Finite Group Databasebeta

← #14295650

Subgroups — #14295650

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5522C211yesmaximal, characteristic
2#11748611C211yesmaximal, characteristic, complemented
3C2×#90979C211yesmaximal, characteristic
4C2×#90978C211yesmaximal, characteristic
5Dic3×#681C211yesmaximal, characteristic, complemented
6#11746353C211yesmaximal, characteristic, complemented
7C3×#64752C211yesmaximal, characteristic, complemented
8#10602436—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C2×C4×Dic15C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C5×#893C412yescomplemented
15C5×#887C412yescomplemented
16C5×#67⋊C4—22no—
17C5×#872—21no—
18C42×C5×D3—21no—
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21#90994—22no—
22#90979C414yescomplemented
23#90978C414yescomplemented
24#90953—21no—
25Dic3×#168—21no—
26#68729—51nomaximal, Hall
27C4×C5×#41—31no—
28C2×#3969—31no—
29C2×#3969—31no—
30#65221—31no—
31C4×#681—31no—
32#64752—31no—
33#64752D311yescharacteristic, complemented
34C22×C3×C4×C5C2311yescharacteristic
35C22×Dic15—21no—
36C2×C5×#67—21no—
37C22×C5×Dic3D412yes—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235C2×C412yes—
41C5×#235C2×C412yes—
42C2×C5×D12C2×C411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C4×C5×D3—22no—
45C2×C5×Dic6C2×C411yescharacteristic
46C22×C3×Dic5—21no—
47C3×C5×C4⋊C4C2×C412yes—
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C4×Dic15—21no—
51C4×Dic15C2×C412yescomplemented
52#6710—21no—
53#6710C2×C412yescomplemented
54C2×C3⋊#168—21no—
55C5×(C2×D3)⋊C4—22no—
56C5×Dic3⋊C4—22no—
57C4×C5×Dic3—22no—
58C4×C5×Dic3C2×C412yes—
59C4×C5×Dic3D411yescharacteristic
60C4×C5×Dic3D411yescharacteristic
61C3×C4×Dic5—21no—
62C3×C4×Dic5C2×C412yescomplemented
63C3×#681—21no—
64C3×#681C2×C412yescomplemented
65C2×C3×#168—21no—
66#5522D511yescharacteristic
67C2×#825—51no—
68Dic3×M16—51no—
69#4932—51no—
70C2×#805—51no—
71C3×#339—51no—
72#4532—51no—
73C2×C5×#41—31no—
74C4×C5×Q8—32no—
75C4×C5×D4—32no—
76C4×C5×D4—62no—
77C5×#117—61no—
78C5×#117C2×D311yescharacteristic
79C2×C42×C5—31no—
80C2×C42×C5—61no—
81C2×C4×Dic5—31no—
82C2×C4×Dic5C2×D311yescharacteristic
83C2×#681—31no—
84C2×#681C2×D311yescharacteristic
85#3969—34no—
86#3969—34no—
87#3963—62no—
88#3935—61no—
89C4×#168—61no—
90#1759—151noSylow
91C23×C3×C5C2×D411yescharacteristic
92C22×C5×D3—21no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5C22×C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×C4×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5#4111yescharacteristic
99C2×C3×C4×C5#4111yescharacteristic
100C2×Dic15—22no—
101C2×Dic15—42no—
102C5×#67—22no—
103C5×#67—22no—
104C2×C5×Dic3—21no—
105C2×C5×Dic3—42no—
106C2×C5×Dic3C22⋊C412yes—
107C2×C5×Dic3C22⋊C412yes—
108C5×D12—21no—
109C5×D12C4212yes—
110C4×C5×D3—22no—
111C4×C5×D3—22no—
112C4×C5×D3—44no—
113C5×Dic6—21no—
114C5×Dic6C4212yes—
115C2×C3×Dic5—22no—
116C2×C3×Dic5—42no—
117C3⋊#168—22no—
118C3×#168—22no—
119C2×#235C2×D511yescharacteristic
120C2×C3×M16—51no—
121C3×#117C2×D511yescharacteristic
122C2×C3×C42—51no—
123C2×C4×Dic3—51no—
124C2×C4×Dic3C2×D511yescharacteristic
125C2×#208—51no—
126#893Dic512yescomplemented
127#887Dic512yescomplemented
128#67⋊C4—22no—
129#872—21no—
130C42×D3—21no—
131C3⋊#95—102no—
132#825—54no—
133#815—101no—
134C8×Dic3—101no—
135#805—54no—
136C5×#41—32no—
137C5×#41—32no—
138C5×#41—61no—
139C5×#41—61no—
140C2×C5×Q8—31no—
141C2×C5×D4—31no—
142C2×C5×D4—61no—
143C22×C4×C5—32no—
144C22×C4×C5—62no—
145C22×C4×C5—61no—
146C22×C4×C5C22×D311yescharacteristic
147C22×Dic5—21no—
148C22×Dic5—61no—
149C5×C4⋊C4—62no—
150C5×C4⋊C4C4×D312yes—
151C5×C22⋊C4—21no—
152C5×C22⋊C4—62no—
153C42×C5—21no—
154C42×C5—32no—
155C42×C5—31no—
156C42×C5—31no—
157C42×C5—62no—
158C4×Dic5—21no—
159C4×Dic5—32no—
160C4×Dic5—61no—
161C4×Dic5C4×D312yescomplemented
162#681—21no—
163#681—32no—
164#681—61no—
165#681C4×D312yescomplemented
166C2×#168—21no—
167C2×#168—61no—
168C4×#41—31no—
169C2×#104—151no—
170C2×#104—151no—
171C4×M16—151no—
172#376—151no—
173#339—51no—
174#339—151no—
175C22×C3×C5—22no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C22×C3×C5C4×D411yescharacteristic
178C22×C3×C5C4×D411yescharacteristic
179C2×C5×D3—22no—
180C2×C5×D3—42no—
181C3×C4×C5—42no—
182C3×C4×C5C2×C4211yescharacteristic
183C3×C4×C5#11711yescharacteristic
184C3×C4×C5C4×D411yescharacteristic
185C3×C4×C5C4×D411yescharacteristic
186Dic15—42no—
187C3×Dic5—42no—
188C5×Dic3—22no—
189C5×Dic3—22no—
190C5×Dic3—22no—
191C22×C3×C4—101no—
192C22×C3×C4C22×D511yescharacteristic
193C2×#67—21no—
194C22×Dic3—101no—
195C22×Dic3C22⋊D512yes—
196#235—21no—
197#235—21no—
198#235C4×D512yes—
199#235C4×D512yes—
200C2×D12C2×Dic511yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—22no—
203C2×Dic6C2×Dic511yescharacteristic
204C3×M16—52no—
205C3×M16—101no—
206C2×C3×C8—101no—
207C3×C4⋊C4C2×Dic512yes—
208C3×C22⋊C4—21no—
209C3×C42—21no—
210C3×C42—52no—
211C3×C42—101no—
212(C2×D3)⋊C4—22no—
213Dic3⋊C4—22no—
214C4×Dic3—22no—
215C4×Dic3—52no—
216C4×Dic3—101no—
217C4×Dic3C2×Dic512yes—
218C4×Dic3C22⋊D511yescharacteristic
219C4×Dic3C22⋊D511yescharacteristic
220#208—52no—
221#208—101no—
222C2×C3⋊C8—101no—
223C23×C5—61no—
224C23×C5D3×D411yescharacteristic
225C5×Q8—32no—
226C5×Q8—61no—
227C5×D4—32no—
228C5×D4—62no—
229C5×D4—62no—
230C5×D4—61no—
231C2×C4×C5—22no—
232C2×C4×C5—32no—
233C2×C4×C5—32no—
234C2×C4×C5—61no—
235C2×C4×C5—62no—
236C2×C4×C5—62no—
237C2×C4×C5—124no—
238C2×C4×C5—122no—
239C2×C4×C5C2×C4×D311yescharacteristic
240C2×C4×C5C2×C4×D311yescharacteristic
241C2×C4×C5C2×C4×D311yescharacteristic
242C2×C4×C5D3×D411yescharacteristic
243C2×C4×C5#23711yescharacteristic
244C2×C4×C5#23911yescharacteristic
245C2×Dic5—22no—
246C2×Dic5—42no—
247C2×Dic5—62no—
248C2×Dic5—122no—
249#168—22no—
250#168—62no—
251C2×#41—31no—
252C2×M16—51no—
253C2×M16—151no—
254C4×Q8—32no—
255C4×D4—32no—
256C4×D4—62no—
257#117—61no—
258#117D3×D511yescharacteristic
259C2×C42—31no—
260C2×C42—51no—
261C2×C42—61no—
262C2×C42—151no—
263#104—154no—
264#104—154no—
265#97—301no—
266C4×C8—301no—
267#95—302no—
268C2×C3×C5—21no—
269C2×C3×C5—21no—
270C2×C3×C5C4×C22⋊C411yescharacteristic
271C2×C3×C5#42112yes—
272C5×D3—42no—
273C23×C3C2×C22⋊D511yescharacteristic
274C22×D3—21no—
275C2×C3×C4—22no—
276C2×C3×C4—102no—
277C2×C3×C4—202no—
278C2×C3×C4C2×C4×D511yescharacteristic
279C2×C3×C4C2×C4×D511yescharacteristic
280C2×C3×C4#70911yescharacteristic
281C2×C3×C4#70911yescharacteristic
282C2×C3×C4C22×Dic511yescharacteristic
283C2×C3×C4C2×C22⋊D511yescharacteristic
284#67—22no—
285#67—22no—
286C2×Dic3—21no—
287C2×Dic3—42no—
288C2×Dic3—102no—
289C2×Dic3—202no—
290C2×Dic3C22⋊Dic512yes—
291C2×Dic3C22⋊Dic512yes—
292D12—21no—
293D12C4×Dic512yes—
294C4×D3—22no—
295C4×D3—22no—
296C4×D3—44no—
297Dic6—21no—
298Dic6C4×Dic512yes—
299C3×C8—102no—
300C3⋊C8—102no—
301C22×C5—22no—
302C22×C5—62no—
303C22×C5—122no—
304C22×C5D3×C22⋊C411yescharacteristic
305C22×C5#67⋊C411yescharacteristic
306C22×C5#89311yescharacteristic
307C4×C5—42no—
308C4×C5—62no—
309C4×C5—62no—
310C4×C5—62no—
311C4×C5C42×D311yescharacteristic
312C4×C5#67⋊C411yescharacteristic
313C4×C5#89211yescharacteristic
314C4×C5#89311yescharacteristic
315Dic5—42no—
316Dic5—122no—
317#41—32no—
318#41—32no—
319#41—61no—
320#41—61no—
321C2×Q8—31no—
322C2×D4—31no—
323C2×D4—61no—
324C22×C4—32no—
325C22×C4—62no—
326C22×C4—61no—
327C22×C4—101no—
328C22×C4—301no—
329C22×C4C2×D3×D511yescharacteristic
330M16—52no—
331M16—101no—
332M16—152no—
333M16—301no—
334C2×C8—101no—
335C2×C8—301no—
336C4⋊C4—62no—
337C4⋊C4D3×Dic512yes—
338C22⋊C4—21no—
339C22⋊C4—62no—
340C42—21no—
341C42—32no—
342C42—31no—
343C42—31no—
344C42—52no—
345C42—62no—
346C42—101no—
347C42—152no—
348C42—301no—
349C3×C5#175911yescharacteristic, complemented, Hall
350C22×C3—22no—
351C22×C3C4×C22⋊D511yescharacteristic
352C22×C3C4×C22⋊D511yescharacteristic
353C22×C3C2×C22⋊Dic511yescharacteristic
354C2×D3—22no—
355C2×D3—42no—
356C3×C4—42no—
357C3×C4—202no—
358C3×C4C2×C4×Dic511yescharacteristic
359C3×C4#407211yescharacteristic
360C3×C4C4×C22⋊D511yescharacteristic
361C3×C4C4×C22⋊D511yescharacteristic
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—202no—
366C2×C5—21no—
367C2×C5—21no—
368C2×C5—122no—
369C2×C5#466611yescharacteristic
370C2×C5#482112yes—
371C23—61no—
372C23D3×C22⋊D511yescharacteristic
373Q8—32no—
374Q8—61no—
375D4—32no—
376D4—62no—
377D4—62no—
378D4—61no—
379C2×C4—22no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—61no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—102no—
386C2×C4—124no—
387C2×C4—122no—
388C2×C4—202no—
389C2×C4—302no—
390C2×C4—602no—
391C2×C4#676511yescharacteristic
392C2×C4#676911yescharacteristic
393C2×C4C4×D3×D511yescharacteristic
394C2×C4C4×D3×D511yescharacteristic
395C2×C4C2×D3×Dic511yescharacteristic
396C2×C4D3×C22⋊D511yescharacteristic
397C8—102no—
398C8—302no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3C4×C22⋊Dic511yescharacteristic
402C2×C3#6552512yes—
403D3—42no—
404C5#6872911yescharacteristic, complemented, Sylow
405C22—22no—
406C22—62no—
407C22—122no—
408C22#9144111yescharacteristic
409C22#9144611yescharacteristic
410C22D3×C22⋊Dic511yescharacteristic
411C4—42no—
412C4—62no—
413C4—62no—
414C4—62no—
415C4—202no—
416C4—602no—
417C4#9133811yescharacteristic
418C4C4×D3×Dic511yescharacteristic
419C4#9144111yescharacteristic
420C4#9144611yescharacteristic
421C3#1060243611yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—122no—
425C2#1174871911yescharacteristic
426C2#1174942912yes—
subgroupG/H
centerC2×C4D3×C22⋊D5
commutatorC3×C4×C5C2×C42
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1759
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11748611C2semidirect
5. Dic3×#681C2semidirect
6. #11746353C2semidirect
7. C3×#64752C2semidirect
14. C5×#893C4semidirect
15. C5×#887C4semidirect
22. #90979C4semidirect
23. #90978C4semidirect
33. #64752D3semidirect
51. C4×Dic15C2×C4semidirect
53. #6710C2×C4semidirect
62. C3×C4×Dic5C2×C4semidirect
64. C3×#681C2×C4semidirect
126. #893Dic5semidirect
127. #887Dic5semidirect
161. C4×Dic5C4×D3semidirect
165. #681C4×D3semidirect
349. C3×C5#1759semidirect
404. C5#68729semidirect
421. C3#10602436semidirect

Maximal subgroups

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

subgroupindex
C5×#55222
#117486112
C2×#909792
C2×#909782
Dic3×#6812
#117463532
C3×#647522
#106024363
#687295