Finite Group Databasebeta

← #11752939

Subgroups — #11752939

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91767C211yesmaximal, characteristic, complemented
2C5×#885C211yesmaximal, characteristic, complemented
3C3×#4031C211yesmaximal, characteristic, complemented
4#91565C211yesmaximal, characteristic
5#91559C211yesmaximal, characteristic, complemented
6#91543C211yesmaximal, characteristic, complemented
7#91530C211yesmaximal, characteristic, complemented
8#91462C211yesmaximal, characteristic, complemented
9#91421C211yesmaximal, characteristic, complemented
10#91414C211yesmaximal, characteristic, complemented
11#91410C211yesmaximal, characteristic, complemented
12#91398C211yesmaximal, characteristic, complemented
13#91376C211yesmaximal, characteristic, complemented
14#91363C211yesmaximal, characteristic, complemented
15Dic3×Dic10C211yesmaximal, characteristic, complemented
16#66003—31nomaximal, Hall
17C22×Dic15C2211yescharacteristic
18C2×Dic30C2211yescharacteristic
19C2×C5×#67C2211yescharacteristic
20C2×C5×Dic6C2211yescharacteristic
21C2×C3×C22⋊D5C2211yescharacteristic
22C2×C3×Dic10C2211yescharacteristic
23C2×#1201C2211yescharacteristic
24C2×D3×Dic5C2211yescharacteristic
25C2×D5×Dic3C2211yescharacteristic
26C3×C5×C22⋊C4C2211yescharacteristic
27#6719C2211yescharacteristic
28C4⋊Dic15C2211yescharacteristic
29#6712C2211yescharacteristic
30#6712C2211yescharacteristic
31C5×#217C2211yescharacteristic
32C5×(C2×D3)⋊C4C2211yescharacteristic
33C5×(C2×D3)⋊C4C2211yescharacteristic
34C4×C5×Dic3C2211yescharacteristic
35C3×C22⋊Dic5C2211yescharacteristic
36C3×#685C2211yescharacteristic
37C3×#685C2211yescharacteristic
38C3×C4×Dic5C2211yescharacteristic
39Dic3⋊Dic5C2211yescharacteristic
40#6669C2211yescharacteristic
41#6668C2211yescharacteristic
42#6668C2211yescharacteristic
43#6668C2211yescharacteristic
44#6668C2211yescharacteristic
45#6668C2211yescharacteristic
46(C2×D3)⋊Dic5C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48#6665C2211yescharacteristic
49#6665C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5682—51nomaximal, Hall
53C5×#124—31no—
54#4092—31no—
55#4090—31no—
56#4085—31no—
57#4081—31no—
58#4031—31no—
59#4031D311yescharacteristic, complemented
60#4029—31no—
61#4027—31no—
62#4025—31no—
63#4024—31no—
64#4024—31no—
65#4023—31no—
66#4022—31no—
67C4×Dic10—31no—
68C23×C3×C5C2311yescharacteristic
69C22×C5×D3C2311yescharacteristic
70C22×C3×D5C2311yescharacteristic
71C2×C3×C4×C5C2311yescharacteristic
72C2×C3×C4×C5C2311yescharacteristic
73C2×Dic15—41no—
74C2×Dic15C2311yescharacteristic
75C2×Dic15C2311yescharacteristic
76C2×Dic15C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78Dic30—41no—
79C5×#67—41no—
80C2×C5×Dic3C2311yescharacteristic
81C2×C5×Dic3C2311yescharacteristic
82C2×C5×Dic3C2311yescharacteristic
83C5×Dic6—41no—
84C3×C22⋊D5—22no—
85C2×C3×Dic5C2311yescharacteristic
86C2×C3×Dic5C2311yescharacteristic
87C2×C3×Dic5C2311yescharacteristic
88C3×Dic10—22no—
89#1201—41no—
90D3×Dic5—22no—
91D5×Dic3—41no—
92C3×#124—51no—
93#946—51no—
94Q8×Dic3—51no—
95#938—51no—
96D4×Dic3—51no—
97#886—51no—
98#886—51no—
99#885—51no—
100#885D511yescharacteristic, complemented
101#879—51no—
102#879—51no—
103#878—51no—
104#878—51no—
105#872—51no—
106#869—51no—
107C2×C5×Q8—31no—
108C2×C5×D4—31no—
109C2×C22⋊D5—31no—
110C2×C22⋊D5C2×D311yescharacteristic
111C22×Dic5—31no—
112C22×Dic5—31no—
113C2×C4×D5—31no—
114C2×Dic10—31no—
115C2×Dic10C2×D311yescharacteristic
116C5×C22⋊C4—31no—
117C5×C22⋊C4—31no—
118C5×C22⋊C4—31no—
119C5×C22⋊C4C2×D311yescharacteristic
120C42×C5—31no—
121C22⋊Dic5—31no—
122C22⋊Dic5—31no—
123C22⋊Dic5—31no—
124C22⋊Dic5C2×D311yescharacteristic
125#685—31no—
126#685—31no—
127#685C2×D311yescharacteristic
128#685C2×D311yescharacteristic
129C4⋊Dic5—31no—
130C4⋊Dic5—31no—
131#683—31no—
132#683—31no—
133#683—31no—
134#683—31no—
135#683—31no—
136#683—31no—
137#683—31no—
138#683—31no—
139C4×Dic5—31no—
140C4×Dic5—31no—
141C4×Dic5C2×D311yescharacteristic
142#542—151noSylow
143C22×C3×C5—21no—
144C22×C3×C5—21no—
145C22×C3×C5—21no—
146C22×C3×C5C2411yescharacteristic
147C2×C5×D3—21no—
148C2×C5×D3—21no—
149C2×C5×D3—21no—
150C2×C3×D5—21no—
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C3×C4×C5—21no—
154C3×C4×C5—21no—
155Dic15—21no—
156Dic15—21no—
157Dic15—21no—
158Dic15—21no—
159C3×Dic5—21no—
160C3×Dic5—21no—
161C3×Dic5#4112yes—
162C5×Dic3—21no—
163C5×Dic3—21no—
164C5×Dic3—21no—
165C2×C3×Q8—51no—
166C2×C3×D4—51no—
167C2×#67—51no—
168C2×#67C2×D511yescharacteristic
169C22×Dic3—51no—
170C22×Dic3—51no—
171C2×C4×D3—51no—
172C2×Dic6—51no—
173C2×Dic6C2×D511yescharacteristic
174C3×C22⋊C4—51no—
175C3×C22⋊C4—51no—
176C3×C22⋊C4—51no—
177C3×C22⋊C4C2×D511yescharacteristic
178C3×C42—51no—
179#217—51no—
180#217—51no—
181#217—51no—
182#217C2×D511yescharacteristic
183(C2×D3)⋊C4—51no—
184(C2×D3)⋊C4—51no—
185(C2×D3)⋊C4C2×D511yescharacteristic
186(C2×D3)⋊C4C2×D511yescharacteristic
187C4⋊Dic3—51no—
188C4⋊Dic3—51no—
189C4⋊Dic3—51no—
190C4⋊Dic3—51no—
191C4⋊Dic3—51no—
192C4⋊Dic3—51no—
193Dic3⋊C4—51no—
194Dic3⋊C4—51no—
195Dic3⋊C4—51no—
196Dic3⋊C4—51no—
197C4×Dic3—51no—
198C4×Dic3—51no—
199C4×Dic3C2×D511yescharacteristic
200C23×C5—31no—
201C23×C5C22×D311yescharacteristic
202C22×D5C22×D311yescharacteristic
203C5×Q8—121no—
204C5×D4—121no—
205C2×C4×C5—31no—
206C2×C4×C5—31no—
207C2×C4×C5—31no—
208C2×C4×C5C22×D311yescharacteristic
209C2×C4×C5C22×D311yescharacteristic
210C22⋊D5—22no—
211C22⋊D5—121no—
212C2×Dic5—31no—
213C2×Dic5—31no—
214C2×Dic5—31no—
215C2×Dic5—31no—
216C2×Dic5—62no—
217C2×Dic5—121no—
218C2×Dic5C22×D311yescharacteristic
219C2×Dic5C22×D311yescharacteristic
220C2×Dic5C22×D311yescharacteristic
221C4×D5—121no—
222Dic10—22no—
223Dic10—121no—
224C4⋊Q8—151no—
225#126—151no—
226#126—151no—
227#124—31no—
228#124—51no—
229#124—151no—
230#123—151no—
231#123—151no—
232C22⋊Q8—151no—
233C22⋊Q8—151no—
234C22⋊Q8—151no—
235#121—151no—
236C4×Q8—151no—
237C4×D4—151no—
238#117—151no—
239C2×C3×C5—41no—
240C2×C3×C5C2×#4111yescharacteristic
241C2×C3×C5H(5,𝔽2)11yescharacteristic
242C2×C3×C52−1+411yescharacteristic
243C3×D5—41no—
244C5×D3—41no—
245C23×C3—51no—
246C23×C3C22×D511yescharacteristic
247C22×D3C22×D511yescharacteristic
248C3×Q8—102no—
249C3×D4—102no—
250C2×C3×C4—51no—
251C2×C3×C4—51no—
252C2×C3×C4—51no—
253C2×C3×C4C22×D511yescharacteristic
254C2×C3×C4C22×D511yescharacteristic
255#67—41no—
256#67—201no—
257C2×Dic3—51no—
258C2×Dic3—51no—
259C2×Dic3—51no—
260C2×Dic3—51no—
261C2×Dic3—201no—
262C2×Dic3—201no—
263C2×Dic3C22×D511yescharacteristic
264C2×Dic3C22×D511yescharacteristic
265C2×Dic3C22×D511yescharacteristic
266C4×D3—102no—
267Dic6—41no—
268Dic6—201no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—61no—
273C22×C5—61no—
274C22×C5—61no—
275C22×C5C23×D311yescharacteristic
276C2×D5—21no—
277C2×D5—21no—
278C2×D5—21no—
279C4×C5—21no—
280C4×C5—21no—
281C4×C5—61no—
282C4×C5—61no—
283C4×C5—61no—
284Dic5—21no—
285Dic5—21no—
286Dic5—61no—
287Dic5—61no—
288Dic5—61no—
289Dic5—61no—
290Dic5#23712yes—
291C2×Q8—31no—
292C2×Q8—51no—
293C2×Q8—151no—
294C2×D4—31no—
295C2×D4—51no—
296C2×D4—151no—
297C22×C4—151no—
298C22×C4—151no—
299C22×C4—151no—
300C4⋊C4—151no—
301C4⋊C4—151no—
302C4⋊C4—151no—
303C4⋊C4—151no—
304C4⋊C4—151no—
305C4⋊C4—151no—
306C4⋊C4—151no—
307C4⋊C4—151no—
308C4⋊C4—151no—
309C4⋊C4—151no—
310C22⋊C4—31no—
311C22⋊C4—31no—
312C22⋊C4—31no—
313C22⋊C4—51no—
314C22⋊C4—51no—
315C22⋊C4—51no—
316C22⋊C4—151no—
317C22⋊C4—151no—
318C22⋊C4—151no—
319C22⋊C4—151no—
320C22⋊C4—151no—
321C22⋊C4D3×D511yescharacteristic
322C42—31no—
323C42—51no—
324C42—151no—
325C42—151no—
326C3×C5#54211yescharacteristic, complemented, Hall
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—101no—
331C22×C3—101no—
332C22×C3—101no—
333C22×C3C23×D511yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—52no—
340C3×C4—101no—
341C3×C4—101no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—101no—
346Dic3—101no—
347Dic3—101no—
348Dic3—101no—
349C2×C5—41no—
350C2×C5—121no—
351C2×C5C2×#23711yescharacteristic
352C2×C5#100911yescharacteristic
353C2×C5#101011yescharacteristic
354D5—41no—
355C23—31no—
356C23—51no—
357C23C2×D3×D511yescharacteristic
358Q8—102no—
359Q8—121no—
360Q8—601no—
361D4—102no—
362D4—121no—
363D4—601no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—51no—
368C2×C4—51no—
369C2×C4—51no—
370C2×C4—151no—
371C2×C4—151no—
372C2×C4—151no—
373C2×C4—151no—
374C2×C4—302no—
375C2×C4—601no—
376C2×C4—601no—
377C2×C4C2×D3×D511yescharacteristic
378C2×C4C2×D3×D511yescharacteristic
379C2×C3—41no—
380C2×C3—201no—
381C2×C3#414411yescharacteristic
382C2×C3D5×#4111yescharacteristic
383C2×C3#415011yescharacteristic
384D3—41no—
385C5#568211yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—101no—
393C22—101no—
394C22—101no—
395C22C22×D3×D511yescharacteristic
396C4—21no—
397C4—21no—
398C4—52no—
399C4—61no—
400C4—61no—
401C4—61no—
402C4—101no—
403C4—101no—
404C4—301no—
405C4—301no—
406C4—301no—
407C4—301no—
408C3#6600311yescharacteristic, complemented, Sylow
409C2—41no—
410C2—121no—
411C2—201no—
412C2#9200111yescharacteristic
413C2D5×#23711yescharacteristic
414C2#9202711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
prime radical#91565C2
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#542
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. #91767C2semidirect
2. C5×#885C2semidirect
3. C3×#4031C2semidirect
5. #91559C2semidirect
6. #91543C2semidirect
7. #91530C2semidirect
8. #91462C2semidirect
9. #91421C2semidirect
10. #91414C2semidirect
11. #91410C2semidirect
12. #91398C2semidirect
13. #91376C2semidirect
14. #91363C2semidirect
15. Dic3×Dic10C2semidirect
59. #4031D3semidirect
100. #885D5semidirect
326. C3×C5#542semidirect
385. C5#5682semidirect
408. C3#66003semidirect

Maximal subgroups

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

subgroupindex
#917672
C5×#8852
C3×#40312
#915652
#915592
#915432
#915302
#914622
#914212
#914142
#914102
#913982
#913762
#913632
Dic3×Dic102
#660033
#56825