Finite Group Databasebeta

← #11753154

Subgroups — #11753154

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
1C22⋊C4×D15C211yesmaximal, characteristic, complemented
2C5×#883C211yesmaximal, characteristic, complemented
3C3×#4025C211yesmaximal, characteristic, complemented
4#91573C211yesmaximal, characteristic
5#91561C211yesmaximal, characteristic, complemented
6#91531C211yesmaximal, characteristic, complemented
7#91525C211yesmaximal, characteristic, complemented
8#91446C211yesmaximal, characteristic, complemented
9C4×#1203C211yesmaximal, characteristic, complemented
10#91438C211yesmaximal, characteristic, complemented
11#91419C211yesmaximal, characteristic, complemented
12#91402C211yesmaximal, characteristic, complemented
13#91377C211yesmaximal, characteristic, complemented
14#91355C211yesmaximal, characteristic, complemented
15#91347C211yesmaximal, characteristic, complemented
16#65993—31nomaximal, Hall
17C23×D15C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C4×D15C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C4×C5×D3C2211yescharacteristic, complemented
22C2×#1204C2211yescharacteristic, complemented
23C2×#1203C2211yescharacteristic
24C2×#1203C2211yescharacteristic
25C2×#1200C2211yescharacteristic
26C2×#1200C2211yescharacteristic
27C3×C5×C22⋊C4C2211yescharacteristic
28#6719C2211yescharacteristic
29#6714C2211yescharacteristic
30#6714C2211yescharacteristic
31C5×#217C2211yescharacteristic
32C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
33C5×C4⋊Dic3C2211yescharacteristic, complemented
34C5×Dic3⋊C4C2211yescharacteristic, complemented
35C3×C22⋊Dic5C2211yescharacteristic
36C3×C22⋊Dic5C2211yescharacteristic
37C3×C4⋊Dic5C2211yescharacteristic, complemented
38C3×#683C2211yescharacteristic, complemented
39C3×#683C2211yescharacteristic, complemented
40C3×C4×Dic5C2211yescharacteristic, complemented
41Dic3⋊Dic5C2211yescharacteristic, complemented
42#6669C2211yescharacteristic, complemented
43#6668C2211yescharacteristic, complemented
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46#6667C2211yescharacteristic
47#6667C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic, complemented
49(C2×D3)⋊Dic5C2211yescharacteristic, complemented
50Dic3×Dic5C2211yescharacteristic, complemented
51Dic3×Dic5C2211yescharacteristic, complemented
52#5705—51nomaximal, Hall
53C5×#123—31no—
54#4083—31no—
55#4082—31no—
56C4×C22⋊D5—31no—
57C4×C22⋊D5—31no—
58#4042—31no—
59#4042—31no—
60#4038—31no—
61#4038—31no—
62#4029—31no—
63#4029—31no—
64D5×C22⋊C4—31no—
65D5×C22⋊C4—31no—
66#4025—31no—
67#4025D311yescharacteristic, complemented
68C23×C3×C5C2311yescharacteristic
69C22×D15—22no—
70C22×D15—22no—
71C22×D15—41no—
72C22×D15C2311yescharacteristic
73C22×D15C2311yescharacteristic
74C22×C5×D3C2311yescharacteristic
75C2×C3×C4×C5C2311yescharacteristic
76C2×C3×C4×C5C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C4×D15—22no—
80C4×D15—41no—
81C5×#67—41no—
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C4×C5×D3—41no—
86C2×C3×Dic5C2311yescharacteristic
87C2×C3×Dic5C2311yescharacteristic
88C2×C3×Dic5C2311yescharacteristic
89C2×C3×Dic5C2311yescharacteristic
90#1204—41no—
91#1203—22no—
92#1203—22no—
93#1200—22no—
94#1200—41no—
95C3×#126—51no—
96#897—51no—
97#896—51no—
98#894—51no—
99#894—51no—
100#893—51no—
101#892—51no—
102#885—51no—
103#883D511yescharacteristic, complemented
104#882—51no—
105D3×C22⋊C4—51no—
106D3×C22⋊C4—51no—
107#879—51no—
108C4×D12—51no—
109#872—51no—
110C23×D5—31no—
111C2×C5×D4—31no—
112C22×C4×C5—31no—
113C2×C22⋊D5—31no—
114C2×C22⋊D5—31no—
115C2×C4×D5—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—
124C5×C22⋊C4C2×D311yescharacteristic
125C22⋊Dic5—31no—
126C22⋊Dic5—31no—
127C22⋊Dic5—31no—
128C22⋊Dic5C2×D311yescharacteristic
129C22⋊Dic5C2×D311yescharacteristic
130#685—31no—
131#685—31no—
132#685—31no—
133#685—31no—
134#685—31no—
135#685—31no—
136C4⋊Dic5—31no—
137C4⋊Dic5C2×D311yescharacteristic, complemented
138#683—31no—
139#683—31no—
140#683C2×D311yescharacteristic, complemented
141#683C2×D311yescharacteristic, complemented
142C4×Dic5—31no—
143C4×Dic5—31no—
144C4×Dic5C2×D311yescharacteristic, complemented
145#551—151noSylow
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5C2411yescharacteristic
150C2×D15—21no—
151C2×D15—21no—
152C2×D15—21no—
153C2×D15—21no—
154C2×D15—44no—
155C2×D15#4112yes—
156C2×D15#4112yes—
157C2×C5×D3—21no—
158C2×C5×D3—21no—
159C2×C5×D3—21no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162Dic15—21no—
163Dic15—21no—
164C3×Dic5—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C5×Dic3—21no—
171C23×D3—51no—
172C2×#67C2×D511yescharacteristic
173C2×D12—51no—
174C2×D12—51no—
175C2×C4×D3—51no—
176C2×C4×D3—51no—
177C2×C4×D3—51no—
178C2×C4×D3—51no—
179C2×C4×D3C2×D511yescharacteristic, complemented
180C2×Dic6—51no—
181C3×C4⋊C4—51no—
182C3×C4⋊C4—51no—
183C3×C4⋊C4—51no—
184C3×C22⋊C4—51no—
185C3×C22⋊C4—51no—
186C3×C22⋊C4C2×D511yescharacteristic
187C3×C42—51no—
188#217—51no—
189#217C2×D511yescharacteristic
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—
197(C2×D3)⋊C4—51no—
198(C2×D3)⋊C4C2×D511yescharacteristic, complemented
199C4⋊Dic3—51no—
200C4⋊Dic3C2×D511yescharacteristic, complemented
201Dic3⋊C4—51no—
202Dic3⋊C4—51no—
203Dic3⋊C4C2×D511yescharacteristic, complemented
204C4×Dic3—51no—
205C4×Dic3—51no—
206C23×C5—31no—
207C23×C5C22×D311yescharacteristic
208C22×D5—31no—
209C22×D5—31no—
210C22×D5—62no—
211C22×D5—62no—
212C22×D5—121no—
213C5×D4—121no—
214C2×C4×C5—31no—
215C2×C4×C5—31no—
216C2×C4×C5—31no—
217C2×C4×C5—121no—
218C2×C4×C5C22×D311yescharacteristic
219C2×C4×C5C22×D311yescharacteristic
220C22⋊D5—62no—
221C22⋊D5—62no—
222C2×Dic5—31no—
223C2×Dic5—31no—
224C2×Dic5C22×D311yescharacteristic
225C2×Dic5C22×D311yescharacteristic
226C2×Dic5C22×D311yescharacteristic
227C2×Dic5C22×D311yescharacteristic
228C4×D5—62no—
229C4×D5—62no—
230C4×D5—121no—
231C4×D5—121no—
232Dic10—121no—
233#126—51no—
234#126—151no—
235#124—151no—
236#123—31no—
237#123—151no—
238#123—151no—
239C22⋊Q8—151no—
240C22⋊Q8—151no—
241#120—151no—
242C4×D4—151no—
243C4×D4—151no—
244#117—151no—
245#117—151no—
246C2×C22⋊C4—151no—
247C2×C22⋊C4—151no—
248C2×C3×C5—41no—
249C2×C3×C5C2×#4111yescharacteristic
250C2×C3×C5C2×#4111yescharacteristic
251C2×C3×C5H(5,𝔽2)11yescharacteristic
252D15—22no—
253D15—22no—
254C5×D3—41no—
255C23×C3C22×D511yescharacteristic
256C22×D3—51no—
257C22×D3—51no—
258C22×D3—102no—
259C22×D3—102no—
260C22×D3—201no—
261C22×D3C22×D511yescharacteristic
262C2×C3×C4—51no—
263C2×C3×C4—51no—
264C2×C3×C4—51no—
265C2×C3×C4—51no—
266C2×C3×C4C22×D511yescharacteristic
267C2×C3×C4C22×D511yescharacteristic
268#67—41no—
269C2×Dic3—51no—
270C2×Dic3—51no—
271C2×Dic3C22×D511yescharacteristic
272C2×Dic3C22×D511yescharacteristic
273C2×Dic3C22×D511yescharacteristic
274D12—102no—
275D12—102no—
276C4×D3—41no—
277C4×D3—102no—
278C4×D3—102no—
279C4×D3—201no—
280C4×D3—201no—
281Dic6—201no—
282C22×C5—21no—
283C22×C5—21no—
284C22×C5—21no—
285C22×C5—61no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5C23×D311yescharacteristic
289C2×D5—32no—
290C2×D5—32no—
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—
301Dic5—21no—
302Dic5—21no—
303Dic5—21no—
304Dic5—21no—
305Dic5—61no—
306Dic5—61no—
307C24—151no—
308C2×Q8—151no—
309C2×D4—31no—
310C2×D4—151no—
311C2×D4—151no—
312C22×C4—31no—
313C22×C4—151no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4—151no—
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—51no—
320C4⋊C4—51no—
321C4⋊C4—51no—
322C4⋊C4—151no—
323C4⋊C4—151no—
324C4⋊C4—151no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—51no—
328C22⋊C4—51no—
329C22⋊C4—151no—
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⋊C4D3×D511yescharacteristic
339C42—51no—
340C42—151no—
341C42—151no—
342C3×C5#55111yescharacteristic, complemented, Hall
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3C23×D511yescharacteristic
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—52no—
351C2×D3—52no—
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—
362C3×C4—101no—
363Dic3—21no—
364Dic3—21no—
365Dic3—21no—
366Dic3—101no—
367Dic3—101no—
368C2×C5—41no—
369C2×C5—121no—
370C2×C5D3×#4111yescharacteristic
371C2×C5D3×#4111yescharacteristic
372C2×C5#100911yescharacteristic
373D5—62no—
374D5—62no—
375C23—31no—
376C23—151no—
377C23—151no—
378C23—302no—
379C23—302no—
380C23—601no—
381C23C2×D3×D511yescharacteristic
382Q8—601no—
383D4—121no—
384D4—302no—
385D4—302no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—51no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—121no—
394C2×C4—151no—
395C2×C4—151no—
396C2×C4—302no—
397C2×C4—302no—
398C2×C4—601no—
399C2×C4—601no—
400C2×C4C2×D3×D511yescharacteristic
401C2×C4C2×D3×D511yescharacteristic
402C2×C3—41no—
403C2×C3#414411yescharacteristic
404C2×C3D5×#4111yescharacteristic
405C2×C3D5×#4111yescharacteristic
406D3—41no—
407D3—102no—
408D3—102no—
409C5#570511yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—21no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—152no—
417C22—152no—
418C22—301no—
419C22—301no—
420C22—301no—
421C22—301no—
422C22—604no—
423C22C22×D3×D511yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—101no—
430C4—101no—
431C4—101no—
432C4—101no—
433C4—301no—
434C4—301no—
435C3#6599311yescharacteristic, complemented, Sylow
436C2—41no—
437C2—121no—
438C2—302no—
439C2—302no—
440C2#9201611yescharacteristic
441C2#9202411yescharacteristic
442C2#9202711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
prime radical#91573C2
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. C22⋊C4×D15C2semidirect
2. C5×#883C2semidirect
3. C3×#4025C2semidirect
5. #91561C2semidirect
6. #91531C2semidirect
7. #91525C2semidirect
8. #91446C2semidirect
9. C4×#1203C2semidirect
10. #91438C2semidirect
11. #91419C2semidirect
12. #91402C2semidirect
13. #91377C2semidirect
14. #91355C2semidirect
15. #91347C2semidirect
21. C2×C4×C5×D3C22semidirect
22. C2×#1204C22semidirect
32. C5×(C2×D3)⋊C4C22semidirect
33. C5×C4⋊Dic3C22semidirect
34. C5×Dic3⋊C4C22semidirect
37. C3×C4⋊Dic5C22semidirect
38. C3×#683C22semidirect
39. C3×#683C22semidirect
40. C3×C4×Dic5C22semidirect
41. Dic3⋊Dic5C22semidirect
42. #6669C22semidirect
43. #6668C22semidirect
48. (C2×D3)⋊Dic5C22semidirect
49. (C2×D3)⋊Dic5C22semidirect
50. Dic3×Dic5C22semidirect
51. Dic3×Dic5C22semidirect
67. #4025D3semidirect
103. #883D5semidirect
137. C4⋊Dic5C2×D3semidirect
140. #683C2×D3semidirect
141. #683C2×D3semidirect
144. C4×Dic5C2×D3semidirect
179. C2×C4×D3C2×D5semidirect
198. (C2×D3)⋊C4C2×D5semidirect
200. C4⋊Dic3C2×D5semidirect
203. Dic3⋊C4C2×D5semidirect
342. C3×C5#551semidirect
409. C5#5705semidirect
435. C3#65993semidirect

Maximal subgroups

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

subgroupindex
C22⋊C4×D152
C5×#8832
C3×#40252
#915732
#915612
#915312
#915252
#914462
C4×#12032
#914382
#914192
#914022
#913772
#913552
#913472
#659933
#57055