Finite Group Databasebeta

← #10609854

Subgroups — #10609854

Order 640.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#399C211yesmaximal, characteristic, complemented
2#65283C211yesmaximal, characteristic, complemented
3#65283C211yesmaximal, characteristic, complemented
4#65276C211yesmaximal, characteristic, complemented
5#65272C211yesmaximal, characteristic, complemented
6#65270C211yesmaximal, characteristic, complemented
7#65267C211yesmaximal, characteristic, complemented
8#65247C211yesmaximal, characteristic
9#65246C211yesmaximal, characteristic
10#65245C211yesmaximal, characteristic
11#65239C211yesmaximal, characteristic, complemented
12#65238C211yesmaximal, characteristic, complemented
13#65235C211yesmaximal, characteristic, complemented
14#64956C211yesmaximal, characteristic, complemented
15#64946C211yesmaximal, 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×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41C5×#95C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2748—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C23×D5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C2×C22⋊D5—41no—
61C2×C22⋊D5—41no—
62C2×C22⋊D5Q814yes—
63C22×Dic5—41no—
64C22×Dic5—41no—
65C22×Dic5—41no—
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C2×C4×D5—22no—
74C2×C4×D5—22no—
75C2×C4×D5—41no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—41no—
79C5×C22⋊C4—22no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C22⋊Dic5—22no—
83C22⋊Dic5—41no—
84#685—41no—
85#685—41no—
86#685—41no—
87#685—41no—
88#685—41no—
89C4⋊Dic5—41no—
90C4⋊Dic5—41no—
91C4⋊Dic5—41no—
92#683—22no—
93#683—41no—
94#683—41no—
95#683—41no—
96#683—41no—
97C4×Dic5—41no—
98#399—51no—
99#399D511yescharacteristic, complemented
100#398—51no—
101#398—51no—
102#396—51no—
103#396—51no—
104#390—51no—
105#387—51no—
106#387—51no—
107#386—51no—
108#386—51no—
109#385—51no—
110#385—51no—
111#380—51no—
112#380—51no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5C2×Q812yes—
120C23×C5C2411yescharacteristic
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5C2×Q812yes—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
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#4112yes—
146C22⋊D5—44no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
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—44no—
167C2×Dic5C2×Q812yes—
168C2×Dic5#4112yes—
169C4×D5—44no—
170C22×D4—51no—
171C23×C4—51no—
172C23×C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
181C2×C4⋊C4C2×D511yescharacteristic
182C2×C4⋊C4C2×D511yescharacteristic
183C2×C4⋊C4C2×D511yescharacteristic
184C2×C22⋊C4—51no—
185C2×C22⋊C4—51no—
186C2×C22⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4C2×D511yescharacteristic
192C2×C22⋊C4C2×D511yescharacteristic
193C2×C22⋊C4C2×D511yescharacteristic
194C2×C42—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95C2×D511yescharacteristic
205C22×C5—22no—
206C22×C5—22no—
207C22×C5—22no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5C22×Q811yescharacteristic
213C22×C5C2×#4111yescharacteristic
214C22×C5C2×#4111yescharacteristic
215C22×C5H(5,𝔽2)11yescharacteristic
216C22×C5H(5,𝔽2)11yescharacteristic
217C22×C5H(5,𝔽2)11yescharacteristic
218C22×C5H(5,𝔽2)11yescharacteristic
219C2×D5—22no—
220C2×D5—22no—
221C2×D5—22no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C4×C5—22no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5—41no—
232Dic5—22no—
233Dic5—22no—
234Dic5—41no—
235Dic5—41no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239C24—51no—
240C24C22×D511yescharacteristic
241C2×D4—54no—
242C2×D4—201no—
243C2×D4—201no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—102no—
252C22×C4—102no—
253C22×C4—201no—
254C22×C4—201no—
255C22×C4—201no—
256C22×C4—201no—
257C22×C4C22×D511yescharacteristic
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—102no—
267C4⋊C4—201no—
268C4⋊C4—201no—
269C4⋊C4—201no—
270C4⋊C4—201no—
271C4⋊C4—201no—
272C4⋊C4—201no—
273C4⋊C4—201no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—102no—
278C22⋊C4—201no—
279C22⋊C4—201no—
280C22⋊C4—201no—
281C22⋊C4—201no—
282C22⋊C4—201no—
283C22⋊C4—201no—
284C42—201no—
285C2×C5—42no—
286C2×C5#54011yescharacteristic
287C2×C5#54011yescharacteristic
288C2×C5#54311yescharacteristic
289C2×C5#55111yescharacteristic
290C2×C5#55411yescharacteristic
291C2×C5#55411yescharacteristic
292C2×C5#56011yescharacteristic
293D5—42no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—52no—
301C23—101no—
302C23—101no—
303C23—101no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23Q8×D512yes—
308C23C23×D511yescharacteristic
309D4—204no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—52no—
328C2×C4—52no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—101no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—204no—
349C2×C4—204no—
350C2×C4#71112yes—
351C5#274811yescharacteristic, complemented, Sylow
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—102no—
360C22—102no—
361C22—102no—
362C22—201no—
363C22—201no—
364C22—201no—
365C22—201no—
366C22C2×#71111yescharacteristic
367C22#414411yescharacteristic
368C22#414411yescharacteristic
369C22C2×Q8×D511yescharacteristic
370C22D5×#4111yescharacteristic
371C22#414911yescharacteristic
372C22#414911yescharacteristic
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—102no—
380C4—102no—
381C4—201no—
382C4—201no—
383C4—201no—
384C4—201no—
385C4—201no—
386C2—42no—
387C2—202no—
388C2#6596811yescharacteristic
389C2#6596911yescharacteristic
390C2#6597011yescharacteristic
391C2#6599211yescharacteristic
392C2#6599711yescharacteristic
393C2#6600011yescharacteristic
394C2#6605311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2748
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C5×#399C2semidirect
2. #65283C2semidirect
3. #65283C2semidirect
4. #65276C2semidirect
5. #65272C2semidirect
6. #65270C2semidirect
7. #65267C2semidirect
11. #65239C2semidirect
12. #65238C2semidirect
13. #65235C2semidirect
14. #64956C2semidirect
15. #64946C2semidirect
99. #399D5semidirect
351. C5#2748semidirect

Maximal subgroups

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

subgroupindex
C5×#3992
#652832
#652762
#652722
#652702
#652672
#652472
#652462
#652452
#652392
#652382
#652352
#649562
#649462
#27485