Finite Group Databasebeta

← #11753428

Subgroups — #11753428

Order 960.

580 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6771C211yesmaximal, characteristic, complemented
2C2×D5×Dic6C211yesmaximal, characteristic, complemented
3#91785C211yesmaximal, characteristic, complemented
4C5×#896C211yesmaximal, characteristic, complemented
5C3×#4039C211yesmaximal, characteristic, complemented
6#91462C211yesmaximal, characteristic, complemented
7#91455C211yesmaximal, characteristic, complemented
8#91448C211yesmaximal, characteristic, complemented
9#91441C211yesmaximal, characteristic, complemented
10#91409C211yesmaximal, characteristic, complemented
11#91408C211yesmaximal, characteristic, complemented
12#91396C211yesmaximal, characteristic, complemented
13#91350C211yesmaximal, characteristic, complemented
14#91349C211yesmaximal, characteristic, complemented
15#91323C211yesmaximal, characteristic, complemented
16#65967—31nomaximal, Hall
17C2×C4×D15C2211yescharacteristic
18C2×Dic30C2211yescharacteristic
19C2×C4×C5×D3C2211yescharacteristic
20C2×C5×Dic6C2211yescharacteristic
21C2×C3×D20C2211yescharacteristic
22C2×C3×C4×D5C2211yescharacteristic
23C2×C3×C4×D5C2211yescharacteristic
24C2×#1204C2211yescharacteristic
25C2×#1204C2211yescharacteristic
26C2×#1203C2211yescharacteristic
27C2×#1202C2211yescharacteristic
28C2×#1202C2211yescharacteristic
29C2×#1201C2211yescharacteristic
30C2×#1201C2211yescharacteristic
31C2×D5×Dic3C2211yescharacteristic
32C2×D5×Dic3C2211yescharacteristic
33#6771—24no—
34D5×Dic6—24no—
35C3×C5×C4⋊C4C2211yescharacteristic
36#6714C2211yescharacteristic
37#6714C2211yescharacteristic
38#6712C2211yescharacteristic
39#6712C2211yescharacteristic
40C5×(C2×D3)⋊C4C2211yescharacteristic
41C5×(C2×D3)⋊C4C2211yescharacteristic
42C5×Dic3⋊C4C2211yescharacteristic
43C5×Dic3⋊C4C2211yescharacteristic
44C3×#685C2211yescharacteristic
45C3×#685C2211yescharacteristic
46C3×C4×Dic5C2211yescharacteristic
47#6668C2211yescharacteristic
48#6667C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50#6665C2211yescharacteristic
51#6665C2211yescharacteristic
52Dic3×Dic5C2211yescharacteristic
53Dic3×Dic5C2211yescharacteristic
54#5552—51nomaximal, Hall
55C2×Q8×D5—31no—
56C2×#709—31no—
57C5×C22⋊Q8—31no—
58#4093—31no—
59C4×C22⋊D5—31no—
60#4042—31no—
61#4042—31no—
62#4041—31no—
63#4039—31no—
64#4039D311yescharacteristic, complemented
65#4033—31no—
66#4031—31no—
67#4031—31no—
68#4030—31no—
69#4030—31no—
70C22×D15C2311yescharacteristic
71C22×C5×D3C2311yescharacteristic
72C22×C3×D5C2311yescharacteristic
73C22×C3×D5C2311yescharacteristic
74C2×C3×C4×C5C2311yescharacteristic
75C2×C3×C4×C5C2311yescharacteristic
76C2×C3×C4×C5C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C2×Dic15C2311yescharacteristic
80C4×D15—22no—
81Dic30—22no—
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C4×C5×D3—22no—
86C5×Dic6—22no—
87C2×C3×Dic5C2311yescharacteristic
88C2×C3×Dic5C2311yescharacteristic
89C3×D20—22no—
90C3×C4×D5—22no—
91C3×C4×D5—22no—
92#1204—22no—
93#1204D414yescomplemented
94#1203—22no—
95#1202—22no—
96#1202—22no—
97#1201—22no—
98#1201—22no—
99D5×Dic3—22no—
100D5×Dic3—22no—
101C2×#235—51no—
102C22×Dic6—51no—
103C3×C4×D4—51no—
104#939—51no—
105#939—51no—
106#930—51no—
107#924—51no—
108#896—51no—
109#896D511yescharacteristic, complemented
110#885—51no—
111#885—51no—
112#67⋊C4—51no—
113#67⋊C4—51no—
114#875—51no—
115C4×Dic6—51no—
116C2×C5×Q8—31no—
117C22×C4×C5—31no—
118C2×C22⋊D5—31no—
119C2×C22⋊D5—31no—
120C2×C22⋊D5—31no—
121Q8×D5—64no—
122#709—64no—
123C2×D20—31no—
124C2×D20—31no—
125C2×D20C2×D311yescharacteristic
126C2×C4×D5—31no—
127C2×C4×D5—31no—
128C2×C4×D5—31no—
129C2×C4×D5C2×D311yescharacteristic
130C2×C4×D5C2×D311yescharacteristic
131C2×Dic10—31no—
132C2×Dic10—31no—
133C2×Dic10—31no—
134C5×C4⋊C4—31no—
135C5×C4⋊C4—31no—
136C5×C4⋊C4C2×D311yescharacteristic
137C5×C22⋊C4—31no—
138C5×C22⋊C4—31no—
139C22⋊Dic5—31no—
140#685—31no—
141#685—31no—
142#685—31no—
143#685—31no—
144#685—31no—
145#685C2×D311yescharacteristic
146#685C2×D311yescharacteristic
147#683—31no—
148#683—31no—
149#683—31no—
150C4×Dic5—31no—
151C4×Dic5—31no—
152C4×Dic5C2×D311yescharacteristic
153#548—151noSylow
154C22×C3×C5C2411yescharacteristic
155C2×D15—21no—
156C2×D15—21no—
157C2×D15—21no—
158C2×C5×D3—21no—
159C2×C5×D3—21no—
160C2×C5×D3—21no—
161C2×C3×D5—21no—
162C2×C3×D5—21no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5#4112yescomplemented
167C3×C4×C5—21no—
168C3×C4×C5—21no—
169C3×C4×C5—21no—
170Dic15—21no—
171Dic15—21no—
172Dic15C2×D412yes—
173C3×Dic5—21no—
174C3×Dic5C2×D412yes—
175C5×Dic3—21no—
176C5×Dic3—21no—
177C5×Dic3C2×D412yes—
178C2×C3×D4—51no—
179C22×C3×C4—51no—
180C22×C3×C4—51no—
181C2×#67—51no—
182C2×#67—51no—
183C2×#67—51no—
184C2×#67—51no—
185C22×Dic3—51no—
186C22×Dic3—51no—
187#235—104no—
188C2×D12—51no—
189C2×C4×D3—51no—
190C2×C4×D3C2×D511yescharacteristic
191C2×Dic6—51no—
192C2×Dic6—51no—
193C2×Dic6—51no—
194C2×Dic6—104no—
195C2×Dic6C2×D511yescharacteristic
196C3×C4⋊C4C2×D511yescharacteristic
197C3×C22⋊C4—51no—
198C3×C22⋊C4—51no—
199C3×C42—51no—
200#217—51no—
201#217—51no—
202(C2×D3)⋊C4—51no—
203(C2×D3)⋊C4—51no—
204(C2×D3)⋊C4—51no—
205(C2×D3)⋊C4—51no—
206(C2×D3)⋊C4C2×D511yescharacteristic
207(C2×D3)⋊C4C2×D511yescharacteristic
208C4⋊Dic3—51no—
209Dic3⋊C4—51no—
210Dic3⋊C4—51no—
211Dic3⋊C4C2×D511yescharacteristic
212Dic3⋊C4C2×D511yescharacteristic
213C4×Dic3—51no—
214C4×Dic3—51no—
215C23×C5—31no—
216C22×D5—31no—
217C22×D5C22×D311yescharacteristic
218C22×D5C22×D311yescharacteristic
219C5×Q8—62no—
220C2×C4×C5—31no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—62no—
224C2×C4×C5C22×D311yescharacteristic
225C2×C4×C5C22×D311yescharacteristic
226C2×C4×C5C22×D311yescharacteristic
227C22⋊D5—62no—
228C22⋊D5—62no—
229C22⋊D5—62no—
230C2×Dic5—31no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5C22×D311yescharacteristic
234C2×Dic5C22×D311yescharacteristic
235D20—22no—
236D20—62no—
237D20—62no—
238C4×D5—22no—
239C4×D5—22no—
240C4×D5—62no—
241C4×D5—62no—
242C4×D5—62no—
243Dic10—34no—
244Dic10—62no—
245Dic10—62no—
246C2×#41—151no—
247C22×Q8—151no—
248#124—151no—
249#124—151no—
250#124—151no—
251C22⋊Q8—31no—
252C22⋊Q8—151no—
253C22⋊Q8—151no—
254#121—151no—
255#121—151no—
256#121—151no—
257C4×Q8—151no—
258C4×D4—51no—
259C4×D4—151no—
260C4×D4—151no—
261C2×C3×C5C22×D411yescharacteristic
262C2×C3×C5C2×#4111yescharacteristic
263C2×C3×C52−1+411yescharacteristic
264D15—41no—
265C3×D5—22no—
266C3×D5—41no—
267C5×D3—41no—
268C23×C3—51no—
269C23×C3—51no—
270C22×D3—51no—
271C22×D3C22×D511yescharacteristic
272C3×D4—102no—
273C2×C3×C4—51no—
274C2×C3×C4—51no—
275C2×C3×C4—102no—
276C2×C3×C4—102no—
277C2×C3×C4C22×D511yescharacteristic
278C2×C3×C4C22×D511yescharacteristic
279C2×C3×C4C22×D511yescharacteristic
280#67—102no—
281#67—102no—
282#67—102no—
283#67—102no—
284C2×Dic3—51no—
285C2×Dic3—51no—
286C2×Dic3—51no—
287C2×Dic3—102no—
288C2×Dic3—102no—
289C2×Dic3C22×D511yescharacteristic
290C2×Dic3C22×D511yescharacteristic
291C2×Dic3C22×D511yescharacteristic
292D12—102no—
293C4×D3—22no—
294C4×D3—102no—
295Dic6—22no—
296Dic6—54no—
297Dic6—102no—
298Dic6—102no—
299C22×C5—61no—
300C22×C5—61no—
301C22×C5—61no—
302C22×C5C23×D311yescharacteristic
303C2×D5—21no—
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—61no—
309C2×D5—61no—
310C2×D5—61no—
311C2×D5#23512yescomplemented
312C4×C5—21no—
313C4×C5—21no—
314C4×C5—21no—
315C4×C5—32no—
316C4×C5—61no—
317C4×C5—61no—
318Dic5—21no—
319Dic5—32no—
320Dic5—61no—
321Dic5—61no—
322Dic5D3×D412yes—
323#41—304no—
324C2×Q8—31no—
325C2×Q8—151no—
326C2×Q8—151no—
327C2×Q8—151no—
328C2×Q8—304no—
329C2×D4—51no—
330C2×D4—151no—
331C2×D4—151no—
332C2×D4—151no—
333C2×D4—151no—
334C2×D4—151no—
335C22×C4—31no—
336C22×C4—51no—
337C22×C4—51no—
338C22×C4—151no—
339C22×C4—151no—
340C22×C4—151no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—151no—
344C4⋊C4—151no—
345C4⋊C4—151no—
346C4⋊C4D3×D511yescharacteristic
347C22⋊C4—31no—
348C22⋊C4—31no—
349C22⋊C4—51no—
350C22⋊C4—51no—
351C22⋊C4—151no—
352C22⋊C4—151no—
353C22⋊C4—151no—
354C22⋊C4—151no—
355C22⋊C4—151no—
356C22⋊C4—151no—
357C42—51no—
358C42—151no—
359C42—151no—
360C3×C5#54811yescharacteristic, complemented, Hall
361C22×C3—52no—
362C22×C3—101no—
363C22×C3—101no—
364C22×C3—101no—
365C22×C3—101no—
366C22×C3—101no—
367C22×C3C23×D511yescharacteristic
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—21no—
371C2×D3—101no—
372C2×D3—101no—
373C2×D3—101no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—52no—
378C3×C4—101no—
379Dic3—21no—
380Dic3—21no—
381Dic3—52no—
382Dic3—101no—
383Dic3—101no—
384Dic3D4×D512yes—
385C2×C5—121no—
386C2×C5C2×#23511yescharacteristic
387C2×C5C2×D3×D411yescharacteristic
388C2×C5#101011yescharacteristic
389D5—22no—
390D5—41no—
391D5—121no—
392C23—31no—
393C23—51no—
394C23—51no—
395C23—151no—
396Q8—62no—
397Q8—154no—
398Q8—302no—
399Q8—302no—
400D4—102no—
401D4—302no—
402D4—302no—
403D4—302no—
404D4—302no—
405D4—302no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—62no—
412C2×C4—102no—
413C2×C4—102no—
414C2×C4—151no—
415C2×C4—151no—
416C2×C4—151no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4—302no—
420C2×C4C2×D3×D511yescharacteristic
421C2×C4C2×D3×D511yescharacteristic
422C2×C4C2×D3×D511yescharacteristic
423C2×C3—102no—
424C2×C3—201no—
425C2×C3C2×D4×D511yescharacteristic
426C2×C3#414711yescharacteristic
427C2×C3D5×#4111yescharacteristic
428D3—41no—
429D3—201no—
430C5#555211yescharacteristic, complemented, Sylow
431C22—52no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—101no—
436C22—101no—
437C22—101no—
438C22—101no—
439C22—101no—
440C22—301no—
441C22—301no—
442C22—301no—
443C22C22×D3×D511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—32no—
448C4—52no—
449C4—61no—
450C4—61no—
451C4—101no—
452C4—152no—
453C4—301no—
454C4—301no—
455C3#6596711yescharacteristic, complemented, Sylow
456C2—102no—
457C2—121no—
458C2—201no—
459C2—601no—
460C2D5×#23511yescharacteristic
461C2D3×D4×D511yescharacteristic
462C2#9202811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#548
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6771C2semidirect
2. C2×D5×Dic6C2semidirect
3. #91785C2semidirect
4. C5×#896C2semidirect
5. C3×#4039C2semidirect
6. #91462C2semidirect
7. #91455C2semidirect
8. #91448C2semidirect
9. #91441C2semidirect
10. #91409C2semidirect
11. #91408C2semidirect
12. #91396C2semidirect
13. #91350C2semidirect
14. #91349C2semidirect
15. #91323C2semidirect
64. #4039D3semidirect
93. #1204D4semidirect
109. #896D5semidirect
166. C2×C3×D5#41semidirect
311. C2×D5#235semidirect
360. C3×C5#548semidirect
430. C5#5552semidirect
455. C3#65967semidirect

Maximal subgroups

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

subgroupindex
C2×#67712
C2×D5×Dic62
#917852
C5×#8962
C3×#40392
#914622
#914552
#914482
#914412
#914092
#914082
#913962
#913502
#913492
#913232
#659673
#55525