Group #18091222 — subgroups

Subgroups — #18091222

Order 4864.

686 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 235 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	#14572225	C₂	1	3	yes	maximal
2	#14563333	C₂	1	3	yes	maximal
3	C₂×C₁₉×#338	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
5	C₂×#10597372	C₂²	1	3	yes	—
6	#11926586	—	2	12	no	—
7	#11926582	—	2	6	no	—
8	#11926298	—	2	6	no	—
9	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
10	C₁₉×#338	C₄	1	4	yes	complemented
11	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	—	2	6	no	—
13	C₂×M₁₆×C₁₉	—	2	6	no	—
14	C₂×M₁₆×C₁₉	C₂×C₄	1	6	yes	—
15	C₂×M₁₆×C₁₉	Q₈	1	6	yes	—
16	C₂²×C₈×C₁₉	D₄	1	6	yes	—
17	C₂×C₂²⋊Dic₁₉	—	2	3	no	—
18	#10597372	—	2	6	no	—
19	#10597372	—	2	6	no	—
20	C₂×C₄⋊Dic₁₉	D₄	1	6	yes	—
21	C₂×C₄×Dic₁₉	—	2	3	no	—
22	C₂²×C₁₉⋊C₈	—	2	3	no	—
23	C₂²×C₁₉⋊C₈	—	2	3	no	—
24	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
25	#10597275	—	2	12	no	—
26	#10597274	—	2	12	no	—
27	C₂⁴×C₁₉	#41	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
30	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
31	C₂²×C₄×C₁₉	C₂×Q₈	1	3	yes	—
32	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	#41	1	3	yes	—
34	C₂²×Dic₁₉	—	2	6	no	—
35	M₁₆×C₁₉	—	2	12	no	—
36	M₁₆×C₁₉	—	2	12	no	—
37	C₂×C₈×C₁₉	—	2	6	no	—
38	C₂×C₈×C₁₉	—	2	6	no	—
39	C₂×C₈×C₁₉	C₄⋊C₄	1	12	yes	—
40	C₂²⋊Dic₁₉	—	4	6	no	—
41	C₄⋊Dic₁₉	—	2	12	no	—
42	C₄×Dic₁₉	—	4	3	no	—
43	C₄×Dic₁₉	—	4	3	no	—
44	C₂×C₁₉⋊C₈	—	2	6	no	—
45	C₂×C₁₉⋊C₈	—	4	3	no	—
46	C₂×C₁₉⋊C₈	—	4	1	no	—
47	#11337	—	19	1	no	maximal, Sylow
48	C₂³×C₁₉	—	4	2	no	—
49	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
50	C₂³×C₁₉	#121	1	3	yes	—
51	C₂³×C₁₉	#125	1	3	yes	—
52	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	3	yes	—
53	C₂×C₄×C₁₉	C₄×D₄	1	3	yes	—
54	C₂×C₄×C₁₉	#120	1	3	yes	—
55	C₂×C₄×C₁₉	C₂²⋊Q₈	1	6	yes	—
56	C₂×C₄×C₁₉	C₂²⋊Q₈	1	3	yes	—
57	C₂×C₄×C₁₉	#123	1	3	yes	—
58	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₄⋊Q₈	1	3	yes	—
60	C₂×C₄×C₁₉	C₄⋊Q₈	1	3	yes	—
61	C₂×Dic₁₉	—	4	12	no	—
62	C₂×Dic₁₉	—	4	6	no	—
63	C₈×C₁₉	—	2	12	no	—
64	C₁₉⋊C₈	—	4	2	no	—
65	#2911	—	19	3	no	—
66	C₂×#338	D₁₉	1	1	yes	characteristic
67	#1729	—	19	3	no	—
68	C₂²×C₁₉	—	2	6	no	—
69	C₂²×C₁₉	—	4	1	no	—
70	C₂²×C₁₉	—	4	3	no	—
71	C₂²×C₁₉	#389	1	3	yes	—
72	C₂²×C₁₉	#392	1	1	yes	characteristic
73	C₂²×C₁₉	#398	1	3	yes	—
74	C₄×C₁₉	#384	1	1	yes	characteristic
75	C₄×C₁₉	#385	1	3	yes	—
76	C₄×C₁₉	#395	1	3	yes	—
77	C₄×C₁₉	#399	1	1	yes	characteristic
78	Dic₁₉	—	8	6	no	—
79	C₂²×M₁₆	C₂×D₁₉	1	3	yes	—
80	C₂³×C₈	—	19	1	no	—
81	C₂×#117	—	19	3	no	—
82	#428	—	38	12	no	—
83	#427	—	38	6	no	—
84	#338	Dic₁₉	1	4	yes	complemented
85	#336	—	38	6	no	—
86	C₂×C₁₉	—	2	3	no	—
87	C₂×C₁₉	—	2	3	no	—
88	C₂×C₁₉	#1444	1	1	yes	characteristic
89	C₂×C₁₉	#2017	1	2	yes	—
90	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
91	C₂×M₁₆	—	2	6	no	—
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	Dic₃₈	1	6	yes	—
94	C₂×M₁₆	C₂×Dic₁₉	1	6	yes	—
95	C₂²×C₈	—	19	2	no	—
96	C₂²×C₈	—	38	3	no	—
97	C₂²×C₈	—	38	3	no	—
98	C₂²×C₈	D₇₆	1	6	yes	—
99	#117	—	38	6	no	—
100	#117	—	38	6	no	—
101	C₂×C₄⋊C₄	—	19	6	no	—
102	C₂×C₂²⋊C₄	—	38	3	no	—
103	C₂×C₄²	—	38	3	no	—
104	#107	—	38	12	no	—
105	#106	—	38	12	no	—
106	C₁₉	#11337	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	#64545	1	1	yes	characteristic
108	C₂²×C₄	—	38	6	no	—
109	C₂²×C₄	C₂×Dic₃₈	1	3	yes	—
110	C₂²×C₄	C₂×D₇₆	1	3	yes	—
111	C₂²×C₄	D₄×D₁₉	1	3	yes	—
112	C₂²×C₄	#64543	1	3	yes	—
113	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	38	6	no	—
120	C₂×C₈	—	76	1	no	—
121	C₂×C₈	—	76	3	no	—
122	C₂×C₈	C₄⋊Dic₁₉	1	12	yes	—
123	C₄⋊C₄	—	38	12	no	—
124	C₂²⋊C₄	—	76	6	no	—
125	C₄²	—	76	3	no	—
126	C₄²	—	76	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	#10597336	1	3	yes	—
129	C₂³	#10597341	1	3	yes	—
130	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
131	C₂×C₄	—	76	12	no	—
132	C₂×C₄	—	76	6	no	—
133	C₂×C₄	#10597315	1	3	yes	—
134	C₂×C₄	#10597320	1	1	yes	characteristic
135	C₂×C₄	#10597324	1	6	yes	—
136	C₂×C₄	#10597328	1	3	yes	—
137	C₂×C₄	#10597332	1	3	yes	—
138	C₂×C₄	#10597334	1	3	yes	—
139	C₂×C₄	#10597343	1	3	yes	—
140	C₂×C₄	C₂×C₄⋊Dic₁₉	1	3	yes	—
141	C₂×C₄	D₄×Dic₁₉	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	76	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	#11926443	1	3	yes	—
148	C₂²	#11926454	1	1	yes	characteristic
149	C₂²	#11926476	1	3	yes	—
150	C₄	—	152	6	no	—
151	C₄	#11926414	1	1	yes	characteristic
152	C₄	#11926419	1	3	yes	—
153	C₄	#11926465	1	3	yes	—
154	C₄	#11926481	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#14561900	1	1	yes	characteristic
158	C₂	#14564740	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14572225

C₂

yes

maximal

#14563333

C₂

yes

maximal

C₂×C₁₉×#338

C₂

yes

maximal, characteristic

C₂³×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×#10597372

C₂²

yes

—

#11926586

—

#11926582

—

#11926298

—

C₂²×M₁₆×C₁₉

C₂²

yes

—

C₁₉×#338

C₄

yes

complemented

C₂³×C₄×C₁₉

C₂³

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

—

C₂×M₁₆×C₁₉

Q₈

yes

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₂²⋊Dic₁₉

—

#10597372

—

#10597372

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

C₂×C₄×Dic₁₉

—

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

Q₈

yes

—

#10597275

—

#10597274

—

C₂⁴×C₁₉

#41

yes

characteristic

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×Q₈

yes

—

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂²×Dic₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#11337

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

C₄×Q₈

yes

characteristic

C₂³×C₁₉

#121

yes

—

C₂³×C₁₉

#125

yes

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

—

C₂×C₄×C₁₉

C₄⋊Q₈

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2911

—

C₂×#338

D₁₉

yes

characteristic

#1729

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#389

yes

—

C₂²×C₁₉

#392

yes

characteristic

C₂²×C₁₉

#398

yes

—

C₄×C₁₉

#384

yes

characteristic

C₄×C₁₉

#385

yes

—

C₄×C₁₉

#395

yes

—

C₄×C₁₉

#399

yes

characteristic

Dic₁₉

—

C₂²×M₁₆

C₂×D₁₉

yes

—

C₂³×C₈

—

—

—

—

Dic₁₉

yes

complemented

#336

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1444

yes

characteristic

C₂×C₁₉

#2017

yes

—

C₂³×C₄

C₂²×D₁₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃₈

yes

—

C₂×M₁₆

C₂×Dic₁₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₇₆

yes

—

#117

—

100

#117

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

—

104

#107

—

105

#106

—

106

C₁₉

#11337

yes

characteristic, complemented, Sylow

107

C₂⁴

#64545

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₃₈

yes

—

110

C₂²×C₄

C₂×D₇₆

yes

—

111

C₂²×C₄

D₄×D₁₉

yes

—

112

C₂²×C₄

#64543

yes

—

113

C₂²×C₄

Q₈×D₁₉

yes

characteristic

114

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄⋊Dic₁₉

yes

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₂³

—

128

C₂³

#10597336

yes

—

129

C₂³

#10597341

yes

—

130

C₂³

Q₈×Dic₁₉

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#10597315

yes

—

134

C₂×C₄

#10597320

yes

characteristic

135

C₂×C₄

#10597324

yes

—

136

C₂×C₄

#10597328

yes

—

137

C₂×C₄

#10597332

yes

—

138

C₂×C₄

#10597334

yes

—

139

C₂×C₄

#10597343

yes

—

140

C₂×C₄

C₂×C₄⋊Dic₁₉

yes

—

141

C₂×C₄

D₄×Dic₁₉

yes

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11926443

yes

—

148

C₂²

#11926454

yes

characteristic

149

C₂²

#11926476

yes

—

150

C₄

—

152

—

151

C₄

#11926414

yes

characteristic

152

C₄

#11926419

yes

—

153

C₄

#11926465

yes

—

154

C₄

#11926481

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#14561900

yes

characteristic

158

C₂

#14564740

yes

—

	subgroup	G/H
center	C₂×C₄	#10597320
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#338	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#338	D₁₉
hypocenter	C₁₉	#11337
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#10597320

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₁₉

Fitting

C₂×C₁₉×#338

C₂

C₂²×C₁₉

C₂⁴×C₁₉

C₂³×C₄×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. C₁₉×#338	C₄	semidirect
84. #338	Dic₁₉	semidirect
106. C₁₉	#11337	semidirect

H ≅ G/N

type

10. C₁₉×#338

C₄

semidirect

84. #338

Dic₁₉

semidirect

106. C₁₉

#11337

semidirect

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

subgroup	index
#14572225	2
#14563333	2
C₂×C₁₉×#338	2
#11337	19

subgroup

index

#14572225

#14563333

C₂×C₁₉×#338

#11337

Subgroups — #18091222

Conjugacy classes of subgroups

Complements

Maximal subgroups