Group #19134459 — subgroups

Subgroups — C₄⋊C₄×C₄₃⋊C₈ #19134459

Order 5504.

296 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 181 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	2	yes	maximal
3	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14582654	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
10	C₄×C₄₃⋊C₈	—	2	4	no	—
11	C₄×C₄₃⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
13	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	#95×C₄₃	C₄	1	2	yes	—
16	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₄₃	C₈	1	8	yes	complemented
21	C₄²×C₄₃	—	2	2	no	—
22	C₄²×C₄₃	—	2	2	no	—
23	C₄²×C₄₃	C₂×C₄	1	2	yes	—
24	C₄²×C₄₃	C₂×C₄	1	2	yes	—
25	C₂×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄₃⋊C₈	—	2	1	no	—
27	C₂×C₄₃⋊C₈	—	2	1	no	—
28	C₂×C₄₃⋊C₈	—	2	2	no	—
29	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
33	C₂×C₄₃⋊C₈	Q₈	1	2	yes	—
34	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₃	—	2	4	no	—
36	C₂×C₄×C₄₃	—	2	2	no	—
37	C₂×C₄×C₄₃	—	2	4	no	—
38	C₂×C₄×C₄₃	C₄²	1	2	yes	—
39	C₂×C₄×C₄₃	C₄²	1	2	yes	—
40	C₂×C₄×C₄₃	C₂×C₈	1	8	yes	—
41	C₂×C₄×C₄₃	C₂×C₈	1	4	yes	—
42	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
48	C₄₃⋊C₈	—	2	2	no	—
49	C₄₃⋊C₈	—	2	4	no	—
50	C₄₃⋊C₈	C₄⋊C₄	1	4	yes	direct product factor
51	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₄₃	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₄₃	C₂²×C₈	1	2	yes	—
55	C₂²×C₄₃	#131	1	2	yes	—
56	C₄×C₄₃	—	2	2	no	—
57	C₄×C₄₃	—	2	8	no	—
58	C₄×C₄₃	C₄×C₈	1	4	yes	complemented
59	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₄₃	#117	1	1	yes	characteristic
61	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
62	C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
63	C₈×C₄⋊C₄	—	43	1	no	maximal, Sylow
64	C₂×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×C₄₃	#405	1	1	yes	characteristic
67	C₂×C₄₃	C₈×D₄	1	2	yes	—
68	C₂×C₄₃	C₈×Q₈	1	2	yes	—
69	C₂×C₄⋊C₈	—	43	1	no	—
70	C₂×C₄×C₈	—	43	1	no	—
71	C₂×C₄×C₈	—	43	2	no	—
72	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
73	#336	—	43	2	no	—
74	C₄₃	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	43	1	no	—
76	C₂²×C₈	—	43	1	no	—
77	C₂²×C₈	—	43	2	no	—
78	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
79	C₂×C₄²	C₂×D₄₃	1	2	yes	—
80	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
81	C₄⋊C₈	—	43	4	no	—
82	C₄×C₈	—	43	4	no	—
83	C₄×C₈	—	86	4	no	—
84	#95	Dic₄₃	1	2	yes	—
85	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
86	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
89	C₂×C₈	—	43	2	no	—
90	C₂×C₈	—	43	2	no	—
91	C₂×C₈	—	43	2	no	—
92	C₂×C₈	—	43	2	no	—
93	C₂×C₈	—	43	4	no	—
94	C₂×C₈	—	86	2	no	—
95	C₂×C₈	—	86	1	no	—
96	C₂×C₈	—	86	1	no	—
97	C₂×C₈	—	86	2	no	—
98	C₄⋊C₄	C₄₃⋊C₈	1	8	yes	direct product factor
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄×D₄₃	1	2	yes	—
102	C₄²	C₄×D₄₃	1	2	yes	—
103	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₂×C₄₃⋊C₈	1	8	yes	—
108	C₂×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
109	C₂×C₄	C₄×Dic₄₃	1	2	yes	—
110	C₂×C₄	C₄×Dic₄₃	1	2	yes	—
111	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
113	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
114	C₂×C₄	#10621900	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
116	C₂×C₄	#10621902	1	1	yes	characteristic
117	C₈	—	43	4	no	—
118	C₈	—	86	2	no	—
119	C₈	—	86	4	no	—
120	C₂²	C₂×C₄×Dic₄₃	1	1	yes	characteristic
121	C₂²	D₄×Dic₄₃	1	1	yes	characteristic
122	C₂²	Q₈×Dic₄₃	1	1	yes	characteristic
123	C₂²	C₂²×C₄₃⋊C₈	1	2	yes	—
124	C₂²	#13063332	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	8	no	—
127	C₄	C₄×C₄₃⋊C₈	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
129	C₄	#13063283	1	1	yes	characteristic
130	C₄	#13063287	1	1	yes	characteristic
131	C₄	#13063290	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
133	C₂	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
134	C₂	#14582855	1	1	yes	characteristic
135	C₂	D₄×C₄₃⋊C₈	1	2	yes	—
136	C₂	Q₈×C₄₃⋊C₈	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

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

#14582654

C₂

yes

maximal

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₄

yes

complemented

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

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

C₈

yes

complemented

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

Q₈

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄²

yes

—

C₂×C₄×C₄₃

C₄²

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₄⋊C₄

yes

direct product factor

C₂²×C₄₃

C₂×C₄²

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

C₄×Q₈

yes

characteristic

C₂²×C₄₃

C₂²×C₈

yes

—

C₂²×C₄₃

#131

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×C₈

yes

complemented

C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

C₄×Q₈

yes

characteristic

C₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄₃

C₂×C₄×C₈

yes

characteristic

C₂×C₄₃

#405

yes

characteristic

C₂×C₄₃

C₈×D₄

yes

—

C₂×C₄₃

C₈×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic

#336

—

C₄₃

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₄₃⋊C₈

yes

direct product factor

C₄²

—

100

C₄²

—

101

C₄²

C₄×D₄₃

yes

—

102

C₄²

C₄×D₄₃

yes

—

103

C₂³

C₂²×Dic₄₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

108

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

109

C₂×C₄

C₄×Dic₄₃

yes

—

110

C₂×C₄

C₄×Dic₄₃

yes

—

111

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

112

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

113

C₂×C₄

D₄×D₄₃

yes

characteristic

114

C₂×C₄

#10621900

yes

characteristic

115

C₂×C₄

Q₈×D₄₃

yes

characteristic

116

C₂×C₄

#10621902

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

C₂×C₄×Dic₄₃

yes

characteristic

121

C₂²

D₄×Dic₄₃

yes

characteristic

122

C₂²

Q₈×Dic₄₃

yes

characteristic

123

C₂²

C₂²×C₄₃⋊C₈

yes

—

124

C₂²

#13063332

yes

—

125

C₄

—

126

C₄

—

127

C₄

C₄×C₄₃⋊C₈

yes

complemented

128

C₄

C₄⋊C₄×D₄₃

yes

characteristic

129

C₄

#13063283

yes

characteristic

130

C₄

#13063287

yes

characteristic

131

C₄

#13063290

yes

characteristic

132

C₂

C₄⋊C₄×Dic₄₃

yes

characteristic

133

C₂

C₂×C₄×C₄₃⋊C₈

yes

characteristic

134

C₂

#14582855

yes

characteristic

135

C₂

D₄×C₄₃⋊C₈

yes

—

136

C₂

Q₈×C₄₃⋊C₈

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator

C₂×C₄₃

C₂×C₄×C₈

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂²×C₄

hypercenter

C₄×C₄⋊C₄

D₄₃

hypocenter

C₄₃

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄⋊(C₄₃⋊C₈)	C₄	semidirect
11. C₄×C₄₃⋊C₈	C₄	semidirect
20. C₄⋊C₄×C₄₃	C₈	semidirect
50. C₄₃⋊C₈	C₄⋊C₄	direct (N × H)
58. C₄×C₄₃	C₄×C₈	semidirect
74. C₄₃	C₈×C₄⋊C₄	semidirect
98. C₄⋊C₄	C₄₃⋊C₈	direct (N × H)
127. C₄	C₄×C₄₃⋊C₈	semidirect

H ≅ G/N

type

9. C₄⋊(C₄₃⋊C₈)

C₄

semidirect

11. C₄×C₄₃⋊C₈

C₄

semidirect

20. C₄⋊C₄×C₄₃

C₈

semidirect

50. C₄₃⋊C₈

C₄⋊C₄

direct (N × H)

58. C₄×C₄₃

C₄×C₈

semidirect

74. C₄₃

C₈×C₄⋊C₄

semidirect

98. C₄⋊C₄

C₄₃⋊C₈

direct (N × H)

127. C₄

C₄×C₄₃⋊C₈

semidirect

Subgroups — C₄⋊C₄×C₄₃⋊C₈ #19134459

Conjugacy classes of subgroups

Complements

Subgroups — C4⋊C4×C43⋊C8 #19134459

Conjugacy classes of subgroups

Complements

Subgroups — C₄⋊C₄×C₄₃⋊C₈ #19134459