Group #19133025 — subgroups

Subgroups — #19133025

Order 5504.

248 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14582923	C₂	1	1	yes	maximal, characteristic
3	#14582631	C₂	1	4	yes	maximal, complemented
4	C₄₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	C₄	1	1	yes	characteristic
6	C₂×#10621877	—	2	1	no	—
7	C₄×Dic₈₆	—	2	1	no	—
8	C₄×Dic₈₆	—	2	1	no	—
9	C₄×Dic₈₆	C₄	1	2	yes	—
10	C₄×Dic₈₆	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
12	C₂×Dic₄₃⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₄₃	—	2	1	no	—
14	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊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	C₂²×Dic₄₃	—	2	1	no	—
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂×Dic₈₆	—	2	1	no	—
26	C₂×Dic₈₆	—	2	4	no	—
27	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₃	D₄	1	1	yes	characteristic
31	C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₄₃	—	2	1	no	—
34	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
35	Dic₄₃⋊C₄	—	2	2	no	—
36	Dic₄₃⋊C₄	—	2	2	no	—
37	C₄×Dic₄₃	—	2	2	no	—
38	C₄×Dic₄₃	—	2	2	no	—
39	#10621877	—	4	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	4	1	no	—
44	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
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	4	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	Dic₈₆	—	2	4	no	—
57	Dic₈₆	—	2	2	no	—
58	Dic₈₆	M₁₆	1	4	yes	complemented
59	C₈×C₄₃	—	4	2	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	#103	1	2	yes	—
64	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	#103	1	2	yes	—
69	C₄×C₄₃	C₂×M₁₆	1	2	yes	—
70	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₄₃	—	2	4	no	—
73	Dic₄₃	—	4	2	no	—
74	#1487	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	—	2	2	no	—
76	C₂×C₄₃	#407	1	1	yes	characteristic
77	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₄₃	#421	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	43	1	no	—
80	#431	D₄₃	1	1	yes	characteristic
81	#423	—	43	1	no	—
82	Q₈⋊C₈	—	43	4	no	—
83	C₄₃	#1487	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	43	1	no	—
85	C₂×M₁₆	—	86	1	no	—
86	C₄×Q₈	—	43	2	no	—
87	C₄×Q₈	—	43	2	no	—
88	C₄×Q₈	—	86	1	no	—
89	C₄×Q₈	—	86	1	no	—
90	C₂×C₄⋊C₄	—	43	1	no	—
91	C₂×C₄⋊C₄	—	86	1	no	—
92	C₂×C₄²	—	86	1	no	—
93	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	43	2	no	—
96	C₄⋊C₈	—	86	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₄×C₈	C₂×D₄₃	1	2	yes	—
99	C₂×Q₈	—	43	2	no	—
100	C₂×Q₈	—	86	4	no	—
101	C₂×Q₈	—	86	1	no	—
102	C₂²×C₄	—	86	1	no	—
103	C₂²×C₄	—	86	1	no	—
104	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
105	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
107	M₁₆	—	172	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	86	2	no	—
110	C₄⋊C₄	—	43	2	no	—
111	C₄⋊C₄	—	86	1	no	—
112	C₄⋊C₄	—	86	2	no	—
113	C₄⋊C₄	—	86	2	no	—
114	C₄²	—	86	2	no	—
115	C₄²	—	86	2	no	—
116	C₄²	C₄×D₄₃	1	1	yes	characteristic
117	C₄²	D₁₇₂	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
119	C₄²	C₂²×D₄₃	1	1	yes	characteristic
120	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
121	Q₈	—	43	4	no	—
122	Q₈	—	86	4	no	—
123	Q₈	—	86	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	86	4	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	—	86	2	no	—
129	C₂×C₄	—	172	2	no	—
130	C₂×C₄	#10621873	1	2	yes	—
131	C₂×C₄	Dic₁₇₂	1	2	yes	—
132	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	172	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#13063250	1	2	yes	—
143	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	86	4	no	—
148	C₄	—	172	2	no	—
149	C₄	#13063250	1	2	yes	—
150	C₄	M₁₆×D₄₃	1	2	yes	—
151	C₄	C₂×#10621873	1	1	yes	characteristic
152	C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	#14582862	1	1	yes	characteristic
155	C₂	C₂×#13063250	1	1	yes	characteristic
156	C₂	#14582917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

#14582923

C₂

yes

maximal, characteristic

#14582631

C₂

yes

maximal, complemented

C₄₃×#431

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₄

yes

characteristic

C₂×#10621877

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

—

C₄×Dic₈₆

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₄

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊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

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

M₁₆

yes

complemented

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

C₂×M₁₆

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#1487

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×Q₈

—

#431

D₄₃

yes

characteristic

#423

—

Q₈⋊C₈

—

C₄₃

#1487

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₃

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₄₃

yes

characteristic

105

C₂²×C₄

D₁₇₂

yes

characteristic

106

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

107

M₁₆

—

172

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₄×D₄₃

yes

characteristic

117

C₄²

D₁₇₂

yes

characteristic

118

C₄²

C₂²⋊D₄₃

yes

characteristic

119

C₄²

C₂²×D₄₃

yes

characteristic

120

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

172

—

130

C₂×C₄

#10621873

yes

—

131

C₂×C₄

Dic₁₇₂

yes

—

132

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

136

C₂×C₄

C₂×D₁₇₂

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

138

C₈

—

139

C₈

—

172

—

140

C₂²

—

141

C₂²

—

142

C₂²

#13063250

yes

—

143

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

172

—

149

C₄

#13063250

yes

—

150

C₄

M₁₆×D₄₃

yes

—

151

C₄

C₂×#10621873

yes

characteristic

152

C₄

C₂×Dic₁₇₂

yes

characteristic

153

C₂

—

154

C₂

#14582862

yes

characteristic

155

C₂

C₂×#13063250

yes

characteristic

156

C₂

#14582917

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1487
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14582631	C₂	semidirect
58. Dic₈₆	M₁₆	semidirect
83. C₄₃	#1487	semidirect

H ≅ G/N

type

3. #14582631

C₂

semidirect

58. Dic₈₆

M₁₆

semidirect

83. C₄₃

#1487

semidirect

Subgroups — #19133025

Conjugacy classes of subgroups

Complements