Finite Group Databasebeta

← #72218524

Subgroups — #72218524

Order 2240.

1324 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D35C211yesmaximal, characteristic, complemented
2C2×#10587963C213yesmaximal, complemented
3C7×#4099C211yesmaximal, characteristic, complemented
4C5×#6516C211yesmaximal, characteristic, complemented
5#11765018C213yesmaximal, complemented
6#11765017C213yesmaximal, complemented
7#11765004C213yesmaximal, complemented
8C24×C5×C7C2211yescharacteristic, complemented
9C23×D35—212no—
10C23×D35C2213yes—
11C23×D35C2213yescomplemented
12C22×D5×D7C2211yescharacteristic, complemented
13C2×C7×C22⋊D5C2213yescomplemented
14C2×C5×C22⋊D7C2213yescomplemented
15#10587963—212no—
16C2×#63307C2213yescomplemented
17C2×#63306C2213yescomplemented
18C2×#63304C2213yescomplemented
19C7×C22⋊Dic5C2213yescomplemented
20C5×C22⋊Dic7C2213yescomplemented
21#10587856C2216yescomplemented
22D7×#120—51nomaximal, Hall
23D5×#120—71nomaximal, Hall
24C23×C5×C7—26no—
25C23×C5×C7C2313yes—
26C22×D35—232no—
27C22×D35—26no—
28C22×D35—212no—
29C22×D35—212no—
30C22×D35D4112yescomplemented
31C22×D35C2313yes—
32C22×D35C2311yescharacteristic
33C22×C7×D5C2311yescharacteristic, complemented
34C22×C5×D7C2311yescharacteristic, complemented
35C2×D5×D7—26no—
36C7×C22⋊D5—26no—
37C2×C7×Dic5C2313yescomplemented
38C5×C22⋊D7—26no—
39C2×C5×Dic7C2313yescomplemented
40#63307—26no—
41#63306—26no—
42#63304—26no—
43C24×D7—51no—
44C2×D4×D7—53no—
45C7×#120—51no—
46#6516D511yescharacteristic, complemented
47#6500—53no—
48#6445—53no—
49D7×C22⋊C4—53no—
50C24×D5—71no—
51C2×D4×D5—73no—
52C5×#120—71no—
53#4099D711yescharacteristic, complemented
54#4083—73no—
55#4028—73no—
56D5×C22⋊C4—73no—
57C22×C5×C7—28no—
58C22×C5×C7—26no—
59C22×C5×C7C2×D416yescomplemented
60C22×C5×C7C2411yescharacteristic
61C2×D35—224no—
62C2×D35—26no—
63C2×D35—224no—
64C2×D35C2×D416yes—
65C2×D35C2×D416yescomplemented
66C2×C7×D5—23no—
67C2×C5×D7—23no—
68D5×D7—44no—
69C5×Dic7—23no—
70C7×Dic5—23no—
71C24×C7C2×D511yescharacteristic, complemented
72C23×D7—53no—
73C23×D7—53no—
74C23×D7—51no—
75C23×D7—1012no—
76C2×C7×D4—53no—
77C2×C22⋊D7—53no—
78C2×C22⋊D7C2×D513yescomplemented
79D4×D7—1012no—
80C2×D28—53no—
81C2×C4×D7—53no—
82C7×C22⋊C4—53no—
83C22⋊Dic7C2×D513yescomplemented
84(C2×D7)⋊C4—56no—
85C24×C5C2×D711yescharacteristic, complemented
86C23×D5—73no—
87C23×D5—73no—
88C23×D5—71no—
89C23×D5—1412no—
90C2×C5×D4—73no—
91C2×C22⋊D5—73no—
92C2×C22⋊D5C2×D713yescomplemented
93D4×D5—1412no—
94C2×D20—73no—
95C2×C4×D5—73no—
96C5×C22⋊C4—73no—
97C22⋊Dic5C2×D713yescomplemented
98#685—76no—
99C2×C5×C7—26no—
100C2×C5×C7C22×D413yes—
101D35—26no—
102D35#12014yescomplemented
103C5×D7—41no—
104C7×D5—41no—
105C2×#120—351noSylow
106C23×C7—26no—
107C23×C7—51no—
108C23×C7C22×D513yes—
109C22×D7—51no—
110C22×D7—512no—
111C22×D7—53no—
112C22×D7—106no—
113C22×D7—1012no—
114C22×D7—1012no—
115C22×D7—1032no—
116C22×D7—106no—
117C22×D7C22×D511yescharacteristic, complemented
118C7×D4—106no—
119C2×C4×C7—53no—
120C22⋊D7—26no—
121C22⋊D7—106no—
122C2×Dic7C22×D513yescomplemented
123D28—106no—
124C4×D7—106no—
125C23×C5—26no—
126C23×C5—71no—
127C23×C5C22×D713yes—
128C22×D5—71no—
129C22×D5—712no—
130C22×D5—73no—
131C22×D5—146no—
132C22×D5—1412no—
133C22×D5—1412no—
134C22×D5—1432no—
135C22×D5—146no—
136C22×D5C22×D711yescharacteristic, complemented
137C5×D4—146no—
138C2×C4×C5—73no—
139C22⋊D5—26no—
140C22⋊D5—146no—
141C2×Dic5C22×D713yescomplemented
142D20—146no—
143C4×D5—146no—
144C5×C7C2×#12011yescharacteristic, complemented, Hall
145C25—351no—
146C22×D4—353no—
147#120—51no—
148#120—71no—
149#120—353no—
150#120—353no—
151C2×C22⋊C4—353no—
152C22×C7—28no—
153C22×C7—26no—
154C22×C7—103no—
155C22×C7D4×D516yescomplemented
156C22×C7C23×D511yescharacteristic
157C2×D7—23no—
158C2×D7—56no—
159C2×D7—56no—
160C2×D7—1024no—
161C2×D7—1024no—
162C2×D7—106no—
163C2×D7—204no—
164C4×C7—103no—
165Dic7—23no—
166C22×C5—28no—
167C22×C5—26no—
168C22×C5—143no—
169C22×C5D4×D716yescomplemented
170C22×C5C23×D711yescharacteristic
171C2×D5—23no—
172C2×D5—76no—
173C2×D5—76no—
174C2×D5—1424no—
175C2×D5—1424no—
176C2×D5—146no—
177C2×D5—284no—
178C4×C5—143no—
179Dic5—23no—
180C24—351no—
181C24—353no—
182C24—353no—
183C24—7012no—
184C24D5×D711yescharacteristic, complemented
185C2×D4—53no—
186C2×D4—73no—
187C2×D4—353no—
188C2×D4—353no—
189C2×D4—7012no—
190C22×C4—353no—
191C22⋊C4—53no—
192C22⋊C4—73no—
193C22⋊C4—356no—
194C2×C7—26no—
195C2×C7—201no—
196C2×C7C2×D4×D513yes—
197D7—41no—
198D7—54no—
199D7—106no—
200C2×C5—26no—
201C2×C5—281no—
202C2×C5C2×D4×D713yes—
203D5—41no—
204D5—74no—
205D5—146no—
206C23—26no—
207C23—51no—
208C23—71no—
209C23—351no—
210C23—3512no—
211C23—353no—
212C23—7012no—
213C23—7012no—
214C23—7032no—
215C23—706no—
216C23—706no—
217C23C2×D5×D713yes—
218D4—106no—
219D4—146no—
220D4—706no—
221D4—706no—
222C2×C4—53no—
223C2×C4—73no—
224C2×C4—706no—
225C7D5×#12011yescharacteristic, complemented, Sylow
226C5D7×#12011yescharacteristic, complemented, Sylow
227C22—28no—
228C22—26no—
229C22—103no—
230C22—143no—
231C22—356no—
232C22—356no—
233C22—7024no—
234C22—7024no—
235C22—706no—
236C22—1404no—
237C22#1058796316yescomplemented
238C22C22×D5×D711yescharacteristic
239C4—103no—
240C4—143no—
241C2—26no—
242C2—201no—
243C2—281no—
244C2—354no—
245C2—706no—
246C2C2×#1058796313yes—
subgroupG/H
center, FrattiniC22C22×D5×D7
commutator, socle, cosocleC22×C5×C7C24
FittingC24×C5×C7C22
hypercenterC24D5×D7
hypocenterC5×C7C2×#120
LCSG ▷ C22×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C24

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C24×D35C2semidirect
2. C2×#10587963C2semidirect
3. C7×#4099C2semidirect
4. C5×#6516C2semidirect
5. #11765018C2semidirect
6. #11765017C2semidirect
7. #11765004C2semidirect
8. C24×C5×C7C22semidirect
11. C23×D35C22semidirect
12. C22×D5×D7C22semidirect
13. C2×C7×C22⋊D5C22semidirect
14. C2×C5×C22⋊D7C22semidirect
16. C2×#63307C22semidirect
17. C2×#63306C22semidirect
18. C2×#63304C22semidirect
19. C7×C22⋊Dic5C22semidirect
20. C5×C22⋊Dic7C22semidirect
21. #10587856C22semidirect
30. C22×D35D4semidirect
33. C22×C7×D5C23semidirect
34. C22×C5×D7C23semidirect
37. C2×C7×Dic5C23semidirect
39. C2×C5×Dic7C23semidirect
46. #6516D5semidirect
53. #4099D7semidirect
59. C22×C5×C7C2×D4semidirect
65. C2×D35C2×D4semidirect
71. C24×C7C2×D5semidirect
78. C2×C22⋊D7C2×D5semidirect
83. C22⋊Dic7C2×D5semidirect
85. C24×C5C2×D7semidirect
92. C2×C22⋊D5C2×D7semidirect
97. C22⋊Dic5C2×D7semidirect
102. D35#120semidirect
117. C22×D7C22×D5semidirect
122. C2×Dic7C22×D5semidirect
136. C22×D5C22×D7semidirect
141. C2×Dic5C22×D7semidirect
144. C5×C7C2×#120semidirect
155. C22×C7D4×D5semidirect
169. C22×C5D4×D7semidirect
184. C24D5×D7semidirect
225. C7D5×#120semidirect
226. C5D7×#120semidirect
237. C22#10587963semidirect

Maximal subgroups

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

subgroupindex
C24×D352
C2×#105879632
C7×#40992
C5×#65162
#117650182
#117650172
#117650042
D7×#1205
D5×#1207