Finite Group Databasebeta

← #10614389

Subgroups — #10614389

Order 640.

710 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#121C211yesmaximal, characteristic, complemented
2C2×#4084C211yesmaximal, characteristic, complemented
3C2×#4081C211yesmaximal, characteristic, complemented
4C2×#4027C211yesmaximal, characteristic, complemented
5#65512C211yesmaximal, characteristic, complemented
6#65512C211yesmaximal, characteristic, complemented
7#65511C211yesmaximal, characteristic, complemented
8#65509C211yesmaximal, characteristic, complemented
9#65502C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
11#65278C211yesmaximal, characteristic, complemented
12#65273C211yesmaximal, characteristic, complemented
13#65246C211yesmaximal, characteristic, complemented
14#65240C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C23×C4×C5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C22×C22⋊D5C2211yescharacteristic
22C22×C22⋊D5C2211yescharacteristic
23C23×Dic5C2211yescharacteristic
24C22×C4×D5C2211yescharacteristic
25C5×#121—24no—
26C2×C5×C4⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C5×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic
37#4084—24no—
38#4081—24no—
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×C4⋊Dic5C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×#683C2211yescharacteristic
46C2×C4×Dic5C2211yescharacteristic
47C2×C4×Dic5C2211yescharacteristic
48#4027—24no—
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2464—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C24×C5C2311yescharacteristic
59C23×D5C2311yescharacteristic
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—22no—
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C2×C5×D4—41no—
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—41no—
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C2×C22⋊D5—22no—
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C2×C22⋊D5—22no—
82C2×C22⋊D5—22no—
83C2×C22⋊D5—41no—
84C2×C22⋊D5—41no—
85C22×Dic5—22no—
86C22×Dic5—22no—
87C22×Dic5—41no—
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
93C22×Dic5C2311yescharacteristic
94C22×Dic5C2311yescharacteristic
95C2×C4×D5—22no—
96C2×C4×D5—22no—
97C2×C4×D5—41no—
98C5×C4⋊C4—22no—
99C5×C22⋊C4—22no—
100C5×C22⋊C4D414yescomplemented
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—41no—
107C22⋊Dic5—41no—
108C22⋊Dic5D414yescomplemented
109#685—22no—
110#685—22no—
111#685—41no—
112C4⋊Dic5—22no—
113#683—22no—
114#683—22no—
115#683—22no—
116C4×Dic5—22no—
117C4×Dic5—41no—
118C2×#123—51no—
119C2×#121—51no—
120C2×#121D511yescharacteristic, complemented
121C2×C4×D4—51no—
122#398—51no—
123#394—51no—
124#392—51no—
125#392—51no—
126#390—51no—
127#388—51no—
128#388—51no—
129#386—51no—
130#386—51no—
131#385—51no—
132C4×C22⋊C4—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5—21no—
146C23×C5—21no—
147C23×C5—21no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5C2×D412yes—
153C23×C5C2×D412yes—
154C23×C5C2411yescharacteristic
155C22×D5—21no—
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—21no—
161C22×D5#4112yes—
162C5×D4—44no—
163C5×D4—44no—
164C5×D4—44no—
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×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—44no—
175C2×C4×C5C2×D412yes—
176C2×C4×C5C2×D412yes—
177C2×C4×C5#4112yes—
178C22⋊D5—28no—
179C22⋊D5—44no—
180C22⋊D5—44no—
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×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—44no—
200C2×Dic5C2×D412yes—
201C2×Dic5C2×D412yes—
202C2×Dic5#4112yes—
203C4×D5—44no—
204C22×D4—51no—
205C22×D4—51no—
206C22×D4—51no—
207C22×D4C2×D511yescharacteristic
208C22×D4C2×D511yescharacteristic
209C22×D4C2×D511yescharacteristic
210C23×C4—51no—
211C23×C4—51no—
212C23×C4C2×D511yescharacteristic
213#123—104no—
214#121—24no—
215#121—104no—
216C4×D4—104no—
217C2×C4⋊C4—51no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4C2×D511yescharacteristic
222C2×C22⋊C4—51no—
223C2×C22⋊C4—51no—
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⋊C4—51no—
233C2×C22⋊C4C2×D511yescharacteristic
234C2×C22⋊C4C2×D511yescharacteristic
235C2×C42—51no—
236C2×C42—51no—
237#95—51no—
238#95—51no—
239#95—51no—
240#95—51no—
241#95—51no—
242#95—51no—
243C22×C5—22no—
244C22×C5—22no—
245C22×C5—22no—
246C22×C5—22no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5—41no—
264C22×C5C22×D411yescharacteristic
265C22×C5C22×D411yescharacteristic
266C22×C5C2×#4111yescharacteristic
267C22×C5C2×#4111yescharacteristic
268C22×C5C2×#4111yescharacteristic
269C22×C5H(5,𝔽2)11yescharacteristic
270C22×C5H(5,𝔽2)11yescharacteristic
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—22no—
274C2×D5—41no—
275C2×D5—41no—
276C2×D5—41no—
277C2×D5—41no—
278C4×C5—22no—
279C4×C5—22no—
280C4×C5—22no—
281C4×C5—41no—
282Dic5—22no—
283Dic5—22no—
284Dic5—22no—
285Dic5—41no—
286Dic5—41no—
287Dic5—41no—
288Dic5—41no—
289C24—51no—
290C24C22×D511yescharacteristic
291C24C22×D511yescharacteristic
292C24C22×D511yescharacteristic
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—102no—
303C2×D4—102no—
304C2×D4—102no—
305C2×D4—102no—
306C2×D4—102no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—201no—
310C2×D4—201no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—51no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—102no—
322C22×C4—102no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—201no—
326C22×C4—201no—
327C22×C4C22×D511yescharacteristic
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C4⋊C4—22no—
332C4⋊C4—102no—
333C4⋊C4—102no—
334C4⋊C4—102no—
335C4⋊C4—102no—
336C22⋊C4—22no—
337C22⋊C4—54no—
338C22⋊C4—102no—
339C22⋊C4—102no—
340C22⋊C4—102no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—201no—
346C22⋊C4—201no—
347C22⋊C4—201no—
348C22⋊C4C22⋊D514yescomplemented
349C42—102no—
350C42—201no—
351C2×C5—42no—
352C2×C5—42no—
353C2×C5—81no—
354C2×C5#52511yescharacteristic
355C2×C5#53211yescharacteristic
356C2×C5#54011yescharacteristic
357C2×C5D4211yescharacteristic
358C2×C5#54611yescharacteristic
359C2×C5#54611yescharacteristic
360C2×C5#55311yescharacteristic
361D5—42no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—52no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23D4×D512yes—
389C23C2×C22⋊D512yes—
390C23C23×D511yescharacteristic
391D4—44no—
392D4—44no—
393D4—44no—
394D4—108no—
395D4—204no—
396D4—204no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—52no—
408C2×C4—52no—
409C2×C4—52no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—101no—
419C2×C4—101no—
420C2×C4—101no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—204no—
429C2×C4—204no—
430C2×C4#71112yes—
431C2×C4C2×C22⋊D512yes—
432C2×C4C2×C22⋊D512yes—
433C5#246411yescharacteristic, complemented, Sylow
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—102no—
456C22—102no—
457C22—102no—
458C22—201no—
459C22—201no—
460C22—201no—
461C22—201no—
462C22C2×D4×D511yescharacteristic
463C22C2×#71111yescharacteristic
464C22#414411yescharacteristic
465C22#414411yescharacteristic
466C22D5×#4111yescharacteristic
467C22D5×#4111yescharacteristic
468C22C22×C22⋊D511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—102no—
474C4—102no—
475C4—102no—
476C4—201no—
477C4—201no—
478C4—201no—
479C4—201no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—202no—
484C2#6593011yescharacteristic
485C2#6594411yescharacteristic
486C2#6594811yescharacteristic
487C2#6594911yescharacteristic
488C2D4×C22⋊D511yescharacteristic
489C2#6614011yescharacteristic
490C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2464
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C5×#121C2semidirect
2. C2×#4084C2semidirect
3. C2×#4081C2semidirect
4. C2×#4027C2semidirect
5. #65512C2semidirect
6. #65512C2semidirect
7. #65511C2semidirect
8. #65509C2semidirect
9. #65502C2semidirect
10. C4×C22⋊Dic5C2semidirect
11. #65278C2semidirect
12. #65273C2semidirect
13. #65246C2semidirect
14. #65240C2semidirect
15. #65233C2semidirect
100. C5×C22⋊C4D4semidirect
108. C22⋊Dic5D4semidirect
120. C2×#121D5semidirect
348. C22⋊C4C22⋊D5semidirect
433. C5#2464semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40842
C2×#40812
C2×#40272
#655122
#655112
#655092
#655022
C4×C22⋊Dic52
#652782
#652732
#652462
#652402
#652332
#24645