Finite Group Databasebeta

← #10607299

Subgroups — #10607299

Order 640.

428 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#388C211yesmaximal, characteristic
2#65273C211yesmaximal, characteristic, complemented
3#65262C211yesmaximal, characteristic, complemented
4#65240C211yesmaximal, characteristic
5#65238C211yesmaximal, characteristic
6#65237C211yesmaximal, characteristic
7#65235C211yesmaximal, characteristic
8#65231C211yesmaximal, characteristic
9C22⋊C4×Dic5C211yesmaximal, characteristic
10#65222C211yesmaximal, characteristic, complemented
11C4×#683C211yesmaximal, characteristic, complemented
12#64946C211yesmaximal, characteristic, complemented
13#64944C211yesmaximal, characteristic, complemented
14#64943C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C23×Dic5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C42×C5C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2522—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C22×C4×C5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×Dic5—22no—
60C22×Dic5—22no—
61C22×Dic5—41no—
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C5×C4⋊C4—41no—
71C5×C22⋊C4—41no—
72C5×C22⋊C4—41no—
73C5×C22⋊C4Q814yes—
74C42×C5—41no—
75C22⋊Dic5—22no—
76C22⋊Dic5—22no—
77C22⋊Dic5—41no—
78C4⋊Dic5—22no—
79C4⋊Dic5—22no—
80C4⋊Dic5—41no—
81#683—22no—
82#683—22no—
83#683—22no—
84#683—22no—
85#683—22no—
86#683—41no—
87#683—41no—
88#683—41no—
89C4×Dic5—22no—
90C4×Dic5—41no—
91C4×Dic5—41no—
92#399—51no—
93#398—51no—
94#398—51no—
95#396—51no—
96#389—51no—
97#389—51no—
98#388—51no—
99#388D511yescharacteristic
100#387—51no—
101#387—51no—
102#387—51no—
103#385—51no—
104#380—51no—
105C4×C4⋊C4—51no—
106C4×C22⋊C4—51no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5C2×Q812yes—
114C23×C5C2411yescharacteristic
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5C2×Q812yes—
132C2×C4×C5C2×Q812yes—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—44no—
154C2×Dic5#4112yes—
155C2×Dic5#4112yes—
156C2×Dic5#4112yes—
157C2×Dic5#4112yes—
158C23×C4—51no—
159C2×C4⋊C4—51no—
160C2×C4⋊C4—51no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4—51no—
167C2×C4⋊C4—51no—
168C2×C4⋊C4—51no—
169C2×C4⋊C4—51no—
170C2×C4⋊C4C2×D511yescharacteristic
171C2×C22⋊C4—51no—
172C2×C22⋊C4—51no—
173C2×C22⋊C4—51no—
174C2×C22⋊C4C2×D511yescharacteristic
175C2×C22⋊C4C2×D511yescharacteristic
176C2×C22⋊C4C2×D511yescharacteristic
177C2×C42—51no—
178C2×C42—51no—
179C2×C42—51no—
180C2×C42C2×D511yescharacteristic
181#95—51no—
182#95—51no—
183#95—51no—
184#95—51no—
185#95—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95—51no—
191#95C2×D511yescharacteristic
192#95C2×D511yescharacteristic
193C22×C5—22no—
194C22×C5—22no—
195C22×C5—22no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5C22×Q811yescharacteristic
201C22×C5C2×#4111yescharacteristic
202C22×C5C2×#4111yescharacteristic
203C22×C5C2×#4111yescharacteristic
204C22×C5C2×#4111yescharacteristic
205C22×C5H(5,𝔽2)11yescharacteristic
206C22×C5H(5,𝔽2)11yescharacteristic
207C4×C5—22no—
208C4×C5—22no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213Dic5—22no—
214Dic5—22no—
215Dic5—22no—
216Dic5—22no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220Dic5—41no—
221C24C22×D511yescharacteristic
222C22×C4—51no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—102no—
231C22×C4—102no—
232C22×C4—201no—
233C22×C4C22×D511yescharacteristic
234C22×C4C22×D511yescharacteristic
235C22×C4C22×D511yescharacteristic
236C22×C4C22×D511yescharacteristic
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C4⋊C4—41no—
240C4⋊C4—102no—
241C4⋊C4—102no—
242C4⋊C4—102no—
243C4⋊C4—102no—
244C4⋊C4—102no—
245C4⋊C4—102no—
246C4⋊C4—102no—
247C4⋊C4—201no—
248C4⋊C4—201no—
249C4⋊C4—201no—
250C4⋊C4—201no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—102no—
254C22⋊C4—102no—
255C22⋊C4—201no—
256C22⋊C4Dic1014yes—
257C42—41no—
258C42—102no—
259C42—201no—
260C42—201no—
261C2×C5—42no—
262C2×C5#52911yescharacteristic
263C2×C5#53311yescharacteristic
264C2×C5#54011yescharacteristic
265C2×C5#55111yescharacteristic
266C2×C5#55311yescharacteristic
267C2×C5#55411yescharacteristic
268C2×C5#55411yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23C2×Dic1012yes—
276C23C23×D511yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—52no—
296C2×C4—52no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—101no—
306C2×C4—101no—
307C2×C4—101no—
308C2×C4—101no—
309C2×C4—101no—
310C2×C4—101no—
311C2×C4—101no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—204no—
318C2×C4C2×Dic1012yes—
319C2×C4C2×Dic1012yes—
320C5#252211yescharacteristic, complemented, Sylow
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C22×Dic1011yescharacteristic
329C22#414411yescharacteristic
330C22#414411yescharacteristic
331C22D5×#4111yescharacteristic
332C22D5×#4111yescharacteristic
333C22D5×#4111yescharacteristic
334C22D5×#4111yescharacteristic
335C4—22no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—102no—
342C4—102no—
343C4—102no—
344C4—102no—
345C4—201no—
346C4—201no—
347C4—201no—
348C4—201no—
349C2—42no—
350C2#6585111yescharacteristic
351C2#6587311yescharacteristic
352C2#6587311yescharacteristic
353C2#6594911yescharacteristic
354C2#6599511yescharacteristic
355C2#6601311yescharacteristic
356C2#6604011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#388C2
prime radicalC24×C5C23
hypercenter#388D5
hypocenterC5#2522
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #65273C2semidirect
3. #65262C2semidirect
10. #65222C2semidirect
11. C4×#683C2semidirect
12. #64946C2semidirect
13. #64944C2semidirect
14. #64943C2semidirect
15. #64936C2semidirect
320. C5#2522semidirect

Maximal subgroups

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

subgroupindex
C5×#3882
#652732
#652622
#652402
#652382
#652372
#652352
#652312
C22⋊C4×Dic52
#652222
C4×#6832
#649462
#649442
#649432
#649362
#25225