Group #21458450 — subgroups

Subgroups — #21458450

Order 7808.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840108	C₂	1	3	yes	maximal, complemented
3	#16840042	C₂	1	3	yes	maximal, complemented
4	#16839954	C₂	1	3	yes	maximal, complemented
5	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11758199	C₂	1	3	yes	maximal, complemented
7	C₆₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	—	2	1	no	—
9	C₂×#11758199	C₂²	1	3	yes	—
10	#14519777	C₂²	1	3	yes	—
11	#14519757	—	2	3	no	—
12	C₄×D₂₄₄	C₂²	1	3	yes	—
13	#14519710	C₂²	1	3	yes	—
14	C₄⋊#92418	C₂²	1	3	yes	—
15	#14519698	C₂²	1	6	yes	—
16	#14519698	C₂²	1	3	yes	—
17	#14519697	C₂²	1	1	yes	characteristic
18	C₄×#92418	C₂²	1	3	yes	—
19	C₂×#41×C₆₁	—	2	1	no	—
20	#127×C₆₁	C₂²	1	3	yes	—
21	#121×C₆₁	—	2	3	no	—
22	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₆₁	C₂²	1	3	yes	—
24	#41×C₆₁	—	4	4	no	—
25	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×D₄×C₆₁	—	2	3	no	—
27	C₂×D₄×C₆₁	—	2	3	no	—
28	C₂×D₄×C₆₁	C₂³	1	3	yes	—
29	C₂²×C₄×C₆₁	—	2	3	no	—
30	Q₈×D₆₁	—	4	4	no	—
31	C₂×D₂₄₄	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₆₁	—	2	3	no	—
33	C₂×Dic₁₂₂	—	2	3	no	—
34	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
35	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₆₁	—	2	3	no	—
37	C₄²×C₆₁	C₂³	1	3	yes	—
38	#11758199	—	2	6	no	—
39	#11758198	—	2	3	no	—
40	C₄⋊Dic₆₁	—	2	3	no	—
41	C₄⋊Dic₆₁	C₂³	1	3	yes	—
42	C₂×#92418	C₂³	1	1	yes	characteristic
43	C₂×#92418	C₂³	1	3	yes	—
44	C₂³×C₆₁	—	2	3	no	—
45	C₂²×D₆₁	—	2	1	no	—
46	Q₈×C₆₁	D₈	1	4	yes	complemented
47	D₄×C₆₁	—	2	3	no	—
48	D₄×C₆₁	—	2	3	no	—
49	D₄×C₆₁	—	4	6	no	—
50	D₄×C₆₁	—	8	3	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	4	6	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	3	no	—
57	D₂₄₄	—	2	1	no	—
58	D₂₄₄	#41	1	2	yes	complemented
59	C₄×D₆₁	—	4	6	no	—
60	Dic₁₂₂	—	4	6	no	—
61	#92418	—	2	3	no	—
62	#92418	—	2	1	no	—
63	C₂²×C₆₁	—	4	3	no	—
64	C₂²×C₆₁	—	4	3	no	—
65	C₂²×C₆₁	—	4	3	no	—
66	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₆₁	—	2	2	no	—
68	C₂×D₆₁	—	4	1	no	—
69	C₂×D₆₁	—	4	1	no	—
70	C₄×C₆₁	—	2	3	no	—
71	C₄×C₆₁	—	4	1	no	—
72	C₄×C₆₁	C₂×D₈	1	6	yes	—
73	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
74	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₆₁	—	4	3	no	—
76	#3358	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	—	8	3	no	—
78	C₂×C₆₁	#548	1	1	yes	characteristic
79	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
80	C₂×C₆₁	#577	1	1	yes	characteristic
81	D₆₁	—	4	2	no	—
82	#550	D₆₁	1	1	yes	characteristic, complemented
83	D₄×Q₈	—	61	1	no	—
84	#486	—	61	3	no	—
85	#474	—	61	3	no	—
86	#459	—	61	3	no	—
87	C₈×Q₈	—	61	1	no	—
88	C₄×D₈	—	61	3	no	—
89	C₆₁	#3358	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	122	1	no	—
92	C₂×D₈	—	61	3	no	—
93	C₄⋊Q₈	—	61	3	no	—
94	#127	C₂×D₆₁	1	3	yes	—
95	C₂²⋊Q₈	—	122	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
98	C₄×D₄	—	61	3	no	—
99	C₄×D₄	C₂×D₆₁	1	3	yes	—
100	#107	—	61	3	no	—
101	C₄⋊C₈	—	61	3	no	—
102	#102	—	61	3	no	—
103	#102	—	61	6	no	—
104	#102	—	61	1	no	—
105	C₄×C₈	—	61	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	122	3	no	—
108	C₂×Q₈	—	244	4	no	—
109	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	61	1	no	—
113	C₂×D₄	C₂²×D₆₁	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	122	3	no	—
116	D₈	—	122	6	no	—
117	C₂×C₈	—	61	1	no	—
118	C₂×C₈	—	61	3	no	—
119	C₄⋊C₄	—	61	3	no	—
120	C₄⋊C₄	—	122	3	no	—
121	C₄⋊C₄	C₂²⋊D₆₁	1	3	yes	—
122	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	122	3	no	—
125	C₄²	C₂²×D₆₁	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	122	1	no	—
128	Q₈	—	244	6	no	—
129	Q₈	#11758199	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	61	2	no	—
135	D₄	—	122	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	122	3	no	—
139	C₂×C₄	—	244	6	no	—
140	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
142	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
143	C₈	—	122	3	no	—
144	C₈	—	122	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	122	2	no	—
149	C₂²	—	244	1	no	—
150	C₂²	—	244	1	no	—
151	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	244	3	no	—
155	C₄	C₂×#11758199	1	6	yes	—
156	C₄	#41×D₆₁	1	1	yes	characteristic
157	C₄	#14519824	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	244	2	no	—
160	C₂	#16840419	1	1	yes	characteristic
161	C₂	C₂²×#11758199	1	1	yes	characteristic
162	C₂	#16840573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₄₄

C₂

yes

maximal, characteristic, complemented

#16840108

C₂

yes

maximal, complemented

#16840042

C₂

yes

maximal, complemented

#16839954

C₂

yes

maximal, complemented

Q₈×#92418

C₂

yes

maximal, characteristic, complemented

C₄×#11758199

C₂

yes

maximal, complemented

C₆₁×#550

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×#11758199

C₂²

yes

—

#14519777

C₂²

yes

—

#14519757

—

C₄×D₂₄₄

C₂²

yes

—

#14519710

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519698

C₂²

yes

—

#14519698

C₂²

yes

—

#14519697

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

—

C₂×#41×C₆₁

—

#127×C₆₁

C₂²

yes

—

#121×C₆₁

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄×D₄×C₆₁

C₂²

yes

—

#41×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

Q₈×D₆₁

—

C₂×D₂₄₄

C₂³

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

#11758199

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

D₈

yes

complemented

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

#41

yes

complemented

C₄×D₆₁

—

Dic₁₂₂

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

#3358

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#548

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

#577

yes

characteristic

D₆₁

—

#550

D₆₁

yes

characteristic, complemented

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

—

C₄×D₈

—

C₆₁

#3358

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

122

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₆₁

yes

—

C₂²⋊Q₈

—

122

—

#121

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₆₁

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

—

106

#41

—

107

C₂×Q₈

—

122

—

108

C₂×Q₈

—

244

—

109

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²×D₆₁

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

122

—

116

D₈

—

122

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

C₂²⋊D₆₁

yes

—

122

C₄⋊C₄

C₂²×D₆₁

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

122

—

125

C₄²

C₂²×D₆₁

yes

—

126

C₂³

—

127

C₂³

—

122

—

128

Q₈

—

244

—

129

Q₈

#11758199

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

122

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

122

—

139

C₂×C₄

—

244

—

140

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

141

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

142

C₂×C₄

C₂³×D₆₁

yes

characteristic

143

C₈

—

122

—

144

C₈

—

122

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

122

—

149

C₂²

—

244

—

150

C₂²

—

244

—

151

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

244

—

155

C₄

C₂×#11758199

yes

—

156

C₄

#41×D₆₁

yes

characteristic

157

C₄

#14519824

yes

characteristic

158

C₂

—

159

C₂

—

244

—

160

C₂

#16840419

yes

characteristic

161

C₂

C₂²×#11758199

yes

characteristic

162

C₂

#16840573

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#550	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#550	D₆₁
hypocenter	C₆₁	#3358
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#550

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₂₄₄	C₂	semidirect
2. #16840108	C₂	semidirect
3. #16840042	C₂	semidirect
4. #16839954	C₂	semidirect
5. Q₈×#92418	C₂	semidirect
6. C₄×#11758199	C₂	semidirect
7. C₆₁×#550	C₂	semidirect
46. Q₈×C₆₁	D₈	semidirect
58. D₂₄₄	#41	semidirect
82. #550	D₆₁	semidirect
89. C₆₁	#3358	semidirect
129. Q₈	#11758199	semidirect

H ≅ G/N

type

1. Q₈×D₂₄₄

C₂

semidirect

2. #16840108

C₂

semidirect

3. #16840042

C₂

semidirect

4. #16839954

C₂

semidirect

5. Q₈×#92418

C₂

semidirect

6. C₄×#11758199

C₂

semidirect

7. C₆₁×#550

C₂

semidirect

46. Q₈×C₆₁

D₈

semidirect

58. D₂₄₄

#41

semidirect

82. #550

D₆₁

semidirect

89. C₆₁

#3358

semidirect

129. Q₈

#11758199

semidirect

Subgroups — #21458450

Conjugacy classes of subgroups

Complements