Finite Group Databasebeta

← #11754904

Subgroups — #11754904

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#91828C211yesmaximal, characteristic
2C5×#935C211yesmaximal, characteristic, complemented
3C3×#4082C211yesmaximal, characteristic, complemented
4#91576C211yesmaximal, characteristic, complemented
5#91575C211yesmaximal, characteristic, complemented
6#91562C211yesmaximal, characteristic, complemented
7#91556C211yesmaximal, characteristic, complemented
8#91532C211yesmaximal, characteristic, complemented
9#91531C211yesmaximal, characteristic, complemented
10#91526C211yesmaximal, characteristic, complemented
11#91526C211yesmaximal, characteristic, complemented
12#91411C211yesmaximal, characteristic, complemented
13#91387C211yesmaximal, characteristic, complemented
14#91327C211yesmaximal, characteristic, complemented
15#91323C211yesmaximal, characteristic, complemented
16#65996—31nomaximal, Hall
17C2×C3×C5×D4C2211yescharacteristic
18C2×#1220C2211yescharacteristic
19C2×#1220C2211yescharacteristic
20C22×Dic15C2211yescharacteristic
21C22×C5×Dic3C2211yescharacteristic
22C2×C3×Dic10C2211yescharacteristic
23C2×#1204C2211yescharacteristic
24C2×#1204C2211yescharacteristic
25C2×#1200C2211yescharacteristic
26#6719C2211yescharacteristic
27#6714C2211yescharacteristic
28#6712C2211yescharacteristic
29C5×#217C2211yescharacteristic
30C5×#217C2211yescharacteristic
31C5×#217C2211yescharacteristic
32C5×Dic3⋊C4C2211yescharacteristic
33C5×Dic3⋊C4C2211yescharacteristic
34C3×C22⋊Dic5C2211yescharacteristic
35C3×C22⋊Dic5C2211yescharacteristic
36C3×C22⋊Dic5C2211yescharacteristic
37C3×C22⋊Dic5C2211yescharacteristic
38C3×C4×Dic5C2211yescharacteristic
39Dic3⋊Dic5C2211yescharacteristic
40Dic3⋊Dic5C2211yescharacteristic
41#6669C2211yescharacteristic
42#6669C2211yescharacteristic
43#6669C2211yescharacteristic
44#6668C2211yescharacteristic
45#6668C2211yescharacteristic
46#6667C2211yescharacteristic
47#6667C2211yescharacteristic
48#6667C2211yescharacteristic
49Dic3×Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5669—51nomaximal, Hall
53C5×#123—31no—
54#4085—31no—
55#4082D311yescharacteristic, complemented
56C4×C22⋊D5—31no—
57#4070—31no—
58#4044—31no—
59#4042—31no—
60#4034—31no—
61#4033—31no—
62#4031—31no—
63#4031—31no—
64#4029—31no—
65#4025—31no—
66#4024—31no—
67#4023—31no—
68C23×C3×C5C2311yescharacteristic
69C23×C3×C5C2311yescharacteristic
70C22×D15C2311yescharacteristic
71C3×C5×D4—41no—
72C2×C3×C4×C5C2311yescharacteristic
73#1220—22no—
74#1220—41no—
75C2×Dic15—22no—
76C2×Dic15C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C2×C5×Dic3—41no—
80C2×C5×Dic3C2311yescharacteristic
81C2×C5×Dic3C2311yescharacteristic
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C2×C3×Dic5C2311yescharacteristic
85C2×C3×Dic5C2311yescharacteristic
86C2×C3×Dic5C2311yescharacteristic
87C2×C3×Dic5C2311yescharacteristic
88C3×Dic10—41no—
89#1204—22no—
90#1204—41no—
91#1200—41no—
92C3×#124—51no—
93#946—51no—
94#944—51no—
95#939—51no—
96#935D511yescharacteristic, complemented
97#885—51no—
98#885—51no—
99#883—51no—
100#67⋊C4—51no—
101#879—51no—
102#878—51no—
103#878—51no—
104#877—51no—
105#876—51no—
106C4×Dic6—51no—
107C2×C5×D4C2×D311yescharacteristic
108C22×C4×C5—31no—
109C2×C22⋊D5—31no—
110C2×C22⋊D5—31no—
111C22×Dic5—31no—
112C2×C4×D5—31no—
113C2×Dic10—31no—
114C2×Dic10—31no—
115C2×Dic10C2×D311yescharacteristic
116C5×C4⋊C4—31no—
117C5×C4⋊C4—31no—
118C5×C22⋊C4—31no—
119C5×C22⋊C4—31no—
120C5×C22⋊C4—31no—
121C22⋊Dic5—31no—
122C22⋊Dic5C2×D311yescharacteristic
123C22⋊Dic5C2×D311yescharacteristic
124C22⋊Dic5C2×D311yescharacteristic
125C22⋊Dic5C2×D311yescharacteristic
126#685—31no—
127#685—31no—
128#685—31no—
129#685—31no—
130C4⋊Dic5—31no—
131C4⋊Dic5—31no—
132#683—31no—
133#683—31no—
134#683—31no—
135#683—31no—
136#683—31no—
137#683—31no—
138C4×Dic5—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×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5C2411yescharacteristic
150C2×D15—21no—
151C2×D15—21no—
152C2×D15—21no—
153C3×C4×C5—21no—
154Dic15—21no—
155Dic15—21no—
156Dic15#4112yes—
157C3×Dic5—21no—
158C3×Dic5—21no—
159C3×Dic5—21no—
160C3×Dic5—21no—
161C5×Dic3—21no—
162C5×Dic3—21no—
163C5×Dic3—21no—
164C5×Dic3—21no—
165C2×C3×Q8—51no—
166C2×C3×D4C2×D511yescharacteristic
167C2×#67—51no—
168C2×#67—51no—
169C22×Dic3—51no—
170C22×Dic3C2×D511yescharacteristic
171C2×C4×D3—51no—
172C2×Dic6—51no—
173C2×Dic6—51no—
174C3×C22⋊C4—51no—
175C3×C22⋊C4—51no—
176C3×C22⋊C4—51no—
177C3×C22⋊C4—51no—
178C3×C42—51no—
179#217—51no—
180#217C2×D511yescharacteristic
181#217C2×D511yescharacteristic
182#217C2×D511yescharacteristic
183(C2×D3)⋊C4—51no—
184(C2×D3)⋊C4—51no—
185(C2×D3)⋊C4—51no—
186(C2×D3)⋊C4—51no—
187C4⋊Dic3—51no—
188C4⋊Dic3—51no—
189Dic3⋊C4—51no—
190Dic3⋊C4—51no—
191Dic3⋊C4—51no—
192Dic3⋊C4—51no—
193Dic3⋊C4—51no—
194Dic3⋊C4—51no—
195Dic3⋊C4C2×D511yescharacteristic
196Dic3⋊C4C2×D511yescharacteristic
197C4×Dic3—51no—
198C4×Dic3—51no—
199C4×Dic3—51no—
200C23×C5C22×D311yescharacteristic
201C23×C5C22×D311yescharacteristic
202C22×D5—31no—
203C5×D4—41no—
204C2×C4×C5—31no—
205C2×C4×C5—31no—
206C2×C4×C5—31no—
207C2×C4×C5—31no—
208C2×C4×C5—121no—
209C2×C4×C5C22×D311yescharacteristic
210C22⋊D5—62no—
211C22⋊D5—121no—
212C2×Dic5—31no—
213C2×Dic5—31no—
214C2×Dic5—31no—
215C2×Dic5—62no—
216C2×Dic5C22×D311yescharacteristic
217C2×Dic5C22×D311yescharacteristic
218C2×Dic5C22×D311yescharacteristic
219C2×Dic5C22×D311yescharacteristic
220C4×D5—121no—
221Dic10—41no—
222Dic10—62no—
223Dic10—121no—
224C4⋊Q8—151no—
225#126—151no—
226#126—151no—
227#124—51no—
228#124—151no—
229#124—151no—
230#123—31no—
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×C5—41no—
241C2×C3×C5C2×#4111yescharacteristic
242C2×C3×C5H(5,𝔽2)11yescharacteristic
243C2×C3×C52−1+411yescharacteristic
244D15—41no—
245C23×C3C22×D511yescharacteristic
246C23×C3C22×D511yescharacteristic
247C22×D3—51no—
248C3×Q8—201no—
249C3×D4—41no—
250C2×C3×C4—51no—
251C2×C3×C4—51no—
252C2×C3×C4—51no—
253C2×C3×C4—51no—
254C2×C3×C4C22×D511yescharacteristic
255#67—102no—
256#67—201no—
257C2×Dic3—41no—
258C2×Dic3—51no—
259C2×Dic3—51no—
260C2×Dic3—51no—
261C2×Dic3—102no—
262C2×Dic3C22×D511yescharacteristic
263C2×Dic3C22×D511yescharacteristic
264C2×Dic3C22×D511yescharacteristic
265C2×Dic3C22×D511yescharacteristic
266C4×D3—201no—
267Dic6—102no—
268Dic6—201no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—21no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5C23×D311yescharacteristic
276C2×D5—61no—
277C2×D5—61no—
278C2×D5—61no—
279C4×C5—21no—
280C4×C5—61no—
281C4×C5—61no—
282C4×C5—61no—
283C4×C5—61no—
284Dic5—21no—
285Dic5—21no—
286Dic5—21no—
287Dic5—21no—
288Dic5—32no—
289Dic5—61no—
290Dic5—61no—
291C2×Q8—51no—
292C2×Q8—151no—
293C2×Q8—151no—
294C2×D4—151no—
295C2×D4—151no—
296C2×D4D3×D511yescharacteristic
297C22×C4—31no—
298C22×C4—151no—
299C22×C4—151no—
300C4⋊C4—31no—
301C4⋊C4—31no—
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—51no—
317C22⋊C4—151no—
318C22⋊C4—151no—
319C22⋊C4—151no—
320C22⋊C4—151no—
321C22⋊C4—151no—
322C42—51no—
323C42—151no—
324C42—151no—
325C42—151no—
326C3×C5#54211yescharacteristic, complemented, Hall
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3C23×D511yescharacteristic
334C2×D3—101no—
335C2×D3—101no—
336C2×D3—101no—
337C3×C4—21no—
338C3×C4—101no—
339C3×C4—101no—
340C3×C4—101no—
341C3×C4—101no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—21no—
346Dic3—52no—
347Dic3—101no—
348Dic3—101no—
349C2×C5—41no—
350C2×C5—41no—
351C2×C5#100411yescharacteristic
352C2×C5D3×#4111yescharacteristic
353C2×C5#101011yescharacteristic
354D5—121no—
355C23—151no—
356C23C2×D3×D511yescharacteristic
357C23C2×D3×D511yescharacteristic
358Q8—201no—
359Q8—302no—
360Q8—601no—
361D4—41no—
362D4—302no—
363D4—601no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—51no—
369C2×C4—51no—
370C2×C4—51no—
371C2×C4—51no—
372C2×C4—121no—
373C2×C4—151no—
374C2×C4—151no—
375C2×C4—151no—
376C2×C4—302no—
377C2×C4—601no—
378C2×C4C2×D3×D511yescharacteristic
379C2×C3—41no—
380C2×C3—41no—
381C2×C3#414411yescharacteristic
382C2×C3D5×#4111yescharacteristic
383C2×C3#415011yescharacteristic
384D3—201no—
385C5#566911yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—301no—
393C22—301no—
394C22—301no—
395C22C22×D3×D511yescharacteristic
396C4—21no—
397C4—61no—
398C4—61no—
399C4—61no—
400C4—61no—
401C4—101no—
402C4—101no—
403C4—101no—
404C4—101no—
405C4—152no—
406C4—301no—
407C4—301no—
408C3#6599611yescharacteristic, complemented, Sylow
409C2—41no—
410C2—41no—
411C2—601no—
412C2#9202011yescharacteristic
413C2#9202411yescharacteristic
414C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
prime radical#91828C2
hypercenterC2×D4D3×D5
hypocenterC3×C5#542
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C5×#935C2semidirect
3. C3×#4082C2semidirect
4. #91576C2semidirect
5. #91575C2semidirect
6. #91562C2semidirect
7. #91556C2semidirect
8. #91532C2semidirect
9. #91531C2semidirect
10. #91526C2semidirect
11. #91526C2semidirect
12. #91411C2semidirect
13. #91387C2semidirect
14. #91327C2semidirect
15. #91323C2semidirect
55. #4082D3semidirect
96. #935D5semidirect
326. C3×C5#542semidirect
385. C5#5669semidirect
408. C3#65996semidirect

Maximal subgroups

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

subgroupindex
#918282
C5×#9352
C3×#40822
#915762
#915752
#915622
#915562
#915322
#915312
#915262
#914112
#913872
#913272
#913232
#659963
#56695