Finite Group Databasebeta

← #10614413

Subgroups — #10614413

Order 640.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#121C211yesmaximal, characteristic
2C2×#4082C211yesmaximal, characteristic, complemented
3C2×#4081C211yesmaximal, characteristic, complemented
4C2×#4024C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65511C211yesmaximal, characteristic, complemented
8#65509C211yesmaximal, characteristic, complemented
9#65501C211yesmaximal, characteristic, complemented
10#65499C211yesmaximal, characteristic, complemented
11#65268C211yesmaximal, characteristic, complemented
12#65261C211yesmaximal, characteristic, complemented
13#65240C211yesmaximal, characteristic, complemented
14#65238C211yesmaximal, characteristic, complemented
15#65237C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C23×C4×C5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C22×Dic10C2211yescharacteristic
22C5×#121—24no—
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
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#4082—24no—
38#4081—24no—
39C2×C4⋊Dic5C2211yescharacteristic
40C2×C4⋊Dic5C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×#683C2211yescharacteristic
46C2×#683C2211yescharacteristic
47C2×C4×Dic5C2211yescharacteristic
48#4024—24no—
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2672—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C24×C5C2311yescharacteristic
59C2×C5×D4—22no—
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—41no—
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—41no—
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×Dic5—22no—
76C22×Dic5—22no—
77C22×Dic5—41no—
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C2×Dic10—22no—
87C2×Dic10—22no—
88C2×Dic10—41no—
89C5×C4⋊C4—22no—
90C5×C22⋊C4—22no—
91C5×C22⋊C4D414yes—
92C22⋊Dic5—22no—
93C22⋊Dic5—22no—
94C22⋊Dic5—22no—
95C22⋊Dic5—22no—
96C22⋊Dic5—22no—
97C22⋊Dic5—22no—
98C22⋊Dic5—41no—
99C22⋊Dic5—41no—
100C22⋊Dic5—41no—
101C22⋊Dic5—41no—
102C22⋊Dic5D414yescomplemented
103C4⋊Dic5—22no—
104C4⋊Dic5—41no—
105#683—22no—
106#683—22no—
107#683—22no—
108#683—22no—
109#683—22no—
110#683—41no—
111C4×Dic5—22no—
112C2×#124—51no—
113C2×#123—51no—
114C2×C22⋊Q8—51no—
115C2×#121D511yescharacteristic
116#399—51no—
117#396—51no—
118#395—51no—
119#394—51no—
120#394—51no—
121#394—51no—
122#389—51no—
123#388—51no—
124#386—51no—
125#385—51no—
126#380—51no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
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×C5C2×D412yes—
147C23×C5C2×D412yes—
148C23×C5C2411yescharacteristic
149C5×D4—44no—
150C5×D4—44no—
151C5×D4—44no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—44no—
162C2×C4×C5C2×D412yes—
163C2×C4×C5C2×D412yes—
164C2×C4×C5#4112yes—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
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—44no—
188C2×Dic5C2×D412yes—
189C2×Dic5C2×D412yes—
190Dic10—44no—
191C22×Q8—51no—
192C22×D4C2×D511yescharacteristic
193C22×D4C2×D511yescharacteristic
194C22×D4C2×D511yescharacteristic
195C23×C4—51no—
196C23×C4C2×D511yescharacteristic
197#124—104no—
198#123—104no—
199C22⋊Q8—104no—
200#121—24no—
201C2×C4⋊C4—51no—
202C2×C4⋊C4—51no—
203C2×C4⋊C4—51no—
204C2×C4⋊C4—51no—
205C2×C4⋊C4—51no—
206C2×C4⋊C4—51no—
207C2×C4⋊C4—51no—
208C2×C4⋊C4—51no—
209C2×C4⋊C4C2×D511yescharacteristic
210C2×C22⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4—51no—
215C2×C22⋊C4—51no—
216C2×C22⋊C4—51no—
217C2×C22⋊C4—51no—
218C2×C22⋊C4—51no—
219C2×C22⋊C4—51no—
220C2×C22⋊C4—51no—
221C2×C22⋊C4C2×D511yescharacteristic
222C2×C22⋊C4C2×D511yescharacteristic
223C2×C42—51no—
224#95—51no—
225#95—51no—
226#95—51no—
227#95—51no—
228#95—51no—
229#95—51no—
230C22×C5—22no—
231C22×C5—22no—
232C22×C5—22no—
233C22×C5—22no—
234C22×C5—22no—
235C22×C5—22no—
236C22×C5—41no—
237C22×C5—41no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—41no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5C22×D411yescharacteristic
252C22×C5C22×D411yescharacteristic
253C22×C5C2×#4111yescharacteristic
254C22×C5H(5,𝔽2)11yescharacteristic
255C22×C5H(5,𝔽2)11yescharacteristic
256C22×C52−1+411yescharacteristic
257C22×C52−1+411yescharacteristic
258C4×C5—22no—
259C4×C5—22no—
260C4×C5—22no—
261C4×C5—41no—
262Dic5—22no—
263Dic5—22no—
264Dic5—41no—
265Dic5—41no—
266Dic5—41no—
267Dic5—41no—
268Dic5—41no—
269Dic5—41no—
270C24C22×D511yescharacteristic
271C24C22×D511yescharacteristic
272C24C22×D511yescharacteristic
273C2×Q8—102no—
274C2×Q8—102no—
275C2×Q8—201no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—51no—
294C22×C4—51no—
295C22×C4—51no—
296C22×C4—102no—
297C22×C4—102no—
298C22×C4—201no—
299C22×C4C22×D511yescharacteristic
300C22×C4C22×D511yescharacteristic
301C22×C4C22×D511yescharacteristic
302C22×C4C22×D511yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—102no—
305C4⋊C4—102no—
306C4⋊C4—102no—
307C4⋊C4—102no—
308C4⋊C4—102no—
309C4⋊C4—102no—
310C4⋊C4—201no—
311C4⋊C4—201no—
312C22⋊C4—22no—
313C22⋊C4—54no—
314C22⋊C4—102no—
315C22⋊C4—102no—
316C22⋊C4—102no—
317C22⋊C4—102no—
318C22⋊C4—102no—
319C22⋊C4—102no—
320C22⋊C4—201no—
321C22⋊C4—201no—
322C22⋊C4—201no—
323C22⋊C4—201no—
324C22⋊C4C22⋊D514yes—
325C42—102no—
326C2×C5—42no—
327C2×C5—42no—
328C2×C5—81no—
329C2×C5#53611yescharacteristic
330C2×C5#53711yescharacteristic
331C2×C5#54211yescharacteristic
332C2×C5D4211yescharacteristic
333C2×C5#54611yescharacteristic
334C2×C5#54711yescharacteristic
335C2×C5#56211yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23D4×D512yes—
356C23C2×C22⋊D512yes—
357C23C23×D511yescharacteristic
358Q8—204no—
359D4—44no—
360D4—44no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—52no—
373C2×C4—52no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—101no—
383C2×C4—101no—
384C2×C4—101no—
385C2×C4—101no—
386C2×C4—101no—
387C2×C4—101no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—101no—
395C2×C4—101no—
396C2×C4—204no—
397C2×C4#71112yes—
398C2×C4C2×C22⋊D512yes—
399C2×C4C2×C22⋊D512yes—
400C5#267211yescharacteristic, complemented, Sylow
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22C2×D4×D511yescharacteristic
423C22C2×#71111yescharacteristic
424C22#414411yescharacteristic
425C22#414411yescharacteristic
426C22#415011yescharacteristic
427C22#415011yescharacteristic
428C22C22×C22⋊D511yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—102no—
434C4—102no—
435C4—201no—
436C4—201no—
437C4—201no—
438C4—201no—
439C4—201no—
440C4—201no—
441C2—42no—
442C2—42no—
443C2—81no—
444C2#6593111yescharacteristic
445C2#6593711yescharacteristic
446C2#6594511yescharacteristic
447C2#6595511yescharacteristic
448C2D4×C22⋊D511yescharacteristic
449C2#6614011yescharacteristic
450C2#6615911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2672
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4082C2semidirect
3. C2×#4081C2semidirect
4. C2×#4024C2semidirect
5. #65511C2semidirect
6. #65511C2semidirect
7. #65511C2semidirect
8. #65509C2semidirect
9. #65501C2semidirect
10. #65499C2semidirect
11. #65268C2semidirect
12. #65261C2semidirect
13. #65240C2semidirect
14. #65238C2semidirect
15. #65237C2semidirect
102. C22⋊Dic5D4semidirect
400. C5#2672semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40822
C2×#40812
C2×#40242
#655112
#655092
#655012
#654992
#652682
#652612
#652402
#652382
#652372
#26725