Group #19144001 — subgroups

Subgroups — #19144001

Order 5504.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#14583470	C₂	1	3	yes	maximal, complemented
2	#14583089	C₂	1	8	yes	maximal, complemented
3	#14582873	C₂	1	3	yes	maximal, complemented
4	C₄₃×#585	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic, complemented
6	#13063323	—	2	3	no	—
7	#13063312	—	2	3	no	—
8	#13063306	C₂²	1	12	yes	complemented
9	#13063304	—	2	6	no	—
10	#13063284	C₂²	1	3	yes	complemented
11	#13063240	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₄₃	—	2	3	no	—
13	H(5,𝔽₂)×C₄₃	—	2	3	no	—
14	H(5,𝔽₂)×C₄₃	C₂²	1	4	yes	—
15	C₂×#41×C₄₃	—	2	6	no	—
16	C₂×#41×C₄₃	C₂²	1	3	yes	—
17	#41×C₄₃	—	2	4	no	—
18	#41×C₄₃	—	2	12	no	—
19	#41×C₄₃	—	2	6	no	—
20	#41×C₄₃	—	2	12	no	—
21	#41×C₄₃	—	2	6	no	—
22	C₂×Q₈×C₄₃	—	2	6	no	—
23	C₂×Q₈×C₄₃	D₄	1	3	yes	—
24	C₂×D₄×C₄₃	—	2	6	no	—
25	C₂×D₄×C₄₃	—	2	12	no	—
26	C₂×D₄×C₄₃	D₄	1	3	yes	—
27	C₂×D₄×C₄₃	C₂³	1	6	yes	—
28	C₂²×C₄×C₄₃	—	2	4	no	—
29	C₂²×C₄×C₄₃	D₄	1	6	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₃	C₂³	1	2	yes	—
32	C₂×Dic₈₆	—	2	3	no	—
33	C₂×Dic₈₆	—	2	3	no	—
34	C₂²⋊Dic₄₃	—	2	12	no	—
35	C₂²⋊Dic₄₃	C₂³	1	6	yes	—
36	C₄⋊Dic₄₃	—	2	3	no	—
37	Dic₄₃⋊C₄	—	2	12	no	—
38	C₄×Dic₄₃	—	2	3	no	—
39	C₂³×C₄₃	—	2	4	no	—
40	C₂³×C₄₃	C₂×D₄	1	6	yes	—
41	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
42	Q₈×C₄₃	—	2	4	no	—
43	Q₈×C₄₃	—	2	6	no	—
44	D₄×C₄₃	—	2	12	no	—
45	D₄×C₄₃	—	2	12	no	—
46	D₄×C₄₃	—	2	6	no	—
47	C₂×C₄×C₄₃	—	2	12	no	—
48	C₂×C₄×C₄₃	—	2	6	no	—
49	C₂×C₄×C₄₃	—	2	6	no	—
50	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
53	C₂×Dic₄₃	—	2	6	no	—
54	C₂×Dic₄₃	—	2	6	no	—
55	Dic₈₆	—	4	1	no	—
56	Dic₈₆	—	4	3	no	—
57	C₂²×C₄₃	—	2	6	no	—
58	C₂²×C₄₃	—	4	1	no	—
59	C₂²×C₄₃	—	4	6	no	—
60	C₂²×C₄₃	—	8	4	no	—
61	C₂²×C₄₃	C₂²×D₄	1	3	yes	—
62	C₄×C₄₃	—	2	6	no	—
63	C₄×C₄₃	#120	1	1	yes	characteristic
64	C₄×C₄₃	#120	1	3	yes	—
65	Dic₄₃	—	4	6	no	—
66	Dic₄₃	—	4	2	no	—
67	#3028	—	43	1	no	maximal, Sylow
68	C₂×C₄₃	—	2	3	no	—
69	C₂×C₄₃	—	4	6	no	—
70	C₂×C₄₃	C₂×#120	1	1	yes	characteristic
71	#585	D₄₃	1	1	yes	characteristic
72	#536	—	43	3	no	—
73	#458	—	43	8	no	—
74	#410	—	43	3	no	—
75	C₄₃	#3028	1	1	yes	characteristic, complemented, Sylow
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₄₃	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₄₃	1	3	yes	—
81	C₂²×Q₈	—	43	1	no	—
82	C₄⋊Q₈	—	86	3	no	—
83	#124	—	86	3	no	—
84	#123	—	43	12	no	—
85	C₂²⋊Q₈	—	86	6	no	—
86	#117	—	43	3	no	—
87	#99	—	43	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	86	3	no	—
95	C₂×Q₈	—	86	3	no	—
96	C₂×Q₈	C₂²⋊D₄₃	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₄₃	1	3	yes	—
100	C₂×D₄	C₂²×D₄₃	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	43	2	no	—
103	C₂²×C₄	C₂²⋊D₄₃	1	6	yes	—
104	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
105	C₄⋊C₄	—	86	3	no	—
106	C₄⋊C₄	—	86	12	no	—
107	C₂²⋊C₄	—	43	6	no	—
108	C₂²⋊C₄	—	86	12	no	—
109	C₄²	—	86	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₄₃	1	6	yes	—
112	C₂³	C₂³×D₄₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	172	1	no	—
116	Q₈	—	172	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	86	6	no	—
124	C₂×C₄	—	86	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₄₃	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	172	6	no	—
135	C₄	—	172	2	no	—
136	C₄	#13063294	1	1	yes	characteristic
137	C₄	#13063294	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#13063294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583470

C₂

yes

maximal, complemented

#14583089

C₂

yes

maximal, complemented

#14582873

C₂

yes

maximal, complemented

C₄₃×#585

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic, complemented

#13063323

—

#13063312

—

#13063306

C₂²

yes

complemented

#13063304

—

#13063284

C₂²

yes

complemented

#13063240

C₂²

yes

complemented

2₋¹⁺⁴×C₄₃

—

H(5,𝔽₂)×C₄₃

—

H(5,𝔽₂)×C₄₃

C₂²

yes

—

C₂×#41×C₄₃

—

C₂×#41×C₄₃

C₂²

yes

—

#41×C₄₃

—

#41×C₄₃

—

#41×C₄₃

—

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

D₄

yes

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#120

yes

—

Dic₄₃

—

Dic₄₃

—

#3028

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂×#120

yes

characteristic

#585

D₄₃

yes

—

—

—

C₄₃

#3028

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₄₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₄₃

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₄₃

yes

—

100

C₂×D₄

C₂²×D₄₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₄₃

yes

—

104

C₂²×C₄

C₂²×D₄₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₄₃

yes

—

112

C₂³

C₂³×D₄₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

172

—

116

Q₈

—

172

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

127

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₄₃

yes

—

133

C₄

—

134

C₄

—

172

—

135

C₄

—

172

—

136

C₄

#13063294

yes

characteristic

137

C₄

#13063294

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#13063294

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13063294
commutator	C₂³×C₄₃	C₂⁴
Frattini	C₂³	C₂³×D₄₃
Fitting	C₄₃×#585	C₂
socle	C₂×C₄₃	C₂×#120
hypercenter	#585	D₄₃
hypocenter	C₄₃	#3028
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#13063294

commutator

C₂³×C₄₃

C₂⁴

Frattini

C₂³

C₂³×D₄₃

Fitting

C₄₃×#585

C₂

C₂×C₄₃

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #14583470	C₂	semidirect
2. #14583089	C₂	semidirect
3. #14582873	C₂	semidirect
5. C₂²×Dic₈₆	C₂²	semidirect
8. #13063306	C₂²	semidirect
10. #13063284	C₂²	semidirect
11. #13063240	C₂²	semidirect
75. C₄₃	#3028	semidirect

H ≅ G/N

type

1. #14583470

C₂

semidirect

2. #14583089

C₂

semidirect

3. #14582873

C₂

semidirect

5. C₂²×Dic₈₆

C₂²

semidirect

8. #13063306

C₂²

semidirect

10. #13063284

C₂²

semidirect

11. #13063240

C₂²

semidirect

75. C₄₃

#3028

semidirect

Subgroups — #19144001

Conjugacy classes of subgroups

Complements