Finite Group Databasebeta

← #11754682

Subgroups — #11754682

Order 960.

500 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1220C211yesmaximal, characteristic, complemented
2C5×#930C211yesmaximal, characteristic, complemented
3C3×#4075C211yesmaximal, characteristic, complemented
4#91577C211yesmaximal, characteristic, complemented
5#91565C211yesmaximal, characteristic, complemented
6#91557C211yesmaximal, characteristic, complemented
7#91525C211yesmaximal, characteristic, complemented
8#91475C211yesmaximal, characteristic, complemented
9#91473C211yesmaximal, characteristic, complemented
10#91464C211yesmaximal, characteristic, complemented
11#91423C211yesmaximal, characteristic, complemented
12#91419C211yesmaximal, characteristic, complemented
13#91406C211yesmaximal, characteristic, complemented
14#91367C211yesmaximal, characteristic, complemented
15#91361C211yesmaximal, characteristic, complemented
16#65947—31nomaximal, Hall
17C22×D3×D5C2211yescharacteristic
18C22×C3×C4×C5C2211yescharacteristic, complemented
19C2×#1220C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic
21C2×C5×#67C2211yescharacteristic
22C2×C5×#67C2211yescharacteristic, complemented
23C2×C5×D12C2211yescharacteristic
24C2×C3×C22⋊D5C2211yescharacteristic
25C2×#1204C2211yescharacteristic, complemented
26C2×#1203C2211yescharacteristic
27C2×#1201C2211yescharacteristic
28C2×#1200C2211yescharacteristic
29C2×D3×Dic5C2211yescharacteristic, complemented
30#6719C2211yescharacteristic, complemented
31#6714C2211yescharacteristic
32#6712C2211yescharacteristic, complemented
33C4×Dic15C2211yescharacteristic, complemented
34C5×(C2×D3)⋊C4C2211yescharacteristic
35C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
36C5×C4⋊Dic3C2211yescharacteristic, complemented
37C3×C22⋊Dic5C2211yescharacteristic, complemented
38C3×#685C2211yescharacteristic
39C3×#685C2211yescharacteristic
40C3×#683C2211yescharacteristic, complemented
41C3×#683C2211yescharacteristic, complemented
42Dic3⋊Dic5C2211yescharacteristic, complemented
43#6668C2211yescharacteristic, complemented
44#6667C2211yescharacteristic
45(C2×D3)⋊Dic5C2211yescharacteristic
46(C2×D3)⋊Dic5C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic, complemented
49#6665C2211yescharacteristic
50#6665C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic, complemented
52#5657—51nomaximal, Hall
53C5×#121—31no—
54#4085—31no—
55#4084—31no—
56#4083—31no—
57#4083—31no—
58#4082—31no—
59#4081—31no—
60#4075D311yescharacteristic, complemented
61C4×C22⋊D5—31no—
62#4044—31no—
63#4042—31no—
64#4031—31no—
65#4027—31no—
66D5×C22⋊C4—31no—
67#4025—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C22×C5×D3C2311yescharacteristic
72C22×C3×D5C2311yescharacteristic
73C2×D3×D5—22no—
74C2×D3×D5—41no—
75C2×D3×D5—41no—
76C2×C3×C4×C5—41no—
77C2×C3×C4×C5C2311yescharacteristic
78C2×C3×C4×C5C2311yescharacteristic
79#1220—22no—
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C2×Dic15C2311yescharacteristic
83C4×D15—41no—
84C5×#67—41no—
85C5×#67—41no—
86C2×C5×Dic3C2311yescharacteristic
87C2×C5×Dic3C2311yescharacteristic
88C5×D12—41no—
89C3×C22⋊D5—41no—
90C2×C3×Dic5C2311yescharacteristic
91C2×C3×Dic5C2311yescharacteristic
92C2×C3×Dic5C2311yescharacteristic
93#1204—41no—
94#1203—22no—
95#1201—41no—
96#1200—41no—
97D3×Dic5—41no—
98C3×#123—51no—
99#938—51no—
100#937—51no—
101#930D511yescharacteristic, complemented
102C4×#67—51no—
103#898—51no—
104#897—51no—
105#894—51no—
106#893—51no—
107#885—51no—
108#885—51no—
109#884—51no—
110#882—51no—
111D3×C22⋊C4—51no—
112#879—51no—
113C23×D5—31no—
114C2×C5×D4—31no—
115C2×C5×D4—31no—
116C2×C5×D4—31no—
117C22×C4×C5C2×D311yescharacteristic, complemented
118C2×C22⋊D5—31no—
119C2×C22⋊D5—31no—
120C2×C22⋊D5—31no—
121C2×C22⋊D5C2×D311yescharacteristic
122C22×Dic5—31no—
123C2×C4×D5—31no—
124C2×C4×D5—31no—
125C2×Dic10—31no—
126C5×C4⋊C4—31no—
127C5×C22⋊C4—31no—
128C5×C22⋊C4—31no—
129C22⋊Dic5—31no—
130C22⋊Dic5—31no—
131C22⋊Dic5—31no—
132C22⋊Dic5—31no—
133C22⋊Dic5—31no—
134C22⋊Dic5C2×D311yescharacteristic, complemented
135#685—31no—
136#685—31no—
137#685—31no—
138#685—31no—
139#685C2×D311yescharacteristic
140#685C2×D311yescharacteristic
141C4⋊Dic5—31no—
142#683—31no—
143#683—31no—
144#683C2×D311yescharacteristic, complemented
145#683C2×D311yescharacteristic, complemented
146C4×Dic5—31no—
147C4×Dic5—31no—
148#538—151noSylow
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—21no—
152C22×C3×C5C2411yescharacteristic
153C2×D15—21no—
154C2×D15—21no—
155C2×D15#4112yes—
156C2×C5×D3—21no—
157C2×C5×D3—21no—
158C2×C5×D3—21no—
159C2×C5×D3—21no—
160C2×C5×D3—21no—
161C2×C5×D3—21no—
162C2×C3×D5—21no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165D3×D5—44no—
166C3×C4×C5—21no—
167C3×C4×C5—21no—
168Dic15—21no—
169Dic15—21no—
170Dic15—21no—
171C3×Dic5—21no—
172C3×Dic5—21no—
173C3×Dic5—21no—
174C5×Dic3—21no—
175C5×Dic3—21no—
176C23×D3—51no—
177C2×C3×D4—51no—
178C22×C3×C4C2×D511yescharacteristic, complemented
179C2×#67—51no—
180C2×#67—51no—
181C2×#67C2×D511yescharacteristic
182C2×#67C2×D511yescharacteristic, complemented
183C2×D12—51no—
184C2×D12C2×D511yescharacteristic
185C2×C4×D3—51no—
186C2×C4×D3—51no—
187C2×C4×D3—51no—
188C2×Dic6—51no—
189C3×C4⋊C4—51no—
190C3×C4⋊C4—51no—
191C3×C22⋊C4—51no—
192C3×C22⋊C4—51no—
193C3×C22⋊C4—51no—
194#217—51no—
195#217—51no—
196#217—51no—
197(C2×D3)⋊C4—51no—
198(C2×D3)⋊C4—51no—
199(C2×D3)⋊C4—51no—
200(C2×D3)⋊C4—51no—
201(C2×D3)⋊C4—51no—
202(C2×D3)⋊C4—51no—
203(C2×D3)⋊C4C2×D511yescharacteristic
204(C2×D3)⋊C4C2×D511yescharacteristic, complemented
205C4⋊Dic3—51no—
206C4⋊Dic3C2×D511yescharacteristic, complemented
207Dic3⋊C4—51no—
208Dic3⋊C4—51no—
209C4×Dic3—51no—
210C4×Dic3—51no—
211C23×C5—31no—
212C23×C5—31no—
213C23×C5C22×D311yescharacteristic
214C22×D5—31no—
215C22×D5—62no—
216C22×D5—121no—
217C22×D5—121no—
218C22×D5C22×D311yescharacteristic
219C5×D4—121no—
220C5×D4—121no—
221C5×D4—121no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—41no—
225C2×C4×C5C22×D311yescharacteristic
226C2×C4×C5C22×D311yescharacteristic
227C22⋊D5—41no—
228C22⋊D5—62no—
229C22⋊D5—62no—
230C22⋊D5—121no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5—121no—
235C2×Dic5C22×D311yescharacteristic
236C2×Dic5C22×D311yescharacteristic
237C2×Dic5C22×D311yescharacteristic
238C4×D5—121no—
239C4×D5—121no—
240Dic10—121no—
241#126—151no—
242#126—151no—
243#124—151no—
244#124—151no—
245#123—51no—
246#123—151no—
247C22⋊Q8—151no—
248#121—31no—
249#121—151no—
250#121—151no—
251#120—151no—
252#120—151no—
253C4×D4—151no—
254C4×D4—151no—
255C2×C22⋊C4—151no—
256C2×C3×C5—41no—
257C2×C3×C5C2×#4111yescharacteristic
258C2×C3×C5H(5,𝔽2)11yescharacteristic
259C2×C3×C5H(5,𝔽2)11yescharacteristic
260D15—22no—
261C3×D5—41no—
262C5×D3—41no—
263C5×D3—41no—
264C23×C3—51no—
265C23×C3C22×D511yescharacteristic
266C22×D3—51no—
267C22×D3—102no—
268C22×D3—201no—
269C22×D3—201no—
270C22×D3C22×D511yescharacteristic
271C22×D3C22×D511yescharacteristic
272C3×D4—201no—
273C2×C3×C4—41no—
274C2×C3×C4—51no—
275C2×C3×C4—51no—
276C2×C3×C4—51no—
277C2×C3×C4C22×D511yescharacteristic
278C2×C3×C4C22×D511yescharacteristic
279#67—41no—
280#67—41no—
281#67—102no—
282#67—201no—
283C2×Dic3—51no—
284C2×Dic3—51no—
285C2×Dic3—51no—
286C2×Dic3C22×D511yescharacteristic
287C2×Dic3C22×D511yescharacteristic
288D12—41no—
289D12—102no—
290C4×D3—201no—
291C4×D3—201no—
292C4×D3—201no—
293Dic6—201no—
294C22×C5—21no—
295C22×C5—21no—
296C22×C5—21no—
297C22×C5—61no—
298C22×C5—61no—
299C22×C5—61no—
300C22×C5—61no—
301C22×C5—61no—
302C22×C5—61no—
303C22×C5C23×D311yescharacteristic
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—32no—
308C2×D5—61no—
309C2×D5—61no—
310C2×D5—124no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—61no—
314C4×C5—61no—
315Dic5—21no—
316Dic5—21no—
317Dic5—21no—
318Dic5—61no—
319Dic5—61no—
320Dic5—61no—
321C24—151no—
322C2×Q8—151no—
323C2×D4—31no—
324C2×D4—31no—
325C2×D4—31no—
326C2×D4—51no—
327C2×D4—151no—
328C2×D4—151no—
329C2×D4—151no—
330C22×C4—151no—
331C22×C4—151no—
332C22×C4—151no—
333C22×C4D3×D511yescharacteristic, complemented
334C4⋊C4—31no—
335C4⋊C4—51no—
336C4⋊C4—51no—
337C4⋊C4—151no—
338C4⋊C4—151no—
339C4⋊C4—151no—
340C22⋊C4—31no—
341C22⋊C4—31no—
342C22⋊C4—51no—
343C22⋊C4—51no—
344C22⋊C4—51no—
345C22⋊C4—151no—
346C22⋊C4—151no—
347C22⋊C4—151no—
348C22⋊C4—151no—
349C22⋊C4—151no—
350C22⋊C4—151no—
351C22⋊C4—151no—
352C22⋊C4—151no—
353C22⋊C4—151no—
354C42—151no—
355C42—151no—
356C3×C5#53811yescharacteristic, complemented, Hall
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3C23×D511yescharacteristic
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—21no—
370C2×D3—52no—
371C2×D3—101no—
372C2×D3—101no—
373C2×D3—204no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—101no—
377C3×C4—101no—
378C3×C4—101no—
379Dic3—21no—
380Dic3—21no—
381Dic3—101no—
382Dic3—101no—
383Dic3—101no—
384C2×C5—41no—
385C2×C5—121no—
386C2×C5—121no—
387C2×C5#100411yescharacteristic
388C2×C5D3×#4111yescharacteristic
389C2×C5#100911yescharacteristic
390D5—41no—
391D5—62no—
392C23—31no—
393C23—31no—
394C23—51no—
395C23—151no—
396C23—302no—
397C23—601no—
398C23—601no—
399C23C2×D3×D511yescharacteristic
400Q8—601no—
401D4—121no—
402D4—121no—
403D4—121no—
404D4—201no—
405D4—302no—
406D4—302no—
407D4—601no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—41no—
411C2×C4—51no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—151no—
415C2×C4—151no—
416C2×C4—151no—
417C2×C4—601no—
418C2×C4—601no—
419C2×C4—601no—
420C2×C4C2×D3×D511yescharacteristic
421C2×C4C2×D3×D511yescharacteristic
422C2×C3—41no—
423C2×C3—201no—
424C2×C3#414411yescharacteristic
425C2×C3#414411yescharacteristic
426C2×C3D5×#4111yescharacteristic
427D3—41no—
428D3—41no—
429D3—102no—
430C5#565711yescharacteristic, complemented, Sylow
431C22—21no—
432C22—21no—
433C22—21no—
434C22—61no—
435C22—61no—
436C22—61no—
437C22—61no—
438C22—61no—
439C22—61no—
440C22—101no—
441C22—101no—
442C22—101no—
443C22—152no—
444C22—301no—
445C22—301no—
446C22—604no—
447C22C22×D3×D511yescharacteristic
448C4—21no—
449C4—21no—
450C4—61no—
451C4—61no—
452C4—101no—
453C4—101no—
454C4—101no—
455C4—301no—
456C4—301no—
457C4—301no—
458C3#6594711yescharacteristic, complemented, Sylow
459C2—41no—
460C2—121no—
461C2—121no—
462C2—201no—
463C2—302no—
464C2#9201611yescharacteristic
465C2#9201811yescharacteristic
466C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#538
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×#930C2semidirect
3. C3×#4075C2semidirect
4. #91577C2semidirect
5. #91565C2semidirect
6. #91557C2semidirect
7. #91525C2semidirect
8. #91475C2semidirect
9. #91473C2semidirect
10. #91464C2semidirect
11. #91423C2semidirect
12. #91419C2semidirect
13. #91406C2semidirect
14. #91367C2semidirect
15. #91361C2semidirect
18. C22×C3×C4×C5C22semidirect
22. C2×C5×#67C22semidirect
25. C2×#1204C22semidirect
29. C2×D3×Dic5C22semidirect
30. #6719C22semidirect
32. #6712C22semidirect
33. C4×Dic15C22semidirect
35. C5×(C2×D3)⋊C4C22semidirect
36. C5×C4⋊Dic3C22semidirect
37. C3×C22⋊Dic5C22semidirect
40. C3×#683C22semidirect
41. C3×#683C22semidirect
42. Dic3⋊Dic5C22semidirect
43. #6668C22semidirect
48. (C2×D3)⋊Dic5C22semidirect
51. Dic3×Dic5C22semidirect
60. #4075D3semidirect
101. #930D5semidirect
117. C22×C4×C5C2×D3semidirect
134. C22⋊Dic5C2×D3semidirect
144. #683C2×D3semidirect
145. #683C2×D3semidirect
178. C22×C3×C4C2×D5semidirect
182. C2×#67C2×D5semidirect
204. (C2×D3)⋊C4C2×D5semidirect
206. C4⋊Dic3C2×D5semidirect
333. C22×C4D3×D5semidirect
356. C3×C5#538semidirect
430. C5#5657semidirect
458. C3#65947semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9302
C3×#40752
#915772
#915652
#915572
#915252
#914752
#914732
#914642
#914232
#914192
#914062
#913672
#913612
#659473
#56575