Group #19135328 — subgroups

Subgroups — #19135328

Order 5504.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#13063259	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063259	C₂	1	1	yes	maximal, characteristic
3	#14582797	C₂	1	2	yes	maximal
4	#14582662	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
8	C₂×Dic₄₃⋊C₄	—	2	2	no	—
9	#13063259	—	2	2	no	—
10	#13063259	C₄	1	4	yes	complemented
11	C₄₃⋊#95	—	2	2	no	—
12	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₄₃	—	2	2	no	—
16	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₃	—	2	2	no	—
20	C₂²×Dic₄₃	—	2	2	no	—
21	C₂×C₈×C₄₃	—	2	1	no	—
22	C₂×C₈×C₄₃	—	2	1	no	—
23	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₄₃	D₄	1	2	yes	—
25	C₂×C₈×C₄₃	Q₈	1	2	yes	—
26	C₂×C₈×C₄₃	Q₈	1	2	yes	—
27	C₄²×C₄₃	—	2	1	no	—
28	C₄²×C₄₃	—	2	1	no	—
29	C₄⋊Dic₄₃	—	2	2	no	—
30	C₄⋊Dic₄₃	—	2	2	no	—
31	C₄⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
33	Dic₄₃⋊C₄	—	4	4	no	—
34	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₄₃	—	2	1	no	—
36	C₂×C₄×C₄₃	—	2	1	no	—
37	C₂×C₄×C₄₃	—	2	1	no	—
38	C₂×C₄×C₄₃	—	2	1	no	—
39	C₂×C₄×C₄₃	D₈	1	2	yes	—
40	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
41	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	—	4	2	no	—
49	C₂×Dic₄₃	—	4	2	no	—
50	C₂×Dic₄₃	—	4	2	no	—
51	C₂×Dic₄₃	—	4	2	no	—
52	C₂×Dic₄₃	—	4	2	no	—
53	C₈×C₄₃	—	2	2	no	—
54	C₈×C₄₃	C₄⋊C₄	1	4	yes	complemented
55	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
56	C₂²×C₄₃	#121	1	1	yes	characteristic
57	C₂²×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₄₃	#135	1	1	yes	characteristic
61	C₂²×C₄₃	#135	1	1	yes	characteristic
62	C₄×C₄₃	—	2	2	no	—
63	C₄×C₄₃	—	2	2	no	—
64	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
66	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₄₃	#125	1	1	yes	characteristic
68	Dic₄₃	—	4	4	no	—
69	Dic₄₃	—	8	2	no	—
70	#1942	—	43	1	no	maximal, Sylow
71	C₂×C₄₃	#389	1	1	yes	characteristic
72	C₂×C₄₃	C₄×D₈	1	1	yes	characteristic
73	C₂×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
74	C₂×C₄₃	#466	1	1	yes	characteristic
75	C₂×C₄₃	#467	1	1	yes	characteristic
76	C₂×C₄₃	#499	1	1	yes	characteristic
77	C₂×C₄₃	#500	1	1	yes	characteristic
78	C₂×#107	—	43	1	no	—
79	C₂×#107	—	43	1	no	—
80	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic
81	#389	—	43	2	no	—
82	#340	—	43	2	no	—
83	C₄₃	#1942	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	43	2	no	—
87	C₂×C₄⋊C₄	—	43	2	no	—
88	C₂×C₄⋊C₄	—	86	2	no	—
89	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
90	#107	—	43	4	no	—
91	#107	—	86	2	no	—
92	C₄×C₈	—	2	2	no	—
93	#95	—	86	2	no	—
94	C₂²×C₄	—	86	2	no	—
95	C₂²×C₄	—	86	2	no	—
96	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
97	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	Dic₈₆	1	2	yes	—
102	C₂×C₈	Dic₈₆	1	2	yes	—
103	C₂×C₈	C₄×D₄₃	1	2	yes	—
104	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
105	C₄⋊C₄	—	43	4	no	—
106	C₄⋊C₄	—	86	2	no	—
107	C₄⋊C₄	—	86	2	no	—
108	C₄⋊C₄	—	86	2	no	—
109	C₄⋊C₄	—	172	4	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	86	4	no	—
118	C₂×C₄	—	172	2	no	—
119	C₂×C₄	—	172	2	no	—
120	C₂×C₄	—	172	2	no	—
121	C₂×C₄	—	172	2	no	—
122	C₂×C₄	—	172	2	no	—
123	C₂×C₄	D₃₄₄	1	2	yes	—
124	C₂×C₄	Dic₁₇₂	1	2	yes	—
125	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
126	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
128	C₂×C₄	#10621898	1	1	yes	characteristic
129	C₂×C₄	#10621898	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	Dic₄₃⋊C₄	1	4	yes	complemented
133	C₂²	C₄×D₁₇₂	1	1	yes	characteristic
134	C₂²	#13063299	1	1	yes	characteristic
135	C₂²	#13063325	1	1	yes	characteristic
136	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
137	C₂²	C₂×Dic₁₇₂	1	1	yes	characteristic
138	C₂²	#13063347	1	1	yes	characteristic
139	C₂²	#13063347	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	344	2	no	—
143	C₄	—	172	4	no	—
144	C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
145	C₄	C₄×Dic₈₆	1	1	yes	characteristic
146	C₄	#13063304	1	1	yes	characteristic
147	C₄	#13063317	1	1	yes	characteristic
148	C₂	#14582797	1	1	yes	characteristic
149	C₂	C₄×D₃₄₄	1	1	yes	characteristic
150	C₂	C₄×Dic₁₇₂	1	1	yes	characteristic
151	C₂	#14583135	1	1	yes	characteristic
152	C₂	#14583140	1	1	yes	characteristic
153	C₂	#14583301	1	1	yes	characteristic
154	C₂	#14583304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063259

C₂

yes

maximal, characteristic

C₂×#13063259

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal

#14582662

C₂

yes

maximal

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

—

#13063259

—

#13063259

C₄

yes

complemented

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

Q₈

yes

—

C₂×C₈×C₄₃

Q₈

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₄⋊C₄

yes

complemented

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₄⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

C₄×Q₈

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

#125

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₄×D₈

yes

characteristic

C₂×C₄₃

C₄×Q₁₆

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₂×C₄×C₈

D₄₃

yes

characteristic

#389

—

#340

—

C₄₃

#1942

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#107

—

#107

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₈₆

yes

—

102

C₂×C₈

Dic₈₆

yes

—

103

C₂×C₈

C₄×D₄₃

yes

—

104

C₂×C₈

C₂²⋊D₄₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

172

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×D₁₇₂

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

172

—

119

C₂×C₄

—

172

—

120

C₂×C₄

—

172

—

121

C₂×C₄

—

172

—

122

C₂×C₄

—

172

—

123

C₂×C₄

D₃₄₄

yes

—

124

C₂×C₄

Dic₁₇₂

yes

—

125

C₂×C₄

C₂×Dic₈₆

yes

characteristic

126

C₂×C₄

C₂×Dic₈₆

yes

characteristic

127

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

128

C₂×C₄

#10621898

yes

characteristic

129

C₂×C₄

#10621898

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

131

C₈

—

132

C₈

Dic₄₃⋊C₄

yes

complemented

133

C₂²

C₄×D₁₇₂

yes

characteristic

134

C₂²

#13063299

yes

characteristic

135

C₂²

#13063325

yes

characteristic

136

C₂²

C₂×D₃₄₄

yes

characteristic

137

C₂²

C₂×Dic₁₇₂

yes

characteristic

138

C₂²

#13063347

yes

characteristic

139

C₂²

#13063347

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

344

—

143

C₄

—

172

—

144

C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

145

C₄

C₄×Dic₈₆

yes

characteristic

146

C₄

#13063304

yes

characteristic

147

C₄

#13063317

yes

characteristic

148

C₂

#14582797

yes

characteristic

149

C₂

C₄×D₃₄₄

yes

characteristic

150

C₂

C₄×Dic₁₇₂

yes

characteristic

151

C₂

#14583135

yes

characteristic

152

C₂

#14583140

yes

characteristic

153

C₂

#14583301

yes

characteristic

154

C₂

#14583304

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	#1942
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

#1942

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
10. #13063259	C₄	semidirect
54. C₈×C₄₃	C₄⋊C₄	semidirect
83. C₄₃	#1942	semidirect
132. C₈	Dic₄₃⋊C₄	semidirect

H ≅ G/N

type

10. #13063259

C₄

semidirect

54. C₈×C₄₃

C₄⋊C₄

semidirect

83. C₄₃

#1942

semidirect

132. C₈

Dic₄₃⋊C₄

semidirect

Subgroups — #19135328

Conjugacy classes of subgroups

Complements