Group #18070178 — subgroups

Subgroups — #18070178

Order 4864.

136 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 51 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	C₄⋊(C₁₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#14561709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2249	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉⋊C₃₂	—	2	1	no	—
5	#11926297	—	2	1	no	—
6	C₂×C₁₉×Q₃₂	C₄	1	1	yes	characteristic
7	C₂×C₁₉×SD₃₂	—	2	1	no	—
8	C₂×C₁₉×D₁₆	C₄	1	1	yes	characteristic
9	C₁₉×#495	—	2	1	no	—
10	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₉	—	2	1	no	—
12	C₂×SD₁₆×C₁₉	—	4	1	no	—
13	C₂×D₈×C₁₉	—	2	1	no	—
14	C₁₉×#124	—	4	1	no	—
15	C₁₉×Q₃₂	—	4	1	no	—
16	C₁₉×SD₃₂	—	4	1	no	—
17	C₁₉×SD₃₂	—	4	1	no	—
18	C₁₉×D₁₆	—	4	1	no	—
19	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
23	C₁₉⋊C₃₂	—	4	1	no	—
24	C₂×Q₈×C₁₉	—	4	1	no	—
25	C₂×D₄×C₁₉	—	4	1	no	—
26	Q₁₆×C₁₉	—	4	1	no	—
27	Q₁₆×C₁₉	—	4	1	no	—
28	SD₁₆×C₁₉	—	8	2	no	—
29	D₈×C₁₉	—	4	1	no	—
30	D₈×C₁₉	—	4	1	no	—
31	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₂²⋊C₄×C₁₉	—	8	1	no	—
34	C₂²⋊C₄×C₁₉	—	8	1	no	—
35	C₄²×C₁₉	D₈	1	1	yes	characteristic
36	C₂×C₁₉⋊C₈	—	8	1	no	—
37	C₁₆×C₁₉	—	2	1	no	—
38	C₁₆×C₁₉	D₈	1	1	yes	characteristic
39	C₁₆×C₁₉	SD₁₆	1	1	yes	characteristic
40	#7446	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	—	8	1	no	—
42	Q₈×C₁₉	—	8	1	no	—
43	Q₈×C₁₉	—	8	1	no	—
44	D₄×C₁₉	—	8	1	no	—
45	D₄×C₁₉	—	8	1	no	—
46	C₂×C₄×C₁₉	—	8	1	no	—
47	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
48	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
50	C₈×C₁₉	—	2	1	no	—
51	C₈×C₁₉	#100	1	1	yes	characteristic
52	C₈×C₁₉	#102	1	1	yes	characteristic
53	C₁₉⋊C₈	—	16	1	no	—
54	#2249	D₁₉	1	1	yes	characteristic
55	C₄⋊C₃₂	—	19	1	no	—
56	#1372	—	19	1	no	—
57	C₂²×C₁₉	—	16	1	no	—
58	C₂²×C₁₉	—	16	1	no	—
59	C₂²×C₁₉	—	16	1	no	—
60	C₂²×C₁₉	#357	1	1	yes	characteristic
61	C₄×C₁₉	—	2	1	no	—
62	C₄×C₁₉	—	2	1	no	—
63	C₄×C₁₉	—	16	1	no	—
64	C₄×C₁₉	#331	1	1	yes	characteristic
65	C₄×C₁₉	#361	1	1	yes	characteristic
66	C₂×Q₃₂	Dic₁₉	1	1	yes	characteristic
67	C₂×SD₃₂	—	2	1	no	—
68	C₂×D₁₆	Dic₁₉	1	1	yes	characteristic
69	#495	—	2	1	no	—
70	C₂×C₃₂	—	38	1	no	—
71	C₄×C₁₆	C₂×D₁₉	1	1	yes	characteristic
72	#335	—	38	1	no	—
73	C₂×C₁₉	—	32	1	no	—
74	C₂×C₁₉	#1361	1	1	yes	characteristic
75	C₂×C₁₉	#1417	1	1	yes	characteristic
76	C₂×C₁₉	#1418	1	1	yes	characteristic
77	C₂×Q₁₆	—	2	1	no	—
78	C₂×SD₁₆	—	4	1	no	—
79	C₂×D₈	—	2	1	no	—
80	#124	—	4	1	no	—
81	Q₃₂	—	4	1	no	—
82	SD₃₂	—	4	1	no	—
83	SD₃₂	—	4	1	no	—
84	D₁₆	—	4	1	no	—
85	C₂×C₁₆	C₂×Dic₁₉	1	1	yes	characteristic
86	C₂×C₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
87	C₄⋊C₈	—	76	1	no	—
88	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
89	C₃₂	—	76	1	no	—
90	C₁₉	#7446	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	4	1	no	—
92	C₂×D₄	—	4	1	no	—
93	Q₁₆	—	4	1	no	—
94	Q₁₆	—	4	1	no	—
95	SD₁₆	—	8	2	no	—
96	D₈	—	4	1	no	—
97	D₈	—	4	1	no	—
98	C₂×C₈	—	152	1	no	—
99	C₂×C₈	#64526	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
101	C₂²⋊C₄	—	8	1	no	—
102	C₂²⋊C₄	—	8	1	no	—
103	C₄²	#64525	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	#64525	1	1	yes	characteristic
106	C₁₆	#64526	1	1	yes	characteristic
107	C₂³	—	8	1	no	—
108	Q₈	—	8	1	no	—
109	Q₈	—	8	1	no	—
110	D₄	—	8	1	no	—
111	D₄	—	8	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	#10597283	1	1	yes	characteristic
114	C₂×C₄	#10597284	1	1	yes	characteristic
115	C₂×C₄	#10597289	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	304	1	no	—
118	C₈	#10597289	1	1	yes	characteristic
119	C₈	#10597290	1	1	yes	characteristic
120	C₂²	—	16	1	no	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	1	no	—
123	C₂²	#11926345	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	16	1	no	—
127	C₄	#11926288	1	1	yes	characteristic
128	C₄	#11926357	1	1	yes	characteristic
129	C₂	—	32	1	no	—
130	C₂	#14561681	1	1	yes	characteristic
131	C₂	#14561819	1	1	yes	characteristic
132	C₂	#14561822	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₁₉⋊C₃₂)

C₂

yes

maximal, characteristic, complemented

#14561709

C₂

yes

maximal, characteristic, complemented

C₁₉×#2249

C₂

yes

maximal, characteristic

C₂×C₁₉⋊C₃₂

—

#11926297

—

C₂×C₁₉×Q₃₂

C₄

yes

characteristic

C₂×C₁₉×SD₃₂

—

C₂×C₁₉×D₁₆

C₄

yes

characteristic

C₁₉×#495

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂×D₈×C₁₉

—

C₁₉×#124

—

C₁₉×Q₃₂

—

C₁₉×SD₃₂

—

C₁₉×SD₃₂

—

C₁₉×D₁₆

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄⋊(C₁₉⋊C₈)

—

C₁₉⋊C₃₂

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

D₈

yes

characteristic

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₆×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

SD₁₆

yes

characteristic

#7446

—

maximal, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#102

yes

characteristic

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₁₉⋊C₈

—

#2249

D₁₉

yes

characteristic

C₄⋊C₃₂

—

#1372

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#357

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×Q₃₂

Dic₁₉

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

Dic₁₉

yes

characteristic

#495

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₁₉

yes

characteristic

#335

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#124

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

C₂×Dic₁₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

C₃₂

—

C₁₉

#7446

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

152

—

C₂×C₈

#64526

yes

characteristic

100

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

#64525

yes

characteristic

104

C₁₆

—

105

C₁₆

#64525

yes

characteristic

106

C₁₆

#64526

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

#10597283

yes

characteristic

114

C₂×C₄

#10597284

yes

characteristic

115

C₂×C₄

#10597289

yes

characteristic

116

C₈

—

117

C₈

—

304

—

118

C₈

#10597289

yes

characteristic

119

C₈

#10597290

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11926345

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#11926288

yes

characteristic

128

C₄

#11926357

yes

characteristic

129

C₂

—

130

C₂

#14561681

yes

characteristic

131

C₂

#14561819

yes

characteristic

132

C₂

#14561822

yes

characteristic

	subgroup	G/H
center	C₂²	#11926345
commutator	C₂×C₁₆×C₁₉	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₁₉
Fitting	C₁₉×#2249	C₂
socle	C₂²×C₁₉	#357
prime radical	C₂×C₁₉×D₁₆	C₄
cosocle	C₄×C₁₆×C₁₉	C₂²
hypercenter	#2249	D₁₉
hypocenter	C₁₉	#7446
LCS	G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

subgroup

G/H

center

C₂²

#11926345

commutator

C₂×C₁₆×C₁₉

C₂×C₄

Frattini

C₄×C₁₆

C₂×D₁₉

Fitting

C₁₉×#2249

C₂

socle

C₂²×C₁₉

#357

prime radical

C₂×C₁₉×D₁₆

C₄

cosocle

C₄×C₁₆×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

N	H ≅ G/N	type
1. C₄⋊(C₁₉⋊C₃₂)	C₂	semidirect
2. #14561709	C₂	semidirect
90. C₁₉	#7446	semidirect

H ≅ G/N

type

1. C₄⋊(C₁₉⋊C₃₂)

C₂

semidirect

2. #14561709

C₂

semidirect

90. C₁₉

#7446

semidirect

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

subgroup	index
C₄⋊(C₁₉⋊C₃₂)	2
#14561709	2
C₁₉×#2249	2
#7446	19

subgroup

index

C₄⋊(C₁₉⋊C₃₂)

#14561709

C₁₉×#2249

#7446

Subgroups — #18070178

Conjugacy classes of subgroups

Complements

Maximal subgroups