Finite Group Databasebeta

← C2×#11756336 #14482980

Subgroups — C2×#11756336 #14482980

Order 1920.

3472 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6821C212yesmaximal, complemented
2C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
3#11756336C2116yesmaximal, direct product factor
4C2×#91784C216yesmaximal, complemented
5C2×#91763C213yesmaximal, complemented
6C2×C4×D60C213yesmaximal, complemented
7C2×#65911—31nomaximal, Hall
8C22×C3×C5×Q8C2211yescharacteristic
9C2×#6821—216no—
10C2×#6821C22116yescomplemented
11C22×D60C2216yes—
12C22×D60C2213yes—
13C22×D60C2216yes—
14C22×C4×D15C2216yes—
15C3×C4×C5×Q8C2218yescomplemented
16C2×C3×C5×C4⋊C4C2213yes—
17C2×C3×C42×C5C2213yes—
18C2×#6714C2216yes—
19C2×C4⋊Dic15C2211yescharacteristic
20#91784C22148yescomplemented
21#91763C22124yescomplemented
22C4×D60C22124yescomplemented
23C2×#5577—51nomaximal, Hall
24C22×#713—32no—
25C2×C4×C5×Q8D311yescharacteristic, complemented
26#65911—316no—
27C2×#4041—36no—
28C2×#4020—33no—
29C2×C4×D20—33no—
30C23×D15C2316yes—
31C2×C3×C5×Q8D418yescomplemented
32C2×C3×C5×Q8C2314yes—
33C22×C3×C4×C5C2313yes—
34C22×C3×C4×C5C2313yes—
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15C2312yes—
37#6821—264no—
38C2×D60—224no—
39C2×D60—212no—
40C2×D60—412no—
41C2×D60C23112yes—
42C2×D60C23124yes—
43C2×D60C23124yes—
44C2×C4×D15—224no—
45C2×C4×D15C23124yes—
46C3×C5×C4⋊C4C23112yes—
47C3×C42×C5C23112yes—
48#6714C23124yes—
49C4⋊Dic15C2314yes—
50C22×#239—52no—
51C2×C3×C4×Q8D511yescharacteristic, complemented
52#5577—516no—
53C2×#895—56no—
54C2×#874—53no—
55C2×C4×D12—53no—
56C22×C5×Q8C2×D311yescharacteristic
57C2×#713—316no—
58C2×#713—616no—
59C22×D20—33no—
60C22×D20—36no—
61C22×D20—36no—
62C22×C4×D5—36no—
63C4×C5×Q8C2×D318yescomplemented
64C2×C5×C4⋊C4C2×D313yes—
65C2×C42×C5C2×D313yes—
66C2×#685—36no—
67C2×C4⋊Dic5—31no—
68#4041—348no—
69#4020—324no—
70C4×D20—324no—
71C2×#550—151noSylow
72C23×C3×C5C2411yescharacteristic
73C22×D15—212no—
74C22×D15—212no—
75C22×D15—212no—
76C22×D15C24112yes—
77C3×C5×Q8C2×D4116yescomplemented
78C2×C3×C4×C5—26no—
79C2×C3×C4×C5C2×D4112yes—
80C2×C3×C4×C5#4114yescomplemented
81C2×C3×C4×C5C2416yes—
82C2×C3×C4×C5C2416yes—
83C2×C3×C4×C5C2412yes—
84C2×Dic15—24no—
85C2×Dic15C2414yes—
86D60—248no—
87D60—224no—
88D60—424no—
89C4×D15—248no—
90C22×C3×Q8C2×D511yescharacteristic
91C2×#239—516no—
92C2×#239—1016no—
93C22×D12—56no—
94C22×D12—53no—
95C22×D12—56no—
96C22×C4×D3—56no—
97C3×C4×Q8C2×D518yescomplemented
98C2×C3×C4⋊C4C2×D513yes—
99C2×C3×C42C2×D513yes—
100C2×(C2×D3)⋊C4—56no—
101C2×C4⋊Dic3—51no—
102#895—548no—
103#874—524no—
104C4×D12—524no—
105C23×D5—36no—
106C2×C5×Q8D1218yescomplemented
107C2×C5×Q8C22×D314yes—
108C22×C4×C5C22×D313yes—
109C22×C4×C5C22×D313yes—
110C22×C4×C5C22×D311yescharacteristic
111C22×Dic5—32no—
112#713—664no—
113C2×D20—324no—
114C2×D20—312no—
115C2×D20—324no—
116C2×D20—624no—
117C2×D20—612no—
118C2×D20—1212no—
119C2×C4×D5—324no—
120C2×C4×D5—624no—
121C5×C4⋊C4C22×D3112yes—
122C42×C5C22×D3112yes—
123#685—324no—
124C4⋊Dic5—34no—
125C22×#41—152no—
126#550—1516no—
127C2×#127—153no—
128C2×#121—156no—
129C2×C4×Q8D1511yescharacteristic, complemented
130C2×C4×D4—153no—
131C22×C3×C5C22×D412yes—
132C22×C3×C5C2×#4112yes—
133C22×C3×C5H(5,𝔽2)12yes—
134C22×C3×C5C2511yescharacteristic
135C2×D15—212no—
136C2×D15—212no—
137C2×D15—212no—
138C2×D15—424no—
139C3×C4×C5—26no—
140C3×C4×C5C22×D4112yes—
141C3×C4×C5C2×#4114yescomplemented
142Dic15—24no—
143C23×D3—56no—
144C2×C3×Q8D2018yescomplemented
145C2×C3×Q8C22×D514yes—
146C22×C3×C4C22×D513yes—
147C22×C3×C4C22×D513yes—
148C22×C3×C4C22×D511yescharacteristic
149C22×Dic3—52no—
150#239—1064no—
151C2×D12—512no—
152C2×D12—524no—
153C2×D12—524no—
154C2×D12—1024no—
155C2×D12—1012no—
156C2×D12—2012no—
157C2×C4×D3—524no—
158C2×C4×D3—1024no—
159C3×C4⋊C4C22×D5112yes—
160C3×C42C22×D5112yes—
161(C2×D3)⋊C4—524no—
162C4⋊Dic3—54no—
163C23×C5C23×D311yescharacteristic
164C22×D5—312no—
165C22×D5—612no—
166C22×D5—612no—
167C22×D5—612no—
168C5×Q8C2×D12116yescomplemented
169C2×C4×C5—26no—
170C2×C4×C5C2×D12112yes—
171C2×C4×C5#23914yescomplemented
172C2×C4×C5C23×D316yes—
173C2×C4×C5C23×D316yes—
174C2×C4×C5C23×D312yes—
175C2×Dic5—34no—
176C2×Dic5—64no—
177D20—648no—
178D20—624no—
179D20—1224no—
180C4×D5—648no—
181C2×#41—1516no—
182C2×#41—3016no—
183C22×Q8C2×D1511yescharacteristic
184C22×D4—153no—
185C22×D4—156no—
186C22×D4—156no—
187C23×C4—156no—
188#127—1524no—
189#121—1548no—
190C4×Q8C2×D1518yescomplemented
191C4×D4—1524no—
192C2×C4⋊C4—151no—
193C2×C4⋊C4C2×D1513yes—
194C2×C22⋊C4—156no—
195C2×C42C2×D1513yes—
196C2×C3×C5#55014yescomplemented
197C2×C3×C5C23×D411yescharacteristic
198C2×C3×C5C22×#4111yescharacteristic
199C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
200D15—412no—
201C23×C3C23×D511yescharacteristic
202C22×D3—512no—
203C22×D3—1012no—
204C22×D3—1012no—
205C22×D3—1012no—
206C3×Q8C2×D20116yescomplemented
207C2×C3×C4—26no—
208C2×C3×C4C2×D20112yes—
209C2×C3×C4#71314yescomplemented
210C2×C3×C4C23×D516yes—
211C2×C3×C4C23×D516yes—
212C2×C3×C4C23×D512yes—
213C2×Dic3—54no—
214C2×Dic3—104no—
215D12—1048no—
216D12—1024no—
217D12—2024no—
218C4×D3—1048no—
219C22×C5C22×D1212yes—
220C22×C5C2×#23912yes—
221C22×C5#100912yes—
222C22×C5C24×D311yescharacteristic
223C2×D5—612no—
224C2×D5—612no—
225C2×D5—612no—
226C2×D5—1224no—
227C4×C5—26no—
228C4×C5C22×D12112yes—
229C4×C5C2×#23914yescomplemented
230Dic5—64no—
231C24—156no—
232#41—3064no—
233C2×Q8D6018yescomplemented
234C2×Q8C22×D1514yes—
235C2×D4—1512no—
236C2×D4—1524no—
237C2×D4—1524no—
238C2×D4—3024no—
239C2×D4—3012no—
240C2×D4—6012no—
241C22×C4—152no—
242C22×C4—1524no—
243C22×C4—3024no—
244C22×C4C22×D1513yes—
245C22×C4C22×D1513yes—
246C22×C4C22×D1511yescharacteristic
247C4⋊C4—154no—
248C4⋊C4C22×D15112yes—
249C22⋊C4—1524no—
250C42C22×D15112yes—
251C3×C5C2×#55011yescharacteristic, complemented, Hall
252C22×C3C22×D2012yes—
253C22×C3C2×#71312yes—
254C22×C3#414912yes—
255C22×C3C24×D511yescharacteristic
256C2×D3—1012no—
257C2×D3—1012no—
258C2×D3—1012no—
259C2×D3—2024no—
260C3×C4—26no—
261C3×C4C22×D20112yes—
262C3×C4C2×#71314yescomplemented
263Dic3—104no—
264C2×C5#557714yescomplemented
265C2×C5C23×D1211yescharacteristic
266C2×C5C22×#23911yescharacteristic
267C2×C5C2×#100911yescharacteristic
268D5—1212no—
269C23—1512no—
270C23—3012no—
271C23—3012no—
272C23—3012no—
273C23C23×D1511yescharacteristic
274Q8C2×D60116yescomplemented
275D4—3048no—
276D4—3024no—
277D4—6024no—
278C2×C4—26no—
279C2×C4—154no—
280C2×C4—304no—
281C2×C4—3048no—
282C2×C4C2×D60112yes—
283C2×C4#682114yescomplemented
284C2×C4C23×D1516yes—
285C2×C4C23×D1516yes—
286C2×C4C23×D1512yes—
287C2×C3#6591114yescomplemented
288C2×C3C23×D2011yescharacteristic
289C2×C3C22×#71311yescharacteristic
290C2×C3C2×#414911yescharacteristic
291D3—2012no—
292C5C2×#557711yescharacteristic, complemented, Sylow
293C22—3012no—
294C22—3012no—
295C22—3012no—
296C22—6024no—
297C22C22×D6012yes—
298C22C2×#682112yes—
299C22#9210012yes—
300C22C24×D1511yescharacteristic
301C4—26no—
302C4—304no—
303C4C22×D60112yes—
304C4C2×#682114yescomplemented
305C3C2×#6591111yescharacteristic, complemented, Sylow
306C2—6012no—
307C2#1175633614yesdirect product factor
308C2C23×D6011yescharacteristic
309C2C22×#682111yescharacteristic
310C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C24
hypercenterC2×C4×Q8D15
hypocenterC3×C5C2×#550
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#6821C2semidirect
2. C2×C3×C4×C5×Q8C2semidirect
3. #11756336C2direct (N × H)
4. C2×#91784C2semidirect
5. C2×#91763C2semidirect
6. C2×C4×D60C2semidirect
10. C2×#6821C22semidirect
15. C3×C4×C5×Q8C22semidirect
20. #91784C22semidirect
21. #91763C22semidirect
22. C4×D60C22semidirect
25. C2×C4×C5×Q8D3semidirect
31. C2×C3×C5×Q8D4semidirect
51. C2×C3×C4×Q8D5semidirect
63. C4×C5×Q8C2×D3semidirect
77. C3×C5×Q8C2×D4semidirect
80. C2×C3×C4×C5#41semidirect
97. C3×C4×Q8C2×D5semidirect
106. C2×C5×Q8D12semidirect
129. C2×C4×Q8D15semidirect
141. C3×C4×C5C2×#41semidirect
144. C2×C3×Q8D20semidirect
168. C5×Q8C2×D12semidirect
171. C2×C4×C5#239semidirect
190. C4×Q8C2×D15semidirect
196. C2×C3×C5#550semidirect
206. C3×Q8C2×D20semidirect
209. C2×C3×C4#713semidirect
229. C4×C5C2×#239semidirect
233. C2×Q8D60semidirect
251. C3×C5C2×#550semidirect
262. C3×C4C2×#713semidirect
264. C2×C5#5577semidirect
274. Q8C2×D60semidirect
283. C2×C4#6821semidirect
287. C2×C3#65911semidirect
292. C5C2×#5577semidirect
304. C4C2×#6821semidirect
305. C3C2×#65911semidirect
307. C2#11756336direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#68212
C2×C3×C4×C5×Q82
#117563362
C2×#917842
C2×#917632
C2×C4×D602
C2×#659113
C2×#55775