Finite Group Databasebeta

← #10606331

Subgroups — #10606331

Order 640.

864 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#709C211yesmaximal, characteristic
2C2×#4040C211yesmaximal, characteristic, complemented
3C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×#4029C211yesmaximal, characteristic, complemented
5C5×#385C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8#65268C211yesmaximal, characteristic, complemented
9#65250C211yesmaximal, characteristic, complemented
10#65248C211yesmaximal, characteristic, complemented
11#65245C211yesmaximal, characteristic, complemented
12#64957C211yesmaximal, characteristic, complemented
13#64954C211yesmaximal, characteristic, complemented
14#64953C211yesmaximal, characteristic, complemented
15#64945C211yesmaximal, characteristic, complemented
16C23×C4×C5C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C2×#709—24no—
20C2×#709—24no—
21C2×#709—24no—
22C22×D20C2211yescharacteristic
23C22×C4×D5C2211yescharacteristic
24C22×C4×D5C2211yescharacteristic
25C22×C4×D5C2211yescharacteristic
26C22×C4×D5C2211yescharacteristic
27C22×Dic10C2211yescharacteristic
28C2×C5×C4⋊C4C2211yescharacteristic
29C2×C5×C4⋊C4C2211yescharacteristic
30C2×C5×C22⋊C4C2211yescharacteristic
31C2×C5×C22⋊C4C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×C4⋊Dic5C2211yescharacteristic
41C2×C4⋊Dic5C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×#683C2211yescharacteristic
46#4040—24no—
47D5×C4⋊C4—24no—
48#4029—24no—
49C5×#95C2211yescharacteristic
50C5×#95C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#3963C2211yescharacteristic
56#3963C2211yescharacteristic
57#2406—51nomaximal, Sylow
58C24×C5C2311yescharacteristic
59C23×D5C2311yescharacteristic
60C23×D5C2311yescharacteristic
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5—22no—
73C2×C22⋊D5—22no—
74C2×C22⋊D5—22no—
75C2×C22⋊D5D414yes—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82#709—232no—
83C2×D20—22no—
84C2×D20—22no—
85C2×D20D414yes—
86C2×C4×D5—22no—
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C2×C4×D5—22no—
90C2×C4×D5—22no—
91C2×C4×D5—22no—
92C2×C4×D5—22no—
93C2×C4×D5—22no—
94C2×C4×D5—22no—
95C2×C4×D5—22no—
96C2×C4×D5—22no—
97C2×C4×D5D414yes—
98C2×Dic10—22no—
99C2×Dic10—22no—
100C2×Dic10—22no—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—22no—
103C5×C22⋊C4—22no—
104C5×C22⋊C4—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—22no—
112#685—22no—
113C4⋊Dic5—22no—
114C4⋊Dic5—22no—
115#683—22no—
116#683—22no—
117#683—22no—
118#683—22no—
119C22×#41—51no—
120C2×#123—51no—
121C2×#123—51no—
122C22×C4⋊C4—51no—
123#395—51no—
124#395—51no—
125#394—51no—
126#394—51no—
127#386—51no—
128#386—51no—
129#385—51no—
130#385—51no—
131#385—51no—
132#385D511yescharacteristic, complemented
133#381—51no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5C2×D412yes—
141C23×C5C2411yescharacteristic
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5C2×D412yes—
153C22×D5C2×D412yes—
154C22×D5C2×D412yes—
155C22×D5C2×D412yes—
156C2×C4×C5—28no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5C2×D412yes—
173C2×C4×C5C2×D412yes—
174C2×C4×C5#4112yescomplemented
175C22⋊D5—28no—
176C22⋊D5—28no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—21no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5C2×D412yes—
193C2×Dic5C2×D412yes—
194C2×Dic5#4112yescomplemented
195D20—28no—
196C4×D5—28no—
197C4×D5—28no—
198C4×D5—28no—
199C4×D5—44no—
200Dic10—28no—
201C2×#41—104no—
202C2×#41—104no—
203C2×#41—104no—
204C22×Q8—51no—
205C22×D4—51no—
206C22×D4—51no—
207C22×D4—51no—
208C23×C4—51no—
209C23×C4—51no—
210C23×C4—51no—
211C23×C4—51no—
212C23×C4C2×D511yescharacteristic
213#123—104no—
214#123—104no—
215C2×C4⋊C4—51no—
216C2×C4⋊C4—51no—
217C2×C4⋊C4—51no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—104no—
222C2×C4⋊C4C2×D511yescharacteristic
223C2×C4⋊C4C2×D511yescharacteristic
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4C2×D511yescharacteristic
233C2×C22⋊C4C2×D511yescharacteristic
234#95—51no—
235#95—51no—
236#95—51no—
237#95—51no—
238#95—51no—
239#95—51no—
240#95C2×D511yescharacteristic
241#95C2×D511yescharacteristic
242C22×C5—22no—
243C22×C5—22no—
244C22×C5—22no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5C22×D411yescharacteristic
250C22×C5C22×D411yescharacteristic
251C22×C5C22×D411yescharacteristic
252C22×C5C2×#4111yescharacteristic
253C22×C5C2×#4111yescharacteristic
254C22×C52−1+411yescharacteristic
255C22×C52−1+411yescharacteristic
256C2×D5—22no—
257C2×D5—22no—
258C2×D5—22no—
259C2×D5—22no—
260C2×D5—22no—
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—41no—
266C2×D5—41no—
267C2×D5—41no—
268C2×D5—41no—
269C2×D5#12014yes—
270C4×C5—22no—
271C4×C5—22no—
272C4×C5—22no—
273C4×C5—41no—
274C4×C5—41no—
275C4×C5—41no—
276Dic5—22no—
277Dic5—22no—
278Dic5—22no—
279Dic5—41no—
280Dic5—41no—
281Dic5—41no—
282C24—51no—
283C24—51no—
284C24C22×D511yescharacteristic
285#41—1032no—
286C2×Q8—102no—
287C2×Q8—102no—
288C2×Q8—102no—
289C2×D4—54no—
290C2×D4—54no—
291C2×D4—102no—
292C2×D4—102no—
293C2×D4—102no—
294C2×D4—102no—
295C2×D4—102no—
296C2×D4—102no—
297C2×D4—102no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—51no—
304C22×C4—54no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—102no—
309C22×C4—102no—
310C22×C4—102no—
311C22×C4—102no—
312C22×C4—102no—
313C22×C4—102no—
314C22×C4—102no—
315C22×C4—102no—
316C22×C4—102no—
317C22×C4—102no—
318C22×C4—102no—
319C22×C4C22×D511yescharacteristic
320C22×C4C22×D511yescharacteristic
321C22×C4C22×D511yescharacteristic
322C22×C4C22×D511yescharacteristic
323C22×C4C22×D511yescharacteristic
324C22×C4C22×D511yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—102no—
328C4⋊C4—102no—
329C4⋊C4—102no—
330C4⋊C4—102no—
331C4⋊C4—102no—
332C4⋊C4—102no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—102no—
336C22⋊C4—102no—
337C22⋊C4—102no—
338C22⋊C4—102no—
339C22⋊C4—102no—
340C22⋊C4—102no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C2×C5—42no—
344C2×C5C2×#12011yescharacteristic
345C2×C5#52511yescharacteristic
346C2×C5#53611yescharacteristic
347C2×C5#54711yescharacteristic
348C2×C5#54711yescharacteristic
349C2×C5#54711yescharacteristic
350C2×C5#54711yescharacteristic
351D5—24no—
352D5—42no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—52no—
360C23—52no—
361C23—52no—
362C23—52no—
363C23—101no—
364C23—101no—
365C23—101no—
366C23—101no—
367C23—101no—
368C23—101no—
369C23—101no—
370C23—101no—
371C23—101no—
372C23—101no—
373C23D4×D512yes—
374C23C23×D511yescharacteristic
375Q8—108no—
376D4—108no—
377D4—108no—
378D4—108no—
379C2×C4—28no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—52no—
396C2×C4—52no—
397C2×C4—52no—
398C2×C4—108no—
399C2×C4—108no—
400C2×C4—108no—
401C2×C4—101no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—101no—
408C2×C4—101no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—204no—
417C2×C4#70912yescomplemented
418C2×C4D4×D512yes—
419C2×C4D4×D512yes—
420C5#240611yescharacteristic, complemented, Sylow
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—54no—
429C22—102no—
430C22—102no—
431C22—102no—
432C22—102no—
433C22—102no—
434C22—102no—
435C22—102no—
436C22—102no—
437C22—102no—
438C22—201no—
439C22—201no—
440C22—201no—
441C22—201no—
442C22C2×#70911yescharacteristic
443C22C2×D4×D511yescharacteristic
444C22C2×D4×D511yescharacteristic
445C22C2×D4×D511yescharacteristic
446C22#414711yescharacteristic
447C22D5×#4111yescharacteristic
448C22#415011yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—102no—
456C4—102no—
457C4—102no—
458C4—201no—
459C4—201no—
460C4—201no—
461C2—42no—
462C2—104no—
463C2—202no—
464C2#6584111yescharacteristic
465C2#6588111yescharacteristic
466C2#6588511yescharacteristic
467C2D5×#12011yescharacteristic
468C2#6596211yescharacteristic
469C2#6596511yescharacteristic
470C2#6598911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC22×#709C2
hypercenter#385D5
hypocenterC5#2406
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#4040C2semidirect
3. C2×D5×C4⋊C4C2semidirect
4. C2×#4029C2semidirect
5. C5×#385C2semidirect
6. #65499C2semidirect
7. #65278C2semidirect
8. #65268C2semidirect
9. #65250C2semidirect
10. #65248C2semidirect
11. #65245C2semidirect
12. #64957C2semidirect
13. #64954C2semidirect
14. #64953C2semidirect
15. #64945C2semidirect
132. #385D5semidirect
174. C2×C4×C5#41semidirect
194. C2×Dic5#41semidirect
417. C2×C4#709semidirect
420. C5#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#7092
C2×#40402
C2×D5×C4⋊C42
C2×#40292
C5×#3852
#654992
#652782
#652682
#652502
#652482
#652452
#649572
#649542
#649532
#649452
#24065