Group #18069197 — subgroups

Subgroups — #18069197

Order 4864.

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

normal subgroups only

This group has 140 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	#14561699	C₂	1	1	yes	maximal, characteristic
2	#14561553	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
5	C₄×C₁₉⋊C₁₆	—	2	1	no	—
6	C₈×Q₈×C₁₉	—	2	1	no	—
7	C₄×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
8	C₁₉×#335	C₄	1	1	yes	characteristic
9	C₁₉×Q₈⋊C₈	—	2	1	no	—
10	C₈²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₉	C₈	1	1	yes	characteristic
12	C₄×Q₈×C₁₉	—	2	1	no	—
13	C₁₉×#107	C₈	1	1	yes	characteristic
14	C₁₉×C₄⋊C₈	—	2	1	no	—
15	C₁₉×C₄⋊C₈	—	4	1	no	—
16	C₁₉×#103	—	2	1	no	—
17	C₄×C₈×C₁₉	—	4	1	no	—
18	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₁₉⋊C₁₆	—	2	2	no	—
22	C₂×C₁₉⋊C₁₆	—	4	1	no	—
23	C₂×Q₈×C₁₉	—	2	1	no	—
24	Q₁₆×C₁₉	—	2	1	no	—
25	Q₁₆×C₁₉	C₁₆	1	2	yes	complemented
26	C₂×C₈×C₁₉	—	2	1	no	—
27	C₂×C₈×C₁₉	—	4	1	no	—
28	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
30	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
31	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
32	C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₄⋊C₄×C₁₉	—	4	1	no	—
34	C₄²×C₁₉	—	4	1	no	—
35	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₉⋊C₁₆	—	4	2	no	—
37	C₁₉⋊C₁₆	—	8	1	no	—
38	#7071	—	19	1	no	maximal, Sylow
39	Q₈×C₁₉	—	2	2	no	—
40	Q₈×C₁₉	—	4	1	no	—
41	C₂×C₄×C₁₉	—	4	1	no	—
42	C₂×C₄×C₁₉	—	4	1	no	—
43	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
45	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
46	C₈×C₁₉	—	2	2	no	—
47	C₈×C₁₉	—	2	1	no	—
48	C₈×C₁₉	—	2	1	no	—
49	C₈×C₁₉	—	2	1	no	—
50	C₈×C₁₉	—	8	1	no	—
51	C₈×C₁₉	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₁₉	M₃₂	1	1	yes	characteristic
53	C₈×Q₁₆	D₁₉	1	1	yes	characteristic
54	#1367	—	19	1	no	—
55	#1312	—	19	1	no	—
56	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
57	C₄×C₁₉	—	2	1	no	—
58	C₄×C₁₉	—	4	2	no	—
59	C₄×C₁₉	—	8	1	no	—
60	C₄×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
61	C₄×C₁₉	#350	1	1	yes	characteristic
62	C₄×C₁₉	#360	1	1	yes	characteristic
63	C₄×C₁₉	#362	1	1	yes	characteristic
64	C₈×Q₈	—	2	1	no	—
65	C₄×Q₁₆	Dic₁₉	1	1	yes	characteristic
66	C₄⋊C₁₆	—	38	1	no	—
67	C₄×C₁₆	—	38	1	no	—
68	#335	Dic₁₉	1	1	yes	characteristic
69	Q₈⋊C₈	—	2	1	no	—
70	C₈²	C₂×D₁₉	1	1	yes	characteristic
71	C₂×C₁₉	D₄⋊C₁₆	1	1	yes	characteristic
72	C₂×C₁₉	#1334	1	1	yes	characteristic
73	C₂×C₁₉	#1336	1	1	yes	characteristic
74	C₂×Q₁₆	C₁₉⋊C₈	1	1	yes	characteristic
75	C₄×Q₈	—	2	1	no	—
76	C₂×C₁₆	—	38	2	no	—
77	C₂×C₁₆	—	76	1	no	—
78	#107	C₁₉⋊C₈	1	1	yes	characteristic
79	C₄⋊C₈	—	2	1	no	—
80	C₄⋊C₈	—	4	1	no	—
81	#103	—	2	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
84	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
85	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
86	C₁₉	#7071	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	C₁₉⋊C₁₆	1	2	yes	complemented
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
93	C₂×C₈	#64520	1	1	yes	characteristic
94	C₂×C₈	#64525	1	1	yes	characteristic
95	C₂×C₈	#64526	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
100	C₁₆	—	76	2	no	—
101	C₁₆	—	152	1	no	—
102	Q₈	—	2	2	no	—
103	Q₈	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
107	C₂×C₄	#10597289	1	1	yes	characteristic
108	C₂×C₄	#10597294	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	8	1	no	—
114	C₈	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
115	C₈	#10597269	1	1	yes	characteristic
116	C₂²	D₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	4	2	no	—
119	C₄	—	8	1	no	—
120	C₄	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
121	C₄	#11926328	1	1	yes	characteristic
122	C₄	#11926354	1	1	yes	characteristic
123	C₄	#11926360	1	1	yes	characteristic
124	C₂	D₄⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
125	C₂	#14561605	1	1	yes	characteristic
126	C₂	#14561611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561699

C₂

yes

maximal, characteristic

#14561553

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₁₉

C₂

yes

maximal, characteristic

C₄⋊(C₁₉⋊C₁₆)

—

C₄×C₁₉⋊C₁₆

—

C₈×Q₈×C₁₉

—

C₄×Q₁₆×C₁₉

C₄

yes

characteristic

C₁₉×#335

C₄

yes

characteristic

C₁₉×Q₈⋊C₈

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

C₈

yes

characteristic

C₄×Q₈×C₁₉

—

C₁₉×#107

C₈

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×Q₈×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

C₁₆

yes

complemented

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

#7071

—

maximal, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×C₁₆

yes

characteristic

C₈×C₁₉

M₃₂

yes

characteristic

C₈×Q₁₆

D₁₉

yes

characteristic

#1367

—

#1312

—

C₂²×C₁₉

D₄⋊C₈

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂²⋊C₁₆

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₈×Q₈

—

C₄×Q₁₆

Dic₁₉

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

Dic₁₉

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂×D₁₉

yes

characteristic

C₂×C₁₉

D₄⋊C₁₆

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

C₁₉⋊C₈

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

C₁₉⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×Dic₁₉

yes

characteristic

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

C₁₉

#7071

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₁₉⋊C₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₁₉⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²⋊Dic₁₉

yes

characteristic

100

C₁₆

—

101

C₁₆

—

152

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

107

C₂×C₄

#10597289

yes

characteristic

108

C₂×C₄

#10597294

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

C₂×C₁₉⋊C₁₆

yes

characteristic

115

C₈

#10597269

yes

characteristic

116

C₂²

D₄⋊(C₁₉⋊C₈)

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

121

C₄

#11926328

yes

characteristic

122

C₄

#11926354

yes

characteristic

123

C₄

#11926360

yes

characteristic

124

C₂

D₄⋊(C₁₉⋊C₁₆)

yes

characteristic

125

C₂

#14561605

yes

characteristic

126

C₂

#14561611

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597289
commutator	C₈×C₁₉	C₂×C₁₆
Frattini	C₈²	C₂×D₁₉
Fitting	C₈×Q₁₆×C₁₉	C₂
socle, prime radical	C₂²×C₁₉	D₄⋊C₈
cosocle	C₈²×C₁₉	C₂²
hypercenter	C₈×Q₁₆	D₁₉
hypocenter	C₁₉	#7071
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₄

#10597289

commutator

C₈×C₁₉

C₂×C₁₆

Frattini

C₈²

C₂×D₁₉

Fitting

C₈×Q₁₆×C₁₉

C₂

socle, prime radical

C₂²×C₁₉

D₄⋊C₈

cosocle

C₈²×C₁₉

C₂²

hypercenter

C₈×Q₁₆

D₁₉

hypocenter

C₁₉

#7071

LCS

G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×Q₁₆

N	H ≅ G/N	type
25. Q₁₆×C₁₉	C₁₆	semidirect
86. C₁₉	#7071	semidirect
89. Q₁₆	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

25. Q₁₆×C₁₉

C₁₆

semidirect

86. C₁₉

#7071

semidirect

89. Q₁₆

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561699	2
#14561553	2
C₈×Q₁₆×C₁₉	2
#7071	19

subgroup

index

#14561699

#14561553

C₈×Q₁₆×C₁₉

#7071

Subgroups — #18069197

Conjugacy classes of subgroups

Complements

Maximal subgroups