Finite Group Databasebeta

← #11754402

Subgroups — #11754402

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
1C4×#1220C211yesmaximal, characteristic, complemented
2C5×#924C211yesmaximal, characteristic, complemented
3C2×C3×#685C211yesmaximal, characteristic, complemented
4#91564C211yesmaximal, characteristic
5D5×#217C211yesmaximal, characteristic, complemented
6#91546C211yesmaximal, characteristic, complemented
7#91528C211yesmaximal, characteristic, complemented
8#91455C211yesmaximal, characteristic, complemented
9#91421C211yesmaximal, characteristic, complemented
10#91413C211yesmaximal, characteristic, complemented
11#91396C211yesmaximal, characteristic, complemented
12#91394C211yesmaximal, characteristic, complemented
13#91366C211yesmaximal, characteristic, complemented
14#91361C211yesmaximal, characteristic, complemented
15#91354C211yesmaximal, characteristic, complemented
16#65971—31nomaximal, Hall
17C23×C3×D5C2211yescharacteristic
18C22×C3×C4×C5C2211yescharacteristic
19C2×#1220C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic, complemented
21C2×C5×Dic6C2211yescharacteristic, complemented
22C22×C3×Dic5C2211yescharacteristic
23C2×#1202C2211yescharacteristic
24C2×#1202C2211yescharacteristic
25C2×D5×Dic3C2211yescharacteristic
26C2×D5×Dic3C2211yescharacteristic
27#6719C2211yescharacteristic
28#6714C2211yescharacteristic, complemented
29#6712C2211yescharacteristic, complemented
30C4×Dic15C2211yescharacteristic, complemented
31C5×#217C2211yescharacteristic
32C5×#217C2211yescharacteristic
33C5×C4⋊Dic3C2211yescharacteristic, complemented
34C5×Dic3⋊C4C2211yescharacteristic, complemented
35C5×Dic3⋊C4C2211yescharacteristic, complemented
36C3×#685C2211yescharacteristic
37C3×#685C2211yescharacteristic
38C3×#685C2211yescharacteristic
39C3×#685C2211yescharacteristic
40Dic3⋊Dic5C2211yescharacteristic, complemented
41Dic3⋊Dic5C2211yescharacteristic, complemented
42#6668C2211yescharacteristic, complemented
43#6668C2211yescharacteristic, complemented
44#6667C2211yescharacteristic, complemented
45#6667C2211yescharacteristic, complemented
46#6665C2211yescharacteristic
47#6665C2211yescharacteristic
48#6665C2211yescharacteristic
49#6665C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic, complemented
51Dic3×Dic5C2211yescharacteristic, complemented
52#5495—51nomaximal, Hall
53C5×C22⋊Q8—31no—
54#4093—31no—
55#4092—31no—
56C4×C22⋊D5—31no—
57C2×#685D311yescharacteristic, complemented
58#4044—31no—
59#4044—31no—
60#4040—31no—
61#4040—31no—
62#4039—31no—
63#4038—31no—
64#4032—31no—
65#4028—31no—
66D5×C22⋊C4—31no—
67#4023—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C3×D5—22no—
71C22×C3×D5—22no—
72C22×C3×D5—41no—
73C22×C3×D5C2311yescharacteristic
74C22×C3×D5C2311yescharacteristic
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78#1220—22no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C4×D15—41no—
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C5×Dic6—41no—
88C2×C3×Dic5—22no—
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91#1202—22no—
92#1202—41no—
93D5×Dic3—22no—
94D5×Dic3—41no—
95C2×C3×C22⋊C4—51no—
96#953—51no—
97C2×#217—51no—
98#929—51no—
99C4×#67—51no—
100#926—51no—
101#924D511yescharacteristic, complemented
102#886—51no—
103#885—51no—
104#883—51no—
105#67⋊C4—51no—
106#879—51no—
107#879—51no—
108#878—51no—
109#877—51no—
110C23×D5C2×D311yescharacteristic
111C2×C5×Q8—31no—
112C22×C4×C5C2×D311yescharacteristic
113C2×C22⋊D5—31no—
114C22×Dic5C2×D311yescharacteristic
115C2×D20—31no—
116C2×D20—31no—
117C2×C4×D5—31no—
118C2×C4×D5—31no—
119C2×C4×D5—31no—
120C5×C4⋊C4—31no—
121C5×C4⋊C4—31no—
122C5×C4⋊C4—31no—
123C5×C22⋊C4—31no—
124C5×C22⋊C4—31no—
125C22⋊Dic5—31no—
126#685—31no—
127#685—31no—
128#685—31no—
129#685—31no—
130#685—31no—
131#685—31no—
132#685—31no—
133#685C2×D311yescharacteristic
134#685C2×D311yescharacteristic
135#685C2×D311yescharacteristic
136#685C2×D311yescharacteristic
137C4⋊Dic5—31no—
138C4⋊Dic5—31no—
139#683—31no—
140#683—31no—
141#683—31no—
142C4×Dic5—31no—
143C4×Dic5—31no—
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×C3×D5—21no—
154C2×C3×D5—21no—
155C2×C3×D5—21no—
156C2×C3×D5—21no—
157C2×C3×D5—21no—
158C2×C3×D5—44no—
159C2×C3×D5#4112yes—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162Dic15—21no—
163Dic15—21no—
164Dic15—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C5×Dic3—21no—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C5×Dic3—21no—
171C24×C3—51no—
172C22×C3×C4—51no—
173C22×C3×C4C2×D511yescharacteristic
174C2×#67—51no—
175C2×#67—51no—
176C2×#67—51no—
177C22×Dic3—51no—
178C22×Dic3—51no—
179C2×C4×D3—51no—
180C2×Dic6C2×D511yescharacteristic, complemented
181C3×C22⋊C4—51no—
182C3×C22⋊C4—51no—
183C3×C22⋊C4—51no—
184C3×C22⋊C4—51no—
185#217—51no—
186#217—51no—
187#217—51no—
188#217—51no—
189#217—51no—
190#217C2×D511yescharacteristic
191#217C2×D511yescharacteristic
192(C2×D3)⋊C4—51no—
193(C2×D3)⋊C4—51no—
194(C2×D3)⋊C4—51no—
195C4⋊Dic3—51no—
196C4⋊Dic3—51no—
197C4⋊Dic3C2×D511yescharacteristic, complemented
198Dic3⋊C4—51no—
199Dic3⋊C4—51no—
200Dic3⋊C4—51no—
201Dic3⋊C4C2×D511yescharacteristic, complemented
202Dic3⋊C4C2×D511yescharacteristic, complemented
203C4×Dic3—51no—
204C4×Dic3—51no—
205C4×Dic3—51no—
206C23×C5C22×D311yescharacteristic
207C22×D5—22no—
208C22×D5—22no—
209C22×D5—31no—
210C22×D5—41no—
211C22×D5C22×D311yescharacteristic
212C22×D5C22×D311yescharacteristic
213C5×Q8—121no—
214C2×C4×C5—31no—
215C2×C4×C5—31no—
216C2×C4×C5—31no—
217C2×C4×C5—31no—
218C2×C4×C5—41no—
219C2×C4×C5C22×D311yescharacteristic
220C2×C4×C5C22×D311yescharacteristic
221C22⋊D5—62no—
222C2×Dic5—22no—
223C2×Dic5—31no—
224C2×Dic5—31no—
225C2×Dic5—31no—
226C2×Dic5C22×D311yescharacteristic
227C2×Dic5C22×D311yescharacteristic
228D20—62no—
229D20—121no—
230C4×D5—62no—
231C4×D5—121no—
232C4×D5—121no—
233#126—151no—
234#126—151no—
235#124—151no—
236#123—151no—
237#123—151no—
238#123—151no—
239C22⋊Q8—31no—
240C22⋊Q8—151no—
241#120—151no—
242C4×D4—151no—
243C4×D4—151no—
244#117—151no—
245#117—151no—
246C2×C22⋊C4—51no—
247C2×C22⋊C4—151no—
248C2×C3×C5—22no—
249C2×C3×C5C2×#4111yescharacteristic
250C2×C3×C5C2×#4111yescharacteristic
251C2×C3×C5H(5,𝔽2)11yescharacteristic
252D15—41no—
253C3×D5—22no—
254C3×D5—41no—
255C23×C3—51no—
256C23×C3—51no—
257C23×C3—102no—
258C23×C3—102no—
259C23×C3—201no—
260C23×C3C22×D511yescharacteristic
261C22×D3—51no—
262C2×C3×C4—41no—
263C2×C3×C4—51no—
264C2×C3×C4—51no—
265C2×C3×C4—102no—
266C2×C3×C4C22×D511yescharacteristic
267C2×C3×C4C22×D511yescharacteristic
268#67—102no—
269#67—102no—
270#67—201no—
271C2×Dic3—51no—
272C2×Dic3—51no—
273C2×Dic3—51no—
274C2×Dic3—102no—
275C2×Dic3—201no—
276C2×Dic3C22×D511yescharacteristic
277C2×Dic3C22×D511yescharacteristic
278C2×Dic3C22×D511yescharacteristic
279C2×Dic3C22×D511yescharacteristic
280C4×D3—201no—
281Dic6—41no—
282C22×C5—21no—
283C22×C5—21no—
284C22×C5#23712yes—
285C22×C5C23×D311yescharacteristic
286C2×D5—21no—
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—21no—
290C2×D5—21no—
291C2×D5—44no—
292C2×D5—61no—
293C2×D5—61no—
294C2×D5—61no—
295C2×D5#23512yes—
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—61no—
299C4×C5—61no—
300C4×C5—61no—
301C4×C5—61no—
302Dic5—21no—
303Dic5—21no—
304Dic5—61no—
305Dic5—61no—
306Dic5—61no—
307C24—51no—
308C2×Q8—31no—
309C2×D4—151no—
310C2×D4—151no—
311C2×D4—151no—
312C22×C4—51no—
313C22×C4—151no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4D3×D511yescharacteristic
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—151no—
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—51no—
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—151no—
340C42—151no—
341C42—151no—
342C3×C5#55111yescharacteristic, complemented, Hall
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—52no—
346C22×C3—101no—
347C22×C3—101no—
348C22×C3—101no—
349C22×C3—101no—
350C22×C3—101no—
351C22×C3—204no—
352C22×C3#71312yes—
353C22×C3C23×D511yescharacteristic
354C2×D3—101no—
355C2×D3—101no—
356C2×D3—101no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—101no—
360C3×C4—101no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—101no—
366Dic3—101no—
367Dic3—101no—
368C2×C5—22no—
369C2×C5C2×#23511yescharacteristic
370C2×C5C2×#23711yescharacteristic
371C2×C5#100411yescharacteristic
372D5—22no—
373D5—41no—
374D5—121no—
375C23—51no—
376C23—51no—
377C23—102no—
378C23—102no—
379C23—151no—
380C23—201no—
381C23C2×D3×D511yescharacteristic
382Q8—121no—
383D4—302no—
384D4—302no—
385D4—601no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—41no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—102no—
394C2×C4—151no—
395C2×C4—151no—
396C2×C4—151no—
397C2×C4—302no—
398C2×C4—601no—
399C2×C4—601no—
400C2×C4C2×D3×D511yescharacteristic
401C2×C4C2×D3×D511yescharacteristic
402C2×C3—22no—
403C2×C3—102no—
404C2×C3—201no—
405C2×C3#414411yescharacteristic
406C2×C3C2×#71311yescharacteristic
407C2×C3D5×#4111yescharacteristic
408D3—201no—
409C5#549511yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—52no—
413C22—101no—
414C22—101no—
415C22—101no—
416C22—101no—
417C22—101no—
418C22—204no—
419C22—301no—
420C22—301no—
421C22—301no—
422C22#676612yes—
423C22C22×D3×D511yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—61no—
430C4—101no—
431C4—101no—
432C4—301no—
433C4—301no—
434C4—301no—
435C3#6597111yescharacteristic, complemented, Sylow
436C2—22no—
437C2—102no—
438C2—201no—
439C2—601no—
440C2C2×#676611yescharacteristic
441C2D5×#23511yescharacteristic
442C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radical#91564C2
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. C4×#1220C2semidirect
2. C5×#924C2semidirect
3. C2×C3×#685C2semidirect
5. D5×#217C2semidirect
6. #91546C2semidirect
7. #91528C2semidirect
8. #91455C2semidirect
9. #91421C2semidirect
10. #91413C2semidirect
11. #91396C2semidirect
12. #91394C2semidirect
13. #91366C2semidirect
14. #91361C2semidirect
15. #91354C2semidirect
20. C2×C4×D15C22semidirect
21. C2×C5×Dic6C22semidirect
28. #6714C22semidirect
29. #6712C22semidirect
30. C4×Dic15C22semidirect
33. C5×C4⋊Dic3C22semidirect
34. C5×Dic3⋊C4C22semidirect
35. C5×Dic3⋊C4C22semidirect
40. Dic3⋊Dic5C22semidirect
41. Dic3⋊Dic5C22semidirect
42. #6668C22semidirect
43. #6668C22semidirect
44. #6667C22semidirect
45. #6667C22semidirect
50. Dic3×Dic5C22semidirect
51. Dic3×Dic5C22semidirect
57. C2×#685D3semidirect
101. #924D5semidirect
180. C2×Dic6C2×D5semidirect
197. C4⋊Dic3C2×D5semidirect
201. Dic3⋊C4C2×D5semidirect
202. Dic3⋊C4C2×D5semidirect
342. C3×C5#551semidirect
409. C5#5495semidirect
435. C3#65971semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9242
C2×C3×#6852
#915642
D5×#2172
#915462
#915282
#914552
#914212
#914132
#913962
#913942
#913662
#913612
#913542
#659713
#54955