Group #21444223 — subgroups

Subgroups — #21444223

Order 7808.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#16839494	C₂	1	4	yes	maximal, complemented
4	C₆₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	—	2	1	no	—
6	#14519777	C₄	1	1	yes	characteristic
7	#14519777	C₄	1	1	yes	characteristic
8	#14519769	—	2	1	no	—
9	#14519756	—	2	1	no	—
10	#14519756	C₄	1	2	yes	—
11	C₄×Dic₁₂₂	—	2	1	no	—
12	C₄×Dic₁₂₂	C₂²	1	2	yes	—
13	#14519736	—	2	1	no	—
14	C₄⋊#92418	—	2	1	no	—
15	C₄⋊#92418	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₁	—	2	1	no	—
18	C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₄×C₈×C₆₁	C₂²	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂×Dic₁₂₂	—	2	1	no	—
24	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₆₁	—	2	2	no	—
26	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₆₁	D₄	1	1	yes	characteristic
28	C₄²×C₆₁	D₄	1	1	yes	characteristic
29	C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	1	no	—
32	C₄⋊Dic₆₁	—	2	1	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	4	2	no	—
40	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	4	1	no	—
43	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
47	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	2	no	—
52	C₂×Dic₆₁	—	2	2	no	—
53	Dic₁₂₂	—	2	4	no	—
54	Dic₁₂₂	—	4	2	no	—
55	C₈×C₆₁	—	4	2	no	—
56	#92418	—	4	2	no	—
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	#103	1	2	yes	—
60	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₆₁	—	2	1	no	—
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	#103	1	2	yes	—
65	C₄×C₆₁	#131	1	2	yes	—
66	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
68	Dic₆₁	—	4	2	no	—
69	Dic₆₁	—	4	2	no	—
70	#1481	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	2	no	—
72	C₂×C₆₁	#408	1	1	yes	characteristic
73	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
74	C₂×C₆₁	#421	1	1	yes	characteristic
75	#533	—	61	1	no	—
76	#431	D₆₁	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	61	1	no	—
78	Q₈⋊C₈	—	61	4	no	—
79	C₆₁	#1481	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	122	1	no	—
81	C₄⋊Q₈	—	61	1	no	—
82	C₄⋊Q₈	—	61	1	no	—
83	#125	—	122	1	no	—
84	C₂²⋊Q₈	—	61	2	no	—
85	C₂²⋊Q₈	—	122	1	no	—
86	C₄×Q₈	—	61	2	no	—
87	C₄×Q₈	—	122	1	no	—
88	#117	—	122	1	no	—
89	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	61	2	no	—
92	C₄⋊C₈	—	122	1	no	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₄×C₈	C₂×D₆₁	1	2	yes	—
95	C₂×Q₈	—	61	2	no	—
96	C₂×Q₈	—	122	1	no	—
97	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
98	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	122	2	no	—
102	C₂×C₈	—	244	2	no	—
103	C₄⋊C₄	—	61	2	no	—
104	C₄⋊C₄	—	122	2	no	—
105	C₄⋊C₄	—	122	1	no	—
106	C₄⋊C₄	—	122	1	no	—
107	C₄⋊C₄	—	122	1	no	—
108	C₄⋊C₄	—	122	2	no	—
109	C₂²⋊C₄	—	122	2	no	—
110	C₄²	—	122	2	no	—
111	C₄²	C₄×D₆₁	1	1	yes	characteristic
112	C₄²	D₂₄₄	1	1	yes	characteristic
113	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
114	C₄²	C₂²×D₆₁	1	1	yes	characteristic
115	C₂³	#11758198	1	1	yes	characteristic
116	Q₈	—	122	4	no	—
117	Q₈	—	244	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	122	2	no	—
121	C₂×C₄	—	122	2	no	—
122	C₂×C₄	#11758190	1	2	yes	—
123	C₂×C₄	Dic₂₄₄	1	2	yes	—
124	C₂×C₄	#11758198	1	1	yes	characteristic
125	C₂×C₄	#11758198	1	1	yes	characteristic
126	C₂×C₄	#11758198	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
128	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	244	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#14519702	1	2	yes	—
135	C₂²	C₂×#11758198	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	244	2	no	—
140	C₄	—	244	2	no	—
141	C₄	#14519702	1	2	yes	—
142	C₄	#14519785	1	2	yes	—
143	C₄	C₂×#11758190	1	1	yes	characteristic
144	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#16839730	1	1	yes	characteristic
147	C₂	C₂×#14519702	1	1	yes	characteristic
148	C₂	#16839780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

#16839494

C₂

yes

maximal, complemented

C₆₁×#431

C₂

yes

maximal, characteristic

C₂²×#92418

—

#14519777

C₄

yes

characteristic

#14519777

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×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×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#103

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#1481

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

characteristic

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

C₆₁

#1481

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

122

—

102

C₂×C₈

—

244

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₂²⋊C₄

—

122

—

110

C₄²

—

122

—

111

C₄²

C₄×D₆₁

yes

characteristic

112

C₄²

D₂₄₄

yes

characteristic

113

C₄²

C₂²⋊D₆₁

yes

characteristic

114

C₄²

C₂²×D₆₁

yes

characteristic

115

C₂³

#11758198

yes

characteristic

116

Q₈

—

122

—

117

Q₈

—

244

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

#11758190

yes

—

123

C₂×C₄

Dic₂₄₄

yes

—

124

C₂×C₄

#11758198

yes

characteristic

125

C₂×C₄

#11758198

yes

characteristic

126

C₂×C₄

#11758198

yes

characteristic

127

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

128

C₂×C₄

C₂×D₂₄₄

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

130

C₈

—

131

C₈

—

244

—

132

C₂²

—

133

C₂²

—

134

C₂²

#14519702

yes

—

135

C₂²

C₂×#11758198

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

244

—

140

C₄

—

244

—

141

C₄

#14519702

yes

—

142

C₄

#14519785

yes

—

143

C₄

C₂×#11758190

yes

characteristic

144

C₄

C₂×Dic₂₄₄

yes

characteristic

145

C₂

—

146

C₂

#16839730

yes

characteristic

147

C₂

C₂×#14519702

yes

characteristic

148

C₂

#16839780

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#431	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#431	D₆₁
hypocenter	C₆₁	#1481
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#431

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16839494	C₂	semidirect
79. C₆₁	#1481	semidirect

H ≅ G/N

type

3. #16839494

C₂

semidirect

79. C₆₁

#1481

semidirect

Subgroups — #21444223

Conjugacy classes of subgroups

Complements