Group #19137508 — subgroups

Subgroups — Q₈×Dic₄₃⋊C₄ #19137508

Order 5504.

568 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 287 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×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582797	C₂	1	3	yes	maximal
4	#14582795	C₂	1	3	yes	maximal
5	#14582768	C₂	1	3	yes	maximal
6	C₄×Dic₄₃⋊C₄	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₃	—	2	4	no	—
9	Q₈×Dic₄₃	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₄₃	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	3	yes	—
12	C₂×Dic₄₃⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₄₃⋊C₄	C₂²	1	6	yes	—
14	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₄₃	C₂²	1	3	yes	—
16	C₂×C₄×Dic₄₃	C₂²	1	3	yes	—
17	C₄₃⋊#95	C₂²	1	3	yes	—
18	C₄₃⋊#95	C₂²	1	3	yes	—
19	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₄₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
22	C₂×C₄²×C₄₃	C₂²	1	3	yes	—
23	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	—
24	C₂×Q₈×C₄₃	D₄	1	4	yes	—
25	C₂×Q₈×C₄₃	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
29	C₂²×Dic₄₃	C₂³	1	3	yes	—
30	C₂²×Dic₄₃	C₂³	1	3	yes	—
31	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₄₃	—	2	6	no	—
34	C₄²×C₄₃	—	2	6	no	—
35	C₄⋊Dic₄₃	—	2	6	no	—
36	C₄⋊Dic₄₃	C₂×C₄	1	12	yes	—
37	Dic₄₃⋊C₄	—	2	6	no	—
38	Dic₄₃⋊C₄	—	4	6	no	—
39	Dic₄₃⋊C₄	Q₈	1	4	yes	direct product factor
40	C₄×Dic₄₃	—	2	6	no	—
41	C₄×Dic₄₃	C₂×C₄	1	12	yes	—
42	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
43	Q₈×C₄₃	C₄⋊C₄	1	16	yes	complemented
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	1	no	—
46	C₂×C₄×C₄₃	—	2	3	no	—
47	C₂×C₄×C₄₃	—	2	3	no	—
48	C₂×C₄×C₄₃	—	2	3	no	—
49	C₂×C₄×C₄₃	C₂²×C₄	1	6	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₃	C₂×Q₈	1	6	yes	—
52	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
53	C₂×Dic₄₃	—	2	1	no	—
54	C₂×Dic₄₃	—	2	3	no	—
55	C₂×Dic₄₃	—	2	3	no	—
56	C₂×Dic₄₃	—	2	1	no	—
57	C₂×Dic₄₃	—	2	3	no	—
58	C₂×Dic₄₃	—	2	3	no	—
59	C₂×Dic₄₃	—	2	3	no	—
60	C₂×Dic₄₃	C₂²×C₄	1	6	yes	—
61	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
62	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
63	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
64	C₂×Dic₄₃	#41	1	2	yes	—
65	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
66	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
70	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
72	C₄×C₄₃	—	2	2	no	—
73	C₄×C₄₃	—	4	3	no	—
74	C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
75	Dic₄₃	—	2	6	no	—
76	Dic₄₃	—	2	2	no	—
77	Dic₄₃	—	4	3	no	—
78	Dic₄₃	C₄×Q₈	1	4	yes	complemented
79	Q₈×C₄⋊C₄	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄₃	#520	1	1	yes	characteristic
83	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
84	C₂×C₄₃	#550	1	1	yes	characteristic
85	C₂×C₄₃	#557	1	1	yes	characteristic
86	C₂×C₄₃	Q₈²	1	1	yes	characteristic
87	C₂×C₄×Q₈	—	43	1	no	—
88	C₂×C₄×Q₈	—	43	1	no	—
89	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic
90	#389	—	43	3	no	—
91	#389	—	43	3	no	—
92	#384	—	43	3	no	—
93	C₄×C₄⋊C₄	—	43	3	no	—
94	C₄₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
96	C₄×Q₈	—	2	4	no	—
97	C₄×Q₈	—	43	8	no	—
98	C₄×Q₈	—	86	4	no	—
99	C₂×C₄⋊C₄	—	43	3	no	—
100	C₂×C₄⋊C₄	—	43	1	no	—
101	C₂×C₄⋊C₄	—	43	3	no	—
102	C₂×C₄⋊C₄	—	43	3	no	—
103	C₂×C₄⋊C₄	—	43	6	no	—
104	C₂×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
105	C₂×C₄²	—	43	3	no	—
106	C₂×C₄²	—	43	3	no	—
107	C₂×C₄²	C₂×D₄₃	1	3	yes	—
108	#95	—	43	3	no	—
109	#95	—	43	3	no	—
110	C₂×Q₈	Dic₈₆	1	4	yes	—
111	C₂×Q₈	C₄×D₄₃	1	4	yes	—
112	C₂×Q₈	C₂²⋊D₄₃	1	4	yes	—
113	C₂²×C₄	—	43	1	no	—
114	C₂²×C₄	—	43	1	no	—
115	C₂²×C₄	—	43	3	no	—
116	C₂²×C₄	—	43	3	no	—
117	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
118	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
120	C₄⋊C₄	—	2	6	no	—
121	C₄⋊C₄	—	43	12	no	—
122	C₄⋊C₄	—	43	4	no	—
123	C₄⋊C₄	—	86	6	no	—
124	C₄⋊C₄	—	86	6	no	—
125	C₄⋊C₄	—	172	6	no	—
126	C₄²	—	2	6	no	—
127	C₄²	—	43	12	no	—
128	C₄²	—	86	6	no	—
129	C₂³	C₂³×D₄₃	1	1	yes	characteristic
130	Q₈	Dic₄₃⋊C₄	1	16	yes	direct product factor
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	43	2	no	—
137	C₂×C₄	—	43	2	no	—
138	C₂×C₄	—	43	2	no	—
139	C₂×C₄	—	43	6	no	—
140	C₂×C₄	—	43	2	no	—
141	C₂×C₄	—	86	1	no	—
142	C₂×C₄	—	86	3	no	—
143	C₂×C₄	—	86	1	no	—
144	C₂×C₄	—	86	3	no	—
145	C₂×C₄	—	86	3	no	—
146	C₂×C₄	—	86	3	no	—
147	C₂×C₄	—	86	3	no	—
148	C₂×C₄	C₂×Dic₈₆	1	6	yes	—
149	C₂×C₄	C₂×C₄×D₄₃	1	6	yes	—
150	C₂×C₄	Q₈×D₄₃	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
152	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
155	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
156	C₂²	#41×D₄₃	1	1	yes	characteristic
157	C₂²	#13063371	1	1	yes	characteristic
158	C₂²	#13063373	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	—	43	4	no	—
162	C₄	—	86	2	no	—
163	C₄	—	86	6	no	—
164	C₄	—	172	3	no	—
165	C₄	C₂×Dic₄₃⋊C₄	1	12	yes	—
166	C₂	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
167	C₂	C₄×Q₈×D₄₃	1	1	yes	characteristic
168	C₂	#14583381	1	1	yes	characteristic
169	C₂	Q₈×C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂	#14583568	1	1	yes	characteristic
171	C₂	#14583625	1	1	yes	characteristic
172	C₂	Q₈×Dic₈₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₃

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₃

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal

#14582795

C₂

yes

maximal

#14582768

C₂

yes

maximal

C₄×Dic₄₃⋊C₄

C₂

yes

maximal

C₂×C₄×Q₈×C₄₃

C₂

yes

maximal, characteristic

Q₈×Dic₄₃

—

Q₈×Dic₄₃

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

C₂×C₄

yes

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

Q₈

yes

direct product factor

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂×C₄

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

C₂×Dic₄₃

C₂²×C₄

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

#41

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₄×Q₈

yes

complemented

Q₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₃

#520

yes

characteristic

C₂×C₄₃

D₄×Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

Q₈²

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₄₃

Q₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂×D₄₃

yes

—

108

#95

—

109

#95

—

110

C₂×Q₈

Dic₈₆

yes

—

111

C₂×Q₈

C₄×D₄₃

yes

—

112

C₂×Q₈

C₂²⋊D₄₃

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₃

yes

—

118

C₂²×C₄

C₂²×D₄₃

yes

characteristic

119

C₂²×C₄

C₂²×D₄₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

172

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂³×D₄₃

yes

characteristic

130

Q₈

Dic₄₃⋊C₄

yes

direct product factor

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×Dic₈₆

yes

—

149

C₂×C₄

C₂×C₄×D₄₃

yes

—

150

C₂×C₄

Q₈×D₄₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

152

C₂²

C₂²×C₄×D₄₃

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

154

C₂²

C₂×Q₈×D₄₃

yes

characteristic

155

C₂²

C₂²×Dic₈₆

yes

characteristic

156

C₂²

#41×D₄₃

yes

characteristic

157

C₂²

#13063371

yes

characteristic

158

C₂²

#13063373

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

172

—

165

C₄

C₂×Dic₄₃⋊C₄

yes

—

166

C₂

C₂²×Dic₄₃⋊C₄

yes

characteristic

167

C₂

C₄×Q₈×D₄₃

yes

characteristic

168

C₂

#14583381

yes

characteristic

169

C₂

Q₈×C₂²⋊D₄₃

yes

characteristic

170

C₂

#14583568

yes

characteristic

171

C₂

#14583625

yes

characteristic

172

C₂

Q₈×Dic₈₆

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₃
hypocenter	C₄₃	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₃

hypocenter

C₄₃

Q₈×C₄⋊C₄

LCS

G ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
9. Q₈×Dic₄₃	C₄	semidirect
39. Dic₄₃⋊C₄	Q₈	direct (N × H)
43. Q₈×C₄₃	C₄⋊C₄	semidirect
78. Dic₄₃	C₄×Q₈	semidirect
94. C₄₃	Q₈×C₄⋊C₄	semidirect
130. Q₈	Dic₄₃⋊C₄	direct (N × H)

H ≅ G/N

type

9. Q₈×Dic₄₃

C₄

semidirect

39. Dic₄₃⋊C₄

Q₈

direct (N × H)

43. Q₈×C₄₃

C₄⋊C₄

semidirect

78. Dic₄₃

C₄×Q₈

semidirect

94. C₄₃

Q₈×C₄⋊C₄

semidirect

130. Q₈

Dic₄₃⋊C₄

direct (N × H)

Subgroups — Q₈×Dic₄₃⋊C₄ #19137508

Conjugacy classes of subgroups

Complements

Subgroups — Q8×Dic43⋊C4 #19137508

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₄₃⋊C₄ #19137508