Group #19139811 — subgroups

Subgroups — #19139811

Order 5504.

468 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#14582837	C₂	1	4	yes	maximal
3	#14582806	C₂	1	4	yes	maximal
4	#14582804	C₂	1	2	yes	maximal
5	#14582764	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₄×Dic₈₆	—	2	4	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
15	C₄₃⋊#95	C₂²	1	8	yes	—
16	C₄₃⋊#95	C₂²	1	8	yes	—
17	C₄₃⋊#95	C₂²	1	2	yes	—
18	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₄₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₃	—	2	4	no	—
24	C₂×Q₈×C₄₃	—	2	8	no	—
25	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	C₂³	1	4	yes	—
30	C₂²×Dic₄₃	C₂³	1	2	yes	—
31	C₂²×Dic₄₃	C₂³	1	2	yes	—
32	C₂×Dic₈₆	—	2	2	no	—
33	C₂×Dic₈₆	—	2	2	no	—
34	C₂×Dic₈₆	Q₈	1	4	yes	complemented
35	C₄⋊C₄×C₄₃	—	2	4	no	—
36	C₄⋊C₄×C₄₃	—	2	4	no	—
37	C₄²×C₄₃	D₄	1	4	yes	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	C₄⋊Dic₄₃	—	2	4	no	—
40	Dic₄₃⋊C₄	—	2	4	no	—
41	C₄×Dic₄₃	—	2	4	no	—
42	C₄×Dic₄₃	—	2	4	no	—
43	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
44	Q₈×C₄₃	—	4	8	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	4	no	—
47	C₂×C₄×C₄₃	—	2	8	no	—
48	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₄₃	#41	1	4	yes	—
53	C₂×C₄×C₄₃	#41	1	2	yes	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	2	2	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	2	4	no	—
58	C₂×Dic₄₃	—	2	4	no	—
59	C₂×Dic₄₃	—	2	4	no	—
60	C₂×Dic₄₃	—	2	4	no	—
61	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
62	Dic₈₆	—	2	8	no	—
63	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₄₃	C₂×#41	1	2	yes	—
66	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
67	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₄₃	—	2	2	no	—
69	C₄×C₄₃	—	4	4	no	—
70	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
71	C₄×C₄₃	#124	1	4	yes	—
72	Dic₄₃	—	2	4	no	—
73	Dic₄₃	—	4	2	no	—
74	Dic₄₃	—	4	4	no	—
75	#2568	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
78	C₂×C₄₃	#536	1	1	yes	characteristic
79	C₂×C₄₃	#556	1	2	yes	—
80	C₂×C₄₃	#557	1	2	yes	—
81	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
82	C₂×C₄×Q₈	—	43	1	no	—
83	#400	—	43	4	no	—
84	#391	—	43	4	no	—
85	#391	—	43	2	no	—
86	#382	—	43	2	no	—
87	C₄×C₄⋊C₄	—	43	1	no	—
88	C₄₃	#2568	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	43	1	no	—
90	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
91	C₄⋊Q₈	—	2	4	no	—
92	C₄×Q₈	—	86	4	no	—
93	C₂×C₄⋊C₄	—	43	2	no	—
94	C₂×C₄⋊C₄	—	43	1	no	—
95	C₂×C₄⋊C₄	—	43	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
98	C₂×C₄²	—	43	2	no	—
99	C₂×C₄²	—	43	2	no	—
100	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
101	#95	—	43	2	no	—
102	#95	—	43	8	no	—
103	#95	—	43	8	no	—
104	C₂×Q₈	—	2	8	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	43	4	no	—
107	C₂×Q₈	—	86	2	no	—
108	C₂×Q₈	—	86	2	no	—
109	C₂²×C₄	—	43	2	no	—
110	C₂²×C₄	—	43	4	no	—
111	C₂²×C₄	—	43	2	no	—
112	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	86	4	no	—
119	C₄⋊C₄	—	86	4	no	—
120	C₄⋊C₄	—	86	2	no	—
121	C₄²	—	86	4	no	—
122	C₄²	—	86	4	no	—
123	C₄²	C₂²⋊D₄₃	1	4	yes	—
124	C₂³	C₂³×D₄₃	1	1	yes	characteristic
125	Q₈	—	4	8	no	—
126	Q₈	—	86	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	43	4	no	—
131	C₂×C₄	—	86	2	no	—
132	C₂×C₄	—	86	2	no	—
133	C₂×C₄	—	86	2	no	—
134	C₂×C₄	—	86	4	no	—
135	C₂×C₄	—	86	4	no	—
136	C₂×C₄	—	86	4	no	—
137	C₂×C₄	—	86	4	no	—
138	C₂×C₄	#10621900	1	4	yes	—
139	C₂×C₄	Q₈×D₄₃	1	2	yes	—
140	C₂×C₄	#10621902	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
146	C₂²	C₂×#10621900	1	2	yes	—
147	C₂²	C₂×#10621902	1	1	yes	characteristic
148	C₂²	#13063373	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	86	4	no	—
152	C₄	—	172	4	no	—
153	C₄	—	172	2	no	—
154	C₄	#13063301	1	4	yes	—
155	C₄	#13063312	1	4	yes	—
156	C₂	C₂×#13063301	1	1	yes	characteristic
157	C₂	C₂×#13063312	1	1	yes	characteristic
158	C₂	#14583473	1	1	yes	characteristic
159	C₂	#14583623	1	2	yes	—
160	C₂	#14583624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

#14582837

C₂

yes

maximal

#14582806

C₂

yes

maximal

#14582804

C₂

yes

maximal

#14582764

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

Q₈

yes

complemented

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×Q₈

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

C₄×C₄₃

#124

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2568

—

maximal, Sylow

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#557

yes

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

—

C₄₃

#2568

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂×D₄₃

yes

characteristic

101

#95

—

102

#95

—

103

#95

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₄₃

yes

—

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

C₂²×C₄

C₂²×D₄₃

yes

characteristic

115

C₂²×C₄

C₂²×D₄₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂²⋊D₄₃

yes

—

124

C₂³

C₂³×D₄₃

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#10621900

yes

—

139

C₂×C₄

Q₈×D₄₃

yes

—

140

C₂×C₄

#10621902

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

143

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

144

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

145

C₂²

C₂×Q₈×D₄₃

yes

characteristic

146

C₂²

C₂×#10621900

yes

—

147

C₂²

C₂×#10621902

yes

characteristic

148

C₂²

#13063373

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

172

—

153

C₄

—

172

—

154

C₄

#13063301

yes

—

155

C₄

#13063312

yes

—

156

C₂

C₂×#13063301

yes

characteristic

157

C₂

C₂×#13063312

yes

characteristic

158

C₂

#14583473

yes

characteristic

159

C₂

#14583623

yes

—

160

C₂

#14583624

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2568
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
34. C₂×Dic₈₆	Q₈	semidirect
88. C₄₃	#2568	semidirect

H ≅ G/N

type

34. C₂×Dic₈₆

Q₈

semidirect

88. C₄₃

#2568

semidirect

Subgroups — #19139811

Conjugacy classes of subgroups

Complements