Group #19132715 — subgroups

Subgroups — #19132715

Order 5504.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14582873	C₂	1	1	yes	maximal, characteristic
2	#14582866	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#410	C₂	1	1	yes	maximal, characteristic
4	C₂×#10621877	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₄₃⋊C₈	C₄	1	1	yes	characteristic, complemented
6	#13063284	—	2	1	no	—
7	#13063240	—	2	1	no	—
8	#13063240	C₄	1	2	yes	complemented
9	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
10	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
11	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
12	#117×C₄₃	—	2	1	no	—
13	#99×C₄₃	—	2	1	no	—
14	#99×C₄₃	C₄	1	2	yes	complemented
15	#41×C₄₃	—	4	1	no	—
16	#41×C₄₃	—	4	1	no	—
17	C₂×Q₈×C₄₃	Q₈	1	1	yes	characteristic
18	C₂×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₄₃	—	4	1	no	—
25	C₂²⋊C₄×C₄₃	—	2	2	no	—
26	C₄²×C₄₃	—	4	1	no	—
27	C₂²⋊Dic₄₃	—	2	2	no	—
28	C₄⋊Dic₄₃	—	4	1	no	—
29	C₄×Dic₄₃	—	4	1	no	—
30	#10621877	—	4	1	no	—
31	C₂×C₄₃⋊C₈	—	2	1	no	—
32	C₂×C₄₃⋊C₈	—	2	1	no	—
33	C₂×C₄₃⋊C₈	—	4	1	no	—
34	C₂³×C₄₃	C₄²	1	1	yes	characteristic
35	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
37	Q₈×C₄₃	—	4	1	no	—
38	D₄×C₄₃	—	4	1	no	—
39	D₄×C₄₃	—	4	1	no	—
40	D₄×C₄₃	—	4	1	no	—
41	C₂×C₄×C₄₃	—	2	1	no	—
42	C₂×C₄×C₄₃	—	2	1	no	—
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	1	no	—
46	C₂×C₄×C₄₃	—	2	1	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₄₃	—	4	2	no	—
53	C₄₃⋊C₈	—	4	1	no	—
54	C₄₃⋊C₈	—	4	1	no	—
55	C₂²×C₄₃	—	2	1	no	—
56	C₂²×C₄₃	—	2	1	no	—
57	C₂²×C₄₃	—	2	1	no	—
58	C₂²×C₄₃	—	4	1	no	—
59	C₂²×C₄₃	—	4	1	no	—
60	C₂²×C₄₃	—	4	1	no	—
61	C₂²×C₄₃	#95	1	1	yes	characteristic
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	2	1	no	—
64	C₄×C₄₃	—	2	1	no	—
65	C₄×C₄₃	—	8	2	no	—
66	C₄×C₄₃	#99	1	1	yes	characteristic
67	C₄×C₄₃	#99	1	1	yes	characteristic
68	Dic₄₃	—	8	2	no	—
69	#1391	—	43	1	no	maximal, Sylow
70	C₂×C₄₃	—	2	1	no	—
71	C₂×C₄₃	—	4	1	no	—
72	C₂×C₄₃	—	4	1	no	—
73	C₂×C₄₃	—	4	1	no	—
74	C₂×C₄₃	#342	1	1	yes	characteristic
75	#410	—	43	1	no	—
76	#410	D₄₃	1	1	yes	characteristic
77	#408	—	43	1	no	—
78	C₄₃	#1391	1	1	yes	characteristic, complemented, Sylow
79	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
80	C₂×M₁₆	—	43	1	no	—
81	C₂²×C₈	—	43	1	no	—
82	#117	—	2	1	no	—
83	#117	—	86	1	no	—
84	#99	—	2	1	no	—
85	#99	—	43	2	no	—
86	#99	—	86	1	no	—
87	#99	Dic₄₃	1	2	yes	complemented
88	C₂²⋊C₈	—	86	1	no	—
89	C₂²⋊C₈	—	86	1	no	—
90	#41	—	4	1	no	—
91	#41	—	4	1	no	—
92	C₂×Q₈	Dic₈₆	1	1	yes	characteristic
93	C₂×D₄	C₄×D₄₃	1	1	yes	characteristic
94	C₂×D₄	C₂×Dic₄₃	1	1	yes	characteristic
95	C₂×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
97	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
99	M₁₆	—	172	1	no	—
100	C₂×C₈	—	86	1	no	—
101	C₂×C₈	—	86	1	no	—
102	C₂×C₈	—	172	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	172	1	no	—
105	C₂²⋊C₄	—	2	2	no	—
106	C₂²⋊C₄	—	86	2	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	172	1	no	—
109	C₂³	C₄×Dic₄₃	1	1	yes	characteristic
110	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
111	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	D₄	—	4	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	172	2	no	—
124	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
125	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
127	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
128	C₈	—	172	1	no	—
129	C₈	—	172	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	1	no	—
136	C₂²	C₄₃⋊#95	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	344	2	no	—
142	C₄	#13063239	1	1	yes	characteristic
143	C₄	#13063239	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	4	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	4	1	no	—
148	C₂	#14582665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582873

C₂

yes

maximal, characteristic

#14582866

C₂

yes

maximal, characteristic

C₄₃×#410

C₂

yes

maximal, characteristic

C₂×#10621877

C₄

yes

characteristic, complemented

C₂²×C₄₃⋊C₈

C₄

yes

characteristic, complemented

#13063284

—

#13063240

—

#13063240

C₄

yes

complemented

C₂²⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₂×#41×C₄₃

C₂²

yes

characteristic

#117×C₄₃

—

#99×C₄₃

—

#99×C₄₃

C₄

yes

complemented

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

Q₈

yes

characteristic

C₂×D₄×C₄₃

C₂×C₄

yes

characteristic

C₂×D₄×C₄₃

C₂×C₄

yes

characteristic

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄²

yes

characteristic

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#95

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#1391

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

—

D₄₃

yes

characteristic

#408

—

C₄₃

#1391

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

Dic₄₃

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₈₆

yes

characteristic

C₂×D₄

C₄×D₄₃

yes

characteristic

C₂×D₄

C₂×Dic₄₃

yes

characteristic

C₂×D₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₄×D₄₃

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

M₁₆

—

172

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

172

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

172

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

—

172

—

109

C₂³

C₄×Dic₄₃

yes

characteristic

110

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

111

C₂³

C₂²⋊Dic₄₃

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

172

—

124

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

125

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

126

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

127

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

128

C₈

—

172

—

129

C₈

—

172

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₄₃⋊#95

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

344

—

142

C₄

#13063239

yes

characteristic

143

C₄

#13063239

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#14582665

yes

characteristic

	subgroup	G/H
center	C₄	#13063239
commutator	C₂³×C₄₃	C₄²
Frattini	C₂×#41	C₂×D₄₃
Fitting	C₄₃×#410	C₂
socle	C₂×C₄₃	#342
hypercenter	#410	D₄₃
hypocenter	C₄₃	#1391
LCS	G ▷ C₂³×C₄₃ ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #410

subgroup

G/H

center

C₄

#13063239

commutator

C₂³×C₄₃

C₄²

Frattini

C₂×#41

C₂×D₄₃

Fitting

C₄₃×#410

C₂

C₂×C₄₃

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #410

N	H ≅ G/N	type
4. C₂×#10621877	C₄	semidirect
5. C₂²×C₄₃⋊C₈	C₄	semidirect
8. #13063240	C₄	semidirect
14. #99×C₄₃	C₄	semidirect
78. C₄₃	#1391	semidirect
87. #99	Dic₄₃	semidirect

H ≅ G/N

type

4. C₂×#10621877

C₄

semidirect

5. C₂²×C₄₃⋊C₈

C₄

semidirect

8. #13063240

C₄

semidirect

14. #99×C₄₃

C₄

semidirect

78. C₄₃

#1391

semidirect

87. #99

Dic₄₃

semidirect

Subgroups — #19132715

Conjugacy classes of subgroups

Complements