Group #19147226 — subgroups

Subgroups — #19147226

Order 5504.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583245	C₂	1	3	yes	maximal, complemented
3	#14583179	C₂	1	3	yes	maximal, complemented
4	#14583091	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10621882	C₂	1	3	yes	maximal, complemented
7	C₄₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₃	—	2	1	no	—
9	C₂×#10621882	C₂²	1	3	yes	—
10	#13063325	C₂²	1	3	yes	—
11	#13063305	—	2	3	no	—
12	C₄×D₁₇₂	C₂²	1	3	yes	—
13	#13063258	C₂²	1	3	yes	—
14	C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
15	#13063246	C₂²	1	6	yes	—
16	#13063246	C₂²	1	3	yes	—
17	#13063245	C₂²	1	1	yes	characteristic
18	C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₄₃	—	2	1	no	—
20	#127×C₄₃	C₂²	1	3	yes	—
21	#121×C₄₃	—	2	3	no	—
22	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₃	C₂²	1	3	yes	—
24	#41×C₄₃	—	4	4	no	—
25	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₄₃	—	2	3	no	—
27	C₂×D₄×C₄₃	—	2	3	no	—
28	C₂×D₄×C₄₃	C₂³	1	3	yes	—
29	C₂²×C₄×C₄₃	—	2	3	no	—
30	Q₈×D₄₃	—	4	4	no	—
31	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₄₃	—	2	3	no	—
33	C₂×Dic₈₆	—	2	3	no	—
34	C₄⋊C₄×C₄₃	D₄	1	3	yes	—
35	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₄₃	—	2	3	no	—
37	C₄²×C₄₃	C₂³	1	3	yes	—
38	#10621882	—	2	6	no	—
39	(C₂×D₄₃)⋊C₄	—	2	3	no	—
40	C₄⋊Dic₄₃	—	2	3	no	—
41	C₄⋊Dic₄₃	C₂³	1	3	yes	—
42	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄₃⋊C₈	C₂³	1	3	yes	—
44	C₂³×C₄₃	—	2	3	no	—
45	C₂²×D₄₃	—	2	1	no	—
46	Q₈×C₄₃	D₈	1	4	yes	complemented
47	D₄×C₄₃	—	2	3	no	—
48	D₄×C₄₃	—	2	3	no	—
49	D₄×C₄₃	—	4	6	no	—
50	D₄×C₄₃	—	8	3	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	4	6	no	—
53	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₄₃	—	2	3	no	—
57	D₁₇₂	—	2	1	no	—
58	D₁₇₂	#41	1	2	yes	complemented
59	C₄×D₄₃	—	4	6	no	—
60	Dic₈₆	—	4	6	no	—
61	C₄₃⋊C₈	—	2	3	no	—
62	C₄₃⋊C₈	—	2	1	no	—
63	C₂²×C₄₃	—	4	3	no	—
64	C₂²×C₄₃	—	4	3	no	—
65	C₂²×C₄₃	—	4	3	no	—
66	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₄₃	—	2	2	no	—
68	C₂×D₄₃	—	4	1	no	—
69	C₂×D₄₃	—	4	1	no	—
70	C₄×C₄₃	—	2	3	no	—
71	C₄×C₄₃	—	4	1	no	—
72	C₄×C₄₃	C₂×D₈	1	6	yes	—
73	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
74	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₄₃	—	4	3	no	—
76	#3358	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	—	8	3	no	—
78	C₂×C₄₃	#548	1	1	yes	characteristic
79	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
80	C₂×C₄₃	#577	1	1	yes	characteristic
81	D₄₃	—	4	2	no	—
82	#550	D₄₃	1	1	yes	characteristic, complemented
83	D₄×Q₈	—	43	1	no	—
84	#486	—	43	3	no	—
85	#474	—	43	3	no	—
86	#459	—	43	3	no	—
87	C₈×Q₈	—	43	1	no	—
88	C₄×D₈	—	43	3	no	—
89	C₄₃	#3358	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	86	1	no	—
92	C₂×D₈	—	43	3	no	—
93	C₄⋊Q₈	—	43	3	no	—
94	#127	C₂×D₄₃	1	3	yes	—
95	C₂²⋊Q₈	—	86	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
98	C₄×D₄	—	43	3	no	—
99	C₄×D₄	C₂×D₄₃	1	3	yes	—
100	#107	—	43	3	no	—
101	C₄⋊C₈	—	43	3	no	—
102	#102	—	43	3	no	—
103	#102	—	43	6	no	—
104	#102	—	43	1	no	—
105	C₄×C₈	—	43	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	86	3	no	—
108	C₂×Q₈	—	172	4	no	—
109	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	43	1	no	—
113	C₂×D₄	C₂²×D₄₃	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	86	3	no	—
116	D₈	—	86	6	no	—
117	C₂×C₈	—	43	1	no	—
118	C₂×C₈	—	43	3	no	—
119	C₄⋊C₄	—	43	3	no	—
120	C₄⋊C₄	—	86	3	no	—
121	C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
122	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	86	3	no	—
125	C₄²	C₂²×D₄₃	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	86	1	no	—
128	Q₈	—	172	6	no	—
129	Q₈	#10621882	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	43	2	no	—
135	D₄	—	86	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	86	3	no	—
139	C₂×C₄	—	172	6	no	—
140	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
142	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
143	C₈	—	86	3	no	—
144	C₈	—	86	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	86	2	no	—
149	C₂²	—	172	1	no	—
150	C₂²	—	172	1	no	—
151	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	172	3	no	—
155	C₄	C₂×#10621882	1	6	yes	—
156	C₄	#41×D₄₃	1	1	yes	characteristic
157	C₄	#13063372	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	172	2	no	—
160	C₂	#14583556	1	1	yes	characteristic
161	C₂	C₂²×#10621882	1	1	yes	characteristic
162	C₂	#14583710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₇₂

C₂

yes

maximal, characteristic, complemented

#14583245

C₂

yes

maximal, complemented

#14583179

C₂

yes

maximal, complemented

#14583091

C₂

yes

maximal, complemented

Q₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#10621882

C₂

yes

maximal, complemented

C₄₃×#550

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₃

—

C₂×#10621882

C₂²

yes

—

#13063325

C₂²

yes

—

#13063305

—

C₄×D₁₇₂

C₂²

yes

—

#13063258

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063246

C₂²

yes

—

#13063246

C₂²

yes

—

#13063245

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

—

C₂×#41×C₄₃

—

#127×C₄₃

C₂²

yes

—

#121×C₄₃

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄×D₄×C₄₃

C₂²

yes

—

#41×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

—

Q₈×D₄₃

—

C₂×D₁₇₂

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

—

#10621882

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

D₈

yes

complemented

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

#41

yes

complemented

C₄×D₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂×#41

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

#3358

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#548

yes

characteristic

C₂×C₄₃

C₂²×D₈

yes

characteristic

C₂×C₄₃

#577

yes

characteristic

D₄₃

—

#550

D₄₃

yes

characteristic, complemented

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

—

C₄×D₈

—

C₄₃

#3358

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₄₃

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

C₂×D₄₃

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₄₃

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

—

106

#41

—

107

C₂×Q₈

—

108

C₂×Q₈

—

172

—

109

C₂×Q₈

C₂²⋊D₄₃

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²×D₄₃

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

D₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²⋊D₄₃

yes

—

122

C₄⋊C₄

C₂²×D₄₃

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

C₂²×D₄₃

yes

—

126

C₂³

—

127

C₂³

—

128

Q₈

—

172

—

129

Q₈

#10621882

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

172

—

140

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

142

C₂×C₄

C₂³×D₄₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

172

—

150

C₂²

—

172

—

151

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

172

—

155

C₄

C₂×#10621882

yes

—

156

C₄

#41×D₄₃

yes

characteristic

157

C₄

#13063372

yes

characteristic

158

C₂

—

159

C₂

—

172

—

160

C₂

#14583556

yes

characteristic

161

C₂

C₂²×#10621882

yes

characteristic

162

C₂

#14583710

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#550	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#550	D₄₃
hypocenter	C₄₃	#3358
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#550

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₁₇₂	C₂	semidirect
2. #14583245	C₂	semidirect
3. #14583179	C₂	semidirect
4. #14583091	C₂	semidirect
5. Q₈×C₄₃⋊C₈	C₂	semidirect
6. C₄×#10621882	C₂	semidirect
7. C₄₃×#550	C₂	semidirect
46. Q₈×C₄₃	D₈	semidirect
58. D₁₇₂	#41	semidirect
82. #550	D₄₃	semidirect
89. C₄₃	#3358	semidirect
129. Q₈	#10621882	semidirect

H ≅ G/N

type

1. Q₈×D₁₇₂

C₂

semidirect

2. #14583245

C₂

semidirect

3. #14583179

C₂

semidirect

4. #14583091

C₂

semidirect

5. Q₈×C₄₃⋊C₈

C₂

semidirect

6. C₄×#10621882

C₂

semidirect

7. C₄₃×#550

C₂

semidirect

46. Q₈×C₄₃

D₈

semidirect

58. D₁₇₂

#41

semidirect

82. #550

D₄₃

semidirect

89. C₄₃

#3358

semidirect

129. Q₈

#10621882

semidirect

Subgroups — #19147226

Conjugacy classes of subgroups

Complements