Group #14583584 — subgroups

Subgroups — #14583584

Order 2752.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13063320	C₂	1	2	yes	maximal, complemented
2	#13063316	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063315	C₂	1	2	yes	maximal, complemented
4	#13063308	C₂	1	2	yes	maximal
5	#13063305	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063304	C₂	1	1	yes	maximal, characteristic, complemented
7	#13063290	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊D₄₃	C₂	1	1	yes	maximal, characteristic
9	#13063284	C₂	1	2	yes	maximal, complemented
10	#13063283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₄₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
22	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₄₃	C₂²	1	2	yes	—
27	C₄⋊Dic₄₃	C₂²	1	2	yes	—
28	Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₄₃⋊C₄	C₂²	1	2	yes	—
30	Dic₄₃⋊C₄	C₂²	1	2	yes	—
31	Dic₄₃⋊C₄	C₂²	1	2	yes	—
32	C₄×Dic₄₃	C₂²	1	2	yes	—
33	C₄×Dic₄₃	C₂²	1	2	yes	—
34	C₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₂³×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×D₄₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₃	—	2	4	no	—
38	C₂×C₄×C₄₃	—	2	2	no	—
39	C₂×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₄₃	—	2	2	no	—
44	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₄₃	C₂³	1	2	yes	—
49	C₄×D₄₃	—	2	2	no	—
50	Dic₈₆	—	2	2	no	—
51	C₂²×C₄₃	—	2	1	no	—
52	C₂²×C₄₃	—	2	1	no	—
53	C₂²×C₄₃	#41	1	2	yes	—
54	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₄₃	—	2	1	no	—
56	C₂×D₄₃	—	2	1	no	—
57	C₂×D₄₃	—	2	1	no	—
58	C₄×C₄₃	—	2	2	no	—
59	C₄×C₄₃	—	2	2	no	—
60	C₄×C₄₃	—	2	1	no	—
61	C₄×C₄₃	#41	1	2	yes	—
62	Dic₄₃	—	2	2	no	—
63	Dic₄₃	—	2	1	no	—
64	Dic₄₃	—	2	1	no	—
65	Dic₄₃	—	2	1	no	—
66	Dic₄₃	—	2	2	no	—
67	C₂×C₄₃	—	2	2	no	—
68	C₂×C₄₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₄₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₄₃	—	4	1	no	—
72	#552	—	43	1	no	maximal, Sylow
73	C₄₃	#552	1	1	yes	characteristic, complemented, Sylow
74	#126	—	43	2	no	—
75	#125	—	43	2	no	—
76	#125	—	43	1	no	—
77	#123	—	43	2	no	—
78	C₂²⋊Q₈	—	43	1	no	—
79	C₂²⋊Q₈	—	43	1	no	—
80	C₄×Q₈	—	43	1	no	—
81	C₄×D₄	—	43	1	no	—
82	#117	—	43	2	no	—
83	#117	—	43	1	no	—
84	C₂×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
85	C₂×Q₈	—	43	1	no	—
86	C₂×D₄	—	43	1	no	—
87	C₂²×C₄	—	43	1	no	—
88	C₂²×C₄	C₂×D₄₃	1	2	yes	—
89	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
90	C₄⋊C₄	—	43	2	no	—
91	C₄⋊C₄	—	43	1	no	—
92	C₄⋊C₄	—	43	2	no	—
93	C₄⋊C₄	—	43	1	no	—
94	C₄⋊C₄	—	43	2	no	—
95	C₄⋊C₄	—	43	2	no	—
96	C₄⋊C₄	—	43	2	no	—
97	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
100	C₂²⋊C₄	—	43	1	no	—
101	C₂²⋊C₄	—	43	2	no	—
102	C₂²⋊C₄	—	43	2	no	—
103	C₂²⋊C₄	—	43	1	no	—
104	C₂²⋊C₄	—	43	2	no	—
105	C₄²	—	43	2	no	—
106	C₄²	—	43	2	no	—
107	C₄²	—	43	1	no	—
108	C₂³	—	43	1	no	—
109	C₂³	C₂²×D₄₃	1	1	yes	characteristic
110	Q₈	—	86	2	no	—
111	D₄	—	86	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	43	1	no	—
115	C₂×C₄	—	43	1	no	—
116	C₂×C₄	—	43	2	no	—
117	C₂×C₄	—	43	2	no	—
118	C₂×C₄	—	43	1	no	—
119	C₂×C₄	—	86	2	no	—
120	C₂×C₄	C₂²×D₄₃	1	2	yes	—
121	C₂×C₄	C₂²×D₄₃	1	2	yes	—
122	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	86	1	no	—
127	C₂²	—	86	1	no	—
128	C₂²	—	86	1	no	—
129	C₂²	#10621900	1	2	yes	—
130	C₂²	C₂³×D₄₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	86	1	no	—
135	C₄	—	86	1	no	—
136	C₄	—	86	2	no	—
137	C₄	—	86	1	no	—
138	C₄	—	86	2	no	—
139	C₄	#10621898	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	172	1	no	—
142	C₂	C₂×#10621900	1	1	yes	characteristic
143	C₂	C₂×#10621898	1	1	yes	characteristic
144	C₂	#13063373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063320

C₂

yes

maximal, complemented

#13063316

C₂

yes

maximal, characteristic, complemented

#13063315

C₂

yes

maximal, complemented

#13063308

C₂

yes

maximal

#13063305

C₂

yes

maximal, characteristic, complemented

#13063304

C₂

yes

maximal, characteristic, complemented

#13063290

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₄₃

C₂

yes

maximal, characteristic

#13063284

C₂

yes

maximal, complemented

#13063283

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃

C₂²

yes

characteristic

C₂×C₄×D₄₃

C₂²

yes

characteristic

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

characteristic

C₂³×C₄₃

C₂³

yes

characteristic

C₂²×D₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₄×D₄₃

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#41

yes

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#41

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂×#41

yes

characteristic

C₂×C₄₃

C₂×#41

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

characteristic

D₄₃

—

#552

—

maximal, Sylow

C₄₃

#552

yes

characteristic, complemented, Sylow

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄⋊C₄

D₄₃

yes

characteristic, complemented

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₃

yes

—

C₂²×C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₄⋊C₄

C₂×D₄₃

yes

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

—

109

C₂³

C₂²×D₄₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₄₃

yes

—

121

C₂×C₄

C₂²×D₄₃

yes

—

122

C₂×C₄

C₂²×D₄₃

yes

characteristic

123

C₂×C₄

C₂²×D₄₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#10621900

yes

—

130

C₂²

C₂³×D₄₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#10621898

yes

—

140

C₂

—

141

C₂

—

172

—

142

C₂

C₂×#10621900

yes

characteristic

143

C₂

C₂×#10621898

yes

characteristic

144

C₂

#13063373

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	C₂×C₄⋊C₄×C₄₃	C₂
prime radical	C₂×C₂²⋊D₄₃	C₂²
hypercenter	C₂×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#552
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

C₂×C₄⋊C₄×C₄₃

C₂

prime radical

C₂×C₂²⋊D₄₃

C₂²

hypercenter

C₂×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #13063320	C₂	semidirect
2. #13063316	C₂	semidirect
3. #13063315	C₂	semidirect
5. #13063305	C₂	semidirect
6. #13063304	C₂	semidirect
7. #13063290	C₂	semidirect
9. #13063284	C₂	semidirect
10. #13063283	C₂	semidirect
11. C₂×C₄⋊C₄×C₄₃	C₂	semidirect
73. C₄₃	#552	semidirect
84. C₂×C₄⋊C₄	D₄₃	semidirect

H ≅ G/N

type

1. #13063320

C₂

semidirect

2. #13063316

C₂

semidirect

3. #13063315

C₂

semidirect

5. #13063305

C₂

semidirect

6. #13063304

C₂

semidirect

7. #13063290

C₂

semidirect

9. #13063284

C₂

semidirect

10. #13063283

C₂

semidirect

11. C₂×C₄⋊C₄×C₄₃

C₂

semidirect

73. C₄₃

#552

semidirect

84. C₂×C₄⋊C₄

D₄₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13063320	2
#13063316	2
#13063315	2
#13063308	2
#13063305	2
#13063304	2
#13063290	2
C₄×C₂²⋊D₄₃	2
#13063284	2
#13063283	2
C₂×C₄⋊C₄×C₄₃	2
#552	43

subgroup

index

C₄×C₂²⋊D₄₃

C₂×C₄⋊C₄×C₄₃

Subgroups — #14583584

Conjugacy classes of subgroups

Complements

Maximal subgroups