—	2	1	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₂²⋊C₄×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758201	—	4	2	no	—
36	#11758201	—	4	2	no	—
37	#11758199	—	4	2	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	4	2	no	—
41	C₂×#92418	D₄	1	1	yes	characteristic
42	C₂×#92418	D₄	1	1	yes	characteristic
43	C₂³×C₆₁	—	2	1	no	—
44	C₂²×D₆₁	—	4	1	no	—
45	Q₈×C₆₁	—	4	1	no	—
46	Q₈×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	2	1	no	—
48	D₄×C₆₁	SD₁₆	1	2	yes	complemented
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	4	1	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
56	D₂₄₄	—	4	1	no	—
57	D₂₄₄	—	4	1	no	—
58	D₂₄₄	—	4	1	no	—
59	D₂₄₄	—	4	1	no	—
60	D₂₄₄	—	8	1	no	—
61	C₈×C₆₁	—	4	1	no	—
62	#92418	—	2	1	no	—
63	#92418	—	4	1	no	—
64	#92418	—	4	1	no	—
65	#92418	D₈	1	2	yes	complemented
66	C₂²×C₆₁	—	2	2	no	—
67	C₂²×C₆₁	—	4	1	no	—
68	C₂²×C₆₁	—	4	1	no	—
69	C₂²×C₆₁	#121	1	1	yes	characteristic
70	C₂×D₆₁	—	8	1	no	—
71	C₂×D₆₁	—	8	1	no	—
72	C₂×D₆₁	—	8	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	4	1	no	—
75	C₄×C₆₁	—	8	1	no	—
76	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
77	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₄×C₆₁	#135	1	1	yes	characteristic
79	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
80	#1665	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	4	2	no	—
82	C₂×C₆₁	#459	1	1	yes	characteristic
83	C₂×C₆₁	#465	1	1	yes	characteristic
84	C₂×C₆₁	#471	1	1	yes	characteristic
85	D₆₁	—	16	1	no	—
86	#492	—	61	1	no	—
87	#474	D₆₁	1	1	yes	characteristic, complemented
88	#459	—	61	1	no	—
89	C₈×D₄	—	61	1	no	—
90	#334	—	61	1	no	—
91	#332	—	61	1	no	—
92	#331	—	61	1	no	—
93	C₆₁	#1665	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	122	1	no	—
95	C₂×SD₁₆	—	122	1	no	—
96	C₂×D₈	—	122	1	no	—
97	C₂²×C₈	—	122	1	no	—
98	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
99	#127	—	61	1	no	—
100	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	C₄⋊C₈	—	61	1	no	—
103	C₄⋊C₈	—	61	1	no	—
104	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
105	#102	—	2	1	no	—
106	#102	—	122	1	no	—
107	C₂²⋊C₈	—	122	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	122	1	no	—
111	C₂×D₄	—	122	1	no	—
112	C₂×D₄	—	122	1	no	—
113	C₂×D₄	C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	SD₁₆	—	244	2	no	—
116	SD₁₆	—	244	2	no	—
117	D₈	—	244	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	122	1	no	—
122	C₂×C₈	—	122	1	no	—
123	C₂×C₈	—	244	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	C₂²×D₆₁	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	—	244	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	488	1	no	—
135	D₄	—	244	1	no	—
136	D₄	—	244	1	no	—
137	D₄	—	244	1	no	—
138	D₄	—	244	1	no	—
139	D₄	#11758201	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
145	C₂×C₄	#11758224	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	61	2	no	—
149	C₈	—	122	1	no	—
150	C₈	—	244	1	no	—
151	C₈	—	244	1	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	488	1	no	—
156	C₂²	—	488	1	no	—
157	C₂²	—	488	1	no	—
158	C₂²	#14519749	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	8	1	no	—
162	C₄	D₈×D₆₁	1	1	yes	characteristic
163	C₄	C₂×#11758201	1	1	yes	characteristic
164	C₄	#14519797	1	1	yes	characteristic
165	C₄	#14519801	1	1	yes	characteristic
166	C₂	—	4	2	no	—
167	C₂	—	976	1	no	—
168	C₂	#16839955	1	1	yes	characteristic
169	C₂	#16839993	1	1	yes	characteristic
170	C₂	#16840027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840136

C₂

yes

maximal, characteristic, complemented

#16839957

C₂

yes

maximal, characteristic

D₄×#92418

C₂

yes

maximal, characteristic, complemented

#16839512

C₂

yes

maximal, characteristic, complemented

#16839508

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, characteristic, complemented

C₆₁×#474

C₂

yes

maximal, characteristic, complemented

C₂×#11758201

—

C₂×#11758201

—

C₂×#11758199

—

C₂²×#92418

—

#14519774

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄×D₄×C₆₁

C₂²

yes

characteristic

#107×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

#102×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₂³×C₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

SD₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

—

—

—

D₈

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#492

—

#474

D₆₁

yes

characteristic, complemented

#459

—

C₈×D₄

—

#334

—

#332

—

#331

—

C₆₁

#1665

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₂²×C₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

—

100

C₄×D₄

C₂×D₆₁

yes

characteristic

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₆₁

yes

characteristic

105

#102

—

106

#102

—

122

—

107

C₂²⋊C₈

—

122

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

122

—

111

C₂×D₄

—

122

—

112

C₂×D₄

—

122

—

113

C₂×D₄

C₂²⋊D₆₁

yes

characteristic

114

C₂²×C₄

—

115

SD₁₆

—

244

—

116

SD₁₆

—

244

—

117

D₈

—

244

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

—

122

C₂×C₈

—

122

—

123

C₂×C₈

—

244

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

C₂²⋊D₆₁

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₆₁

yes

characteristic

129

C₂³

—

130

C₂³

—

244

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

488

—

135

D₄

—

244

—

136

D₄

—

244

—

137

D₄

—

244

—

138

D₄

—

244

—

139

D₄

#11758201

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

D₄×D₆₁

yes

characteristic

145

C₂×C₄

#11758224

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

122

—

150

C₈

—

244

—

151

C₈

—

244

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

488

—

156

C₂²

—

488

—

157

C₂²

—

488

—

158

C₂²

#14519749

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

D₈×D₆₁

yes

characteristic

163

C₄

C₂×#11758201

yes

characteristic

164

C₄

#14519797

yes

characteristic

165

C₄

#14519801

yes

characteristic

166

C₂

—

167

C₂

—

976

—

168

C₂

#16839955

yes

characteristic

169

C₂

#16839993

yes

characteristic

170

C₂

#16840027

yes

characteristic

	subgroup	G/H
center	C₂²	#14519749
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#474	C₂
socle	C₂²×C₆₁	#121
hypercenter	#474	D₆₁
hypocenter	C₆₁	#1665
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#14519749

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#474

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #16840136	C₂	semidirect
3. D₄×#92418	C₂	semidirect
4. #16839512	C₂	semidirect
5. #16839508	C₂	semidirect
6. #16839505	C₂	semidirect
7. C₆₁×#474	C₂	semidirect
48. D₄×C₆₁	SD₁₆	semidirect
65. #92418	D₈	semidirect
87. #474	D₆₁	semidirect
93. C₆₁	#1665	semidirect
139. D₄	#11758201	semidirect

H ≅ G/N

type

1. #16840136

C₂

semidirect

3. D₄×#92418

C₂

semidirect

4. #16839512

C₂

semidirect

5. #16839508

C₂

semidirect

6. #16839505

C₂

semidirect

7. C₆₁×#474

C₂

semidirect

48. D₄×C₆₁

SD₁₆

semidirect

65. #92418

D₈

semidirect

87. #474

D₆₁

semidirect

93. C₆₁

#1665

semidirect

139. D₄

#11758201

semidirect

Subgroups — #21445162

Conjugacy classes of subgroups

Complements