Finite Group Databasebeta

← #11754452

Subgroups — #11754452

Order 960.

496 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6714C211yesmaximal, characteristic, complemented
2C5×#926C211yesmaximal, characteristic, complemented
3C3×#4075C211yesmaximal, characteristic, complemented
4#91576C211yesmaximal, characteristic, complemented
5#91572C211yesmaximal, characteristic
6#91561C211yesmaximal, characteristic, complemented
7#91559C211yesmaximal, characteristic, complemented
8#91419C211yesmaximal, characteristic, complemented
9#91413C211yesmaximal, characteristic, complemented
10#91403C211yesmaximal, characteristic, complemented
11#91395C211yesmaximal, characteristic, complemented
12#91380C211yesmaximal, characteristic, complemented
13#91356C211yesmaximal, characteristic, complemented
14#91353C211yesmaximal, characteristic, complemented
15#91337C211yesmaximal, characteristic, complemented
16#65861—31nomaximal, Hall
17C23×D15C2211yescharacteristic
18C22×C3×C4×C5C2211yescharacteristic
19C22×Dic15C2211yescharacteristic
20C2×C3×C22⋊D5C2211yescharacteristic
21C2×#1204C2211yescharacteristic, complemented
22C2×#1202C2211yescharacteristic
23C2×#1202C2211yescharacteristic
24C2×#1200C2211yescharacteristic
25C2×#1200C2211yescharacteristic
26C2×D5×Dic3C2211yescharacteristic, complemented
27#6714C2211yescharacteristic
28#6714C2211yescharacteristic
29#6714C2211yescharacteristic
30#6714C2211yescharacteristic
31C5×#217C2211yescharacteristic
32C5×#217C2211yescharacteristic
33C5×C4⋊Dic3C2211yescharacteristic, complemented
34C5×C4⋊Dic3C2211yescharacteristic, complemented
35C4×C5×Dic3C2211yescharacteristic, complemented
36C4×C5×Dic3C2211yescharacteristic, complemented
37C3×C22⋊Dic5C2211yescharacteristic
38C3×#685C2211yescharacteristic, complemented
39C3×#685C2211yescharacteristic, complemented
40C3×#683C2211yescharacteristic, complemented
41C3×#683C2211yescharacteristic, complemented
42Dic3⋊Dic5C2211yescharacteristic, complemented
43#6669C2211yescharacteristic, complemented
44#6668C2211yescharacteristic, complemented
45#6668C2211yescharacteristic, complemented
46#6667C2211yescharacteristic
47#6667C2211yescharacteristic
48#6667C2211yescharacteristic
49#6667C2211yescharacteristic
50#6665C2211yescharacteristic, complemented
51Dic3×Dic5C2211yescharacteristic, complemented
52#5658—51nomaximal, Hall
53C5×#117—31no—
54#4075D311yescharacteristic, complemented
55C2×#685—31no—
56#4044—31no—
57#4042—31no—
58#4040—31no—
59#4040—31no—
60#4028—31no—
61#4027—31no—
62D5×C22⋊C4—31no—
63#4024—31no—
64#4022—31no—
65#4021—31no—
66C4×D20—31no—
67#4018—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15—22no—
70C22×D15—22no—
71C22×D15—41no—
72C22×D15C2311yescharacteristic
73C22×D15C2311yescharacteristic
74C22×C3×D5C2311yescharacteristic
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78C2×Dic15—41no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×C5×Dic3C2311yescharacteristic
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C3×C22⋊D5—22no—
86C2×C3×Dic5C2311yescharacteristic
87C2×C3×Dic5C2311yescharacteristic
88C2×C3×Dic5C2311yescharacteristic
89#1204—41no—
90#1202—22no—
91#1202—41no—
92#1200—22no—
93#1200—41no—
94D5×Dic3—41no—
95C3×#123—51no—
96#937—51no—
97D4×Dic3—51no—
98C2×(C2×D3)⋊C4—51no—
99#926D511yescharacteristic, complemented
100#898—51no—
101#898—51no—
102#897—51no—
103#892—51no—
104#886—51no—
105#885—51no—
106#883—51no—
107#67⋊C4—51no—
108D3×C22⋊C4—51no—
109#878—51no—
110C23×D5—31no—
111C22×C4×C5C2×D311yescharacteristic
112C2×C22⋊D5C2×D311yescharacteristic
113C22×Dic5—31no—
114C2×D20—31no—
115C2×D20—31no—
116C2×C4×D5—31no—
117C2×C4×D5—31no—
118C2×C4×D5—31no—
119C2×Dic10—31no—
120C5×C4⋊C4—31no—
121C5×C4⋊C4—31no—
122C5×C22⋊C4—31no—
123C5×C22⋊C4—31no—
124C42×C5—31no—
125C42×C5—31no—
126C22⋊Dic5C2×D311yescharacteristic
127#685—31no—
128#685—31no—
129#685—31no—
130#685—31no—
131#685—31no—
132#685—31no—
133#685—31no—
134#685—31no—
135#685—31no—
136#685C2×D311yescharacteristic, complemented
137#685C2×D311yescharacteristic, complemented
138C4⋊Dic5—31no—
139#683—31no—
140#683—31no—
141#683—31no—
142#683C2×D311yescharacteristic, complemented
143#683C2×D311yescharacteristic, complemented
144C4×Dic5—31no—
145#551—151noSylow
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5#4112yes—
149C22×C3×C5C2411yescharacteristic
150C2×D15—21no—
151C2×D15—21no—
152C2×D15—21no—
153C2×D15—21no—
154C2×D15—21no—
155C2×D15—44no—
156C2×D15#4112yes—
157C2×C3×D5—21no—
158C2×C3×D5—21no—
159C2×C3×D5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162Dic15—21no—
163Dic15—21no—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C5×Dic3—21no—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C5×Dic3—21no—
171C23×D3—51no—
172C2×C3×D4—51no—
173C22×C3×C4C2×D511yescharacteristic
174C2×#67—51no—
175C2×#67—51no—
176C22×Dic3—51no—
177C22×Dic3—51no—
178C2×C4×D3—51no—
179C2×C4×D3—51no—
180C2×Dic6—51no—
181C3×C4⋊C4—51no—
182C3×C4⋊C4—51no—
183C3×C22⋊C4—51no—
184C3×C22⋊C4—51no—
185C3×C22⋊C4—51no—
186#217—51no—
187#217C2×D511yescharacteristic
188#217C2×D511yescharacteristic
189(C2×D3)⋊C4—51no—
190(C2×D3)⋊C4—51no—
191(C2×D3)⋊C4—51no—
192(C2×D3)⋊C4—51no—
193(C2×D3)⋊C4—51no—
194(C2×D3)⋊C4—51no—
195(C2×D3)⋊C4—51no—
196(C2×D3)⋊C4—51no—
197C4⋊Dic3—51no—
198C4⋊Dic3—51no—
199C4⋊Dic3C2×D511yescharacteristic, complemented
200C4⋊Dic3C2×D511yescharacteristic, complemented
201Dic3⋊C4—51no—
202Dic3⋊C4—51no—
203C4×Dic3—51no—
204C4×Dic3C2×D511yescharacteristic, complemented
205C4×Dic3C2×D511yescharacteristic, complemented
206C23×C5C22×D311yescharacteristic
207C22×D5—31no—
208C22×D5—31no—
209C22×D5—62no—
210C22×D5—62no—
211C22×D5—121no—
212C22×D5C22×D311yescharacteristic
213C2×C4×C5—22no—
214C2×C4×C5—31no—
215C2×C4×C5—31no—
216C2×C4×C5—31no—
217C2×C4×C5—31no—
218C2×C4×C5C22×D311yescharacteristic
219C2×C4×C5C22×D311yescharacteristic
220C22⋊D5—22no—
221C2×Dic5—31no—
222C2×Dic5—31no—
223C2×Dic5—121no—
224C2×Dic5C22×D311yescharacteristic
225C2×Dic5C22×D311yescharacteristic
226C2×Dic5C22×D311yescharacteristic
227D20—62no—
228D20—121no—
229C4×D5—62no—
230C4×D5—121no—
231C4×D5—121no—
232Dic10—121no—
233#126—151no—
234#126—151no—
235#124—151no—
236#123—51no—
237#123—151no—
238#123—151no—
239C22⋊Q8—151no—
240C22⋊Q8—151no—
241#120—151no—
242C4×D4—151no—
243C4×D4—151no—
244#117—31no—
245#117—151no—
246C2×C22⋊C4—151no—
247C2×C22⋊C4—151no—
248C2×C3×C5—22no—
249C2×C3×C5C2×#4111yescharacteristic
250C2×C3×C5C2×#4111yescharacteristic
251C2×C3×C5H(5,𝔽2)11yescharacteristic
252D15—22no—
253D15—41no—
254C3×D5—41no—
255C23×C3—51no—
256C23×C3C22×D511yescharacteristic
257C22×D3—51no—
258C22×D3—51no—
259C22×D3—102no—
260C22×D3—102no—
261C22×D3—201no—
262C3×D4—102no—
263C2×C3×C4—22no—
264C2×C3×C4—51no—
265C2×C3×C4—51no—
266C2×C3×C4—51no—
267C2×C3×C4C22×D511yescharacteristic
268C2×C3×C4C22×D511yescharacteristic
269#67—102no—
270#67—201no—
271C2×Dic3—51no—
272C2×Dic3—51no—
273C2×Dic3—201no—
274C2×Dic3—201no—
275C2×Dic3C22×D511yescharacteristic
276C2×Dic3C22×D511yescharacteristic
277C2×Dic3C22×D511yescharacteristic
278C2×Dic3C22×D511yescharacteristic
279C4×D3—102no—
280C4×D3—201no—
281Dic6—201no—
282C22×C5—21no—
283C22×C5—21no—
284C22×C5#23712yes—
285C22×C5C23×D311yescharacteristic
286C2×D5—21no—
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—32no—
290C2×D5—61no—
291C2×D5—61no—
292C2×D5—61no—
293C2×D5—61no—
294C2×D5—61no—
295C2×D5—124no—
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—61no—
299C4×C5—61no—
300C4×C5—61no—
301C4×C5—61no—
302Dic5—21no—
303Dic5—21no—
304Dic5—21no—
305Dic5—61no—
306Dic5—61no—
307C24—151no—
308C2×Q8—151no—
309C2×D4—51no—
310C2×D4—151no—
311C2×D4—151no—
312C22×C4—151no—
313C22×C4—151no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4D3×D511yescharacteristic
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—51no—
320C4⋊C4—51no—
321C4⋊C4—151no—
322C4⋊C4—151no—
323C4⋊C4—151no—
324C4⋊C4—151no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—51no—
328C22⋊C4—51no—
329C22⋊C4—51no—
330C22⋊C4—151no—
331C22⋊C4—151no—
332C22⋊C4—151no—
333C22⋊C4—151no—
334C22⋊C4—151no—
335C22⋊C4—151no—
336C22⋊C4—151no—
337C22⋊C4—151no—
338C22⋊C4—151no—
339C42—31no—
340C42—31no—
341C42—151no—
342C3×C5#55111yescharacteristic, complemented, Hall
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—101no—
346C22×C3—101no—
347C22×C3—101no—
348C22×C3#70912yes—
349C22×C3C23×D511yescharacteristic
350C2×D3—52no—
351C2×D3—101no—
352C2×D3—101no—
353C2×D3—101no—
354C2×D3—101no—
355C2×D3—101no—
356C2×D3—204no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—101no—
360C3×C4—101no—
361C3×C4—101no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—21no—
366Dic3—101no—
367Dic3—101no—
368C2×C5—22no—
369C2×C5C2×#23711yescharacteristic
370C2×C5D3×#4111yescharacteristic
371C2×C5#100911yescharacteristic
372D5—41no—
373D5—62no—
374D5—121no—
375C23—51no—
376C23—151no—
377C23—151no—
378C23—302no—
379C23—302no—
380C23—601no—
381C23C2×D3×D511yescharacteristic
382Q8—601no—
383D4—102no—
384D4—302no—
385D4—601no—
386C2×C4—22no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—151no—
395C2×C4—151no—
396C2×C4—302no—
397C2×C4—601no—
398C2×C4—601no—
399C2×C4—601no—
400C2×C4C2×D3×D511yescharacteristic
401C2×C4C2×D3×D511yescharacteristic
402C2×C3—22no—
403C2×C3—201no—
404C2×C3C2×#70911yescharacteristic
405C2×C3D5×#4111yescharacteristic
406C2×C3#414911yescharacteristic
407D3—102no—
408D3—201no—
409C5#565811yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—101no—
413C22—101no—
414C22—101no—
415C22—152no—
416C22—301no—
417C22—301no—
418C22—301no—
419C22—301no—
420C22—301no—
421C22—604no—
422C22#677912yes—
423C22C22×D3×D511yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—61no—
430C4—101no—
431C4—101no—
432C4—101no—
433C4—301no—
434C4—301no—
435C3#6586111yescharacteristic, complemented, Sylow
436C2—22no—
437C2—201no—
438C2—302no—
439C2—601no—
440C2#9201611yescharacteristic
441C2#9201911yescharacteristic
442C2C2×#677911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radical#91572C2
hypercenterC22×C4D3×D5
hypocenterC3×C5#551
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6714C2semidirect
2. C5×#926C2semidirect
3. C3×#4075C2semidirect
4. #91576C2semidirect
6. #91561C2semidirect
7. #91559C2semidirect
8. #91419C2semidirect
9. #91413C2semidirect
10. #91403C2semidirect
11. #91395C2semidirect
12. #91380C2semidirect
13. #91356C2semidirect
14. #91353C2semidirect
15. #91337C2semidirect
21. C2×#1204C22semidirect
26. C2×D5×Dic3C22semidirect
33. C5×C4⋊Dic3C22semidirect
34. C5×C4⋊Dic3C22semidirect
35. C4×C5×Dic3C22semidirect
36. C4×C5×Dic3C22semidirect
38. C3×#685C22semidirect
39. C3×#685C22semidirect
40. C3×#683C22semidirect
41. C3×#683C22semidirect
42. Dic3⋊Dic5C22semidirect
43. #6669C22semidirect
44. #6668C22semidirect
45. #6668C22semidirect
50. #6665C22semidirect
51. Dic3×Dic5C22semidirect
54. #4075D3semidirect
99. #926D5semidirect
136. #685C2×D3semidirect
137. #685C2×D3semidirect
142. #683C2×D3semidirect
143. #683C2×D3semidirect
199. C4⋊Dic3C2×D5semidirect
200. C4⋊Dic3C2×D5semidirect
204. C4×Dic3C2×D5semidirect
205. C4×Dic3C2×D5semidirect
342. C3×C5#551semidirect
409. C5#5658semidirect
435. C3#65861semidirect

Maximal subgroups

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

subgroupindex
C2×#67142
C5×#9262
C3×#40752
#915762
#915722
#915612
#915592
#914192
#914132
#914032
#913952
#913802
#913562
#913532
#913372
#658613
#56585