Group #19135543 — subgroups

Subgroups — #19135543

Order 5504.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13063325	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	—	2	1	no	—
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	#13063325	—	2	2	no	—
8	#13063325	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	—	2	1	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₄₃	—	2	1	no	—
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	#95×C₄₃	—	2	1	no	—
16	C₂²×C₄×C₄₃	—	2	2	no	—
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	2	2	no	—
21	C₂×Dic₈₆	—	2	4	no	—
22	C₂×Dic₈₆	—	4	4	no	—
23	C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₄⋊C₄×C₄₃	Q₈	1	4	yes	complemented
25	C₄²×C₄₃	—	4	2	no	—
26	C₄²×C₄₃	C₂×C₄	1	2	yes	—
27	C₄²×C₄₃	D₄	1	2	yes	—
28	C₄⋊Dic₄₃	—	2	4	no	—
29	C₄⋊Dic₄₃	—	2	2	no	—
30	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
31	C₂×C₄₃⋊C₈	—	2	2	no	—
32	C₂×C₄₃⋊C₈	—	2	4	no	—
33	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₄₃	—	2	2	no	—
35	C₂×C₄×C₄₃	—	2	4	no	—
36	C₂×C₄×C₄₃	—	4	2	no	—
37	C₂×C₄×C₄₃	—	4	2	no	—
38	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
41	C₂×C₄×C₄₃	Q₁₆	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	2	yes	—
45	C₂×C₄×C₄₃	#41	1	2	yes	—
46	C₂×Dic₄₃	—	2	4	no	—
47	C₂×Dic₄₃	—	4	4	no	—
48	Dic₈₆	—	4	8	no	—
49	C₄₃⋊C₈	—	4	2	no	—
50	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₄₃	—	2	2	no	—
56	C₄×C₄₃	—	4	4	no	—
57	C₄×C₄₃	#103	1	4	yes	—
58	C₄×C₄₃	C₄×Q₈	1	2	yes	—
59	C₄×C₄₃	#124	1	1	yes	characteristic
60	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₄₃	—	4	4	no	—
62	#1987	—	43	1	no	maximal, Sylow
63	C₂×C₄₃	#391	1	1	yes	characteristic
64	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
65	C₂×C₄₃	#418	1	1	yes	characteristic
66	C₂×C₄₃	#476	1	2	yes	—
67	C₂×C₄₃	#477	1	2	yes	—
68	C₂×C₄⋊Q₈	—	43	1	no	—
69	C₂×C₄⋊C₈	—	43	1	no	—
70	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
71	#340	—	43	4	no	—
72	C₄₃	#1987	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	86	1	no	—
74	C₂²×C₈	—	43	2	no	—
75	C₄⋊Q₈	—	43	4	no	—
76	C₄⋊Q₈	—	86	2	no	—
77	C₂×C₄⋊C₄	—	43	2	no	—
78	C₂×C₄⋊C₄	—	86	1	no	—
79	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
82	C₄⋊C₈	—	86	2	no	—
83	#95	—	2	1	no	—
84	C₂×Q₈	—	86	4	no	—
85	C₂×Q₈	—	172	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	86	2	no	—
88	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
91	C₂×C₈	—	86	2	no	—
92	C₂×C₈	—	86	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	43	4	no	—
95	C₄⋊C₄	—	86	4	no	—
96	C₄⋊C₄	—	86	2	no	—
97	C₄⋊C₄	Dic₈₆	1	4	yes	complemented
98	C₄²	—	4	2	no	—
99	C₄²	C₄×D₄₃	1	2	yes	—
100	C₄²	D₁₇₂	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
102	Q₈	—	172	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	86	4	no	—
108	C₂×C₄	—	172	4	no	—
109	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
110	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
111	C₂×C₄	#10621883	1	2	yes	—
112	C₂×C₄	#10621885	1	2	yes	—
113	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
115	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
116	C₂×C₄	#10621898	1	2	yes	—
117	C₈	—	172	2	no	—
118	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
119	C₂²	#13063304	1	2	yes	—
120	C₂²	C₂×#10621883	1	1	yes	characteristic
121	C₂²	C₂×#10621885	1	1	yes	characteristic
122	C₂²	#13063350	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	172	4	no	—
126	C₄	#13063248	1	4	yes	—
127	C₄	C₄×Dic₈₆	1	2	yes	—
128	C₄	#13063313	1	1	yes	characteristic
129	C₄	#13063325	1	1	yes	characteristic
130	C₂	#14582805	1	1	yes	characteristic
131	C₂	C₂×#13063248	1	1	yes	characteristic
132	C₂	#14582903	1	1	yes	characteristic
133	C₂	#14583189	1	2	yes	—
134	C₂	#14583194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

#13063325

—

#13063325

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

Q₈

yes

complemented

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

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

SD₁₆

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

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

—

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

C₄×Q₈

yes

—

C₄×C₄₃

#124

yes

characteristic

C₄×C₄₃

C₄⋊Q₈

yes

characteristic

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#477

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic

#340

—

C₄₃

#1987

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₈₆

yes

complemented

C₄²

—

C₄²

C₄×D₄₃

yes

—

100

C₄²

D₁₇₂

yes

—

101

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

102

Q₈

—

172

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

172

—

109

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

110

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

111

C₂×C₄

#10621883

yes

—

112

C₂×C₄

#10621885

yes

—

113

C₂×C₄

C₂×Dic₈₆

yes

—

114

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

115

C₂×C₄

C₂×D₁₇₂

yes

characteristic

116

C₂×C₄

#10621898

yes

—

117

C₈

—

172

—

118

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

119

C₂²

#13063304

yes

—

120

C₂²

C₂×#10621883

yes

characteristic

121

C₂²

C₂×#10621885

yes

characteristic

122

C₂²

#13063350

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

172

—

126

C₄

#13063248

yes

—

127

C₄

C₄×Dic₈₆

yes

—

128

C₄

#13063313

yes

characteristic

129

C₄

#13063325

yes

characteristic

130

C₂

#14582805

yes

characteristic

131

C₂

C₂×#13063248

yes

characteristic

132

C₂

#14582903

yes

characteristic

133

C₂

#14583189

yes

—

134

C₂

#14583194

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1987
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₄₃

hypocenter

C₄₃

#1987

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #13063325	C₄	semidirect
24. C₄⋊C₄×C₄₃	Q₈	semidirect
72. C₄₃	#1987	semidirect
97. C₄⋊C₄	Dic₈₆	semidirect

H ≅ G/N

type

8. #13063325

C₄

semidirect

24. C₄⋊C₄×C₄₃

Q₈

semidirect

72. C₄₃

#1987

semidirect

97. C₄⋊C₄

Dic₈₆

semidirect

Subgroups — #19135543

Conjugacy classes of subgroups

Complements