Finite Group Databasebeta

← C2×C5×#5576 #14482814

Subgroups — C2×C5×#5576 #14482814

Order 1920.

3240 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#239C211yesmaximal, characteristic, complemented
2C22×C5×D3×Q8C211yesmaximal, characteristic, complemented
3C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
4C5×#5576C2116yesmaximal, direct product factor
5C2×C5×#897C213yesmaximal, complemented
6C2×C5×#895C213yesmaximal, complemented
7C2×C5×#875C213yesmaximal, complemented
8C2×C4×C5×D12C213yesmaximal, complemented
9C2×C5×#548—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C2×C5×#239—28no—
12C2×C5×#239C2218yescomplemented
13C2×C5×D3×Q8—28no—
14C2×C5×D3×Q8C2218yescomplemented
15C22×C5×D12C2213yes—
16C22×C5×D12C2213yes—
17C22×C4×C5×D3C2213yes—
18C22×C4×C5×D3C2213yes—
19C22×C5×Dic6C2213yes—
20C3×C4×C5×Q8C2218yescomplemented
21C2×C3×C5×C4⋊C4C2213yes—
22C2×C3×C42×C5C2213yes—
23C2×C5×(C2×D3)⋊C4C2211yescharacteristic
24C2×C5×(C2×D3)⋊C4C2213yes—
25C2×C5×(C2×D3)⋊C4C2216yes—
26C2×C5×C4⋊Dic3C2213yes—
27C5×#897C22124yescomplemented
28C5×#895C22124yescomplemented
29C5×#875C22124yescomplemented
30C4×C5×D12C22124yescomplemented
31C2×#5576C511yesmaximal, characteristic, direct product factor, Hall
32C22×C5×#41—31no—
33C23×C5×Q8—31no—
34C5×#548—316no—
35C2×C5×#124—33no—
36C2×C5×C22⋊Q8—33no—
37C2×C5×#121—33no—
38C2×C4×C5×Q8D311yescharacteristic, complemented
39C2×C4×C5×D4—33no—
40C23×C5×D3C2313yes—
41C23×C5×D3C2311yescharacteristic
42C2×C3×C5×Q8D418yescomplemented
43C2×C3×C5×Q8C2314yes—
44C22×C3×C4×C5C2313yes—
45C22×C3×C4×C5C2311yescharacteristic
46C22×C3×C4×C5C2313yes—
47C22×C5×Dic3C2311yescharacteristic
48C22×C5×Dic3C2313yes—
49C5×#239—232no—
50C5×D3×Q8—232no—
51C2×C5×D12—212no—
52C2×C5×D12—212no—
53C2×C5×D12C23112yes—
54C2×C5×D12C23112yes—
55C2×C4×C5×D3—212no—
56C2×C4×C5×D3—212no—
57C2×C4×C5×D3C23112yes—
58C2×C4×C5×D3C23112yes—
59C2×C5×Dic6—212no—
60C2×C5×Dic6C23112yes—
61C3×C5×C4⋊C4C23112yes—
62C3×C42×C5C23112yes—
63C5×(C2×D3)⋊C4C2314yes—
64C5×(C2×D3)⋊C4C23112yes—
65C5×(C2×D3)⋊C4C23124yes—
66C5×C4⋊Dic3C23112yes—
67C22×#239C2×C511yescharacteristic, complemented
68C22×D3×Q8C2×C511yescharacteristic, complemented
69C2×C3×C4×Q8C2×C511yescharacteristic, complemented
70#5576C2×C5116yesdirect product factor
71C2×#897C2×C513yescomplemented
72C2×#895C2×C513yescomplemented
73C2×#875C2×C513yescomplemented
74C2×C4×D12C2×C513yescomplemented
75C2×C5×#41—38no—
76C2×C5×#41—68no—
77C22×C5×Q8—38no—
78C22×C5×Q8—33no—
79C22×C5×Q8—68no—
80C22×C5×Q8C2×D311yescharacteristic
81C22×C5×D4—33no—
82C22×C5×D4—33no—
83C23×C4×C5—33no—
84C23×C4×C5—33no—
85C5×#124—324no—
86C5×C22⋊Q8—324no—
87C5×#121—324no—
88C4×C5×Q8C2×D318yescomplemented
89C4×C5×D4—324no—
90C2×C5×C4⋊C4—33no—
91C2×C5×C4⋊C4C2×D313yes—
92C2×C5×C22⋊C4—31no—
93C2×C5×C22⋊C4—33no—
94C2×C5×C22⋊C4—36no—
95C2×C42×C5C2×D313yes—
96C2×#548—31noSylow
97C23×C3×C5C2411yescharacteristic
98C22×C5×D3—22no—
99C22×C5×D3—22no—
100C22×C5×D3—26no—
101C22×C5×D3—26no—
102C22×C5×D3—26no—
103C22×C5×D3#4114yescomplemented
104C22×C5×D3C2416yes—
105C22×C5×D3C2412yes—
106C3×C5×Q8C2×D4116yescomplemented
107C2×C3×C4×C5—26no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5C2×D4112yes—
110C2×C3×C4×C5C2416yes—
111C2×C3×C4×C5C2412yes—
112C2×C3×C4×C5C2416yes—
113C2×C5×Dic3—26no—
114C2×C5×Dic3—22no—
115C2×C5×Dic3C2412yes—
116C2×C5×Dic3C2416yes—
117C5×D12—224no—
118C5×D12—224no—
119C4×C5×D3—224no—
120C4×C5×D3—224no—
121C5×Dic6—224no—
122C22×C3×Q8C22×C511yescharacteristic
123C2×#239—28no—
124C2×#239C22×C518yescomplemented
125C2×D3×Q8—28no—
126C2×D3×Q8C22×C518yescomplemented
127C22×D12C22×C513yes—
128C22×D12C22×C513yes—
129C22×C4×D3C22×C513yes—
130C22×C4×D3C22×C513yes—
131C22×Dic6C22×C513yes—
132C3×C4×Q8C22×C518yescomplemented
133C2×C3×C4⋊C4C22×C513yes—
134C2×C3×C42C22×C513yes—
135C2×(C2×D3)⋊C4C22×C511yescharacteristic
136C2×(C2×D3)⋊C4C22×C513yes—
137C2×(C2×D3)⋊C4C22×C516yes—
138C2×C4⋊Dic3C22×C513yes—
139#897C22×C5124yescomplemented
140#895C22×C5124yescomplemented
141#875C22×C5124yescomplemented
142C4×D12C22×C5124yescomplemented
143C24×C5—31no—
144C24×C5—33no—
145C5×#41—632no—
146C2×C5×Q8—312no—
147C2×C5×Q8—632no—
148C2×C5×Q8—612no—
149C2×C5×Q8D1218yescomplemented
150C2×C5×Q8C22×D314yes—
151C2×C5×D4—312no—
152C2×C5×D4—312no—
153C2×C5×D4—612no—
154C2×C5×D4—612no—
155C22×C4×C5—33no—
156C22×C4×C5—31no—
157C22×C4×C5—312no—
158C22×C4×C5—312no—
159C22×C4×C5—612no—
160C22×C4×C5—612no—
161C22×C4×C5C22×D311yescharacteristic
162C22×C4×C5C22×D313yes—
163C22×C4×C5C22×D313yes—
164C5×C4⋊C4—312no—
165C5×C4⋊C4C22×D3112yes—
166C5×C22⋊C4—34no—
167C5×C22⋊C4—312no—
168C5×C22⋊C4—324no—
169C42×C5C22×D3112yes—
170C22×#41—31no—
171C23×Q8—31no—
172#548—316no—
173C2×#124—33no—
174C2×C22⋊Q8—33no—
175C2×#121—33no—
176C2×C4×Q8C5×D311yescharacteristic, complemented
177C2×C4×D4—33no—
178C22×C3×C5C22×D412yes—
179C22×C3×C5C2×#4112yes—
180C22×C3×C52−1+412yes—
181C22×C3×C5C2511yescharacteristic
182C2×C5×D3—28no—
183C2×C5×D3—22no—
184C2×C5×D3—22no—
185C2×C5×D3—26no—
186C2×C5×D3—26no—
187C2×C5×D3—26no—
188C2×C5×D3—412no—
189C2×C5×D3C2×#4114yescomplemented
190C3×C4×C5—26no—
191C3×C4×C5—22no—
192C3×C4×C5C22×D4112yes—
193C5×Dic3—26no—
194C5×Dic3—22no—
195C23×D3C23×C513yes—
196C23×D3C23×C511yescharacteristic
197C2×C3×Q8C5×D418yescomplemented
198C2×C3×Q8C23×C514yes—
199C22×C3×C4C23×C513yes—
200C22×C3×C4C23×C511yescharacteristic
201C22×C3×C4C23×C513yes—
202C22×Dic3C23×C511yescharacteristic
203C22×Dic3C23×C513yes—
204#239—232no—
205D3×Q8—232no—
206C2×D12—212no—
207C2×D12—212no—
208C2×D12C23×C5112yes—
209C2×D12C23×C5112yes—
210C2×C4×D3—212no—
211C2×C4×D3—212no—
212C2×C4×D3C23×C5112yes—
213C2×C4×D3C23×C5112yes—
214C2×Dic6—212no—
215C2×Dic6C23×C5112yes—
216C3×C4⋊C4C23×C5112yes—
217C3×C42C23×C5112yes—
218(C2×D3)⋊C4C23×C514yes—
219(C2×D3)⋊C4C23×C5112yes—
220(C2×D3)⋊C4C23×C5124yes—
221C4⋊Dic3C23×C5112yes—
222C23×C5—34no—
223C23×C5—32no—
224C23×C5—36no—
225C23×C5—62no—
226C23×C5—62no—
227C23×C5—66no—
228C23×C5—66no—
229C23×C5—66no—
230C23×C5C23×D311yescharacteristic
231C5×Q8—624no—
232C5×Q8C2×D12116yescomplemented
233C5×D4—624no—
234C5×D4—624no—
235C2×C4×C5—22no—
236C2×C4×C5—26no—
237C2×C4×C5—36no—
238C2×C4×C5—32no—
239C2×C4×C5—62no—
240C2×C4×C5—66no—
241C2×C4×C5—624no—
242C2×C4×C5—624no—
243C2×C4×C5C2×D12112yes—
244C2×C4×C5C23×D312yes—
245C2×C4×C5C23×D316yes—
246C2×C4×C5C23×D316yes—
247C2×#41—38no—
248C2×#41—68no—
249C22×Q8—38no—
250C22×Q8—33no—
251C22×Q8—68no—
252C22×Q8C2×C5×D311yescharacteristic
253C22×D4—33no—
254C22×D4—33no—
255C23×C4—33no—
256C23×C4—33no—
257#124—324no—
258C22⋊Q8—324no—
259#121—324no—
260C4×Q8C2×C5×D318yescomplemented
261C4×D4—324no—
262C2×C4⋊C4—33no—
263C2×C4⋊C4C2×C5×D313yes—
264C2×C22⋊C4—31no—
265C2×C22⋊C4—33no—
266C2×C22⋊C4—36no—
267C2×C42C2×C5×D313yes—
268C2×C3×C5#54814yescomplemented
269C2×C3×C5C23×D411yescharacteristic
270C2×C3×C5C22×#4111yescharacteristic
271C2×C3×C5C2×2−1+411yescharacteristic
272C5×D3—24no—
273C5×D3—46no—
274C23×C3C24×C511yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—26no—
278C22×D3—26no—
279C22×D3—26no—
280C22×D3C5×#4114yescomplemented
281C22×D3C24×C516yes—
282C22×D3C24×C512yes—
283C3×Q8C2×C5×D4116yescomplemented
284C2×C3×C4—26no—
285C2×C3×C4—22no—
286C2×C3×C4C2×C5×D4112yes—
287C2×C3×C4C24×C516yes—
288C2×C3×C4C24×C512yes—
289C2×C3×C4C24×C516yes—
290C2×Dic3—26no—
291C2×Dic3—22no—
292C2×Dic3C24×C512yes—
293C2×Dic3C24×C516yes—
294D12—224no—
295D12—224no—
296C4×D3—224no—
297C4×D3—224no—
298Dic6—224no—
299C22×C5—34no—
300C22×C5—62no—
301C22×C5—62no—
302C22×C5—68no—
303C22×C5—66no—
304C22×C5—66no—
305C22×C5—66no—
306C22×C5—1212no—
307C22×C5C22×D1212yes—
308C22×C5#100712yes—
309C22×C5D3×#4112yes—
310C22×C5C24×D311yescharacteristic
311C4×C5—22no—
312C4×C5—26no—
313C4×C5—62no—
314C4×C5—66no—
315C4×C5C22×D12112yes—
316C24—31no—
317C24—33no—
318#41—632no—
319C2×Q8—312no—
320C2×Q8—632no—
321C2×Q8—612no—
322C2×Q8C5×D1218yescomplemented
323C2×Q8C22×C5×D314yes—
324C2×D4—312no—
325C2×D4—312no—
326C2×D4—612no—
327C2×D4—612no—
328C22×C4—33no—
329C22×C4—31no—
330C22×C4—312no—
331C22×C4—312no—
332C22×C4—612no—
333C22×C4—612no—
334C22×C4C22×C5×D311yescharacteristic
335C22×C4C22×C5×D313yes—
336C22×C4C22×C5×D313yes—
337C4⋊C4—312no—
338C4⋊C4C22×C5×D3112yes—
339C22⋊C4—34no—
340C22⋊C4—312no—
341C22⋊C4—324no—
342C42C22×C5×D3112yes—
343C3×C5C2×#54811yescharacteristic, complemented, Hall
344C22×C3C22×C5×D412yes—
345C22×C3C2×C5×#4112yes—
346C22×C3C5×2−1+412yes—
347C22×C3C25×C511yescharacteristic
348C2×D3—28no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—26no—
352C2×D3—26no—
353C2×D3—26no—
354C2×D3—412no—
355C2×D3C2×C5×#4114yescomplemented
356C3×C4—26no—
357C3×C4—22no—
358C3×C4C22×C5×D4112yes—
359Dic3—26no—
360Dic3—22no—
361C2×C5—64no—
362C2×C5—126no—
363C2×C5#557614yesdirect product factor
364C2×C5C23×D1211yescharacteristic
365C2×C5C2×#100711yescharacteristic
366C2×C5C2×D3×#4111yescharacteristic
367C23—34no—
368C23—32no—
369C23—36no—
370C23—62no—
371C23—62no—
372C23—66no—
373C23—66no—
374C23—66no—
375C23C23×C5×D311yescharacteristic
376Q8—624no—
377Q8C2×C5×D12116yescomplemented
378D4—624no—
379D4—624no—
380C2×C4—22no—
381C2×C4—26no—
382C2×C4—36no—
383C2×C4—32no—
384C2×C4—62no—
385C2×C4—66no—
386C2×C4—624no—
387C2×C4—624no—
388C2×C4C2×C5×D12112yes—
389C2×C4C23×C5×D312yes—
390C2×C4C23×C5×D316yes—
391C2×C4C23×C5×D316yes—
392C2×C3C5×#54814yescomplemented
393C2×C3C23×C5×D411yescharacteristic
394C2×C3C22×C5×#4111yescharacteristic
395C2×C3C2×C5×2−1+411yescharacteristic
396D3—24no—
397D3—46no—
398C5C2×#557611yescharacteristic, direct product factor, Sylow
399C22—34no—
400C22—62no—
401C22—62no—
402C22—68no—
403C22—66no—
404C22—66no—
405C22—66no—
406C22—1212no—
407C22C22×C5×D1212yes—
408C22C5×#100712yes—
409C22C5×D3×#4112yes—
410C22C24×C5×D311yescharacteristic
411C4—22no—
412C4—26no—
413C4—62no—
414C4—66no—
415C4C22×C5×D12112yes—
416C3C2×C5×#54811yescharacteristic, complemented, Sylow
417C2—64no—
418C2—126no—
419C2C5×#557614yesdirect product factor
420C2C23×C5×D1211yescharacteristic
421C2C2×C5×#100711yescharacteristic
422C2C2×C5×D3×#4111yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C24
hypercenterC2×C4×C5×Q8D3
hypocenterC3C2×C5×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C4×C5×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C5×#239C2semidirect
2. C22×C5×D3×Q8C2semidirect
3. C2×C3×C4×C5×Q8C2semidirect
4. C5×#5576C2direct (N × H)
5. C2×C5×#897C2semidirect
6. C2×C5×#895C2semidirect
7. C2×C5×#875C2semidirect
8. C2×C4×C5×D12C2semidirect
12. C2×C5×#239C22semidirect
14. C2×C5×D3×Q8C22semidirect
20. C3×C4×C5×Q8C22semidirect
27. C5×#897C22semidirect
28. C5×#895C22semidirect
29. C5×#875C22semidirect
30. C4×C5×D12C22semidirect
31. C2×#5576C5direct (N × H)
38. C2×C4×C5×Q8D3semidirect
42. C2×C3×C5×Q8D4semidirect
67. C22×#239C2×C5semidirect
68. C22×D3×Q8C2×C5semidirect
69. C2×C3×C4×Q8C2×C5semidirect
70. #5576C2×C5direct (N × H)
71. C2×#897C2×C5semidirect
72. C2×#895C2×C5semidirect
73. C2×#875C2×C5semidirect
74. C2×C4×D12C2×C5semidirect
88. C4×C5×Q8C2×D3semidirect
103. C22×C5×D3#41semidirect
106. C3×C5×Q8C2×D4semidirect
124. C2×#239C22×C5semidirect
126. C2×D3×Q8C22×C5semidirect
132. C3×C4×Q8C22×C5semidirect
139. #897C22×C5semidirect
140. #895C22×C5semidirect
141. #875C22×C5semidirect
142. C4×D12C22×C5semidirect
149. C2×C5×Q8D12semidirect
176. C2×C4×Q8C5×D3semidirect
189. C2×C5×D3C2×#41semidirect
197. C2×C3×Q8C5×D4semidirect
232. C5×Q8C2×D12semidirect
260. C4×Q8C2×C5×D3semidirect
268. C2×C3×C5#548semidirect
280. C22×D3C5×#41semidirect
283. C3×Q8C2×C5×D4semidirect
322. C2×Q8C5×D12semidirect
343. C3×C5C2×#548semidirect
355. C2×D3C2×C5×#41semidirect
363. C2×C5#5576direct (N × H)
377. Q8C2×C5×D12semidirect
392. C2×C3C5×#548semidirect
398. C5C2×#5576direct (N × H)
416. C3C2×C5×#548semidirect
419. C2C5×#5576direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#2392
C22×C5×D3×Q82
C2×C3×C4×C5×Q82
C5×#55762
C2×C5×#8972
C2×C5×#8952
C2×C5×#8752
C2×C4×C5×D122
C2×C5×#5483
C2×#55765