Group #21451024 — subgroups

Subgroups — #21451024

Order 7808.

500 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#14519777	C₂	1	1	yes	maximal, characteristic
2	#16839682	C₂	1	4	yes	maximal
3	#16839669	C₂	1	2	yes	maximal
4	#16839660	C₂	1	4	yes	maximal
5	#16839631	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
9	#14519777	—	2	4	no	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×#11758196	C₂²	1	8	yes	—
14	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
15	#14519684	C₂²	1	2	yes	—
16	#14519684	C₂²	1	8	yes	—
17	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₆₁	—	2	4	no	—
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	2	no	—
24	C₂×Q₈×C₆₁	—	2	2	no	—
25	C₂×Q₈×C₆₁	Q₈	1	4	yes	complemented
26	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂³	1	4	yes	—
32	C₂²×Dic₆₁	C₂³	1	4	yes	—
33	C₂×Dic₁₂₂	—	2	4	no	—
34	C₂×Dic₁₂₂	—	4	4	no	—
35	C₄⋊C₄×C₆₁	—	2	4	no	—
36	C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	—	2	4	no	—
38	C₄²×C₆₁	D₄	1	4	yes	—
39	C₄⋊Dic₆₁	—	2	4	no	—
40	C₄⋊Dic₆₁	—	2	4	no	—
41	C₄⋊Dic₆₁	Q₈	1	8	yes	complemented
42	#11758196	—	4	8	no	—
43	C₄×Dic₆₁	—	2	4	no	—
44	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
45	Q₈×C₆₁	—	2	8	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₆₁	#41	1	2	yes	—
58	C₂×Dic₆₁	—	2	4	no	—
59	C₂×Dic₆₁	—	2	4	no	—
60	C₂×Dic₆₁	—	2	4	no	—
61	C₂×Dic₆₁	—	2	8	no	—
62	C₂×Dic₆₁	C₂×Q₈	1	8	yes	—
63	Dic₁₂₂	—	4	8	no	—
64	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₆₁	C₂²×Q₈	1	2	yes	—
66	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
68	C₂²×C₆₁	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	4	2	no	—
72	C₄×C₆₁	C₂²⋊Q₈	1	4	yes	—
73	C₄×C₆₁	C₄⋊Q₈	1	4	yes	—
74	Dic₆₁	—	2	8	no	—
75	Dic₆₁	—	4	4	no	—
76	#2567	—	61	1	no	maximal, Sylow
77	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₆₁	#535	1	1	yes	characteristic
80	C₂×C₆₁	#554	1	2	yes	—
81	C₂×C₆₁	Q₈²	1	2	yes	—
82	C₂×C₄⋊Q₈	—	61	1	no	—
83	C₂×C₄×Q₈	D₆₁	1	1	yes	characteristic
84	#395	—	61	4	no	—
85	#391	—	61	2	no	—
86	#389	—	61	4	no	—
87	#384	—	61	2	no	—
88	C₄×C₄⋊C₄	—	61	1	no	—
89	C₆₁	#2567	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	61	2	no	—
91	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊Q₈	—	122	4	no	—
93	C₄×Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	—	61	2	no	—
95	C₂×C₄⋊C₄	—	61	2	no	—
96	C₂×C₄⋊C₄	—	61	2	no	—
97	C₂×C₄⋊C₄	—	61	8	no	—
98	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
100	C₂×C₄²	—	61	2	no	—
101	C₂×C₄²	C₂×D₆₁	1	2	yes	—
102	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
103	#95	—	61	2	no	—
104	#95	—	61	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	122	4	no	—
108	C₂×Q₈	—	244	4	no	—
109	C₂×Q₈	Dic₁₂₂	1	4	yes	complemented
110	C₂²×C₄	—	61	4	no	—
111	C₂²×C₄	—	61	4	no	—
112	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
113	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
114	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	61	8	no	—
120	C₄⋊C₄	—	122	4	no	—
121	C₄⋊C₄	—	122	4	no	—
122	C₄⋊C₄	—	244	8	no	—
123	C₄²	—	2	4	no	—
124	C₄²	—	122	4	no	—
125	C₄²	C₂²⋊D₆₁	1	4	yes	—
126	C₂³	C₂³×D₆₁	1	1	yes	characteristic
127	Q₈	—	2	8	no	—
128	Q₈	—	244	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	61	8	no	—
135	C₂×C₄	—	122	4	no	—
136	C₂×C₄	—	122	4	no	—
137	C₂×C₄	—	122	4	no	—
138	C₂×C₄	—	122	8	no	—
139	C₂×C₄	C₂×Dic₁₂₂	1	4	yes	—
140	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
141	C₂×C₄	#11758222	1	2	yes	—
142	C₂×C₄	Q₈×D₆₁	1	4	yes	—
143	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₆₁	1	2	yes	—
148	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
149	C₂²	C₂×#11758222	1	1	yes	characteristic
150	C₂²	#14519823	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	122	8	no	—
155	C₄	—	244	4	no	—
156	C₄	#14519756	1	4	yes	—
157	C₄	#14519775	1	4	yes	—
158	C₂	C₂×#14519756	1	1	yes	characteristic
159	C₂	C₂×#14519775	1	1	yes	characteristic
160	C₂	#16840326	1	1	yes	characteristic
161	C₂	#16840472	1	2	yes	—
162	C₂	Q₈×Dic₁₂₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519777

C₂

yes

maximal, characteristic

#16839682

C₂

yes

maximal

#16839669

C₂

yes

maximal

#16839660

C₂

yes

maximal

#16839631

C₂

yes

maximal

C₄×C₄⋊Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

#14519777

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

Q₈

yes

complemented

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

Q₈

yes

complemented

#11758196

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

#2567

—

maximal, Sylow

C₂×C₆₁

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×C₄⋊Q₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

Q₈²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₆₁

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₆₁

#2567

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

122

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₆₁

yes

—

102

C₂×C₄²

C₂×D₆₁

yes

characteristic

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

122

—

108

C₂×Q₈

—

244

—

109

C₂×Q₈

Dic₁₂₂

yes

complemented

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₆₁

yes

—

113

C₂²×C₄

C₂²×D₆₁

yes

—

114

C₂²×C₄

C₂²×D₆₁

yes

characteristic

115

C₂²×C₄

C₂²×D₆₁

yes

characteristic

116

C₂²×C₄

C₂²×D₆₁

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

244

—

123

C₄²

—

124

C₄²

—

122

—

125

C₄²

C₂²⋊D₆₁

yes

—

126

C₂³

C₂³×D₆₁

yes

characteristic

127

Q₈

—

128

Q₈

—

244

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

122

—

136

C₂×C₄

—

122

—

137

C₂×C₄

—

122

—

138

C₂×C₄

—

122

—

139

C₂×C₄

C₂×Dic₁₂₂

yes

—

140

C₂×C₄

C₂×Dic₁₂₂

yes

—

141

C₂×C₄

#11758222

yes

—

142

C₂×C₄

Q₈×D₆₁

yes

—

143

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

144

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

145

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

146

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

147

C₂²

C₂×Q₈×D₆₁

yes

—

148

C₂²

C₂²×Dic₁₂₂

yes

characteristic

149

C₂²

C₂×#11758222

yes

characteristic

150

C₂²

#14519823

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

122

—

155

C₄

—

244

—

156

C₄

#14519756

yes

—

157

C₄

#14519775

yes

—

158

C₂

C₂×#14519756

yes

characteristic

159

C₂

C₂×#14519775

yes

characteristic

160

C₂

#16840326

yes

characteristic

161

C₂

#16840472

yes

—

162

C₂

Q₈×Dic₁₂₂

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂×C₄×Q₈×C₆₁	C₂
hypercenter	C₂×C₄×Q₈	D₆₁
hypocenter	C₆₁	#2567
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂×C₄×Q₈×C₆₁

C₂

hypercenter

C₂×C₄×Q₈

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
25. C₂×Q₈×C₆₁	Q₈	semidirect
41. C₄⋊Dic₆₁	Q₈	semidirect
89. C₆₁	#2567	semidirect
109. C₂×Q₈	Dic₁₂₂	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₆₁

Q₈

semidirect

41. C₄⋊Dic₆₁

Q₈

semidirect

89. C₆₁

#2567

semidirect

109. C₂×Q₈

Dic₁₂₂

semidirect

Subgroups — #21451024

Conjugacy classes of subgroups

Complements