Finite Group Databasebeta

← #10609022

Subgroups — #10609022

Order 640.

548 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4035C211yesmaximal, characteristic, complemented
2C2×#4027C211yesmaximal, characteristic
3C5×#396C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65278C211yesmaximal, characteristic, complemented
6#65276C211yesmaximal, characteristic, complemented
7#65261C211yesmaximal, characteristic, complemented
8#65261C211yesmaximal, characteristic, complemented
9#65246C211yesmaximal, characteristic
10#65246C211yesmaximal, characteristic
11#65238C211yesmaximal, characteristic, complemented
12#65235C211yesmaximal, characteristic, complemented
13#65231C211yesmaximal, characteristic, complemented
14#64965C211yesmaximal, characteristic, complemented
15#64938C211yesmaximal, characteristic, complemented
16C22×C22⋊D5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45#4035—24no—
46#4027—24no—
47C5×#95C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2454—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C2×C22⋊D5—22no—
63C2×C22⋊D5—22no—
64C2×C22⋊D5Q814yes—
65C22×Dic5—22no—
66C22×Dic5—22no—
67C22×Dic5—41no—
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—41no—
78C5×C4⋊C4—22no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C22⋊C4—22no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—41no—
84C22⋊Dic5—22no—
85C22⋊Dic5—41no—
86#685—22no—
87#685—22no—
88#685—41no—
89#685—41no—
90#685—41no—
91C4⋊Dic5—22no—
92#683—22no—
93#683—22no—
94#683—22no—
95#683—22no—
96#683—22no—
97#683—22no—
98#683—41no—
99#683—41no—
100#683D414yes—
101C4×Dic5—22no—
102C4×Dic5—22no—
103C4×Dic5—22no—
104C2×#125—51no—
105C2×C4×D4—51no—
106#399—51no—
107#399—51no—
108#396—51no—
109#396D511yescharacteristic, complemented
110#390—51no—
111#390—51no—
112#389—51no—
113#389—51no—
114#385—51no—
115#385—51no—
116#380—51no—
117#380—51no—
118C4×C4⋊C4—51no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5C2×Q812yes—
126C23×C5C2411yescharacteristic
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5C2×Q812yes—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
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×C5C2×D412yes—
152C22⋊D5—28no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—44no—
169C2×Dic5C2×D412yes—
170C2×Dic5C2×D412yes—
171C2×Dic5C2×Q812yes—
172C2×Dic5#4112yes—
173C2×Dic5#4112yes—
174C2×Dic5#4112yes—
175C4×D5—44no—
176C22×D4—51no—
177C23×C4—51no—
178C23×C4—51no—
179#125—104no—
180C4×D4—104no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4C2×D511yescharacteristic
192C2×C4⋊C4C2×D511yescharacteristic
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
203C2×C22⋊C4C2×D511yescharacteristic
204C2×C42—51no—
205C2×C42—51no—
206C2×C42—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95C2×D511yescharacteristic
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5C22×D411yescharacteristic
221C22×C5C22×Q811yescharacteristic
222C22×C5C2×#4111yescharacteristic
223C22×C5C2×#4111yescharacteristic
224C22×C5C2×#4111yescharacteristic
225C22×C5H(5,𝔽2)11yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—41no—
231C2×D5—41no—
232C2×D5—41no—
233C2×D5—41no—
234C4×C5—22no—
235C4×C5—41no—
236C4×C5—41no—
237C4×C5—41no—
238C4×C5—41no—
239C4×C5—41no—
240Dic5—22no—
241Dic5—22no—
242Dic5—22no—
243Dic5—41no—
244Dic5—41no—
245Dic5—41no—
246Dic5C22⋊Q814yes—
247C24—51no—
248C24C22×D511yescharacteristic
249C2×D4—54no—
250C2×D4—102no—
251C2×D4—102no—
252C22×C4—51no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—102no—
260C22×C4—102no—
261C22×C4—102no—
262C22×C4—102no—
263C22×C4—201no—
264C22×C4—201no—
265C22×C4C22×D511yescharacteristic
266C22×C4C22×D511yescharacteristic
267C22×C4C22×D511yescharacteristic
268C22×C4C22×D511yescharacteristic
269C22×C4C22×D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—54no—
275C4⋊C4—102no—
276C4⋊C4—102no—
277C4⋊C4—102no—
278C4⋊C4—102no—
279C4⋊C4—102no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—201no—
283C4⋊C4—201no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—201no—
291C22⋊C4—201no—
292C22⋊C4—201no—
293C22⋊C4—201no—
294C42—102no—
295C42—102no—
296C42—102no—
297C2×C5—42no—
298C2×C5C2×C22⋊Q811yescharacteristic
299C2×C5#52911yescharacteristic
300C2×C5#54011yescharacteristic
301C2×C5#54611yescharacteristic
302C2×C5#55011yescharacteristic
303C2×C5#55411yescharacteristic
304C2×C5#55411yescharacteristic
305D5—42no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—52no—
313C23—101no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23Q8×D512yes—
320C23C23×D511yescharacteristic
321D4—108no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—52no—
340C2×C4—52no—
341C2×C4—52no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—204no—
361C2×C4—204no—
362C2×C4D4×D512yes—
363C5#245411yescharacteristic, complemented, Sylow
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—102no—
372C22—102no—
373C22—102no—
374C22—201no—
375C22—201no—
376C22—201no—
377C22—201no—
378C22C2×D4×D511yescharacteristic
379C22#414411yescharacteristic
380C22#414411yescharacteristic
381C22C2×Q8×D511yescharacteristic
382C22D5×#4111yescharacteristic
383C22D5×#4111yescharacteristic
384C22D5×#4111yescharacteristic
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—54no—
392C4—102no—
393C4—102no—
394C4—102no—
395C4—201no—
396C4—201no—
397C4—201no—
398C2—42no—
399C2—202no—
400C2#6593311yescharacteristic
401C2#6594411yescharacteristic
402C2#6594911yescharacteristic
403C2D5×C22⋊Q811yescharacteristic
404C2#6597011yescharacteristic
405C2#6597011yescharacteristic
406C2#6604011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#396C2
prime radicalC22×C22⋊D5C22
hypercenter#396D5
hypocenterC5#2454
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#4035C2semidirect
3. C5×#396C2semidirect
4. #65281C2semidirect
5. #65278C2semidirect
6. #65276C2semidirect
7. #65261C2semidirect
8. #65261C2semidirect
11. #65238C2semidirect
12. #65235C2semidirect
13. #65231C2semidirect
14. #64965C2semidirect
15. #64938C2semidirect
109. #396D5semidirect
363. C5#2454semidirect

Maximal subgroups

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

subgroupindex
C2×#40352
C2×#40272
C5×#3962
#652812
#652782
#652762
#652612
#652462
#652382
#652352
#652312
#649652
#649382
#24545