Group #19133296 — subgroups

Subgroups — #19133296

Order 5504.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14582646	C₂	1	2	yes	maximal, complemented
4	#14582644	C₂	1	2	yes	maximal, complemented
5	C₄₃×#432	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	—	2	1	no	—
7	#13063325	C₂²	1	1	yes	characteristic
8	#13063325	C₂²	1	1	yes	characteristic
9	#13063317	C₄	1	2	yes	—
10	#13063304	—	2	1	no	—
11	#13063304	—	2	1	no	—
12	C₄×Dic₈₆	—	2	1	no	—
13	C₄×Dic₈₆	—	2	1	no	—
14	#13063284	C₄	1	2	yes	—
15	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
16	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₄₃	—	2	1	no	—
20	#97×C₄₃	—	2	1	no	—
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	C₂×Dic₈₆	—	2	1	no	—
25	C₂×Dic₈₆	—	2	1	no	—
26	C₂×C₈×C₄₃	—	2	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	C₂×C₄	1	2	yes	—
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	C₂×C₄₃⋊C₈	—	4	2	no	—
41	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	1	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	Dic₈₆	—	4	2	no	—
55	Dic₈₆	—	4	1	no	—
56	Dic₈₆	—	4	1	no	—
57	C₈×C₄₃	—	4	2	no	—
58	C₄₃⋊C₈	—	4	2	no	—
59	C₂²×C₄₃	—	2	1	no	—
60	C₂²×C₄₃	—	2	1	no	—
61	C₂²×C₄₃	#101	1	2	yes	—
62	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₄₃	—	2	1	no	—
64	C₄×C₄₃	—	2	1	no	—
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	#135	1	2	yes	—
68	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₄₃	#137	1	1	yes	characteristic
70	Dic₄₃	—	4	2	no	—
71	Dic₄₃	—	4	2	no	—
72	#1552	—	43	1	no	maximal, Sylow
73	C₂×C₄₃	—	2	2	no	—
74	C₂×C₄₃	C₂×#101	1	1	yes	characteristic
75	C₂×C₄₃	#416	1	1	yes	characteristic
76	C₂×C₄₃	#417	1	1	yes	characteristic
77	#533	—	43	1	no	—
78	#432	D₄₃	1	1	yes	characteristic
79	C₂×C₄⋊C₈	—	43	1	no	—
80	#332	—	43	2	no	—
81	#332	—	43	2	no	—
82	C₄₃	#1552	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	86	1	no	—
84	C₄⋊Q₈	—	43	1	no	—
85	C₄⋊Q₈	—	43	1	no	—
86	#125	—	43	2	no	—
87	C₂²⋊Q₈	—	86	1	no	—
88	C₂²⋊Q₈	—	86	1	no	—
89	C₄×Q₈	—	86	1	no	—
90	C₄×Q₈	—	86	1	no	—
91	#117	—	43	2	no	—
92	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
93	C₄⋊C₈	—	43	2	no	—
94	C₄⋊C₈	—	86	1	no	—
95	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	2	1	no	—
98	C₂×Q₈	—	86	1	no	—
99	C₂×Q₈	—	86	1	no	—
100	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
101	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	86	2	no	—
105	C₂×C₈	—	172	2	no	—
106	C₄⋊C₄	—	43	2	no	—
107	C₄⋊C₄	—	43	2	no	—
108	C₄⋊C₄	—	86	2	no	—
109	C₄⋊C₄	—	86	2	no	—
110	C₄⋊C₄	—	86	1	no	—
111	C₄⋊C₄	—	86	1	no	—
112	C₂²⋊C₄	—	86	2	no	—
113	C₄²	—	86	2	no	—
114	C₄²	C₄×D₄₃	1	1	yes	characteristic
115	C₄²	D₁₇₂	1	1	yes	characteristic
116	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
117	C₄²	C₂²×D₄₃	1	1	yes	characteristic
118	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
119	Q₈	—	172	2	no	—
120	Q₈	—	172	1	no	—
121	Q₈	—	172	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	86	2	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	172	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#13063243	1	2	yes	—
139	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	172	2	no	—
145	C₄	—	172	2	no	—
146	C₄	#13063345	1	2	yes	—
147	C₄	#13063353	1	1	yes	characteristic
148	C₄	#13063358	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#13063243	1	1	yes	characteristic
151	C₂	#14582893	1	1	yes	characteristic
152	C₂	#14582901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582646

C₂

yes

maximal, complemented

#14582644

C₂

yes

maximal, complemented

C₄₃×#432

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

#13063325

C₂²

yes

characteristic

#13063325

C₂²

yes

characteristic

#13063317

C₄

yes

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

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

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#101

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#135

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#1552

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

D₄₃

yes

characteristic

C₂×C₄⋊C₈

—

#332

—

#332

—

C₄₃

#1552

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

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

C₄×D₄₃

yes

characteristic

101

C₂²×C₄

D₁₇₂

yes

characteristic

102

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

172

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₄×D₄₃

yes

characteristic

115

C₄²

D₁₇₂

yes

characteristic

116

C₄²

C₂²⋊D₄₃

yes

characteristic

117

C₄²

C₂²×D₄₃

yes

characteristic

118

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

119

Q₈

—

172

—

120

Q₈

—

172

—

121

Q₈

—

172

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

132

C₂×C₄

C₂×D₁₇₂

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

134

C₈

—

135

C₈

—

172

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13063243

yes

—

139

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

172

—

145

C₄

—

172

—

146

C₄

#13063345

yes

—

147

C₄

#13063353

yes

characteristic

148

C₄

#13063358

yes

characteristic

149

C₂

—

150

C₂

C₂×#13063243

yes

characteristic

151

C₂

#14582893

yes

characteristic

152

C₂

#14582901

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#432	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#432	D₄₃
hypocenter	C₄₃	#1552
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#432

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #14582646	C₂	semidirect
4. #14582644	C₂	semidirect
82. C₄₃	#1552	semidirect

H ≅ G/N

type

3. #14582646

C₂

semidirect

4. #14582644

C₂

semidirect

82. C₄₃

#1552

semidirect

Subgroups — #19133296

Conjugacy classes of subgroups

Complements