Group #18069938 — subgroups

Subgroups — #18069938

Order 4864.

138 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₂²⋊(C₁₉⋊C₃₂)	C₂	1	2	yes	maximal
2	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉⋊C₃₂	—	2	2	no	—
4	C₂²×C₁₆×C₁₉	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
7	C₁₉×C₄⋊C₁₆	—	2	2	no	—
8	C₂²×C₈×C₁₉	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₉	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₁₉	—	2	1	no	—
12	C₂×C₁₆×C₁₉	—	4	1	no	—
13	C₂×C₁₆×C₁₉	—	4	1	no	—
14	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
15	C₄×C₈×C₁₉	—	2	1	no	—
16	C₄×C₈×C₁₉	—	2	1	no	—
17	C₁₉⋊C₃₂	—	4	2	no	—
18	C₂²×C₄×C₁₉	C₁₆	1	1	yes	characteristic
19	C₂²×C₄×C₁₉	C₁₆	1	1	yes	characteristic
20	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
21	C₂×C₈×C₁₉	—	2	1	no	—
22	C₂×C₈×C₁₉	—	2	1	no	—
23	C₂×C₈×C₁₉	—	2	1	no	—
24	C₂×C₈×C₁₉	—	4	1	no	—
25	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
27	C₄²×C₁₉	—	2	1	no	—
28	C₄²×C₁₉	—	2	1	no	—
29	C₁₆×C₁₉	—	2	2	no	—
30	C₁₆×C₁₉	—	4	1	no	—
31	#7362	—	19	1	no	maximal, Sylow
32	C₂³×C₁₉	C₂×C₁₆	1	1	yes	characteristic
33	C₂×C₄×C₁₉	—	2	1	no	—
34	C₂×C₄×C₁₉	—	2	1	no	—
35	C₂×C₄×C₁₉	—	2	1	no	—
36	C₂×C₄×C₁₉	—	2	1	no	—
37	C₂×C₄×C₁₉	—	2	1	no	—
38	C₂×C₄×C₁₉	—	4	1	no	—
39	C₂×C₄×C₁₉	C₃₂	1	2	yes	complemented
40	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
42	C₈×C₁₉	—	2	1	no	—
43	C₈×C₁₉	—	4	1	no	—
44	C₈×C₁₉	#99	1	1	yes	characteristic
45	C₈×C₁₉	#101	1	1	yes	characteristic
46	C₂×C₄⋊C₁₆	D₁₉	1	1	yes	characteristic
47	C₂²⋊C₃₂	—	19	2	no	—
48	C₂²×C₁₉	—	2	1	no	—
49	C₂²×C₁₉	—	2	1	no	—
50	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
51	C₂²×C₁₉	C₂×C₃₂	1	1	yes	characteristic
52	C₂²×C₁₉	M₆₄	1	1	yes	characteristic
53	C₄×C₁₉	—	2	1	no	—
54	C₄×C₁₉	—	2	2	no	—
55	C₄×C₁₉	—	4	1	no	—
56	C₄×C₁₉	#324	1	1	yes	characteristic
57	C₄×C₁₉	#349	1	1	yes	characteristic
58	C₂²×C₁₆	Dic₁₉	1	1	yes	characteristic
59	C₂²×C₁₆	C₂×D₁₉	1	1	yes	characteristic
60	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
61	C₂×C₃₂	—	38	2	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₂×C₁₉	—	2	1	no	—
64	C₂×C₁₉	—	2	1	no	—
65	C₂×C₁₉	#1322	1	1	yes	characteristic
66	C₂×C₁₉	C₂²⋊C₃₂	1	1	yes	characteristic
67	C₂×C₁₉	#1400	1	1	yes	characteristic
68	C₂²×C₈	C₁₉⋊C₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
70	C₂×C₄²	C₁₉⋊C₈	1	1	yes	characteristic
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	4	1	no	—
73	C₂×C₁₆	—	4	1	no	—
74	C₂×C₁₆	C₂²⋊D₁₉	1	2	yes	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₃₂	—	76	2	no	—
78	C₁₉	#7362	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	C₁₉⋊C₁₆	1	1	yes	characteristic
80	C₂²×C₄	C₁₉⋊C₁₆	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	#64520	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₁₆	—	2	2	no	—
91	C₁₆	—	4	1	no	—
92	C₂³	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	C₁₉⋊C₃₂	1	2	yes	complemented
100	C₂×C₄	#10597269	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	4	1	no	—
104	C₈	#10597291	1	1	yes	characteristic
105	C₈	#10597293	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
109	C₂²	C₂×C₁₉⋊C₃₂	1	1	yes	characteristic
110	C₂²	#11926378	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	4	1	no	—
114	C₄	#11926270	1	1	yes	characteristic
115	C₄	#11926326	1	1	yes	characteristic
116	C₂	—	2	1	no	—
117	C₂	—	2	1	no	—
118	C₂	#14561575	1	1	yes	characteristic
119	C₂	C₂²⋊(C₁₉⋊C₃₂)	1	1	yes	characteristic
120	C₂	#14561775	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊(C₁₉⋊C₃₂)

C₂

yes

maximal

C₂×C₁₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₁₉⋊C₃₂

—

C₂²×C₁₆×C₁₉

C₄

yes

characteristic

C₂²×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₁₉×C₄⋊C₁₆

—

C₂²×C₈×C₁₉

C₈

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₈

yes

characteristic

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₁₉⋊C₃₂

—

C₂²×C₄×C₁₉

C₁₆

yes

characteristic

C₂²×C₄×C₁₉

C₁₆

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#7362

—

maximal, Sylow

C₂³×C₁₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₃₂

yes

complemented

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×C₄⋊C₁₆

D₁₉

yes

characteristic

C₂²⋊C₃₂

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₉

C₂×C₃₂

yes

characteristic

C₂²×C₁₉

M₆₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂²×C₁₆

Dic₁₉

yes

characteristic

C₂²×C₁₆

C₂×D₁₉

yes

characteristic

C₂×C₄×C₈

Dic₁₉

yes

characteristic

C₂×C₃₂

—

C₄⋊C₁₆

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1322

yes

characteristic

C₂×C₁₉

C₂²⋊C₃₂

yes

characteristic

C₂×C₁₉

#1400

yes

characteristic

C₂²×C₈

C₁₉⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

C₂×C₄²

C₁₉⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₁₉

#7362

yes

characteristic, complemented, Sylow

C₂²×C₄

C₁₉⋊C₁₆

yes

characteristic

C₂²×C₄

C₁₉⋊C₁₆

yes

characteristic

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#64520

yes

characteristic

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₂×C₁₉⋊C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₉⋊C₃₂

yes

complemented

100

C₂×C₄

#10597269

yes

characteristic

101

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

#10597291

yes

characteristic

105

C₈

#10597293

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

109

C₂²

C₂×C₁₉⋊C₃₂

yes

characteristic

110

C₂²

#11926378

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#11926270

yes

characteristic

115

C₄

#11926326

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

#14561575

yes

characteristic

119

C₂

C₂²⋊(C₁₉⋊C₃₂)

yes

characteristic

120

C₂

#14561775

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₁₉
commutator	C₂²×C₁₉	C₂×C₃₂
Frattini	C₂²×C₁₆	C₂×D₁₉
Fitting	C₂×C₁₉×C₄⋊C₁₆	C₂
socle	C₂²×C₁₉	C₂²⋊C₁₆
prime radical	C₂³×C₁₉	C₂×C₁₆
cosocle	C₂²×C₁₆×C₁₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₁₉
hypocenter	C₁₉	#7362
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₁₉

commutator

C₂²×C₁₉

C₂×C₃₂

Frattini

C₂²×C₁₆

C₂×D₁₉

Fitting

C₂×C₁₉×C₄⋊C₁₆

C₂

socle

C₂²×C₁₉

C₂²⋊C₁₆

prime radical

C₂³×C₁₉

C₂×C₁₆

cosocle

C₂²×C₁₆×C₁₉

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₁₉

hypocenter

C₁₉

#7362

LCS

G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
39. C₂×C₄×C₁₉	C₃₂	semidirect
78. C₁₉	#7362	semidirect
99. C₂×C₄	C₁₉⋊C₃₂	semidirect

H ≅ G/N

type

39. C₂×C₄×C₁₉

C₃₂

semidirect

78. C₁₉

#7362

semidirect

99. C₂×C₄

C₁₉⋊C₃₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²⋊(C₁₉⋊C₃₂)	2
C₂×C₁₉×C₄⋊C₁₆	2
#7362	19

subgroup

index

C₂²⋊(C₁₉⋊C₃₂)

C₂×C₁₉×C₄⋊C₁₆

#7362

Subgroups — #18069938

Conjugacy classes of subgroups

Complements

Maximal subgroups