Group #21446652 — subgroups

Subgroups — #21446652

Order 7808.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14519704	C₂	1	4	yes	maximal, complemented
3	#16839608	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	C₂²	1	2	yes	—
6	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
7	C₄⋊#92418	—	2	2	no	—
8	#14519704	—	2	8	no	—
9	#14519684	—	2	2	no	—
10	C₂×#41×C₆₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
12	#127×C₆₁	C₄	1	1	yes	characteristic, complemented
13	#124×C₆₁	C₄	1	2	yes	complemented
14	#121×C₆₁	—	2	2	no	—
15	C₄×D₄×C₆₁	—	2	2	no	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	#41×C₆₁	—	2	4	no	—
18	#41×C₆₁	—	2	4	no	—
19	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₆₁	—	2	2	no	—
21	C₂×D₄×C₆₁	—	2	2	no	—
22	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂²⋊C₄×C₆₁	—	2	4	no	—
29	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	D₄	1	2	yes	—
32	C₄×Dic₆₁	—	2	2	no	—
33	C₄×Dic₆₁	D₄	1	4	yes	complemented
34	#11758194	—	2	4	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂³×C₆₁	—	2	2	no	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₆₁	—	2	4	no	—
39	D₄×C₆₁	—	2	4	no	—
40	D₄×C₆₁	—	4	4	no	—
41	D₄×C₆₁	—	4	4	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	2	yes	—
52	C₂×Dic₆₁	—	2	4	no	—
53	C₂×Dic₆₁	—	4	4	no	—
54	#92418	—	4	2	no	—
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	—	4	2	no	—
57	C₂²×C₆₁	—	4	4	no	—
58	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₆₁	#121	1	2	yes	—
60	C₄×C₆₁	—	2	2	no	—
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	—	4	2	no	—
63	C₄×C₆₁	C₄×D₄	1	2	yes	—
64	C₄×C₆₁	#124	1	1	yes	characteristic
65	C₄×C₆₁	#127	1	1	yes	characteristic
66	Dic₆₁	—	4	4	no	—
67	#1963	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	—	2	2	no	—
69	C₂×C₆₁	—	8	2	no	—
70	C₂×C₆₁	#390	1	1	yes	characteristic
71	C₂×C₆₁	#421	1	2	yes	—
72	#532	D₆₁	1	1	yes	characteristic
73	#423	—	61	1	no	—
74	C₂×#104	—	61	4	no	—
75	#376	—	61	1	no	—
76	C₆₁	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	C₂×D₆₁	1	2	yes	—
78	C₂×M₁₆	—	61	2	no	—
79	C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic, complemented
80	#127	Dic₆₁	1	1	yes	characteristic, complemented
81	#124	Dic₆₁	1	2	yes	complemented
82	#121	—	2	2	no	—
83	C₄×D₄	—	2	2	no	—
84	C₂×C₄²	—	61	2	no	—
85	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
86	C₄⋊C₈	—	122	2	no	—
87	#104	—	122	8	no	—
88	#95	—	122	2	no	—
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₆₁	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	122	2	no	—
97	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
99	M₁₆	—	122	4	no	—
100	C₂×C₈	—	122	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	61	4	no	—
104	C₄²	—	122	2	no	—
105	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
106	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
107	C₄²	C₂²⋊D₆₁	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	122	4	no	—
119	C₂×C₄	—	244	4	no	—
120	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
122	C₂×C₄	D₄×D₆₁	1	2	yes	—
123	C₂×C₄	#11758224	1	2	yes	—
124	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
126	C₈	—	244	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	4	2	no	—
130	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
131	C₂²	#14519748	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	244	4	no	—
136	C₄	D₄×Dic₆₁	1	2	yes	—
137	C₄	#14519764	1	1	yes	characteristic
138	C₄	#14519773	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#16839665	1	1	yes	characteristic
142	C₂	#16839779	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal, characteristic, complemented

C₂×#14519704

C₂

yes

maximal, complemented

#16839608

C₂

yes

maximal, characteristic, complemented

C₆₁×#532

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄⋊#92418

—

#14519704

—

#14519684

—

C₂×#41×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

#127×C₆₁

C₄

yes

characteristic, complemented

#124×C₆₁

C₄

yes

complemented

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

complemented

#11758194

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#121

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1963

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

#421

yes

—

#532

D₆₁

yes

—

—

—

C₆₁

#1963

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₆₁

yes

characteristic, complemented

#127

Dic₆₁

yes

characteristic, complemented

#124

Dic₆₁

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

#104

—

122

—

#95

—

122

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

100

C₂×C₈

—

122

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

122

—

105

C₄²

C₂×Dic₆₁

yes

characteristic

106

C₄²

C₂×Dic₆₁

yes

characteristic

107

C₄²

C₂²⋊D₆₁

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

244

—

120

C₂×C₄

C₂²⋊Dic₆₁

yes

—

121

C₂×C₄

C₂²⋊Dic₆₁

yes

—

122

C₂×C₄

D₄×D₆₁

yes

—

123

C₂×C₄

#11758224

yes

—

124

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

126

C₈

—

244

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

131

C₂²

#14519748

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

244

—

136

C₄

D₄×Dic₆₁

yes

—

137

C₄

#14519764

yes

characteristic

138

C₄

#14519773

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#16839665

yes

characteristic

142

C₂

#16839779

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#532	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#532	D₆₁
hypocenter	C₆₁	#1963
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#532

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16839786	C₂	semidirect
2. C₂×#14519704	C₂	semidirect
3. #16839608	C₂	semidirect
11. C₄⋊Q₈×C₆₁	C₄	semidirect
12. #127×C₆₁	C₄	semidirect
13. #124×C₆₁	C₄	semidirect
33. C₄×Dic₆₁	D₄	semidirect
76. C₆₁	#1963	semidirect
79. C₄⋊Q₈	Dic₆₁	semidirect
80. #127	Dic₆₁	semidirect
81. #124	Dic₆₁	semidirect

H ≅ G/N

type

1. #16839786

C₂

semidirect

2. C₂×#14519704

C₂

semidirect

3. #16839608

C₂

semidirect

11. C₄⋊Q₈×C₆₁

C₄

semidirect

12. #127×C₆₁

C₄

semidirect

13. #124×C₆₁

C₄

semidirect

33. C₄×Dic₆₁

D₄

semidirect

76. C₆₁

#1963

semidirect

79. C₄⋊Q₈

Dic₆₁

semidirect

80. #127

Dic₆₁

semidirect

81. #124

Dic₆₁

semidirect

Subgroups — #21446652

Conjugacy classes of subgroups

Complements