Group #19133399 — subgroups

Subgroups — #19133399

Order 5504.

164 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14583023	C₂	1	1	yes	maximal, characteristic
3	C₄×#10621885	C₂	1	1	yes	maximal, characteristic
4	#14582651	C₂	1	1	yes	maximal, characteristic
5	#14582631	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#10621885	C₄	1	1	yes	characteristic
9	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	#13063258	C₄	1	1	yes	characteristic
11	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	Dic₄₃⋊C₈	—	2	1	no	—
14	#13063249	C₄	1	1	yes	characteristic
15	#13063248	C₄	1	1	yes	characteristic
16	C₄×C₄₃⋊C₈	—	2	1	no	—
17	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₄₃	—	2	1	no	—
20	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₃	—	2	1	no	—
26	C₂×C₈×C₄₃	—	2	1	no	—
27	C₄⋊C₄×C₄₃	—	2	1	no	—
28	C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₃	—	2	1	no	—
30	C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	#10621885	C₈	1	4	yes	complemented
32	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
33	Dic₄₃⋊C₄	—	2	1	no	—
34	C₄×Dic₄₃	—	2	1	no	—
35	C₂×C₄₃⋊C₈	—	2	1	no	—
36	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	Q₈×C₄₃	—	2	1	no	—
41	Q₈×C₄₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₄₃	—	2	1	no	—
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
47	C₂×Dic₄₃	—	2	1	no	—
48	C₂×Dic₄₃	—	2	1	no	—
49	Dic₈₆	—	2	1	no	—
50	Dic₈₆	C₂×C₈	1	2	yes	—
51	C₈×C₄₃	—	2	2	no	—
52	C₈×C₄₃	—	4	1	no	—
53	C₄₃⋊C₈	—	2	1	no	—
54	C₄₃⋊C₈	—	2	1	no	—
55	C₄₃⋊C₈	—	4	1	no	—
56	C₄₃⋊C₈	C₂×C₈	1	2	yes	—
57	C₄₃⋊C₈	Q₁₆	1	2	yes	complemented
58	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₄₃	—	2	2	no	—
60	C₄×C₄₃	—	2	1	no	—
61	C₄×C₄₃	—	4	1	no	—
62	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₄₃	#131	1	1	yes	characteristic
64	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₄₃	#135	1	1	yes	characteristic
66	Dic₄₃	—	2	2	no	—
67	Dic₄₃	—	4	1	no	—
68	C₈×Q₁₆	—	43	1	no	maximal, Sylow
69	C₂×C₄₃	C₈×D₄	1	1	yes	characteristic
70	C₂×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
71	C₂×C₄₃	#443	1	1	yes	characteristic
72	C₈×Q₈	—	43	1	no	—
73	C₈×Q₈	—	43	1	no	—
74	C₄×Q₁₆	—	43	1	no	—
75	#335	—	43	1	no	—
76	Q₈⋊C₈	—	43	1	no	—
77	Q₈⋊C₈	D₄₃	1	1	yes	characteristic
78	C₈²	—	43	1	no	—
79	C₄₃	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	43	1	no	—
81	C₄×Q₈	—	43	1	no	—
82	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
83	#107	—	43	1	no	—
84	C₄⋊C₈	—	43	1	no	—
85	C₄⋊C₈	—	86	1	no	—
86	C₄⋊C₈	—	86	1	no	—
87	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
88	#103	—	43	1	no	—
89	#103	—	43	1	no	—
90	C₄×C₈	—	43	1	no	—
91	C₄×C₈	—	43	1	no	—
92	C₄×C₈	—	86	1	no	—
93	C₄×C₈	—	86	1	no	—
94	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
95	C₂×Q₈	—	43	1	no	—
96	C₂×Q₈	C₄×D₄₃	1	1	yes	characteristic
97	Q₁₆	—	43	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	43	1	no	—
101	C₂×C₈	—	43	1	no	—
102	C₂×C₈	—	43	1	no	—
103	C₂×C₈	—	43	1	no	—
104	C₂×C₈	—	86	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	43	1	no	—
107	C₄⋊C₄	—	86	1	no	—
108	C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	86	1	no	—
111	C₄²	C₂²×D₄₃	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	43	2	no	—
114	Q₈	—	86	1	no	—
115	Q₈	C₈×D₄₃	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	86	1	no	—
119	C₂×C₄	—	86	1	no	—
120	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
122	C₂×C₄	#10621900	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	4	1	no	—
125	C₈	—	43	2	no	—
126	C₈	—	43	2	no	—
127	C₈	—	86	1	no	—
128	C₈	—	86	1	no	—
129	C₈	—	172	1	no	—
130	C₂²	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	86	2	no	—
135	C₄	—	172	1	no	—
136	C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
137	C₄	#13063333	1	1	yes	characteristic
138	C₄	Q₁₆×D₄₃	1	1	yes	characteristic
139	C₄	#13063346	1	1	yes	characteristic
140	C₂	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
141	C₂	#14582991	1	1	yes	characteristic
142	C₂	#14583013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic

#14583023

C₂

yes

maximal, characteristic

C₄×#10621885

C₂

yes

maximal, characteristic

#14582651

C₂

yes

maximal, characteristic

#14582631

C₂

yes

maximal, characteristic

C₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₄₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#10621885

C₄

yes

characteristic

C₄×Dic₈₆

C₂²

yes

characteristic

#13063258

C₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

#13063249

C₄

yes

characteristic

#13063248

C₄

yes

characteristic

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₃

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

C₂×C₄

yes

characteristic

C₂×Dic₈₆

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621885

C₈

yes

complemented

C₄⋊Dic₄₃

C₂×C₄

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

C₂×C₈

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₂×C₈

yes

—

C₄₃⋊C₈

Q₁₆

yes

complemented

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

#131

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₄×C₄₃

#135

yes

characteristic

Dic₄₃

—

Dic₄₃

—

C₈×Q₁₆

—

maximal, Sylow

C₂×C₄₃

C₈×D₄

yes

characteristic

C₂×C₄₃

C₄×Q₁₆

yes

characteristic

C₂×C₄₃

#443

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

D₄₃

yes

characteristic

C₈²

—

C₄₃

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₃

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₄₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₄×D₄₃

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₄₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

C₈×D₄₃

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

121

C₂×C₄

D₄×D₄₃

yes

characteristic

122

C₂×C₄

#10621900

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

172

—

130

C₂²

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

172

—

136

C₄

C₂×C₈×D₄₃

yes

characteristic

137

C₄

#13063333

yes

characteristic

138

C₄

Q₁₆×D₄₃

yes

characteristic

139

C₄

#13063346

yes

characteristic

140

C₂

(C₂²⋊D₄₃)⋊C₈

yes

characteristic

141

C₂

#14582991

yes

characteristic

142

C₂

#14583013

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×Q₈⋊C₈	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	Q₈⋊C₈	D₄₃
hypocenter	C₄₃	C₈×Q₁₆
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×Q₈⋊C₈

C₂

socle

C₂²×C₄₃

C₄×D₄

hypercenter

Q₈⋊C₈

D₄₃

hypocenter

C₄₃

C₈×Q₁₆

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
31. #10621885	C₈	semidirect
57. C₄₃⋊C₈	Q₁₆	semidirect
79. C₄₃	C₈×Q₁₆	semidirect

H ≅ G/N

type

31. #10621885

C₈

semidirect

57. C₄₃⋊C₈

Q₁₆

semidirect

79. C₄₃

C₈×Q₁₆

semidirect

Subgroups — #19133399

Conjugacy classes of subgroups

Complements