Finite Group Databasebeta

← #10607956

Subgroups — #10607956

Order 640.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic, complemented
2C2×#4042C211yesmaximal, characteristic, complemented
3C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×D20C211yesmaximal, characteristic
5C5×#389C211yesmaximal, characteristic, complemented
6#65283C211yesmaximal, characteristic, complemented
7#65281C211yesmaximal, characteristic, complemented
8#65278C211yesmaximal, characteristic, complemented
9#65278C211yesmaximal, characteristic, complemented
10#65278C211yesmaximal, characteristic, complemented
11#65222C211yesmaximal, characteristic, complemented
12#64954C211yesmaximal, characteristic
13#64954C211yesmaximal, characteristic
14#64946C211yesmaximal, characteristic, complemented
15#64945C211yesmaximal, characteristic, complemented
16C22×D20C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×Dic10C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C4⋊C4C2211yescharacteristic
26C2×C42×C5C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43#4042—24no—
44#4042—24no—
45D5×C4⋊C4—24no—
46C4×D20—24no—
47C5×#95C2211yescharacteristic
48C5×#95C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2449—51nomaximal, Sylow
56C23×D5C2311yescharacteristic
57C23×D5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C2×D20—22no—
72C2×D20—22no—
73C2×D20Q814yescomplemented
74C2×C4×D5—22no—
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5—22no—
80C2×C4×D5—22no—
81C2×C4×D5—22no—
82C2×C4×D5—22no—
83C2×C4×D5—41no—
84C2×C4×D5—41no—
85C2×C4×D5—41no—
86C2×Dic10—22no—
87C2×Dic10—22no—
88C2×Dic10—41no—
89C5×C4⋊C4—22no—
90C5×C4⋊C4—22no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4—22no—
93C42×C5—22no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—22no—
98#685—22no—
99#685—22no—
100#685—41no—
101#685D414yes—
102C4⋊Dic5—22no—
103C4⋊Dic5—22no—
104#683—22no—
105#683—22no—
106#683—22no—
107#683—22no—
108#683—41no—
109#683D414yescomplemented
110C2×C22⋊Q8—51no—
111C2×C22⋊Q8—51no—
112C2×C4×D4—51no—
113C22×C4⋊C4—51no—
114#399—51no—
115#398—51no—
116#396—51no—
117#395—51no—
118#389D511yescharacteristic, complemented
119#387—51no—
120#386—51no—
121#386—51no—
122#385—51no—
123#385—51no—
124#385—51no—
125C23×C5C2411yescharacteristic
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5C2×D412yes—
137C22×D5C2×Q812yes—
138C22×D5C2×Q812yes—
139C22×D5#4112yes—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5C2×D412yes—
158C2×C4×C5C2×D412yes—
159C2×C4×C5C2×Q812yes—
160C2×C4×C5#4112yes—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
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×Dic5C2×D412yes—
177C2×Dic5C2×D412yes—
178C2×Dic5C2×D412yes—
179D20—28no—
180C4×D5—28no—
181C4×D5—44no—
182C4×D5—44no—
183C4×D5—44no—
184Dic10—44no—
185C22×Q8—51no—
186C22×D4—51no—
187C23×C4—51no—
188C23×C4—51no—
189C23×C4—51no—
190C23×C4—51no—
191C22⋊Q8—104no—
192C22⋊Q8—104no—
193C4×D4—104no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4—51no—
197C2×C4⋊C4—51no—
198C2×C4⋊C4—51no—
199C2×C4⋊C4—51no—
200C2×C4⋊C4—51no—
201C2×C4⋊C4—51no—
202C2×C4⋊C4—104no—
203C2×C4⋊C4C2×D511yescharacteristic
204C2×C4⋊C4C2×D511yescharacteristic
205C2×C4⋊C4C2×D511yescharacteristic
206C2×C4⋊C4C2×D511yescharacteristic
207C2×C22⋊C4—51no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4—51no—
210C2×C22⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4—51no—
215C2×C42C2×D511yescharacteristic
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95C2×D511yescharacteristic
223#95C2×D511yescharacteristic
224C22×C5C22×D411yescharacteristic
225C22×C5C22×D411yescharacteristic
226C22×C5C22×Q811yescharacteristic
227C22×C5C2×#4111yescharacteristic
228C22×C5C2×#4111yescharacteristic
229C22×C5H(5,𝔽2)11yescharacteristic
230C22×C52−1+411yescharacteristic
231C2×D5—22no—
232C2×D5—22no—
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—22no—
236C2×D5—22no—
237C2×D5—22no—
238C2×D5—22no—
239C2×D5—22no—
240C2×D5—41no—
241C2×D5—41no—
242C2×D5—41no—
243C2×D5—41no—
244C2×D5C22⋊Q814yescomplemented
245C4×C5—22no—
246C4×C5—22no—
247C4×C5—22no—
248C4×C5—22no—
249C4×C5—41no—
250C4×C5—41no—
251C4×C5—41no—
252Dic5—22no—
253Dic5—22no—
254Dic5—22no—
255Dic5—41no—
256Dic5—41no—
257Dic5—41no—
258C24—51no—
259C24—51no—
260C2×Q8—102no—
261C2×Q8—102no—
262C2×Q8—201no—
263C2×D4—54no—
264C2×D4—102no—
265C2×D4—102no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—102no—
273C22×C4—102no—
274C22×C4—102no—
275C22×C4—102no—
276C22×C4—102no—
277C22×C4—102no—
278C22×C4—102no—
279C22×C4—102no—
280C22×C4—102no—
281C22×C4—201no—
282C22×C4—201no—
283C22×C4—201no—
284C22×C4C22×D511yescharacteristic
285C22×C4C22×D511yescharacteristic
286C22×C4C22×D511yescharacteristic
287C22×C4C22×D511yescharacteristic
288C22×C4C22×D511yescharacteristic
289C22×C4C22×D511yescharacteristic
290C22×C4C22×D511yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—54no—
296C4⋊C4—102no—
297C4⋊C4—102no—
298C4⋊C4—102no—
299C4⋊C4—102no—
300C4⋊C4—102no—
301C4⋊C4—102no—
302C4⋊C4—201no—
303C22⋊C4—54no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—102no—
310C22⋊C4—201no—
311C42—22no—
312C2×C5C2×C22⋊Q811yescharacteristic
313C2×C5#52511yescharacteristic
314C2×C5#53911yescharacteristic
315C2×C5D4211yescharacteristic
316C2×C5#54711yescharacteristic
317C2×C5D4×Q811yescharacteristic
318C2×C5#55411yescharacteristic
319D5—24no—
320D5—42no—
321C23—52no—
322C23—52no—
323C23—52no—
324C23—52no—
325C23—101no—
326C23—101no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23—101no—
331C23—101no—
332C23—101no—
333C23—101no—
334C23—101no—
335C23C23×D511yescharacteristic
336Q8—204no—
337D4—108no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—52no—
356C2×C4—52no—
357C2×C4—52no—
358C2×C4—108no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—204no—
375C2×C4—204no—
376C2×C4—204no—
377C2×C4#70912yes—
378C2×C4D4×D512yes—
379C2×C4D4×D512yes—
380C2×C4Q8×D512yes—
381C5#244911yescharacteristic, complemented, Sylow
382C22—54no—
383C22—102no—
384C22—102no—
385C22—102no—
386C22—102no—
387C22—102no—
388C22—102no—
389C22—102no—
390C22—102no—
391C22—102no—
392C22—201no—
393C22—201no—
394C22—201no—
395C22—201no—
396C22C2×#70911yescharacteristic
397C22C2×D4×D511yescharacteristic
398C22C2×D4×D511yescharacteristic
399C22C2×Q8×D511yescharacteristic
400C22#414711yescharacteristic
401C22D5×#4111yescharacteristic
402C22#414911yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—102no—
411C4—102no—
412C4—102no—
413C4—201no—
414C4—201no—
415C4—201no—
416C2—104no—
417C2—202no—
418C2#6584111yescharacteristic
419C2#6588111yescharacteristic
420C2#6591311yescharacteristic
421C2#6594311yescharacteristic
422C2D5×C22⋊Q811yescharacteristic
423C2#6602611yescharacteristic
424C2#6606111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#389C2
prime radicalC22×D20C22
hypercenter#389D5
hypocenterC5#2449
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
2. C2×#4042C2semidirect
3. C2×D5×C4⋊C4C2semidirect
5. C5×#389C2semidirect
6. #65283C2semidirect
7. #65281C2semidirect
8. #65278C2semidirect
9. #65278C2semidirect
10. #65278C2semidirect
11. #65222C2semidirect
14. #64946C2semidirect
15. #64945C2semidirect
73. C2×D20Q8semidirect
109. #683D4semidirect
118. #389D5semidirect
244. C2×D5C22⋊Q8semidirect
381. C5#2449semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×D5×C4⋊C42
C2×C4×D202
C5×#3892
#652832
#652812
#652782
#652222
#649542
#649462
#649452
#24495