Group #19133347 — subgroups

Subgroups — #19133347

Order 5504.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	#14582952	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14582652	C₂	1	2	yes	maximal, complemented
4	#14582650	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
12	C₄×C₄₃⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₄₃	C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₄₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	#97×C₄₃	C₄	1	2	yes	—
17	C₄×C₈×C₄₃	C₂²	1	2	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	Q₈	1	1	yes	characteristic
21	M₁₆×C₄₃	—	2	2	no	—
22	M₁₆×C₄₃	C₈	1	4	yes	complemented
23	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
25	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₄₃	D₄	1	1	yes	characteristic
27	C₄²×C₄₃	Q₈	1	1	yes	characteristic
28	C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×C₄₃⋊C₈	—	2	1	no	—
30	C₂×C₄₃⋊C₈	—	2	1	no	—
31	C₂×C₄₃⋊C₈	—	2	2	no	—
32	C₂×C₄₃⋊C₈	—	4	2	no	—
33	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₃	—	2	1	no	—
35	C₂×C₄×C₄₃	—	2	1	no	—
36	C₂×C₄×C₄₃	—	4	1	no	—
37	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
42	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₄₃	—	2	2	no	—
46	C₈×C₄₃	C₂×C₈	1	4	yes	complemented
47	C₄₃⋊C₈	—	4	1	no	—
48	C₄₃⋊C₈	—	4	2	no	—
49	C₄₃⋊C₈	—	4	1	no	—
50	C₂²×C₄₃	—	2	1	no	—
51	C₂²×C₄₃	—	2	1	no	—
52	C₂²×C₄₃	C₄⋊C₈	1	2	yes	—
53	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₄₃	—	2	1	no	—
55	C₄×C₄₃	—	2	1	no	—
56	C₄×C₄₃	—	2	1	no	—
57	C₄×C₄₃	C₄⋊C₈	1	2	yes	—
58	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₄₃	#137	1	1	yes	characteristic
62	#1565	—	43	1	no	maximal, Sylow
63	C₂×C₄₃	—	2	2	no	—
64	C₂×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄₃	#428	1	1	yes	characteristic
66	C₂×C₄₃	#430	1	1	yes	characteristic
67	#431	—	43	1	no	—
68	C₂×C₄⋊C₈	—	43	1	no	—
69	C₄×M₁₆	D₄₃	1	1	yes	characteristic
70	#335	—	43	2	no	—
71	#334	—	43	2	no	—
72	C₄₃	#1565	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	Dic₄₃	1	1	yes	characteristic
74	C₂×M₁₆	Dic₄₃	1	1	yes	characteristic
75	C₂²×C₈	—	86	1	no	—
76	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
77	C₄⋊C₈	—	43	1	no	—
78	C₄⋊C₈	—	43	2	no	—
79	C₄⋊C₈	—	43	1	no	—
80	C₄⋊C₈	—	86	1	no	—
81	C₂²⋊C₈	—	86	1	no	—
82	#97	Dic₄₃	1	2	yes	—
83	C₄×C₈	—	86	1	no	—
84	C₄×C₈	C₂×D₄₃	1	2	yes	—
85	C₂²×C₄	Dic₈₆	1	1	yes	characteristic
86	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	C₄₃⋊C₈	1	4	yes	complemented
90	C₂×C₈	—	86	1	no	—
91	C₂×C₈	—	86	1	no	—
92	C₂×C₈	—	86	2	no	—
93	C₂×C₈	—	172	2	no	—
94	C₂×C₈	C₂×Dic₄₃	1	2	yes	—
95	C₂×C₈	C₂×Dic₄₃	1	2	yes	—
96	C₄²	Dic₈₆	1	1	yes	characteristic
97	C₄²	D₁₇₂	1	1	yes	characteristic
98	C₄²	C₂×Dic₄₃	1	1	yes	characteristic
99	C₄²	C₂²×D₄₃	1	1	yes	characteristic
100	C₂³	C₄⋊Dic₄₃	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
105	C₂×C₄	#10621877	1	2	yes	—
106	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
107	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
108	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
110	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
111	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	172	2	no	—
114	C₈	—	172	1	no	—
115	C₈	—	172	1	no	—
116	C₈	C₂×C₄₃⋊C₈	1	4	yes	complemented
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
120	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
125	C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
126	C₄	C₂×#10621877	1	1	yes	characteristic
127	C₄	#13063353	1	1	yes	characteristic
128	C₄	#13063358	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
131	C₂	#14582941	1	1	yes	characteristic
132	C₂	#14582948	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582952

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582652

C₂

yes

maximal, complemented

#14582650

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

C₄

yes

characteristic

C₂×M₁₆×C₄₃

C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

C₄

yes

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

Q₈

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

C₈

yes

complemented

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

Q₈

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

C₂×C₈

yes

complemented

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄⋊C₈

yes

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊C₈

yes

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

C₂×M₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₄₃

yes

characteristic

#335

—

#334

—

C₄₃

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

Dic₄₃

yes

characteristic

C₂×M₁₆

Dic₄₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

Dic₄₃

yes

—

C₄×C₈

—

C₄×C₈

C₂×D₄₃

yes

—

C₂²×C₄

Dic₈₆

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

M₁₆

—

M₁₆

C₄₃⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

C₂×Dic₄₃

yes

—

C₂×C₈

C₂×Dic₄₃

yes

—

C₄²

Dic₈₆

yes

characteristic

C₄²

D₁₇₂

yes

characteristic

C₄²

C₂×Dic₄₃

yes

characteristic

C₄²

C₂²×D₄₃

yes

characteristic

100

C₂³

C₄⋊Dic₄₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

105

C₂×C₄

#10621877

yes

—

106

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

107

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

108

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

109

C₂×C₄

C₂×Dic₈₆

yes

characteristic

110

C₂×C₄

C₂×D₁₇₂

yes

characteristic

111

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

112

C₈

—

113

C₈

—

172

—

114

C₈

—

172

—

115

C₈

—

172

—

116

C₈

C₂×C₄₃⋊C₈

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₄⋊(C₄₃⋊C₈)

yes

—

120

C₂²

C₂×C₄⋊Dic₄₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₄⋊(C₄₃⋊C₈)

yes

—

125

C₄

C₂²×C₄₃⋊C₈

yes

characteristic

126

C₄

C₂×#10621877

yes

characteristic

127

C₄

#13063353

yes

characteristic

128

C₄

#13063358

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊(C₄₃⋊C₈)

yes

characteristic

131

C₂

#14582941

yes

characteristic

132

C₂

#14582948

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₄₃
hypocenter	C₄₃	#1565
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄×M₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₄₃

hypocenter

C₄₃

#1565

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14582652	C₂	semidirect
4. #14582650	C₂	semidirect
22. M₁₆×C₄₃	C₈	semidirect
46. C₈×C₄₃	C₂×C₈	semidirect
72. C₄₃	#1565	semidirect
89. M₁₆	C₄₃⋊C₈	semidirect
116. C₈	C₂×C₄₃⋊C₈	semidirect

H ≅ G/N

type

3. #14582652

C₂

semidirect

4. #14582650

C₂

semidirect

22. M₁₆×C₄₃

C₈

semidirect

46. C₈×C₄₃

C₂×C₈

semidirect

72. C₄₃

#1565

semidirect

89. M₁₆

C₄₃⋊C₈

semidirect

116. C₈

C₂×C₄₃⋊C₈

semidirect

Subgroups — #19133347

Conjugacy classes of subgroups

Complements