Group #19132997 — subgroups

Subgroups — #19132997

Order 5504.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14583456	C₂	1	1	yes	maximal, characteristic
2	#14582952	C₂	1	1	yes	maximal, characteristic
3	#14582629	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#13063325	C₄	1	1	yes	characteristic
6	#13063325	C₄	1	1	yes	characteristic
7	#13063317	—	2	1	no	—
8	#13063304	—	2	1	no	—
9	#13063304	C₄	1	2	yes	—
10	C₄×Dic₈₆	—	2	1	no	—
11	C₄×Dic₈₆	C₂²	1	2	yes	—
12	#13063284	—	2	1	no	—
13	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₈×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₄₃	—	2	1	no	—
19	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂×Dic₈₆	—	2	1	no	—
24	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₄₃	—	2	2	no	—
26	C₂×C₈×C₄₃	—	4	2	no	—
27	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₃	D₄	1	1	yes	characteristic
29	C₄²×C₄₃	D₄	1	1	yes	characteristic
30	C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	—	2	1	no	—
33	C₄⋊Dic₄₃	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
36	Dic₄₃⋊C₄	—	2	2	no	—
37	Dic₄₃⋊C₄	—	2	2	no	—
38	C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	4	1	no	—
43	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₄₃	—	2	2	no	—
52	C₂×Dic₄₃	—	2	2	no	—
53	Dic₈₆	—	2	4	no	—
54	Dic₈₆	—	4	2	no	—
55	C₈×C₄₃	—	4	2	no	—
56	C₄₃⋊C₈	—	4	2	no	—
57	C₂²×C₄₃	—	2	1	no	—
58	C₂²×C₄₃	—	2	1	no	—
59	C₂²×C₄₃	#103	1	2	yes	—
60	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	2	1	no	—
64	C₄×C₄₃	#103	1	2	yes	—
65	C₄×C₄₃	#131	1	2	yes	—
66	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
68	Dic₄₃	—	4	2	no	—
69	Dic₄₃	—	4	2	no	—
70	#1481	—	43	1	no	maximal, Sylow
71	C₂×C₄₃	—	2	2	no	—
72	C₂×C₄₃	#408	1	1	yes	characteristic
73	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
74	C₂×C₄₃	#421	1	1	yes	characteristic
75	#533	—	43	1	no	—
76	#431	—	43	1	no	—
77	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
78	Q₈⋊C₈	—	43	4	no	—
79	C₄₃	#1481	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	2	1	no	—
81	C₄⋊Q₈	—	43	1	no	—
82	C₄⋊Q₈	—	43	1	no	—
83	#125	—	86	1	no	—
84	C₂²⋊Q₈	—	43	2	no	—
85	C₂²⋊Q₈	—	86	1	no	—
86	C₄×Q₈	—	43	2	no	—
87	C₄×Q₈	—	86	1	no	—
88	#117	—	86	1	no	—
89	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	86	1	no	—
92	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
93	C₂²⋊C₈	—	86	1	no	—
94	C₄×C₈	—	43	2	no	—
95	C₂×Q₈	—	43	2	no	—
96	C₂×Q₈	—	86	1	no	—
97	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
98	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	86	2	no	—
103	C₄⋊C₄	—	43	2	no	—
104	C₄⋊C₄	—	86	1	no	—
105	C₄⋊C₄	—	86	1	no	—
106	C₄⋊C₄	—	86	2	no	—
107	C₄⋊C₄	—	86	1	no	—
108	C₄⋊C₄	—	86	2	no	—
109	C₂²⋊C₄	—	86	2	no	—
110	C₄²	—	86	2	no	—
111	C₄²	C₄×D₄₃	1	1	yes	characteristic
112	C₄²	D₁₇₂	1	1	yes	characteristic
113	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
114	C₄²	C₂²×D₄₃	1	1	yes	characteristic
115	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
116	Q₈	—	86	4	no	—
117	Q₈	—	172	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	86	2	no	—
121	C₂×C₄	—	86	2	no	—
122	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#10621883	1	2	yes	—
126	C₂×C₄	#10621885	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	172	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#13063248	1	2	yes	—
135	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	172	2	no	—
140	C₄	—	172	2	no	—
141	C₄	#13063248	1	2	yes	—
142	C₄	#13063331	1	2	yes	—
143	C₄	C₂×#10621883	1	1	yes	characteristic
144	C₄	C₂×#10621885	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#14582865	1	1	yes	characteristic
147	C₂	C₂×#13063248	1	1	yes	characteristic
148	C₂	#14582915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal, characteristic

#14582952

C₂

yes

maximal, characteristic

#14582629

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

#13063325

C₄

yes

characteristic

#13063325

C₄

yes

—

—

C₄

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

—

#13063284

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂²

yes

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#103

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

#131

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#1481

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic

Q₈⋊C₈

—

C₄₃

#1481

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₄₃

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₄×D₄₃

yes

characteristic

112

C₄²

D₁₇₂

yes

characteristic

113

C₄²

C₂²⋊D₄₃

yes

characteristic

114

C₄²

C₂²×D₄₃

yes

characteristic

115

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

116

Q₈

—

117

Q₈

—

172

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

124

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

125

C₂×C₄

#10621883

yes

—

126

C₂×C₄

#10621885

yes

—

127

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

128

C₂×C₄

C₂×D₁₇₂

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

130

C₈

—

131

C₈

—

172

—

132

C₂²

—

133

C₂²

—

134

C₂²

#13063248

yes

—

135

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

172

—

140

C₄

—

172

—

141

C₄

#13063248

yes

—

142

C₄

#13063331

yes

—

143

C₄

C₂×#10621883

yes

characteristic

144

C₄

C₂×#10621885

yes

characteristic

145

C₂

—

146

C₂

#14582865

yes

characteristic

147

C₂

C₂×#13063248

yes

characteristic

148

C₂

#14582915

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#1481
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#1481

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #14582629	C₂	semidirect
79. C₄₃	#1481	semidirect

H ≅ G/N

type

3. #14582629

C₂

semidirect

79. C₄₃

#1481

semidirect

Subgroups — #19132997

Conjugacy classes of subgroups

Complements