Group #21456195 — subgroups

Subgroups — #21456195

Order 7808.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#14519774	C₂	1	1	yes	maximal, characteristic
2	#16840128	C₂	1	4	yes	maximal, complemented
3	#16840110	C₂	1	4	yes	maximal, complemented
4	#16839766	C₂	1	4	yes	maximal, complemented
5	C₄×#11758194	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₄₄	—	2	2	no	—
8	C₂²×D₂₄₄	C₂²	1	2	yes	—
9	C₂×#11758194	C₂²	1	2	yes	—
10	#14519774	—	2	2	no	—
11	#14519774	C₂²	1	2	yes	—
12	#14519774	C₂²	1	2	yes	—
13	#14519697	C₂²	1	16	yes	complemented
14	#14519688	C₂²	1	2	yes	complemented
15	C₄×#92418	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	complemented
17	#125×C₆₁	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₆₁	—	2	2	no	—
19	C₄×Q₈×C₆₁	—	2	2	no	—
20	#117×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₂³×D₆₁	—	2	2	no	—
23	C₂×Q₈×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×D₂₄₄	—	2	8	no	—
27	C₂×D₂₄₄	—	2	4	no	—
28	C₂×D₂₄₄	—	2	4	no	—
29	C₂×D₂₄₄	—	2	2	no	—
30	C₂×D₂₄₄	—	4	8	no	—
31	C₂×D₂₄₄	C₂³	1	4	yes	—
32	C₄⋊C₄×C₆₁	—	2	4	no	—
33	C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₆₁	—	2	2	no	—
36	C₄²×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	D₄	1	2	yes	—
38	C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	#11758194	—	2	4	no	—
41	C₂×#92418	C₂³	1	4	yes	—
42	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₆₁	—	2	4	no	—
44	C₂²×D₆₁	—	4	8	no	—
45	C₂²×D₆₁	—	4	4	no	—
46	Q₈×C₆₁	—	4	4	no	—
47	C₂×C₄×C₆₁	—	2	4	no	—
48	C₂×C₄×C₆₁	—	2	4	no	—
49	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₁	#41	1	4	yes	—
53	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
54	D₂₄₄	—	2	8	no	—
55	D₂₄₄	—	4	8	no	—
56	D₂₄₄	—	4	4	no	—
57	D₂₄₄	—	4	8	no	—
58	#92418	—	2	4	no	—
59	C₂²×C₆₁	—	2	2	no	—
60	C₂²×C₆₁	#124	1	2	yes	—
61	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₆₁	—	4	4	no	—
63	C₂×D₆₁	—	4	8	no	—
64	C₂×D₆₁	—	8	8	no	—
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	—	4	4	no	—
67	C₄×C₆₁	#124	1	2	yes	—
68	C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₆₁	C₂×#41	1	2	yes	—
70	#3138	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	2	no	—
72	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
73	C₂×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₆₁	—	8	4	no	—
75	#533	D₆₁	1	1	yes	characteristic, complemented
76	C₂×#127	—	61	1	no	—
77	#490	—	61	4	no	—
78	#486	—	61	4	no	—
79	#418	—	61	4	no	—
80	C₄×M₁₆	—	61	1	no	—
81	C₆₁	#3138	1	1	yes	characteristic, complemented, Sylow
82	C₂²×D₄	—	61	2	no	—
83	C₂²×D₄	—	122	2	no	—
84	C₂×M₁₆	—	61	2	no	—
85	C₄⋊Q₈	C₂×D₆₁	1	2	yes	complemented
86	#127	—	61	2	no	—
87	#127	—	61	2	no	—
88	#127	—	122	2	no	—
89	#125	C₂×D₆₁	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₆₁	1	2	yes	—
93	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
94	#102	—	61	16	no	—
95	#97	—	61	2	no	—
96	C₄×C₈	—	61	2	no	—
97	C₂⁴	—	122	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	61	4	no	—
100	C₂×D₄	—	122	8	no	—
101	C₂×D₄	—	122	4	no	—
102	C₂×D₄	—	122	4	no	—
103	C₂×D₄	—	122	2	no	—
104	C₂×D₄	—	244	8	no	—
105	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	M₁₆	—	122	4	no	—
108	C₂×C₈	—	61	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₆₁	1	2	yes	—
115	C₄²	C₂²×D₆₁	1	1	yes	characteristic
116	C₄²	C₂²×D₆₁	1	1	yes	characteristic
117	C₂³	—	122	4	no	—
118	C₂³	—	244	8	no	—
119	C₂³	—	244	4	no	—
120	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	122	8	no	—
123	D₄	—	244	8	no	—
124	D₄	—	244	4	no	—
125	D₄	—	244	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#11758226	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
132	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
133	C₈	—	122	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	488	8	no	—
136	C₂²	—	244	4	no	—
137	C₂²	—	244	8	no	—
138	C₂²	#14519766	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#14519766	1	2	yes	—
143	C₄	#14519802	1	4	yes	—
144	C₄	C₂×#11758226	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	488	4	no	—
147	C₂	C₂×#14519766	1	1	yes	characteristic
148	C₂	C₂×#14519802	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519774

C₂

yes

maximal, characteristic

#16840128

C₂

yes

maximal, complemented

#16840110

C₂

yes

maximal, complemented

#16839766

C₂

yes

maximal, complemented

C₄×#11758194

C₂

yes

maximal, characteristic, complemented

C₆₁×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

—

C₂²×D₂₄₄

C₂²

yes

—

C₂×#11758194

C₂²

yes

—

#14519774

—

#14519774

C₂²

yes

—

#14519774

C₂²

yes

—

#14519697

C₂²

yes

complemented

#14519688

C₂²

yes

complemented

C₄×#92418

C₂²

yes

complemented

C₄⋊Q₈×C₆₁

C₂²

yes

complemented

#125×C₆₁

C₂²

yes

complemented

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂³×D₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

#124

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#124

yes

—

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₄×C₆₁

C₂×#41

yes

—

#3138

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#124

yes

characteristic

C₂×C₆₁

C₂×AGL(1,ℤ/8)

yes

—

D₆₁

—

#533

D₆₁

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₆₁

#3138

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

122

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₆₁

yes

—

—

—

122

—

#125

C₂×D₆₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

122

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

122

—

101

C₂×D₄

—

122

—

102

C₂×D₄

—

122

—

103

C₂×D₄

—

122

—

104

C₂×D₄

—

244

—

105

C₂²×C₄

C₂²⋊D₆₁

yes

—

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

M₁₆

—

122

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₆₁

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₆₁

yes

—

115

C₄²

C₂²×D₆₁

yes

characteristic

116

C₄²

C₂²×D₆₁

yes

characteristic

117

C₂³

—

122

—

118

C₂³

—

244

—

119

C₂³

—

244

—

120

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

121

Q₈

—

122

D₄

—

122

—

123

D₄

—

244

—

124

D₄

—

244

—

125

D₄

—

244

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#11758226

yes

—

129

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

132

C₂×C₄

C₂³×D₆₁

yes

characteristic

133

C₈

—

122

—

134

C₂²

—

135

C₂²

—

488

—

136

C₂²

—

244

—

137

C₂²

—

244

—

138

C₂²

#14519766

yes

—

139

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#14519766

yes

—

143

C₄

#14519802

yes

—

144

C₄

C₂×#11758226

yes

—

145

C₂

—

146

C₂

—

488

—

147

C₂

C₂×#14519766

yes

characteristic

148

C₂

C₂×#14519802

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#533	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#533	D₆₁
hypocenter	C₆₁	#3138
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#533

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16840128	C₂	semidirect
3. #16840110	C₂	semidirect
4. #16839766	C₂	semidirect
5. C₄×#11758194	C₂	semidirect
6. C₆₁×#533	C₂	semidirect
13. #14519697	C₂²	semidirect
14. #14519688	C₂²	semidirect
15. C₄×#92418	C₂²	semidirect
16. C₄⋊Q₈×C₆₁	C₂²	semidirect
17. #125×C₆₁	C₂²	semidirect
75. #533	D₆₁	semidirect
81. C₆₁	#3138	semidirect
85. C₄⋊Q₈	C₂×D₆₁	semidirect
89. #125	C₂×D₆₁	semidirect

H ≅ G/N

type

2. #16840128

C₂

semidirect

3. #16840110

C₂

semidirect

4. #16839766

C₂

semidirect

5. C₄×#11758194

C₂

semidirect

6. C₆₁×#533

C₂

semidirect

13. #14519697

C₂²

semidirect

14. #14519688

C₂²

semidirect

15. C₄×#92418

C₂²

semidirect

16. C₄⋊Q₈×C₆₁

C₂²

semidirect

17. #125×C₆₁

C₂²

semidirect

75. #533

D₆₁

semidirect

81. C₆₁

#3138

semidirect

85. C₄⋊Q₈

C₂×D₆₁

semidirect

89. #125

C₂×D₆₁

semidirect

Subgroups — #21456195

Conjugacy classes of subgroups

Complements