Group #19135532 — subgroups

Subgroups — #19135532

Order 5504.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14582654	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
8	C₄×C₄₃⋊C₈	—	2	2	no	—
9	C₂²×Q₈×C₄₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₄₃	—	2	2	no	—
11	C₄×Q₈×C₄₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₄₃	—	2	2	no	—
17	C₂×Q₈×C₄₃	—	2	2	no	—
18	C₂×Q₈×C₄₃	C₈	1	4	yes	complemented
19	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₄₃	—	2	4	no	—
25	C₄⋊C₄×C₄₃	—	2	2	no	—
26	C₄²×C₄₃	—	2	2	no	—
27	C₄²×C₄₃	—	2	2	no	—
28	C₄²×C₄₃	D₄	1	2	yes	—
29	C₄²×C₄₃	D₄	1	2	yes	—
30	C₂×C₄₃⋊C₈	—	2	2	no	—
31	C₂×C₄₃⋊C₈	—	2	2	no	—
32	C₂×C₄₃⋊C₈	—	2	2	no	—
33	C₂×C₄₃⋊C₈	—	2	4	no	—
34	C₂×C₄₃⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₄₃	—	2	8	no	—
37	C₂×C₄×C₄₃	—	2	2	no	—
38	C₂×C₄×C₄₃	—	2	2	no	—
39	C₂×C₄×C₄₃	—	2	2	no	—
40	C₂×C₄×C₄₃	—	2	2	no	—
41	C₂×C₄×C₄₃	—	2	4	no	—
42	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
47	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₄₃	#41	1	2	yes	—
51	C₄₃⋊C₈	—	2	4	no	—
52	C₄₃⋊C₈	—	4	2	no	—
53	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄₃	C₄×Q₈	1	2	yes	—
55	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₄₃	#131	1	2	yes	—
58	C₄×C₄₃	—	2	4	no	—
59	C₄×C₄₃	—	2	2	no	—
60	C₄×C₄₃	—	4	2	no	—
61	C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
62	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₄₃	#124	1	1	yes	characteristic
64	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
65	#1984	—	43	1	no	maximal, Sylow
66	C₂×C₄₃	#391	1	1	yes	characteristic
67	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄₃	#408	1	1	yes	characteristic
69	C₂×C₄₃	C₈×Q₈	1	2	yes	—
70	C₂×C₄₃	#446	1	2	yes	—
71	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic
72	C₂×C₄⋊C₈	—	43	1	no	—
73	C₂×C₄×C₈	—	43	1	no	—
74	#336	—	43	4	no	—
75	C₄₃	#1984	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	Dic₄₃	1	1	yes	characteristic
77	C₂²×C₈	—	43	2	no	—
78	C₂²×C₈	—	43	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
83	C₂×C₄²	C₂×D₄₃	1	2	yes	—
84	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
85	C₄⋊C₈	—	86	2	no	—
86	C₄×C₈	—	86	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₄₃⋊C₈	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
91	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
92	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
95	C₂×C₈	—	43	4	no	—
96	C₂×C₈	—	86	2	no	—
97	C₂×C₈	—	86	2	no	—
98	C₂×C₈	—	86	4	no	—
99	C₂×C₈	—	86	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₄₃	1	2	yes	—
105	C₄²	C₂²⋊D₄₃	1	2	yes	—
106	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
114	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
115	C₂×C₄	#10621877	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
118	C₂×C₄	Q₈×D₄₃	1	2	yes	—
119	C₂×C₄	#10621902	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
122	C₈	—	86	4	no	—
123	C₈	—	172	2	no	—
124	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
125	C₂²	Q₈×Dic₄₃	1	2	yes	—
126	C₂²	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×#10621877	1	1	yes	characteristic
128	C₂²	#13063332	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
133	C₄	#13063301	1	2	yes	—
134	C₄	#13063314	1	1	yes	characteristic
135	C₄	#13063323	1	1	yes	characteristic
136	C₂	#14582806	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
138	C₂	#14582866	1	1	yes	characteristic
139	C₂	Q₈×C₄₃⋊C₈	1	2	yes	—
140	C₂	#14583030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

#14582654

C₂

yes

maximal

C₂×C₄×Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

C₄

yes

characteristic

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₈

yes

complemented

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

Q₈

yes

complemented

C₂³×C₄₃

C₂²×C₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×Q₈

yes

—

C₂²×C₄₃

C₂²×C₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

characteristic

C₂²×C₄₃

#131

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

C₄×C₄₃

#124

yes

characteristic

C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄₃

yes

C₂×C₄₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₄₃

#408

yes

characteristic

C₂×C₄₃

C₈×Q₈

yes

—

C₂×C₄₃

#446

yes

—

C₂×C₄×Q₈

D₄₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#336

—

C₄₃

#1984

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₄₃

yes

characteristic

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄₃⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₄₃

yes

—

105

C₄²

C₂²⋊D₄₃

yes

—

106

C₂³

C₂²×Dic₄₃

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

114

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

115

C₂×C₄

#10621877

yes

—

116

C₂×C₄

C₂²⋊Dic₄₃

yes

—

117

C₂×C₄

C₂²⋊Dic₄₃

yes

—

118

C₂×C₄

Q₈×D₄₃

yes

—

119

C₂×C₄

#10621902

yes

—

120

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

122

C₈

—

123

C₈

—

172

—

124

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

125

C₂²

Q₈×Dic₄₃

yes

—

126

C₂²

C₂²×C₄₃⋊C₈

yes

characteristic

127

C₂²

C₂×#10621877

yes

characteristic

128

C₂²

#13063332

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

133

C₄

#13063301

yes

—

134

C₄

#13063314

yes

characteristic

135

C₄

#13063323

yes

characteristic

136

C₂

#14582806

yes

characteristic

137

C₂

C₂×C₂²⋊(C₄₃⋊C₈)

yes

characteristic

138

C₂

#14582866

yes

characteristic

139

C₂

Q₈×C₄₃⋊C₈

yes

—

140

C₂

#14583030

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₂×C₄×Q₈	D₄₃
hypocenter	C₄₃	#1984
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂²×C₄

hypercenter

C₂×C₄×Q₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
18. C₂×Q₈×C₄₃	C₈	semidirect
34. C₂×C₄₃⋊C₈	Q₈	semidirect
75. C₄₃	#1984	semidirect
89. C₂×Q₈	C₄₃⋊C₈	semidirect

H ≅ G/N

type

18. C₂×Q₈×C₄₃

C₈

semidirect

34. C₂×C₄₃⋊C₈

Q₈

semidirect

75. C₄₃

#1984

semidirect

89. C₂×Q₈

C₄₃⋊C₈

semidirect

Subgroups — #19135532

Conjugacy classes of subgroups

Complements