Group #21444547 — subgroups

Subgroups — #21444547

Order 7808.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16839819	C₂	1	1	yes	maximal, characteristic
3	#16839510	C₂	1	2	yes	maximal, complemented
4	#16839509	C₂	1	2	yes	maximal, complemented
5	C₆₁×#423	C₂	1	1	yes	maximal, characteristic
6	#14519777	C₂²	1	1	yes	characteristic
7	#14519777	C₂²	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	C₄	1	2	yes	—
13	#14519736	—	2	1	no	—
14	C₄⋊#92418	C₂²	1	2	yes	—
15	C₂²⋊#92418	—	2	1	no	—
16	#14519688	—	2	1	no	—
17	C₂×M₁₆×C₆₁	—	2	1	no	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	—	2	1	no	—
20	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
26	M₁₆×C₆₁	—	4	2	no	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₄²×C₆₁	D₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₆₁	—	2	2	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	#11758196	—	2	2	no	—
38	#11758196	—	2	2	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆₁	—	2	2	no	—
53	C₂×Dic₆₁	—	2	2	no	—
54	Dic₁₂₂	—	4	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	Dic₁₂₂	—	4	2	no	—
57	C₈×C₆₁	—	4	2	no	—
58	#92418	—	4	2	no	—
59	C₂²×C₆₁	—	2	1	no	—
60	C₂²×C₆₁	—	2	1	no	—
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄×C₆₁	#137	1	1	yes	characteristic
69	C₄×C₆₁	#137	1	2	yes	—
70	Dic₆₁	—	4	2	no	—
71	Dic₆₁	—	4	2	no	—
72	#1559	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	4	1	no	—
74	C₂×C₆₁	#413	1	1	yes	characteristic
75	C₂×C₆₁	#417	1	1	yes	characteristic
76	C₂×C₆₁	#419	1	1	yes	characteristic
77	#533	—	61	1	no	—
78	#432	—	61	1	no	—
79	#423	D₆₁	1	1	yes	characteristic
80	#333	—	61	2	no	—
81	#332	—	61	2	no	—
82	C₆₁	#1559	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₁₆	—	2	1	no	—
84	C₄⋊Q₈	—	61	1	no	—
85	C₄⋊Q₈	—	61	1	no	—
86	#125	—	122	1	no	—
87	C₂²⋊Q₈	—	61	2	no	—
88	C₂²⋊Q₈	—	122	1	no	—
89	C₄×Q₈	—	61	2	no	—
90	C₄×Q₈	—	122	1	no	—
91	#117	—	122	1	no	—
92	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	61	2	no	—
95	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
96	C₂²⋊C₈	—	122	1	no	—
97	#97	—	122	1	no	—
98	C₂×Q₈	—	61	2	no	—
99	C₂×Q₈	—	122	1	no	—
100	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
101	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
103	M₁₆	—	4	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	122	2	no	—
106	C₄⋊C₄	—	61	2	no	—
107	C₄⋊C₄	—	122	2	no	—
108	C₄⋊C₄	—	122	2	no	—
109	C₄⋊C₄	—	122	1	no	—
110	C₄⋊C₄	—	122	1	no	—
111	C₄⋊C₄	—	122	1	no	—
112	C₂²⋊C₄	—	122	2	no	—
113	C₄²	—	122	2	no	—
114	C₄²	C₄×D₆₁	1	1	yes	characteristic
115	C₄²	D₂₄₄	1	1	yes	characteristic
116	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	C₂³	#11758198	1	1	yes	characteristic
119	Q₈	—	244	2	no	—
120	Q₈	—	244	1	no	—
121	Q₈	—	244	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	#11758198	1	1	yes	characteristic
129	C₂×C₄	#11758198	1	1	yes	characteristic
130	C₂×C₄	#11758198	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	244	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	C₂×#11758198	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	244	2	no	—
145	C₄	—	244	2	no	—
146	C₄	#14519803	1	1	yes	characteristic
147	C₄	#14519808	1	1	yes	characteristic
148	C₄	#14519810	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#16839744	1	1	yes	characteristic
151	C₂	#16839761	1	1	yes	characteristic
152	C₂	#16839772	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839819

C₂

yes

maximal, characteristic

#16839510

C₂

yes

maximal, complemented

#16839509

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic

#14519777

C₂²

yes

characteristic

#14519777

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₄

yes

—

#14519736

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

—

#14519688

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊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

—

M₁₆×C₆₁

—

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

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

#1559

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

D₆₁

yes

characteristic

#333

—

#332

—

C₆₁

#1559

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

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

C₂×D₆₁

yes

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

100

C₂²×C₄

C₄×D₆₁

yes

characteristic

101

C₂²×C₄

D₂₄₄

yes

characteristic

102

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

122

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₂²⋊C₄

—

122

—

113

C₄²

—

122

—

114

C₄²

C₄×D₆₁

yes

characteristic

115

C₄²

D₂₄₄

yes

characteristic

116

C₄²

C₂²⋊D₆₁

yes

characteristic

117

C₄²

C₂²×D₆₁

yes

characteristic

118

C₂³

#11758198

yes

characteristic

119

Q₈

—

244

—

120

Q₈

—

244

—

121

Q₈

—

244

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

#11758198

yes

characteristic

129

C₂×C₄

#11758198

yes

characteristic

130

C₂×C₄

#11758198

yes

characteristic

131

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

132

C₂×C₄

C₂×D₂₄₄

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

134

C₈

—

135

C₈

—

244

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂×#11758198

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

244

—

145

C₄

—

244

—

146

C₄

#14519803

yes

characteristic

147

C₄

#14519808

yes

characteristic

148

C₄

#14519810

yes

—

149

C₂

—

150

C₂

#16839744

yes

characteristic

151

C₂

#16839761

yes

characteristic

152

C₂

#16839772

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11758198
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#423	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#423	D₆₁
hypocenter	C₆₁	#1559
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#11758198

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#423

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16839510	C₂	semidirect
4. #16839509	C₂	semidirect
82. C₆₁	#1559	semidirect

H ≅ G/N

type

3. #16839510

C₂

semidirect

4. #16839509

C₂

semidirect

82. C₆₁

#1559

semidirect

Subgroups — #21444547

Conjugacy classes of subgroups

Complements