Finite Group Databasebeta

← #11753142

Subgroups — #11753142

Order 960.

480 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91772C211yesmaximal, characteristic, complemented
2C5×#884C211yesmaximal, characteristic, complemented
3C3×#4032C211yesmaximal, characteristic, complemented
4#91567C211yesmaximal, characteristic, complemented
5#91566C211yesmaximal, characteristic, complemented
6Dic5×#67C211yesmaximal, characteristic, complemented
7#91549C211yesmaximal, characteristic, complemented
8#91466C211yesmaximal, characteristic, complemented
9#91454C211yesmaximal, characteristic, complemented
10#91453C211yesmaximal, characteristic, complemented
11#91427C211yesmaximal, characteristic, complemented
12#91414C211yesmaximal, characteristic, complemented
13#91396C211yesmaximal, characteristic, complemented
14#91346C211yesmaximal, characteristic, complemented
15#91338C211yesmaximal, characteristic, complemented
16#65953—31nomaximal, Hall
17C2×#1220C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C5×#67C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C5×D12C2211yescharacteristic
22C2×C4×C5×D3C2211yescharacteristic
23C2×C3×C22⋊D5C2211yescharacteristic
24C22×C3×Dic5C2211yescharacteristic
25C2×#1203C2211yescharacteristic
26C2×#1203C2211yescharacteristic
27C2×#1202C2211yescharacteristic
28C2×#1201C2211yescharacteristic
29C2×#1201C2211yescharacteristic
30C2×D3×Dic5C2211yescharacteristic
31C2×D5×Dic3C2211yescharacteristic
32C3×C5×C22⋊C4C2211yescharacteristic
33#6719C2211yescharacteristic
34#6714C2211yescharacteristic
35C4⋊Dic15C2211yescharacteristic
36#6712C2211yescharacteristic
37C5×(C2×D3)⋊C4C2211yescharacteristic
38C5×Dic3⋊C4C2211yescharacteristic
39C3×#685C2211yescharacteristic
40C3×#685C2211yescharacteristic
41C3×C4⋊Dic5C2211yescharacteristic
42C3×C4⋊Dic5C2211yescharacteristic
43Dic3⋊Dic5C2211yescharacteristic
44Dic3⋊Dic5C2211yescharacteristic
45#6668C2211yescharacteristic
46#6667C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50Dic3×Dic5C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic
52#5663—51nomaximal, Hall
53C5×#121—31no—
54#4086—31no—
55#4085—31no—
56#4085—31no—
57#4084—31no—
58#4084—31no—
59D4×Dic5—31no—
60#4076—31no—
61#4072—31no—
62#4039—31no—
63#4036—31no—
64#4032—31no—
65#4032D311yescharacteristic, complemented
66#4029—31no—
67#4029—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C22×C5×D3C2311yescharacteristic
72C22×C3×D5C2311yescharacteristic
73C2×C3×C4×C5C2311yescharacteristic
74C2×C3×C4×C5C2311yescharacteristic
75#1220—41no—
76C2×Dic15C2311yescharacteristic
77C2×Dic15C2311yescharacteristic
78C2×Dic15C2311yescharacteristic
79C4×D15—41no—
80C5×#67—22no—
81C5×#67—41no—
82C2×C5×Dic3C2311yescharacteristic
83C2×C5×Dic3C2311yescharacteristic
84C5×D12—41no—
85C4×C5×D3—22no—
86C3×C22⋊D5—41no—
87C2×C3×Dic5—41no—
88C2×C3×Dic5C2311yescharacteristic
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91#1203—41no—
92#1203—41no—
93#1202—22no—
94#1201—41no—
95#1201—41no—
96D3×Dic5—41no—
97D5×Dic3—41no—
98C3×#123—51no—
99#940—51no—
100#939—51no—
101#930—51no—
102C4×#67—51no—
103#895—51no—
104#894—51no—
105#892—51no—
106#889—51no—
107#886—51no—
108#884—51no—
109#884—51no—
110#884D511yescharacteristic, complemented
111#883—51no—
112#67⋊C4—51no—
113C2×C5×D4—31no—
114C2×C5×D4—31no—
115C2×C5×D4—31no—
116C22×C4×C5—31no—
117C2×C22⋊D5—31no—
118C2×C22⋊D5—31no—
119C2×C22⋊D5—31no—
120C2×C22⋊D5—31no—
121C2×C22⋊D5—31no—
122C2×C22⋊D5C2×D311yescharacteristic
123C22×Dic5—31no—
124C22×Dic5C2×D311yescharacteristic
125C2×D20—31no—
126C2×C4×D5—31no—
127C2×C4×D5—31no—
128C5×C4⋊C4—31no—
129C5×C22⋊C4—31no—
130C5×C22⋊C4C2×D311yescharacteristic
131C22⋊Dic5—31no—
132C22⋊Dic5—31no—
133C22⋊Dic5—31no—
134C22⋊Dic5—31no—
135#685—31no—
136#685—31no—
137#685C2×D311yescharacteristic
138#685C2×D311yescharacteristic
139C4⋊Dic5—31no—
140C4⋊Dic5—31no—
141C4⋊Dic5—31no—
142C4⋊Dic5C2×D311yescharacteristic
143C4⋊Dic5C2×D311yescharacteristic
144#683—31no—
145#683—31no—
146C4×Dic5—31no—
147C4×Dic5—31no—
148#540—151noSylow
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—21no—
152C22×C3×C5C2411yescharacteristic
153C2×D15—21no—
154C2×D15—21no—
155C2×D15—21no—
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—
165C3×C4×C5—21no—
166C3×C4×C5—21no—
167Dic15—21no—
168Dic15—21no—
169Dic15—21no—
170C3×Dic5—21no—
171C3×Dic5—21no—
172C3×Dic5—21no—
173C5×Dic3—21no—
174C5×Dic3#4112yes—
175C2×C3×D4—51no—
176C22×C3×C4—51no—
177C2×#67—51no—
178C2×#67—51no—
179C2×#67—51no—
180C2×#67—51no—
181C2×#67C2×D511yescharacteristic
182C2×#67C2×D511yescharacteristic
183C22×Dic3—51no—
184C2×D12—51no—
185C2×D12—51no—
186C2×D12C2×D511yescharacteristic
187C2×C4×D3—51no—
188C2×C4×D3—51no—
189C2×C4×D3C2×D511yescharacteristic
190C3×C4⋊C4—51no—
191C3×C4⋊C4—51no—
192C3×C22⋊C4—51no—
193C3×C22⋊C4—51no—
194C3×C22⋊C4C2×D511yescharacteristic
195#217—51no—
196(C2×D3)⋊C4—51no—
197(C2×D3)⋊C4—51no—
198(C2×D3)⋊C4—51no—
199(C2×D3)⋊C4—51no—
200(C2×D3)⋊C4—51no—
201(C2×D3)⋊C4C2×D511yescharacteristic
202C4⋊Dic3—51no—
203C4⋊Dic3—51no—
204Dic3⋊C4—51no—
205Dic3⋊C4—51no—
206Dic3⋊C4—51no—
207Dic3⋊C4C2×D511yescharacteristic
208C4×Dic3—51no—
209C4×Dic3—51no—
210C23×C5—31no—
211C23×C5—31no—
212C23×C5C22×D311yescharacteristic
213C22×D5—31no—
214C22×D5C22×D311yescharacteristic
215C5×D4—62no—
216C5×D4—121no—
217C5×D4—121no—
218C2×C4×C5—31no—
219C2×C4×C5—31no—
220C2×C4×C5—62no—
221C2×C4×C5C22×D311yescharacteristic
222C2×C4×C5C22×D311yescharacteristic
223C22⋊D5—41no—
224C22⋊D5—121no—
225C22⋊D5—121no—
226C22⋊D5—121no—
227C22⋊D5—121no—
228C22⋊D5—121no—
229C2×Dic5—31no—
230C2×Dic5—31no—
231C2×Dic5—31no—
232C2×Dic5—41no—
233C2×Dic5—121no—
234C2×Dic5C22×D311yescharacteristic
235C2×Dic5C22×D311yescharacteristic
236C2×Dic5C22×D311yescharacteristic
237D20—62no—
238C4×D5—121no—
239C4×D5—121no—
240#127—151no—
241#125—151no—
242#123—51no—
243#123—151no—
244#123—151no—
245#123—151no—
246#121—31no—
247#121—151no—
248#121—151no—
249#121—151no—
250#121—151no—
251#121—151no—
252C4×D4—151no—
253C4×D4—151no—
254#117—151no—
255C2×C3×C5—41no—
256C2×C3×C5C2×#4111yescharacteristic
257C2×C3×C5H(5,𝔽2)11yescharacteristic
258C2×C3×C5H(5,𝔽2)11yescharacteristic
259D15—41no—
260C3×D5—41no—
261C5×D3—41no—
262C5×D3—41no—
263C23×C3—51no—
264C23×C3C22×D511yescharacteristic
265C22×D3—51no—
266C22×D3C22×D511yescharacteristic
267C22×D3C22×D511yescharacteristic
268C3×D4—201no—
269C2×C3×C4—51no—
270C2×C3×C4—51no—
271C2×C3×C4—51no—
272C2×C3×C4—201no—
273C2×C3×C4C22×D511yescharacteristic
274C2×C3×C4C22×D511yescharacteristic
275#67—22no—
276#67—41no—
277#67—102no—
278#67—201no—
279#67—201no—
280#67—201no—
281C2×Dic3—51no—
282C2×Dic3—51no—
283C2×Dic3—51no—
284C2×Dic3—201no—
285C2×Dic3C22×D511yescharacteristic
286C2×Dic3C22×D511yescharacteristic
287D12—41no—
288D12—201no—
289D12—201no—
290C4×D3—22no—
291C4×D3—201no—
292C4×D3—201no—
293C22×C5—21no—
294C22×C5—21no—
295C22×C5—21no—
296C22×C5—61no—
297C22×C5—61no—
298C22×C5—61no—
299C22×C5—61no—
300C22×C5—61no—
301C22×C5—61no—
302C22×C5C23×D311yescharacteristic
303C2×D5—21no—
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—61no—
307C2×D5—61no—
308C2×D5—61no—
309C4×C5—21no—
310C4×C5—21no—
311C4×C5—32no—
312C4×C5—61no—
313Dic5—21no—
314Dic5—21no—
315Dic5—21no—
316Dic5—61no—
317Dic5—61no—
318Dic5—61no—
319C2×D4—31no—
320C2×D4—31no—
321C2×D4—31no—
322C2×D4—51no—
323C2×D4—151no—
324C2×D4—151no—
325C2×D4—151no—
326C2×D4—151no—
327C2×D4—151no—
328C2×D4—151no—
329C22×C4—31no—
330C22×C4—51no—
331C22×C4—151no—
332C22×C4—151no—
333C22×C4—151no—
334C4⋊C4—31no—
335C4⋊C4—51no—
336C4⋊C4—51no—
337C4⋊C4—151no—
338C4⋊C4—151no—
339C4⋊C4—151no—
340C4⋊C4—151no—
341C4⋊C4—151no—
342C22⋊C4—31no—
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⋊C4D3×D511yescharacteristic
352C42—151no—
353C42—151no—
354C3×C5#54011yescharacteristic, complemented, Hall
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—101no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3C23×D511yescharacteristic
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—101no—
369C2×D3—101no—
370C2×D3—101no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—101no—
374C3×C4—101no—
375C3×C4—101no—
376Dic3—21no—
377Dic3—101no—
378Dic3—101no—
379Dic3—101no—
380Dic3#71112yes—
381C2×C5—41no—
382C2×C5—121no—
383C2×C5—121no—
384C2×C5#100411yescharacteristic
385C2×C5D3×#4111yescharacteristic
386C2×C5#100911yescharacteristic
387D5—41no—
388D5—121no—
389C23—31no—
390C23—31no—
391C23—51no—
392C23—151no—
393C23C2×D3×D511yescharacteristic
394D4—62no—
395D4—121no—
396D4—121no—
397D4—201no—
398D4—302no—
399D4—601no—
400D4—601no—
401D4—601no—
402D4—601no—
403D4—601no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—51no—
407C2×C4—51no—
408C2×C4—51no—
409C2×C4—62no—
410C2×C4—151no—
411C2×C4—151no—
412C2×C4—151no—
413C2×C4—201no—
414C2×C4—601no—
415C2×C4—601no—
416C2×C4—601no—
417C2×C4C2×D3×D511yescharacteristic
418C2×C4C2×D3×D511yescharacteristic
419C2×C3—41no—
420C2×C3—201no—
421C2×C3C2×#71111yescharacteristic
422C2×C3#414411yescharacteristic
423C2×C3#414911yescharacteristic
424D3—41no—
425D3—41no—
426D3—201no—
427C5#566311yescharacteristic, complemented, Sylow
428C22—21no—
429C22—21no—
430C22—21no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—61no—
436C22—61no—
437C22—101no—
438C22—101no—
439C22—101no—
440C22—301no—
441C22—301no—
442C22—301no—
443C22C22×D3×D511yescharacteristic
444C4—21no—
445C4—21no—
446C4—32no—
447C4—61no—
448C4—101no—
449C4—101no—
450C4—101no—
451C4—301no—
452C4—301no—
453C4—301no—
454C3#6595311yescharacteristic, complemented, Sylow
455C2—41no—
456C2—121no—
457C2—121no—
458C2—201no—
459C2—601no—
460C2#9201711yescharacteristic
461C2D3×#71111yescharacteristic
462C2#9202711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#540
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. #91772C2semidirect
2. C5×#884C2semidirect
3. C3×#4032C2semidirect
4. #91567C2semidirect
5. #91566C2semidirect
6. Dic5×#67C2semidirect
7. #91549C2semidirect
8. #91466C2semidirect
9. #91454C2semidirect
10. #91453C2semidirect
11. #91427C2semidirect
12. #91414C2semidirect
13. #91396C2semidirect
14. #91346C2semidirect
15. #91338C2semidirect
65. #4032D3semidirect
110. #884D5semidirect
354. C3×C5#540semidirect
427. C5#5663semidirect
454. C3#65953semidirect

Maximal subgroups

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

subgroupindex
#917722
C5×#8842
C3×#40322
#915672
#915662
Dic5×#672
#915492
#914662
#914542
#914532
#914272
#914142
#913962
#913462
#913382
#659533
#56635