Finite Group Databasebeta

← #14282938

Subgroups — #14282938

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5622C211yesmaximal, characteristic
2#11748667C211yesmaximal, characteristic
3#11748329C211yesmaximal, characteristic
4#11746359C211yesmaximal, characteristic
5C3×#64758C211yesmaximal, characteristic
6#11745963C211yesmaximal, characteristic
7#11745962C211yesmaximal, characteristic
8#10602628—31nomaximal, Hall
9C3×C5×C22⋊Q8C2211yescharacteristic
10C5×Q8×Dic3C411yescharacteristic, complemented
11C5×#944C411yescharacteristic, complemented
12C5×#924C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C5×#889—21no—
15C5×#888C411yescharacteristic, complemented
16C5×#887—21no—
17C5×#887C411yescharacteristic, complemented
18C5×#878—21no—
19C5×#877—21no—
20C2×Dic3⋊Dic5—21no—
21C2×Dic3×Dic5—21no—
22#91115C2211yescharacteristic
23#91114C2211yescharacteristic
24C3×#3963C2211yescharacteristic
25C3×C22⋊#168C2211yescharacteristic
26Dic3⋊#168—21no—
27Dic3×#168—21no—
28#68309—51nomaximal, Hall
29C5×#533—31no—
30C4×C4⋊Dic5—31no—
31#65214—31no—
32#64758—31no—
33#64758—31no—
34#64758—31no—
35#64758D311yescharacteristic
36C2×C3×C5×Q8C2×C411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C22×C5×Dic3D411yescharacteristic
40C22×C5×Dic3D411yescharacteristic
41C2×C5×Dic6—21no—
42C2×C5×Dic6C2×C411yescharacteristic
43C22×C3×Dic5—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4C2×C411yescharacteristic
46C3×C5×C22⋊C4—21no—
47C2×C3⋊#168—21no—
48C5×#217—21no—
49C5×C4⋊Dic3—21no—
50C5×C4⋊Dic3C2×C411yescharacteristic
51C5×Dic3⋊C4—21no—
52C5×Dic3⋊C4—21no—
53C5×Dic3⋊C4—21no—
54C5×Dic3⋊C4—21no—
55C5×Dic3⋊C4—21no—
56C4×C5×Dic3—21no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3C2×C411yescharacteristic
59C4×C5×Dic3C2×C411yescharacteristic
60C4×C5×Dic3D411yescharacteristic
61C4×C5×Dic3D411yescharacteristic
62C2×C3×#168—21no—
63Dic3⋊Dic5—24no—
64Dic3×Dic5—42no—
65#5622D511yescharacteristic
66#4718—51no—
67#4649—51no—
68C3×#328—51no—
69#4538—51no—
70#4467—51no—
71#4452—51no—
72C5×C4⋊Q8—31no—
73C5×C4⋊Q8—31no—
74C5×#125—61no—
75C5×C22⋊Q8—31no—
76C5×C22⋊Q8—61no—
77C5×C22⋊Q8C2×D311yescharacteristic
78C4×C5×Q8—31no—
79C4×C5×Q8—31no—
80C4×C5×Q8—61no—
81C5×#117—61no—
82C2×C42×C5—31no—
83C2×C4⋊Dic5—61no—
84C2×C4×Dic5—61no—
85#3963—31no—
86#3963C2×D311yescharacteristic
87C22⋊#168—31no—
88C22⋊#168C2×D311yescharacteristic
89C4⋊#168—61no—
90C4×#168—61no—
91#1503—151noSylow
92C23×C3×C5C2×D411yescharacteristic
93C3×C5×Q8—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×Dic15—22no—
100C2×Dic15—41no—
101C2×Dic15—41no—
102C2×C5×Dic3—22no—
103C2×C5×Dic3—21no—
104C2×C5×Dic3—21no—
105C2×C5×Dic3C22⋊C411yescharacteristic
106C2×C5×Dic3C22⋊C411yescharacteristic
107C2×C5×Dic3C22⋊C411yescharacteristic
108C2×C5×Dic3C22⋊C411yescharacteristic
109C2×C5×Dic3SD1612yes—
110C2×C5×Dic3Q1612yes—
111C5×Dic6—42no—
112C5×Dic6—41no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5—41no—
116C3⋊#168—41no—
117C3×#168—41no—
118C3×C22⋊Q8C2×D511yescharacteristic
119Q8×Dic3Dic511yescharacteristic, complemented
120#944Dic511yescharacteristic, complemented
121#924C2×D511yescharacteristic
122C2×Dic3⋊C4—101no—
123C2×C4×Dic3—101no—
124C2×C4×Dic3C2×D511yescharacteristic
125#889—21no—
126#888Dic511yescharacteristic, complemented
127#887—21no—
128#887Dic511yescharacteristic, complemented
129#878—21no—
130#877—21no—
131C3×C22⋊C8—51no—
132C3×#95—51no—
133C3⋊#95—51no—
134#830—51no—
135Dic3⋊C8—101no—
136C8×Dic3—101no—
137C2×C5×Q8—31no—
138C2×C5×Q8—61no—
139C2×C5×Q8C4×D311yescharacteristic
140C22×C4×C5—31no—
141C22×C4×C5—31no—
142C22×C4×C5C22×D311yescharacteristic
143C22×Dic5—21no—
144C22×Dic5—61no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—31no—
147C5×C4⋊C4—61no—
148C5×C4⋊C4—61no—
149C5×C4⋊C4—61no—
150C5×C4⋊C4—61no—
151C5×C4⋊C4—61no—
152C5×C4⋊C4—61no—
153C5×C4⋊C4C4×D311yescharacteristic
154C5×C22⋊C4—21no—
155C5×C22⋊C4—61no—
156C42×C5—31no—
157C42×C5—31no—
158C42×C5—31no—
159C42×C5—31no—
160C42×C5—61no—
161C42×C5—61no—
162C4⋊Dic5—64no—
163C4×Dic5—122no—
164C2×#168—21no—
165C2×#168—61no—
166#533—31no—
167#431—151no—
168C4×C4⋊C4—151no—
169#328—51no—
170#328—151no—
171#328—151no—
172#328—151no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5C2×C22⋊C411yescharacteristic
176C22×C3×C5C2×SD1611yescharacteristic
177C22×C3×C5C2×Q1611yescharacteristic
178C3×C4×C5—21no—
179C3×C4×C5—21no—
180C3×C4×C5—41no—
181C3×C4×C5—41no—
182Dic15—42no—
183C3×Dic5—42no—
184C5×Dic3—21no—
185C5×Dic3—41no—
186C5×Dic3—41no—
187C5×Dic3#10312yes—
188C5×Dic3#10312yes—
189C5×Dic3#10412yes—
190C2×C3×Q8C2×Dic511yescharacteristic
191C22×C3×C4—101no—
192C22×C3×C4C22×D511yescharacteristic
193C22×Dic3—101no—
194C22×Dic3C22⋊D511yescharacteristic
195C22×Dic3C22⋊D511yescharacteristic
196C2×Dic6—21no—
197C2×Dic6C2×Dic511yescharacteristic
198C2×C3×C8—101no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4C2×Dic511yescharacteristic
201C3×C22⋊C4—21no—
202#217—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×Dic511yescharacteristic
205Dic3⋊C4—21no—
206Dic3⋊C4—21no—
207Dic3⋊C4—21no—
208Dic3⋊C4—21no—
209Dic3⋊C4—21no—
210Dic3⋊C4—104no—
211C4×Dic3—21no—
212C4×Dic3—21no—
213C4×Dic3—202no—
214C4×Dic3C2×Dic511yescharacteristic
215C4×Dic3C2×Dic511yescharacteristic
216C4×Dic3C22⋊D511yescharacteristic
217C4×Dic3C22⋊D511yescharacteristic
218C2×C3⋊C8—101no—
219C23×C5D3×D411yescharacteristic
220C5×Q8—41no—
221C5×Q8—121no—
222C5×Q8—122no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—31no—
226C2×C4×C5—31no—
227C2×C4×C5—31no—
228C2×C4×C5—32no—
229C2×C4×C5—32no—
230C2×C4×C5—31no—
231C2×C4×C5—41no—
232C2×C4×C5—62no—
233C2×C4×C5—61no—
234C2×C4×C5—61no—
235C2×C4×C5C2×C4×D311yescharacteristic
236C2×C4×C5D3×D411yescharacteristic
237C2×Dic5—22no—
238C2×Dic5—41no—
239C2×Dic5—41no—
240C2×Dic5—62no—
241C2×Dic5—121no—
242C2×Dic5—121no—
243#168—41no—
244#168—121no—
245C4⋊Q8—31no—
246C4⋊Q8—31no—
247#125—61no—
248C22⋊Q8—31no—
249C22⋊Q8—61no—
250C22⋊Q8D3×D511yescharacteristic
251C4×Q8—31no—
252C4×Q8—31no—
253C4×Q8—61no—
254#117—61no—
255C2×C4⋊C4—301no—
256C2×C42—31no—
257C2×C42—301no—
258C4⋊C8—301no—
259C22⋊C8—51no—
260C22⋊C8—151no—
261C4×C8—301no—
262#95—51no—
263#95—151no—
264C2×C3×C5—21no—
265C2×C3×C5—21no—
266C2×C3×C5#41011yescharacteristic
267C2×C3×C5C2×#10311yescharacteristic
268C2×C3×C5C2×#10411yescharacteristic
269C23×C3C2×C22⋊D511yescharacteristic
270C3×Q8—41no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—41no—
274C2×C3×C4—102no—
275C2×C3×C4—201no—
276C2×C3×C4—201no—
277C2×C3×C4C22×Dic511yescharacteristic
278C2×C3×C4C2×C22⋊D511yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—102no—
283C2×Dic3—201no—
284C2×Dic3—201no—
285C2×Dic3#68812yes—
286C2×Dic3#68912yes—
287C2×Dic3C22⋊Dic511yescharacteristic
288C2×Dic3C22⋊Dic511yescharacteristic
289C2×Dic3C22⋊Dic511yescharacteristic
290C2×Dic3C22⋊Dic511yescharacteristic
291Dic6—42no—
292Dic6—41no—
293C3×C8—201no—
294C3⋊C8—201no—
295C22×C5—21no—
296C22×C5—21no—
297C22×C5D3×C22⋊C411yescharacteristic
298C22×C5D3×SD1611yescharacteristic
299C22×C5D3×Q1611yescharacteristic
300C4×C5—21no—
301C4×C5—21no—
302C4×C5—32no—
303C4×C5—32no—
304C4×C5—32no—
305C4×C5—41no—
306C4×C5—41no—
307C4×C5—61no—
308C4×C5—121no—
309C4×C5—121no—
310Dic5—42no—
311Dic5—122no—
312C2×Q8—31no—
313C2×Q8—61no—
314C2×Q8D3×Dic511yescharacteristic
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—101no—
318C22×C4—301no—
319C22×C4C2×D3×D511yescharacteristic
320C2×C8—101no—
321C2×C8—301no—
322C4⋊C4—21no—
323C4⋊C4—31no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—304no—
331C4⋊C4D3×Dic511yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—61no—
334C42—31no—
335C42—31no—
336C42—31no—
337C42—31no—
338C42—61no—
339C42—61no—
340C42—602no—
341C3×C5#150311yescharacteristic, complemented, Hall
342C22×C3—21no—
343C22×C3—21no—
344C22×C3C2×#68811yescharacteristic
345C22×C3C2×#68911yescharacteristic
346C22×C3C2×C22⋊Dic511yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—202no—
352Dic3—21no—
353Dic3—41no—
354Dic3—41no—
355Dic3—202no—
356Dic3#396712yes—
357Dic3#396712yes—
358Dic3#396912yes—
359C2×C5—21no—
360C2×C5—21no—
361C2×C5#474011yescharacteristic
362C2×C5D3×#10311yescharacteristic
363C2×C5D3×#10411yescharacteristic
364C23D3×C22⋊D511yescharacteristic
365Q8—41no—
366Q8—121no—
367Q8—122no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—41no—
377C2×C4—62no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—102no—
381C2×C4—201no—
382C2×C4—201no—
383C2×C4—302no—
384C2×C4—601no—
385C2×C4—601no—
386C2×C4C2×D3×Dic511yescharacteristic
387C2×C4D3×C22⋊D511yescharacteristic
388C8—201no—
389C8—601no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3C2×#396711yescharacteristic
393C2×C3C2×#396911yescharacteristic
394C2×C3#6552911yescharacteristic
395C5#6830911yescharacteristic, complemented, Sylow
396C22—21no—
397C22—21no—
398C22D3×#68811yescharacteristic
399C22D3×#68911yescharacteristic
400C22D3×C22⋊Dic511yescharacteristic
401C4—21no—
402C4—21no—
403C4—32no—
404C4—32no—
405C4—32no—
406C4—41no—
407C4—41no—
408C4—61no—
409C4—121no—
410C4—121no—
411C4—202no—
412C4—602no—
413C3#1060262811yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2#1174917311yescharacteristic
417C2D3×#396711yescharacteristic
418C2D3×#396911yescharacteristic
subgroupG/H
centerC22D3×C22⋊Dic5
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×C22⋊Q8C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C2×D4
cosocleC22×C3×C4×C5C23
hypercenterC22⋊Q8D3×D5
hypocenterC3×C5#1503
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
10. C5×Q8×Dic3C4semidirect
11. C5×#944C4semidirect
15. C5×#888C4semidirect
17. C5×#887C4semidirect
119. Q8×Dic3Dic5semidirect
120. #944Dic5semidirect
126. #888Dic5semidirect
128. #887Dic5semidirect
341. C3×C5#1503semidirect
395. C5#68309semidirect
413. C3#10602628semidirect

Maximal subgroups

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

subgroupindex
C5×#56222
#117486672
#117483292
#117463592
C3×#647582
#117459632
#117459622
#106026283
#683095