Group #19137775 — subgroups

Subgroups — #19137775

Order 5504.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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	#14582803	C₂	1	4	yes	maximal
3	#14582784	C₂	1	4	yes	maximal
4	#14582782	C₂	1	4	yes	maximal
5	#14582767	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#383	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	#13063325	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
13	C₄₃⋊#95	C₂²	1	2	yes	—
14	C₄₃⋊#95	C₂²	1	8	yes	—
15	C₄₃⋊#95	C₂²	1	2	yes	—
16	C₄₃⋊#95	C₂²	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	#95×C₄₃	C₂²	1	2	yes	—
19	#95×C₄₃	C₂²	1	2	yes	—
20	#95×C₄₃	C₂²	1	2	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	C₂³	1	4	yes	—
26	C₂²×Dic₄₃	C₂³	1	4	yes	—
27	C₂×Dic₈₆	—	4	2	no	—
28	C₂×Dic₈₆	—	4	2	no	—
29	C₂×Dic₈₆	C₂×C₄	1	8	yes	—
30	C₄²×C₄₃	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
32	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
33	Dic₄₃⋊C₄	—	4	4	no	—
34	C₄×Dic₄₃	—	4	4	no	—
35	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₄₃	—	2	2	no	—
37	C₂×C₄×C₄₃	—	2	2	no	—
38	C₂×C₄×C₄₃	—	2	2	no	—
39	C₂×C₄×C₄₃	—	2	4	no	—
40	C₂×C₄×C₄₃	—	2	4	no	—
41	C₂×C₄×C₄₃	—	2	4	no	—
42	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
45	C₂×Dic₄₃	—	2	4	no	—
46	C₂×Dic₄₃	—	2	4	no	—
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	—	2	4	no	—
49	C₂×Dic₄₃	—	2	4	no	—
50	C₂×Dic₄₃	C₂²×C₄	1	8	yes	—
51	Dic₈₆	—	4	8	no	—
52	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₄₃	—	2	2	no	—
57	C₄×C₄₃	—	2	2	no	—
58	C₄×C₄₃	—	2	2	no	—
59	C₄×C₄₃	—	4	4	no	—
60	Dic₄₃	—	2	8	no	—
61	Dic₄₃	—	4	4	no	—
62	#2381	—	43	1	no	maximal, Sylow
63	C₂×C₄₃	#518	1	1	yes	characteristic
64	C₂×C₄₃	#519	1	1	yes	characteristic
65	C₂×C₄₃	#519	1	1	yes	characteristic
66	C₂×C₄₃	#563	1	4	yes	—
67	C₂×C₄⋊Q₈	—	43	1	no	—
68	#391	—	43	4	no	—
69	#387	—	43	4	no	—
70	#387	—	43	4	no	—
71	#383	—	43	1	no	—
72	#383	D₄₃	1	1	yes	characteristic
73	C₄₃	#2381	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	43	2	no	—
75	C₄⋊Q₈	—	43	8	no	—
76	C₂×C₄⋊C₄	—	43	2	no	—
77	C₂×C₄⋊C₄	—	43	2	no	—
78	C₂×C₄⋊C₄	—	43	4	no	—
79	C₂×C₄²	—	43	4	no	—
80	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
81	#95	—	43	2	no	—
82	#95	—	43	2	no	—
83	#95	—	43	2	no	—
84	#95	—	43	8	no	—
85	#95	C₂×D₄₃	1	2	yes	—
86	#95	C₂×D₄₃	1	2	yes	—
87	#95	C₂×D₄₃	1	2	yes	—
88	C₂×Q₈	—	43	8	no	—
89	C₂×Q₈	—	172	2	no	—
90	C₂×Q₈	—	172	2	no	—
91	C₂²×C₄	—	43	4	no	—
92	C₂²×C₄	—	43	4	no	—
93	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
94	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
97	C₄⋊C₄	—	43	8	no	—
98	C₄⋊C₄	—	43	8	no	—
99	C₄⋊C₄	—	172	4	no	—
100	C₄²	—	172	4	no	—
101	C₄²	C₄×D₄₃	1	4	yes	—
102	C₂³	C₂³×D₄₃	1	1	yes	characteristic
103	Q₈	—	172	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	43	8	no	—
111	C₂×C₄	—	86	4	no	—
112	C₂×C₄	—	86	4	no	—
113	C₂×C₄	—	86	4	no	—
114	C₂×C₄	—	86	4	no	—
115	C₂×C₄	—	86	4	no	—
116	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
119	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
120	C₂²	#13063371	1	2	yes	—
121	C₂²	#13063372	1	2	yes	—
122	C₂²	#13063372	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	86	8	no	—
128	C₄	—	172	4	no	—
129	C₂	#14583373	1	1	yes	characteristic
130	C₂	#14583376	1	1	yes	characteristic
131	C₂	#14583376	1	1	yes	characteristic
132	C₂	#14583651	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

#14582803

C₂

yes

maximal

#14582784

C₂

yes

maximal

#14582782

C₂

yes

maximal

#14582767

C₂

yes

maximal, characteristic

C₄₃×#383

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

#13063325

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

#95×C₄₃

C₂²

yes

—

#95×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

C₂²×C₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₄₃

yes

characteristic

C₄₃

#2381

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

—

—

—

—

C₂×D₄₃

yes

—

#95

C₂×D₄₃

yes

—

#95

C₂×D₄₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×Q₈

—

172

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₂²×C₄

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

172

—

100

C₄²

—

172

—

101

C₄²

C₄×D₄₃

yes

—

102

C₂³

C₂³×D₄₃

yes

characteristic

103

Q₈

—

172

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×D₄₃

yes

—

117

C₂×C₄

C₂×C₄×D₄₃

yes

—

118

C₂×C₄

C₂×C₄×D₄₃

yes

—

119

C₂²

C₂²×C₄×D₄₃

yes

characteristic

120

C₂²

#13063371

yes

—

121

C₂²

#13063372

yes

—

122

C₂²

#13063372

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

172

—

129

C₂

#14583373

yes

characteristic

130

C₂

#14583376

yes

characteristic

131

C₂

#14583376

yes

characteristic

132

C₂

#14583651

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#383	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	#383	D₄₃
hypocenter	C₄₃	#2381
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₄₃×#383

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #13063325	C₄	semidirect
73. C₄₃	#2381	semidirect

H ≅ G/N

type

8. #13063325

C₄

semidirect

73. C₄₃

#2381

semidirect

Subgroups — #19137775

Conjugacy classes of subgroups

Complements