Group #19135555 — subgroups

Subgroups — #19135555

Order 5504.

300 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13063325	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	4	yes	maximal
4	C₄₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	—	2	1	no	—
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	#13063325	—	2	2	no	—
8	#13063325	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	—	2	1	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₄×C₄₃⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₄₃	—	2	2	no	—
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	2	2	no	—
21	C₂×Dic₈₆	—	2	4	no	—
22	C₂×Dic₈₆	—	4	2	no	—
23	C₂×Dic₈₆	—	4	2	no	—
24	C₄⋊C₄×C₄₃	—	2	4	no	—
25	C₄⋊C₄×C₄₃	—	4	2	no	—
26	C₄⋊C₄×C₄₃	—	4	2	no	—
27	C₄²×C₄₃	C₂×C₄	1	2	yes	—
28	C₄²×C₄₃	D₄	1	2	yes	—
29	C₄⋊Dic₄₃	—	2	4	no	—
30	C₄⋊Dic₄₃	—	2	2	no	—
31	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₄₃⋊C₈	—	2	2	no	—
33	C₂×C₄₃⋊C₈	—	2	2	no	—
34	C₂×C₄₃⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₃	—	4	2	no	—
37	C₂×C₄×C₄₃	—	4	2	no	—
38	C₂×C₄×C₄₃	—	4	2	no	—
39	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₄₃	D₈	1	2	yes	—
42	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
45	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₄₃	#41	1	2	yes	—
49	C₂×Dic₄₃	—	2	4	no	—
50	C₂×Dic₄₃	—	4	2	no	—
51	C₂×Dic₄₃	—	4	2	no	—
52	Dic₈₆	—	4	8	no	—
53	C₄₃⋊C₈	—	2	4	no	—
54	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄₃	#124	1	1	yes	characteristic
56	C₂²×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₄₃	—	8	2	no	—
62	C₄×C₄₃	#102	1	4	yes	—
63	C₄×C₄₃	#103	1	4	yes	—
64	C₄×C₄₃	C₄×Q₈	1	2	yes	—
65	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
66	Dic₄₃	—	4	4	no	—
67	#1990	—	43	1	no	maximal, Sylow
68	C₂×C₄₃	#391	1	1	yes	characteristic
69	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
70	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
71	C₂×C₄₃	#486	1	1	yes	characteristic
72	C₂×C₄₃	#487	1	1	yes	characteristic
73	C₂×C₄₃	#498	1	1	yes	characteristic
74	C₂×C₄₃	#500	1	1	yes	characteristic
75	C₂×C₄⋊Q₈	—	43	1	no	—
76	C₂×C₄×C₈	—	43	1	no	—
77	#384	D₄₃	1	1	yes	characteristic
78	#340	—	43	4	no	—
79	C₄₃	#1990	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	86	1	no	—
81	C₂²×C₈	—	43	2	no	—
82	C₄⋊Q₈	—	43	4	no	—
83	C₄⋊Q₈	—	86	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	43	2	no	—
87	C₂×C₄⋊C₄	—	86	1	no	—
88	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
89	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
90	C₄×C₈	—	86	2	no	—
91	C₂×Q₈	—	86	4	no	—
92	C₂×Q₈	—	172	2	no	—
93	C₂×Q₈	—	172	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	86	2	no	—
96	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×C₈	—	43	4	no	—
100	C₂×C₈	—	86	2	no	—
101	C₂×C₈	—	86	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	43	4	no	—
106	C₄⋊C₄	—	86	4	no	—
107	C₄⋊C₄	—	86	2	no	—
108	C₄²	C₄×D₄₃	1	2	yes	—
109	C₄²	D₁₇₂	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
111	Q₈	—	172	8	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	86	4	no	—
116	C₂×C₄	—	172	2	no	—
117	C₂×C₄	—	172	2	no	—
118	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
119	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
120	C₂×C₄	#10621882	1	2	yes	—
121	C₂×C₄	#10621883	1	2	yes	—
122	C₂×C₄	#10621884	1	2	yes	—
123	C₂×C₄	#10621885	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
126	C₂×C₄	#10621900	1	2	yes	—
127	C₂×C₄	Q₈×D₄₃	1	2	yes	—
128	C₈	—	86	4	no	—
129	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
130	C₂²	#13063312	1	1	yes	characteristic
131	C₂²	#13063323	1	1	yes	characteristic
132	C₂²	C₂×#10621882	1	1	yes	characteristic
133	C₂²	C₂×#10621884	1	1	yes	characteristic
134	C₂²	C₂×#10621883	1	1	yes	characteristic
135	C₂²	C₂×#10621885	1	1	yes	characteristic
136	C₄	—	8	2	no	—
137	C₄	—	172	4	no	—
138	C₄	#13063245	1	4	yes	—
139	C₄	#13063248	1	4	yes	—
140	C₄	#13063290	1	2	yes	—
141	C₄	#13063305	1	2	yes	—
142	C₂	#14582804	1	1	yes	characteristic
143	C₂	C₂×#13063245	1	1	yes	characteristic
144	C₂	C₂×#13063248	1	1	yes	characteristic
145	C₂	#14583245	1	1	yes	characteristic
146	C₂	#14583249	1	1	yes	characteristic
147	C₂	#14583295	1	1	yes	characteristic
148	C₂	#14583302	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal

C₄₃×#384

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

#13063325

—

#13063325

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

Q₈

yes

complemented

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#124

yes

characteristic

C₂²×C₄₃

C₄⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

C₄×Q₈

yes

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

—

#384

D₄₃

yes

characteristic

#340

—

C₄₃

#1990

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×Q₈

—

172

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₄×D₄₃

yes

—

109

C₄²

D₁₇₂

yes

—

110

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

111

Q₈

—

172

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

172

—

117

C₂×C₄

—

172

—

118

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

119

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

120

C₂×C₄

#10621882

yes

—

121

C₂×C₄

#10621883

yes

—

122

C₂×C₄

#10621884

yes

—

123

C₂×C₄

#10621885

yes

—

124

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

125

C₂×C₄

C₂×D₁₇₂

yes

characteristic

126

C₂×C₄

#10621900

yes

—

127

C₂×C₄

Q₈×D₄₃

yes

—

128

C₈

—

129

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

130

C₂²

#13063312

yes

characteristic

131

C₂²

#13063323

yes

characteristic

132

C₂²

C₂×#10621882

yes

characteristic

133

C₂²

C₂×#10621884

yes

characteristic

134

C₂²

C₂×#10621883

yes

characteristic

135

C₂²

C₂×#10621885

yes

characteristic

136

C₄

—

137

C₄

—

172

—

138

C₄

#13063245

yes

—

139

C₄

#13063248

yes

—

140

C₄

#13063290

yes

—

141

C₄

#13063305

yes

—

142

C₂

#14582804

yes

characteristic

143

C₂

C₂×#13063245

yes

characteristic

144

C₂

C₂×#13063248

yes

characteristic

145

C₂

#14583245

yes

characteristic

146

C₂

#14583249

yes

characteristic

147

C₂

#14583295

yes

characteristic

148

C₂

#14583302

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#384	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#384	D₄₃
hypocenter	C₄₃	#1990
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#384

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #13063325	C₄	semidirect
34. C₂×C₄₃⋊C₈	Q₈	semidirect
79. C₄₃	#1990	semidirect

H ≅ G/N

type

8. #13063325

C₄

semidirect

34. C₂×C₄₃⋊C₈

Q₈

semidirect

79. C₄₃

#1990

semidirect

Subgroups — #19135555

Conjugacy classes of subgroups

Complements