Group #21445409 — subgroups

Subgroups — #21445409

Order 7808.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16840165	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840138	C₂	1	1	yes	maximal, characteristic
3	#16840135	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840110	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758190	—	2	2	no	—
9	C₂×#11758199	—	2	2	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	#14519774	C₂²	1	1	yes	characteristic
12	#14519710	—	2	2	no	—
13	#14519698	—	2	2	no	—
14	#14519697	—	2	2	no	—
15	C₄×#92418	C₂²	1	1	yes	characteristic
16	C₄×#92418	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₆₁	—	2	2	no	—
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	#127×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	2	1	no	—
22	C₂×D₄×C₆₁	—	2	1	no	—
23	C₂×D₄×C₆₁	—	2	1	no	—
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	1	no	—
29	SD₁₆×C₆₁	—	4	4	no	—
30	C₂×C₈×C₆₁	D₄	1	2	yes	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#11758199	—	4	4	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×#92418	D₄	1	2	yes	—
38	C₂×#92418	D₄	1	2	yes	—
39	#11758190	—	4	4	no	—
40	C₂³×C₆₁	—	4	1	no	—
41	C₂²×D₆₁	—	4	1	no	—
42	Q₈×C₆₁	—	4	2	no	—
43	D₄×C₆₁	—	4	2	no	—
44	D₄×C₆₁	—	4	2	no	—
45	D₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₆₁	—	4	1	no	—
51	D₂₄₄	—	4	2	no	—
52	D₂₄₄	—	4	2	no	—
53	D₂₄₄	—	4	2	no	—
54	Dic₁₂₂	—	4	2	no	—
55	C₈×C₆₁	D₈	1	4	yes	complemented
56	#92418	—	2	2	no	—
57	#92418	SD₁₆	1	4	yes	complemented
58	C₂²×C₆₁	—	8	1	no	—
59	C₂²×C₆₁	—	8	1	no	—
60	C₂²×C₆₁	—	8	1	no	—
61	C₂²×C₆₁	#127	1	1	yes	characteristic
62	C₂×D₆₁	—	8	1	no	—
63	C₂×D₆₁	—	8	1	no	—
64	C₂×D₆₁	—	8	1	no	—
65	C₄×C₆₁	—	8	1	no	—
66	C₄×C₆₁	C₂×D₈	1	2	yes	—
67	C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
68	C₄×C₆₁	#135	1	2	yes	—
69	Dic₆₁	—	8	1	no	—
70	#1706	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	16	1	no	—
72	C₂×C₆₁	#492	1	1	yes	characteristic
73	C₂×C₆₁	#493	1	1	yes	characteristic
74	C₂×C₆₁	#495	1	1	yes	characteristic
75	D₆₁	—	16	1	no	—
76	#500	—	61	1	no	—
77	#493	—	61	1	no	—
78	#492	—	61	1	no	—
79	#492	D₆₁	1	1	yes	characteristic, complemented
80	#486	—	61	1	no	—
81	#486	—	61	1	no	—
82	C₈²	—	61	1	no	—
83	C₆₁	#1706	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	122	2	no	—
86	C₂×D₈	—	122	2	no	—
87	C₄⋊Q₈	—	61	1	no	—
88	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
89	#127	—	61	1	no	—
90	#127	C₂×D₆₁	1	1	yes	characteristic
91	#107	—	122	2	no	—
92	#102	—	122	2	no	—
93	#102	—	122	2	no	—
94	C₄×C₈	—	61	1	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	122	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	122	1	no	—
103	C₂×D₄	—	122	1	no	—
104	C₂×D₄	—	122	1	no	—
105	SD₁₆	—	4	4	no	—
106	SD₁₆	—	244	4	no	—
107	D₈	—	244	4	no	—
108	C₂×C₈	—	61	2	no	—
109	C₂×C₈	—	61	2	no	—
110	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	122	1	no	—
114	C₄⋊C₄	—	122	1	no	—
115	C₄²	C₂²×D₆₁	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	244	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	244	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	244	2	no	—
124	D₄	—	244	2	no	—
125	D₄	—	244	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	244	1	no	—
128	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	C₈	—	61	4	no	—
132	C₈	—	122	2	no	—
133	C₈	#11758199	1	4	yes	complemented
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	488	1	no	—
138	C₂²	—	488	1	no	—
139	C₂²	—	488	1	no	—
140	C₂²	#14519773	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	488	1	no	—
143	C₄	C₂×#11758199	1	2	yes	—
144	C₄	SD₁₆×D₆₁	1	2	yes	—
145	C₄	#14519798	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	976	1	no	—
148	C₂	#16840133	1	1	yes	characteristic
149	C₂	#16840138	1	1	yes	characteristic
150	C₂	#16840143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840165

C₂

yes

maximal, characteristic, complemented

#16840138

C₂

yes

maximal, characteristic

#16840135

C₂

yes

maximal, characteristic, complemented

#16840110

C₂

yes

maximal, characteristic, complemented

#16840108

C₂

yes

maximal, characteristic, complemented

C₈×#92418

C₂

yes

maximal, characteristic, complemented

C₆₁×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11758190

—

C₂×#11758199

—

#14519777

C₂²

yes

characteristic

#14519774

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₂×SD₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

SD₁₆×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758199

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₄

yes

—

C₂×#92418

D₄

yes

—

#11758190

—

C₂³×C₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

D₈

yes

complemented

#92418

—

#92418

SD₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#127

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

#135

yes

—

Dic₆₁

—

#1706

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

#486

—

#486

—

C₈²

—

C₆₁

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

—

#127

C₂×D₆₁

yes

characteristic

#107

—

122

—

#102

—

122

—

#102

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

122

—

103

C₂×D₄

—

122

—

104

C₂×D₄

—

122

—

105

SD₁₆

—

106

SD₁₆

—

244

—

107

D₈

—

244

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₆₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

122

—

114

C₄⋊C₄

—

122

—

115

C₄²

C₂²×D₆₁

yes

characteristic

116

C₂³

—

117

C₂³

—

244

—

118

Q₈

—

119

Q₈

—

244

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

244

—

124

D₄

—

244

—

125

D₄

—

244

—

126

C₂×C₄

—

127

C₂×C₄

—

244

—

128

C₂×C₄

D₄×D₆₁

yes

characteristic

129

C₂×C₄

D₄×D₆₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

C₈

—

132

C₈

—

122

—

133

C₈

#11758199

yes

complemented

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

488

—

138

C₂²

—

488

—

139

C₂²

—

488

—

140

C₂²

#14519773

yes

characteristic

141

C₄

—

142

C₄

—

488

—

143

C₄

C₂×#11758199

yes

—

144

C₄

SD₁₆×D₆₁

yes

—

145

C₄

#14519798

yes

—

146

C₂

—

147

C₂

—

976

—

148

C₂

#16840133

yes

characteristic

149

C₂

#16840138

yes

characteristic

150

C₂

#16840143

yes

characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#492	C₂
socle	C₂²×C₆₁	#127
hypercenter	#492	D₆₁
hypocenter	C₆₁	#1706
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14519773

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#492

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16840165	C₂	semidirect
3. #16840135	C₂	semidirect
4. #16840110	C₂	semidirect
5. #16840108	C₂	semidirect
6. C₈×#92418	C₂	semidirect
7. C₆₁×#492	C₂	semidirect
55. C₈×C₆₁	D₈	semidirect
57. #92418	SD₁₆	semidirect
79. #492	D₆₁	semidirect
83. C₆₁	#1706	semidirect
133. C₈	#11758199	semidirect

H ≅ G/N

type

1. #16840165

C₂

semidirect

3. #16840135

C₂

semidirect

4. #16840110

C₂

semidirect

5. #16840108

C₂

semidirect

6. C₈×#92418

C₂

semidirect

7. C₆₁×#492

C₂

semidirect

55. C₈×C₆₁

D₈

semidirect

57. #92418

SD₁₆

semidirect

79. #492

D₆₁

semidirect

83. C₆₁

#1706

semidirect

133. C₈

#11758199

semidirect

Subgroups — #21445409

Conjugacy classes of subgroups

Complements