Group #21445455 — subgroups

Subgroups — #21445455

Order 7808.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840138	C₂	1	1	yes	maximal, characteristic
2	#16840134	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840112	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840110	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840109	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758200	—	2	2	no	—
9	C₂×#11758199	—	2	2	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	#14519774	C₂²	1	1	yes	characteristic
12	#14519701	—	2	2	no	—
13	#14519699	—	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	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	#127×C₆₁	C₂²	1	1	yes	characteristic
19	#102×C₆₁	—	2	2	no	—
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	C₂×C₈×C₆₁	D₄	1	2	yes	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#11758200	—	4	4	no	—
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	C₂³×C₆₁	—	4	1	no	—
40	C₂²×D₆₁	—	4	1	no	—
41	Q₈×C₆₁	—	4	2	no	—
42	D₄×C₆₁	—	4	2	no	—
43	D₄×C₆₁	—	4	2	no	—
44	D₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₁	—	4	1	no	—
50	D₂₄₄	—	4	2	no	—
51	D₂₄₄	—	4	2	no	—
52	D₂₄₄	—	4	2	no	—
53	Dic₁₂₂	—	4	2	no	—
54	C₈×C₆₁	—	2	2	no	—
55	#92418	—	2	2	no	—
56	#92418	—	2	2	no	—
57	C₂²×C₆₁	—	8	1	no	—
58	C₂²×C₆₁	—	8	1	no	—
59	C₂²×C₆₁	—	8	1	no	—
60	C₂²×C₆₁	#127	1	1	yes	characteristic
61	C₂×D₆₁	—	8	1	no	—
62	C₂×D₆₁	—	8	1	no	—
63	C₂×D₆₁	—	8	1	no	—
64	C₄×C₆₁	—	8	1	no	—
65	C₄×C₆₁	#135	1	2	yes	—
66	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₆₁	—	8	1	no	—
69	#1717	—	61	1	no	maximal, Sylow
70	C₂×C₆₁	—	16	1	no	—
71	C₂×C₆₁	#495	1	1	yes	characteristic
72	C₂×C₆₁	#496	1	1	yes	characteristic
73	C₂×C₆₁	#496	1	1	yes	characteristic
74	D₆₁	—	16	1	no	—
75	#493	—	61	1	no	—
76	#492	—	61	1	no	—
77	#487	—	61	1	no	—
78	#486	—	61	1	no	—
79	#486	—	61	1	no	—
80	#486	D₆₁	1	1	yes	characteristic, complemented
81	#322	—	61	1	no	—
82	C₆₁	#1717	1	1	yes	characteristic, complemented, Sylow
83	C₂×SD₁₆	—	122	2	no	—
84	C₂×D₈	—	122	2	no	—
85	C₄⋊Q₈	—	61	1	no	—
86	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
87	#127	—	61	1	no	—
88	#127	C₂×D₆₁	1	1	yes	characteristic
89	#103	—	122	2	no	—
90	#102	—	2	2	no	—
91	#102	—	122	2	no	—
92	#102	—	122	2	no	—
93	C₄×C₈	—	61	1	no	—
94	C₄×C₈	—	61	1	no	—
95	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	122	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	122	1	no	—
102	C₂×D₄	—	122	1	no	—
103	C₂×D₄	—	122	1	no	—
104	SD₁₆	—	244	4	no	—
105	D₈	—	244	4	no	—
106	C₂×C₈	—	61	2	no	—
107	C₂×C₈	—	61	2	no	—
108	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	122	1	no	—
112	C₄⋊C₄	—	122	1	no	—
113	C₄²	C₂²×D₆₁	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	244	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	244	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	244	2	no	—
122	D₄	—	244	2	no	—
123	D₄	—	244	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	244	1	no	—
126	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	122	2	no	—
131	C₈	—	122	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	488	1	no	—
136	C₂²	—	488	1	no	—
137	C₂²	—	488	1	no	—
138	C₂²	#14519773	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	488	1	no	—
141	C₄	#14519797	1	2	yes	—
142	C₄	#14519801	1	2	yes	—
143	C₄	#14519804	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	976	1	no	—
146	C₂	#16840144	1	1	yes	characteristic
147	C₂	#16840148	1	1	yes	characteristic
148	C₂	#16840150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840138

C₂

yes

maximal, characteristic

#16840134

C₂

yes

maximal, characteristic, complemented

#16840112

C₂

yes

maximal, characteristic, complemented

#16840110

C₂

yes

maximal, characteristic, complemented

#16840109

C₂

yes

maximal, characteristic, complemented

#16839483

C₂

yes

maximal, characteristic, complemented

C₆₁×#486

C₂

yes

maximal, characteristic, complemented

C₂×#11758200

—

C₂×#11758199

—

#14519777

C₂²

yes

characteristic

#14519774

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

#102×C₆₁

—

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

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758200

—

#11758199

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₄

yes

—

C₂×#92418

D₄

yes

—

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

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#127

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

Dic₆₁

—

#1717

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

—

—

—

—

—

D₆₁

yes

characteristic, complemented

#322

—

C₆₁

#1717

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

#127

—

#127

C₂×D₆₁

yes

characteristic

#103

—

122

—

#102

—

#102

—

122

—

#102

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

122

—

102

C₂×D₄

—

122

—

103

C₂×D₄

—

122

—

104

SD₁₆

—

244

—

105

D₈

—

244

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₆₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄²

C₂²×D₆₁

yes

characteristic

114

C₂³

—

115

C₂³

—

244

—

116

Q₈

—

117

Q₈

—

244

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

244

—

122

D₄

—

244

—

123

D₄

—

244

—

124

C₂×C₄

—

125

C₂×C₄

—

244

—

126

C₂×C₄

D₄×D₆₁

yes

characteristic

127

C₂×C₄

D₄×D₆₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

129

C₈

—

130

C₈

—

122

—

131

C₈

—

122

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

488

—

136

C₂²

—

488

—

137

C₂²

—

488

—

138

C₂²

#14519773

yes

characteristic

139

C₄

—

140

C₄

—

488

—

141

C₄

#14519797

yes

—

142

C₄

#14519801

yes

—

143

C₄

#14519804

yes

—

144

C₂

—

145

C₂

—

976

—

146

C₂

#16840144

yes

characteristic

147

C₂

#16840148

yes

characteristic

148

C₂

#16840150

yes

characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#486	C₂
socle	C₂²×C₆₁	#127
hypercenter	#486	D₆₁
hypocenter	C₆₁	#1717
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14519773

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#486

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. #16840134	C₂	semidirect
3. #16840112	C₂	semidirect
4. #16840110	C₂	semidirect
5. #16840109	C₂	semidirect
6. #16839483	C₂	semidirect
7. C₆₁×#486	C₂	semidirect
80. #486	D₆₁	semidirect
82. C₆₁	#1717	semidirect

H ≅ G/N

type

2. #16840134

C₂

semidirect

3. #16840112

C₂

semidirect

4. #16840110

C₂

semidirect

5. #16840109

C₂

semidirect

6. #16839483

C₂

semidirect

7. C₆₁×#486

C₂

semidirect

80. #486

D₆₁

semidirect

82. C₆₁

#1717

semidirect

Subgroups — #21445455

Conjugacy classes of subgroups

Complements