Group #21461169 — subgroups

Subgroups — #21461169

Order 7808.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 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	#16840766	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16840612	C₂	1	4	yes	maximal, complemented
4	#16840317	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758213	—	2	1	no	—
6	C₂×D₆₁⋊C₈	—	2	1	no	—
7	C₄⋊#92420	—	2	1	no	—
8	C₄⋊#92420	C₂²	1	2	yes	—
9	C₄×#92420	—	2	1	no	—
10	C₄×#92420	C₂²	1	2	yes	—
11	#14519777	C₄	1	1	yes	characteristic
12	#14519768	—	2	1	no	—
13	#14519757	C₄	1	2	yes	—
14	#14519753	—	2	1	no	—
15	Q₈×Dic₆₁	—	2	1	no	—
16	#14519742	C₂²	1	2	yes	—
17	#14519735	—	2	1	no	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	—	2	1	no	—
21	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
25	C₂×#92420	—	2	2	no	—
26	C₂×#92420	—	2	2	no	—
27	#11758213	—	4	2	no	—
28	D₆₁⋊C₈	—	2	4	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	#11758198	—	2	2	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₄×Dic₆₁	D₄	1	1	yes	characteristic
39	C₄×Dic₆₁	D₄	1	1	yes	characteristic
40	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	2	no	—
48	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
51	C₄×D₆₁	—	2	2	no	—
52	C₄×D₆₁	—	2	2	no	—
53	C₄×D₆₁	SD₁₆	1	2	yes	—
54	C₄×D₆₁	Q₁₆	1	2	yes	—
55	Dic₁₂₂	—	2	4	no	—
56	#92420	—	2	4	no	—
57	#92420	—	4	2	no	—
58	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₆₁	—	2	1	no	—
60	C₂×D₆₁	—	2	1	no	—
61	C₂×D₆₁	#103	1	2	yes	—
62	C₄×C₆₁	—	4	2	no	—
63	C₄×C₆₁	#103	1	2	yes	—
64	C₄×C₆₁	#104	1	2	yes	—
65	C₄×C₆₁	#131	1	2	yes	—
66	Dic₆₁	—	2	1	no	—
67	Dic₆₁	—	2	1	no	—
68	Dic₆₁	—	4	2	no	—
69	Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
71	#1493	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	#408	1	1	yes	characteristic
73	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
74	C₂×C₆₁	C₂×#104	1	1	yes	characteristic
75	D₆₁	—	2	2	no	—
76	#533	—	61	1	no	—
77	#423	—	61	1	no	—
78	C₂×C₄×C₈	—	61	1	no	—
79	Q₈⋊C₈	—	61	4	no	—
80	C₆₁	#1493	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	122	1	no	—
82	C₂²×C₈	—	122	1	no	—
83	C₄⋊Q₈	—	61	1	no	—
84	C₄⋊Q₈	#6923	1	1	yes	characteristic
85	#125	—	122	1	no	—
86	C₂²⋊Q₈	—	61	2	no	—
87	C₂²⋊Q₈	—	122	1	no	—
88	C₄×Q₈	—	61	2	no	—
89	C₄×Q₈	—	122	1	no	—
90	#117	—	122	1	no	—
91	C₂×C₄²	—	61	1	no	—
92	C₄⋊C₈	—	61	2	no	—
93	C₄⋊C₈	—	122	1	no	—
94	C₄×C₈	—	61	2	no	—
95	C₄×C₈	—	122	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	61	2	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	—	61	1	no	—
100	C₂²×C₄	—	61	1	no	—
101	M₁₆	—	244	2	no	—
102	C₂×C₈	—	122	2	no	—
103	C₂×C₈	—	122	4	no	—
104	C₂×C₈	—	122	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	122	2	no	—
107	C₄⋊C₄	—	122	2	no	—
108	C₄⋊C₄	—	122	1	no	—
109	C₄⋊C₄	—	122	1	no	—
110	C₄⋊C₄	C₂×#6923	1	2	yes	—
111	C₂²⋊C₄	—	122	2	no	—
112	C₄²	—	61	1	no	—
113	C₄²	—	61	1	no	—
114	C₄²	—	61	1	no	—
115	C₄²	—	122	2	no	—
116	C₄²	C₂×#6923	1	1	yes	characteristic
117	C₂³	—	61	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	122	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	2	no	—
123	C₂×C₄	—	61	2	no	—
124	C₂×C₄	—	61	1	no	—
125	C₂×C₄	—	61	1	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	122	2	no	—
129	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
131	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
132	C₈	—	122	4	no	—
133	C₈	—	244	2	no	—
134	C₂²	—	61	2	no	—
135	C₂²	—	122	1	no	—
136	C₂²	—	122	1	no	—
137	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	61	1	no	—
140	C₄	—	61	1	no	—
141	C₄	—	122	1	no	—
142	C₄	—	122	1	no	—
143	C₄	—	244	2	no	—
144	C₄	#14519840	1	2	yes	—
145	C₄	#14519841	1	2	yes	—
146	C₄	#14519865	1	2	yes	—
147	C₂	—	122	2	no	—
148	C₂	#16840737	1	1	yes	characteristic
149	C₂	C₂×#14519840	1	1	yes	characteristic
150	C₂	C₂×#14519841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840766

C₂

yes

maximal, characteristic

C₄×D₆₁⋊C₈

C₂

yes

maximal, characteristic

#16840612

C₂

yes

maximal, complemented

#16840317

C₂

yes

maximal, characteristic

C₂×#11758213

—

C₂×D₆₁⋊C₈

—

C₄⋊#92420

—

C₄⋊#92420

C₂²

yes

—

C₄×#92420

—

C₄×#92420

C₂²

yes

—

#14519777

C₄

yes

characteristic

#14519768

—

#14519757

C₄

yes

—

#14519753

—

Q₈×Dic₆₁

—

#14519742

C₂²

yes

—

#14519735

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₂×#92420

—

#11758213

—

D₆₁⋊C₈

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

SD₁₆

yes

—

C₄×D₆₁

Q₁₆

yes

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#103

yes

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

#104

yes

—

C₄×C₆₁

#131

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

C₂×Q₁₆

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#533

—

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₆₁

#1493

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

#6923

yes

characteristic

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

244

—

102

C₂×C₈

—

122

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

122

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

C₂×#6923

yes

—

111

C₂²⋊C₄

—

122

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

122

—

116

C₄²

C₂×#6923

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

—

122

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

122

—

129

C₂×C₄

C₂²⋊#6923

yes

characteristic

130

C₂×C₄

C₂²⋊#6923

yes

characteristic

131

C₂×C₄

C₂²×#6923

yes

characteristic

132

C₈

—

122

—

133

C₈

—

244

—

134

C₂²

—

135

C₂²

—

122

—

136

C₂²

—

122

—

137

C₂²

C₂×C₂²⋊#6923

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

122

—

142

C₄

—

122

—

143

C₄

—

244

—

144

C₄

#14519840

yes

—

145

C₄

#14519841

yes

—

146

C₄

#14519865

yes

—

147

C₂

—

122

—

148

C₂

#16840737

yes

characteristic

149

C₂

C₂×#14519840

yes

characteristic

150

C₂

C₂×#14519841

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1493
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16840612	C₂	semidirect
80. C₆₁	#1493	semidirect

H ≅ G/N

type

3. #16840612

C₂

semidirect

80. C₆₁

#1493

semidirect

Subgroups — #21461169

Conjugacy classes of subgroups

Complements