Group #21458823 — subgroups

Subgroups — #21458823

Order 7808.

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

normal subgroups only

This group has 348 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	#16840487	C₂	1	2	yes	maximal
2	#16840140	C₂	1	2	yes	maximal
3	#16839970	C₂	1	4	yes	maximal
4	#16839969	C₂	1	2	yes	maximal
5	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₆₁	C₂	1	2	yes	maximal
8	C₆₁×#494	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	C₂×#11758202	C₂²	1	4	yes	—
11	#14519775	C₂²	1	4	yes	—
12	#14519768	—	2	2	no	—
13	#14519767	—	2	4	no	—
14	Q₈×Dic₆₁	C₂²	1	4	yes	—
15	C₄×Dic₁₂₂	C₂²	1	2	yes	—
16	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
17	#14519742	—	2	2	no	—
18	#14519711	C₂²	1	1	yes	characteristic
19	C₄⋊#92418	C₂²	1	1	yes	characteristic
20	#14519706	C₂²	1	2	yes	—
21	#14519702	C₂²	1	2	yes	—
22	#14519701	C₂²	1	4	yes	—
23	C₈×Dic₆₁	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
29	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
30	C₂×Dic₁₂₂	C₂³	1	2	yes	—
31	Q₁₆×C₆₁	—	2	4	no	—
32	Q₁₆×C₆₁	—	4	2	no	—
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	#11758202	—	4	4	no	—
39	C₄⋊Dic₆₁	—	2	4	no	—
40	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
41	C₄⋊Dic₆₁	C₂³	1	2	yes	—
42	#11758196	—	2	2	no	—
43	#11758196	—	2	4	no	—
44	#11758196	—	2	4	no	—
45	#11758196	D₄	1	2	yes	—
46	C₄×Dic₆₁	—	2	2	no	—
47	C₄×Dic₆₁	—	2	4	no	—
48	C₄×Dic₆₁	C₂³	1	2	yes	—
49	C₂×#92418	C₂³	1	2	yes	—
50	Dic₂₄₄	—	2	2	no	—
51	Q₈×C₆₁	—	2	4	no	—
52	Q₈×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₆₁	—	2	2	no	—
58	C₂×Dic₆₁	—	2	2	no	—
59	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
60	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
61	Dic₁₂₂	—	2	2	no	—
62	Dic₁₂₂	—	2	2	no	—
63	Dic₁₂₂	Q₁₆	1	4	yes	complemented
64	C₈×C₆₁	—	2	1	no	—
65	C₈×C₆₁	#41	1	2	yes	—
66	#92418	—	2	2	no	—
67	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	—	4	4	no	—
70	C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
71	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
72	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
73	Dic₆₁	—	2	2	no	—
74	Dic₆₁	—	4	2	no	—
75	Dic₆₁	—	4	2	no	—
76	Dic₆₁	C₂×Q₁₆	1	4	yes	—
77	#3395	—	61	1	no	maximal, Sylow
78	C₂×C₆₁	#550	1	1	yes	characteristic
79	C₂×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂×C₆₁	#578	1	1	yes	characteristic
81	#557	—	61	2	no	—
82	#494	—	61	2	no	—
83	#494	D₆₁	1	1	yes	characteristic
84	#462	—	61	4	no	—
85	#462	—	61	2	no	—
86	C₈×Q₈	—	61	1	no	—
87	C₄×Q₁₆	—	61	2	no	—
88	C₄×Q₁₆	—	61	1	no	—
89	C₆₁	#3395	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₁₆	—	61	4	no	—
91	C₂×Q₁₆	—	61	1	no	—
92	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
93	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
94	C₄⋊Q₈	—	61	4	no	—
95	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
96	#125	—	122	4	no	—
97	#125	—	122	2	no	—
98	C₄×Q₈	—	61	1	no	—
99	C₄×Q₈	—	61	4	no	—
100	C₄×Q₈	—	61	2	no	—
101	C₄×Q₈	—	122	2	no	—
102	#107	—	61	1	no	—
103	C₄⋊C₈	—	61	2	no	—
104	C₄⋊C₈	—	61	1	no	—
105	#103	—	61	4	no	—
106	#103	—	61	2	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₂×Q₈	—	61	1	no	—
110	C₂×Q₈	—	61	2	no	—
111	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
112	Q₁₆	—	2	4	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	122	2	no	—
115	Q₁₆	—	244	4	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	61	2	no	—
122	C₄⋊C₄	—	61	1	no	—
123	C₄⋊C₄	—	61	2	no	—
124	C₄⋊C₄	—	122	4	no	—
125	C₄⋊C₄	—	122	4	no	—
126	C₄⋊C₄	—	122	4	no	—
127	C₄⋊C₄	—	122	2	no	—
128	C₄²	—	61	2	no	—
129	C₄²	—	122	4	no	—
130	C₄²	—	122	2	no	—
131	C₄²	C₂²×D₆₁	1	1	yes	characteristic
132	Q₈	—	2	4	no	—
133	Q₈	—	2	4	no	—
134	Q₈	—	61	4	no	—
135	Q₈	—	122	2	no	—
136	Q₈	—	122	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	122	2	no	—
141	C₂×C₄	—	122	2	no	—
142	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	122	2	no	—
147	C₈	#11758224	1	2	yes	—
148	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	4	no	—
151	C₄	—	61	4	no	—
152	C₄	—	122	2	no	—
153	C₄	—	244	2	no	—
154	C₄	—	244	2	no	—
155	C₄	Q₁₆×D₆₁	1	2	yes	—
156	C₄	C₂×#11758224	1	1	yes	characteristic
157	C₄	#14519822	1	1	yes	characteristic
158	C₂	#16840432	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
160	C₂	#16840575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840487

C₂

yes

maximal

#16840140

C₂

yes

maximal

#16839970

C₂

yes

maximal

#16839969

C₂

yes

maximal

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₁

C₂

yes

maximal

C₆₁×#494

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758202

C₂²

yes

—

#14519775

C₂²

yes

—

#14519768

—

#14519767

—

Q₈×Dic₆₁

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519742

—

#14519711

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519706

C₂²

yes

—

#14519702

C₂²

yes

—

#14519701

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×Dic₁₂₂

C₂³

yes

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

Dic₂₄₄

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

Q₁₆

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

#92418

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×Q₁₆

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×Q₁₆

yes

C₂×C₆₁

yes

—

—

D₆₁

yes

characteristic

#462

—

#462

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₆₁

#3395

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₆₁

yes

—

C₂×Q₁₆

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#125

—

122

—

#125

—

122

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

122

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₆₁

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²×D₆₁

yes

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

122

—

115

Q₁₆

—

244

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₆₁

yes

characteristic

118

C₂×C₈

C₂²×D₆₁

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

122

—

125

C₄⋊C₄

—

122

—

126

C₄⋊C₄

—

122

—

127

C₄⋊C₄

—

122

—

128

C₄²

—

129

C₄²

—

122

—

130

C₄²

—

122

—

131

C₄²

C₂²×D₆₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

122

—

136

Q₈

—

122

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

122

—

141

C₂×C₄

—

122

—

142

C₂×C₄

D₄×D₆₁

yes

characteristic

143

C₂×C₄

D₄×D₆₁

yes

characteristic

144

C₂×C₄

C₂³×D₆₁

yes

characteristic

145

C₈

—

146

C₈

—

122

—

147

C₈

#11758224

yes

—

148

C₂²

C₂×D₄×D₆₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

122

—

153

C₄

—

244

—

154

C₄

—

244

—

155

C₄

Q₁₆×D₆₁

yes

—

156

C₄

C₂×#11758224

yes

characteristic

157

C₄

#14519822

yes

characteristic

158

C₂

#16840432

yes

characteristic

159

C₂

C₂×Q₁₆×D₆₁

yes

characteristic

160

C₂

#16840575

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#494	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#494	D₆₁
hypocenter	C₆₁	#3395
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#494

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
63. Dic₁₂₂	Q₁₆	semidirect
89. C₆₁	#3395	semidirect

H ≅ G/N

type

63. Dic₁₂₂

Q₁₆

semidirect

89. C₆₁

#3395

semidirect

Subgroups — #21458823

Conjugacy classes of subgroups

Complements