Group #21461953 — subgroups

Subgroups — #21461953

Order 7808.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16840766	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14519841	C₂	1	4	yes	maximal, complemented
3	C₄²×#6923	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840310	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758213	C₂²	1	2	yes	—
6	C₂×C₄×#6923	—	2	2	no	—
7	C₂×C₄×#6923	C₂²	1	2	yes	—
8	C₄⋊#92420	—	2	2	no	—
9	#14519841	—	2	8	no	—
10	C₂×#11758224	C₂²	1	2	yes	—
11	#14519777	C₄	1	1	yes	characteristic, complemented
12	#14519764	C₄	1	2	yes	complemented
13	#14519748	—	2	2	no	—
14	D₄×Dic₆₁	—	2	2	no	—
15	C₄²×D₆₁	C₂²	1	1	yes	characteristic
16	#127×C₆₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6923	—	2	2	no	—
18	C₂×D₄×C₆₁	—	2	2	no	—
19	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₂²⋊D₆₁	—	2	2	no	—
21	C₂²×Dic₆₁	—	2	2	no	—
22	#11758224	—	2	4	no	—
23	#11758224	—	2	4	no	—
24	C₂×C₄×D₆₁	D₄	1	2	yes	—
25	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
27	C₂×#92420	—	2	2	no	—
28	C₄×#6923	—	2	8	no	—
29	C₄×#6923	—	2	2	no	—
30	C₄×#6923	D₄	1	4	yes	complemented
31	#11758213	—	2	4	no	—
32	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊Dic₆₁	—	2	4	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₆₁	D₄	1	2	yes	—
37	C₂³×C₆₁	—	2	2	no	—
38	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×#6923	—	2	8	no	—
40	C₂×#6923	—	2	4	no	—
41	D₄×C₆₁	—	2	4	no	—
42	D₄×C₆₁	—	4	4	no	—
43	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²⋊D₆₁	—	4	4	no	—
46	C₂×Dic₆₁	—	2	2	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	—	4	2	no	—
49	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
51	C₄×D₆₁	—	2	4	no	—
52	C₄×D₆₁	C₂×D₄	1	2	yes	—
53	C₄×D₆₁	#41	1	2	yes	—
54	Dic₁₂₂	—	2	4	no	—
55	#92420	—	4	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	C₂×D₆₁	—	2	2	no	—
60	C₂×D₆₁	#121	1	2	yes	—
61	#6923	—	2	8	no	—
62	C₄×C₆₁	#104	1	4	yes	complemented
63	C₄×C₆₁	C₄×D₄	1	2	yes	—
64	Dic₆₁	—	2	2	no	—
65	Dic₆₁	—	4	2	no	—
66	Dic₆₁	#124	1	1	yes	characteristic
67	Dic₆₁	#127	1	1	yes	characteristic
68	#1962	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	—	8	2	no	—
70	C₂×C₆₁	#390	1	1	yes	characteristic
71	C₂×C₆₁	C₂×#104	1	2	yes	—
72	D₆₁	—	2	2	no	—
73	#532	—	61	1	no	—
74	#423	—	61	1	no	—
75	C₂×#104	—	61	4	no	—
76	C₄³	—	61	1	no	—
77	C₆₁	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	—	61	2	no	—
79	C₂×M₁₆	—	61	2	no	—
80	C₄⋊Q₈	—	61	1	no	—
81	#127	#6923	1	1	yes	characteristic, complemented
82	#124	—	61	2	no	—
83	#121	—	122	2	no	—
84	C₄×D₄	—	122	2	no	—
85	C₂×C₄²	—	61	1	no	—
86	C₂×C₄²	—	61	2	no	—
87	C₂×C₄²	—	122	2	no	—
88	C₄⋊C₈	—	122	2	no	—
89	#104	—	122	8	no	—
90	#41	—	122	4	no	—
91	#41	—	122	4	no	—
92	C₂×Q₈	—	61	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	122	2	no	—
95	C₂×D₄	C₂×#6923	1	2	yes	—
96	C₂²×C₄	—	61	1	no	—
97	C₂²×C₄	—	61	2	no	—
98	C₂²×C₄	—	122	2	no	—
99	C₂²×C₄	—	122	2	no	—
100	M₁₆	—	122	4	no	—
101	C₂×C₈	—	122	2	no	—
102	C₄⋊C₄	—	122	2	no	—
103	C₂²⋊C₄	—	122	4	no	—
104	C₄²	—	61	2	no	—
105	C₄²	—	61	1	no	—
106	C₄²	—	61	4	no	—
107	C₄²	—	122	2	no	—
108	C₄²	—	122	8	no	—
109	C₄²	C₂×#6923	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	61	1	no	—
112	Q₈	—	122	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	4	no	—
115	D₄	—	244	4	no	—
116	C₂×C₄	—	61	1	no	—
117	C₂×C₄	—	61	2	no	—
118	C₂×C₄	—	61	2	no	—
119	C₂×C₄	—	61	2	no	—
120	C₂×C₄	—	122	4	no	—
121	C₂×C₄	—	122	8	no	—
122	C₂×C₄	—	122	4	no	—
123	C₂×C₄	—	122	2	no	—
124	C₂×C₄	—	244	2	no	—
125	C₂×C₄	—	244	2	no	—
126	C₂×C₄	C₂²⋊#6923	1	2	yes	—
127	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
128	C₈	—	244	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	61	2	no	—
132	C₂²	—	122	2	no	—
133	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
134	C₄	—	61	1	no	—
135	C₄	—	61	1	no	—
136	C₄	—	122	8	no	—
137	C₄	—	122	2	no	—
138	C₄	—	244	2	no	—
139	C₄	#14519841	1	4	yes	complemented
140	C₄	D₄×#6923	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	—	122	2	no	—
143	C₂	#16840714	1	1	yes	characteristic
144	C₂	C₂×#14519841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840766

C₂

yes

maximal, characteristic, complemented

C₂×#14519841

C₂

yes

maximal, complemented

C₄²×#6923

C₂

yes

maximal, characteristic, complemented

#16840310

C₂

yes

maximal, characteristic

C₂×#11758213

C₂²

yes

—

C₂×C₄×#6923

—

C₂×C₄×#6923

C₂²

yes

—

C₄⋊#92420

—

#14519841

—

C₂×#11758224

C₂²

yes

—

#14519777

C₄

yes

characteristic, complemented

#14519764

C₄

yes

complemented

#14519748

—

D₄×Dic₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

#127×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

#11758224

—

#11758224

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

—

—

—

D₄

yes

complemented

#11758213

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

C₂³×C₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

C₂×#6923

—

C₂×#6923

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

#121

yes

—

#6923

—

C₄×C₆₁

#104

yes

complemented

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

C₂×#104

yes

—

D₆₁

—

#532

—

#423

—

C₂×#104

—

C₄³

—

C₆₁

#1962

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

#6923

yes

characteristic, complemented

#124

—

#121

—

122

—

C₄×D₄

—

122

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

122

—

C₄⋊C₈

—

122

—

#104

—

122

—

#41

—

122

—

#41

—

122

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

122

—

C₂×D₄

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

100

M₁₆

—

122

—

101

C₂×C₈

—

122

—

102

C₄⋊C₄

—

122

—

103

C₂²⋊C₄

—

122

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

122

—

108

C₄²

—

122

—

109

C₄²

C₂×#6923

yes

characteristic

110

C₂³

—

111

C₂³

—

112

Q₈

—

122

—

113

D₄

—

114

D₄

—

115

D₄

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

244

—

125

C₂×C₄

—

244

—

126

C₂×C₄

C₂²⋊#6923

yes

—

127

C₂×C₄

C₂²×#6923

yes

characteristic

128

C₈

—

244

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

122

—

133

C₂²

C₂×C₂²⋊#6923

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

122

—

137

C₄

—

122

—

138

C₄

—

244

—

139

C₄

#14519841

yes

complemented

140

C₄

D₄×#6923

yes

—

141

C₂

—

142

C₂

—

122

—

143

C₂

#16840714

yes

characteristic

144

C₂

C₂×#14519841

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	#127×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#127	#6923
hypocenter	C₆₁	#1962
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

#127×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
1. #16840766	C₂	semidirect
2. C₂×#14519841	C₂	semidirect
3. C₄²×#6923	C₂	semidirect
11. #14519777	C₄	semidirect
12. #14519764	C₄	semidirect
16. #127×C₆₁	C₄	semidirect
30. C₄×#6923	D₄	semidirect
62. C₄×C₆₁	#104	semidirect
77. C₆₁	#1962	semidirect
81. #127	#6923	semidirect
139. C₄	#14519841	semidirect

H ≅ G/N

type

1. #16840766

C₂

semidirect

2. C₂×#14519841

C₂

semidirect

3. C₄²×#6923

C₂

semidirect

11. #14519777

C₄

semidirect

12. #14519764

C₄

semidirect

16. #127×C₆₁

C₄

semidirect

30. C₄×#6923

D₄

semidirect

62. C₄×C₆₁

#104

semidirect

77. C₆₁

#1962

semidirect

81. #127

#6923

semidirect

139. C₄

#14519841

semidirect

Subgroups — #21461953

Conjugacy classes of subgroups

Complements