Group #21460716 — subgroups

Subgroups — #21460716

Order 7808.

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
3	#16839814	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758213	—	2	1	no	—
5	C₂×C₄⋊#6923	—	2	1	no	—
6	C₂×C₄×#6923	—	2	1	no	—
7	C₄⋊#92420	—	2	1	no	—
8	C₄⋊#92420	C₄	1	2	yes	complemented
9	(C₂×#6923)⋊C₄	C₄	1	2	yes	—
10	C₄²×D₆₁	C₂²	1	1	yes	characteristic
11	C₄⋊#92418	C₄	1	1	yes	characteristic, complemented
12	#14519689	—	2	1	no	—
13	C₈×Dic₆₁	—	2	1	no	—
14	C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₂²×#6923	—	2	2	no	—
16	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
19	C₂×#92420	—	2	2	no	—
20	C₄⋊#6923	—	2	2	no	—
21	C₄⋊#6923	—	2	2	no	—
22	C₄×#6923	—	4	1	no	—
23	C₄×#6923	—	4	1	no	—
24	#11758213	—	4	1	no	—
25	#11758213	—	4	1	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₆₁	D₄	1	1	yes	characteristic
30	C₄×Dic₆₁	Q₈	1	1	yes	characteristic
31	C₂×#92418	—	2	1	no	—
32	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×#6923	—	2	4	no	—
34	C₂×#6923	—	4	2	no	—
35	C₂×#6923	—	4	2	no	—
36	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₆₁	—	2	1	no	—
43	C₄×D₆₁	—	2	1	no	—
44	C₄×D₆₁	—	4	1	no	—
45	C₄×D₆₁	D₈	1	1	yes	characteristic
46	C₄×D₆₁	SD₁₆	1	1	yes	characteristic
47	C₄×D₆₁	SD₁₆	1	1	yes	characteristic
48	C₄×D₆₁	Q₁₆	1	1	yes	characteristic
49	#92420	—	4	2	no	—
50	C₈×C₆₁	—	4	1	no	—
51	#92418	—	4	1	no	—
52	C₂²×C₆₁	#95	1	1	yes	characteristic
53	C₂×D₆₁	—	2	1	no	—
54	C₂×D₆₁	—	2	1	no	—
55	C₂×D₆₁	#102	1	1	yes	characteristic
56	C₂×D₆₁	#103	1	1	yes	characteristic
57	#6923	—	4	4	no	—
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	#104	1	1	yes	characteristic
60	C₄×C₆₁	#104	1	1	yes	characteristic
61	C₄×C₆₁	#106	1	1	yes	characteristic
62	C₄×C₆₁	#107	1	1	yes	characteristic
63	Dic₆₁	—	2	1	no	—
64	Dic₆₁	—	2	1	no	—
65	Dic₆₁	#102	1	1	yes	characteristic
66	Dic₆₁	#103	1	1	yes	characteristic
67	#1303	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	#339	1	1	yes	characteristic
69	C₂×C₆₁	#340	1	1	yes	characteristic
70	C₂×C₆₁	#344	1	1	yes	characteristic
71	D₆₁	—	2	1	no	—
72	D₆₁	—	2	1	no	—
73	#431	—	61	1	no	—
74	#423	—	61	1	no	—
75	C₄×C₄⋊C₄	—	61	1	no	—
76	C₆₁	#1303	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	122	1	no	—
78	C₂×C₄⋊C₄	—	122	1	no	—
79	C₂×C₄²	—	61	1	no	—
80	C₂×C₄²	—	122	1	no	—
81	C₄⋊C₈	—	61	2	no	—
82	C₄⋊C₈	—	61	1	no	—
83	C₄⋊C₈	—	122	1	no	—
84	C₄⋊C₈	#6923	1	1	yes	characteristic, complemented
85	C₂²⋊C₈	—	122	1	no	—
86	C₄×C₈	—	122	1	no	—
87	#95	—	61	2	no	—
88	C₂²×C₄	—	61	1	no	—
89	C₂²×C₄	—	61	1	no	—
90	C₂²×C₄	—	61	1	no	—
91	C₂²×C₄	—	122	2	no	—
92	M₁₆	—	244	1	no	—
93	M₁₆	—	244	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	122	2	no	—
96	C₂×C₈	—	122	1	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	—	122	2	no	—
99	C₄²	—	61	1	no	—
100	C₄²	—	61	1	no	—
101	C₄²	—	61	1	no	—
102	C₄²	—	244	1	no	—
103	C₄²	—	244	1	no	—
104	C₄²	C₂×#6923	1	1	yes	characteristic
105	C₂³	—	61	1	no	—
106	C₂×C₄	—	61	1	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	61	1	no	—
109	C₂×C₄	—	61	1	no	—
110	C₂×C₄	—	61	1	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	1	no	—
113	C₂×C₄	—	122	1	no	—
114	C₂×C₄	—	122	1	no	—
115	C₂×C₄	—	122	4	no	—
116	C₂×C₄	—	244	1	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	—	244	2	no	—
119	C₂×C₄	C₄×#6923	1	1	yes	characteristic
120	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
122	C₈	—	4	1	no	—
123	C₈	—	244	2	no	—
124	C₈	—	244	1	no	—
125	C₂²	—	61	1	no	—
126	C₂²	—	61	1	no	—
127	C₂²	—	122	1	no	—
128	C₂²	—	122	1	no	—
129	C₂²	(C₂×#6923)⋊C₄	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	61	1	no	—
132	C₄	—	61	1	no	—
133	C₄	—	122	1	no	—
134	C₄	—	122	1	no	—
135	C₄	—	244	4	no	—
136	C₄	#14519841	1	1	yes	characteristic
137	C₄	#14519842	1	1	yes	characteristic
138	C₄	#14519845	1	1	yes	characteristic
139	C₄	#14519846	1	1	yes	characteristic
140	C₂	—	122	1	no	—
141	C₂	—	122	1	no	—
142	C₂	#16840632	1	1	yes	characteristic
143	C₂	#16840636	1	1	yes	characteristic
144	C₂	#16840645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840766

C₂

yes

maximal, characteristic

C₄×C₄⋊#6923

C₂

yes

maximal, characteristic

#16839814

C₂

yes

maximal, characteristic

C₂×#11758213

—

C₂×C₄⋊#6923

—

C₂×C₄×#6923

—

C₄⋊#92420

—

C₄⋊#92420

C₄

yes

complemented

(C₂×#6923)⋊C₄

C₄

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊#92418

C₄

yes

characteristic, complemented

#14519689

—

C₈×Dic₆₁

—

C₄⋊C₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

—

—

—

—

—

—

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

Q₈

yes

characteristic

C₂×#92418

—

C₂²×D₆₁

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

D₈

yes

characteristic

C₄×D₆₁

SD₁₆

yes

characteristic

C₄×D₆₁

SD₁₆

yes

characteristic

C₄×D₆₁

Q₁₆

yes

characteristic

#92420

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

#95

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

yes

C₂×D₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

#431

—

#423

—

C₄×C₄⋊C₄

—

C₆₁

#1303

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₂×C₄²

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

#6923

yes

characteristic, complemented

C₂²⋊C₈

—

122

—

C₄×C₈

—

122

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

M₁₆

—

244

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

244

—

103

C₄²

—

244

—

104

C₄²

C₂×#6923

yes

characteristic

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

122

—

114

C₂×C₄

—

122

—

115

C₂×C₄

—

122

—

116

C₂×C₄

—

244

—

117

C₂×C₄

—

244

—

118

C₂×C₄

—

244

—

119

C₂×C₄

C₄×#6923

yes

characteristic

120

C₂×C₄

C₄⋊#6923

yes

characteristic

121

C₂×C₄

C₂²⋊#6923

yes

characteristic

122

C₈

—

123

C₈

—

244

—

124

C₈

—

244

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

122

—

128

C₂²

—

122

—

129

C₂²

(C₂×#6923)⋊C₄

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

122

—

134

C₄

—

122

—

135

C₄

—

244

—

136

C₄

#14519841

yes

characteristic

137

C₄

#14519842

yes

characteristic

138

C₄

#14519845

yes

characteristic

139

C₄

#14519846

yes

characteristic

140

C₂

—

122

—

141

C₂

—

122

—

142

C₂

#16840632

yes

characteristic

143

C₂

#16840636

yes

characteristic

144

C₂

#16840645

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#6923)⋊C₄
commutator	C₂×C₄×C₆₁	C₄²
Frattini	C₄²	C₂×#6923
Fitting	C₄⋊C₈×C₆₁	C₄
socle	C₂²×C₆₁	#95
hypercenter	C₄⋊C₈	#6923
hypocenter	C₆₁	#1303
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#6923)⋊C₄

commutator

C₂×C₄×C₆₁

C₄²

Frattini

C₄²

C₂×#6923

Fitting

C₄⋊C₈×C₆₁

C₄

C₂²×C₆₁

C₄⋊C₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
8. C₄⋊#92420	C₄	semidirect
11. C₄⋊#92418	C₄	semidirect
14. C₄⋊C₈×C₆₁	C₄	semidirect
76. C₆₁	#1303	semidirect
84. C₄⋊C₈	#6923	semidirect

H ≅ G/N

type

8. C₄⋊#92420

C₄

semidirect

11. C₄⋊#92418

C₄

semidirect

14. C₄⋊C₈×C₆₁

C₄

semidirect

76. C₆₁

#1303

semidirect

84. C₄⋊C₈

#6923

semidirect

Subgroups — #21460716

Conjugacy classes of subgroups

Complements