Finite Group Databasebeta

← #10609129

Subgroups — #10609129

Order 640.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#396C211yesmaximal, characteristic, complemented
2#65283C211yesmaximal, characteristic, complemented
3#65279C211yesmaximal, characteristic, complemented
4#65279C211yesmaximal, characteristic, complemented
5#65271C211yesmaximal, characteristic, complemented
6#65270C211yesmaximal, characteristic, complemented
7#65265C211yesmaximal, characteristic, complemented
8#65250C211yesmaximal, characteristic
9#65248C211yesmaximal, characteristic
10#65248C211yesmaximal, characteristic
11#65240C211yesmaximal, characteristic, complemented
12#65234C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#64958C211yesmaximal, characteristic, complemented
15#64942C211yesmaximal, characteristic, complemented
16C22×C22⋊D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41C5×#95C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2795—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⋊D5—41no—
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C5×C4⋊C4—22no—
71C5×C4⋊C4—41no—
72C5×C4⋊C4—41no—
73C5×C22⋊C4—22no—
74C5×C22⋊C4—41no—
75C5×C22⋊C4—41no—
76C22⋊Dic5—22no—
77C22⋊Dic5—22no—
78C22⋊Dic5—41no—
79#685—22no—
80#685—22no—
81#685—22no—
82#685—41no—
83#685—41no—
84#685—41no—
85C4⋊Dic5—22no—
86C4⋊Dic5—41no—
87#683—41no—
88#683—41no—
89#683—41no—
90#683—41no—
91C4×Dic5—22no—
92C4×Dic5—41no—
93C4×Dic5—41no—
94#400—51no—
95#396—51no—
96#396D511yescharacteristic, complemented
97#394—51no—
98#394—51no—
99#394—51no—
100#388—51no—
101#388—51no—
102#388—51no—
103#388—51no—
104#388—51no—
105#388—51no—
106#387—51no—
107#387—51no—
108#384—51no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5C2411yescharacteristic
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—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×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#4112yes—
142C22⋊D5—44no—
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—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#4112yes—
163C2×Dic5#4112yes—
164C22×D4—51no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4—51no—
167C2×C4⋊C4—51no—
168C2×C4⋊C4—51no—
169C2×C4⋊C4—51no—
170C2×C4⋊C4—51no—
171C2×C4⋊C4C2×D511yescharacteristic
172C2×C4⋊C4C2×D511yescharacteristic
173C2×C4⋊C4C2×D511yescharacteristic
174C2×C22⋊C4—51no—
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4—51no—
179C2×C22⋊C4—51no—
180C2×C22⋊C4—51no—
181C2×C22⋊C4—51no—
182C2×C22⋊C4—51no—
183C2×C22⋊C4C2×D511yescharacteristic
184C2×C22⋊C4C2×D511yescharacteristic
185C2×C22⋊C4C2×D511yescharacteristic
186C2×C42—51no—
187C2×C42—51no—
188C2×C42—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95C2×D511yescharacteristic
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5C2×#4111yescharacteristic
207C22×C5C2×#4111yescharacteristic
208C22×C5C2×#4111yescharacteristic
209C22×C5H(5,𝔽2)11yescharacteristic
210C22×C5H(5,𝔽2)11yescharacteristic
211C22×C5H(5,𝔽2)11yescharacteristic
212C22×C52−1+411yescharacteristic
213C2×D5—41no—
214C2×D5—41no—
215C2×D5—41no—
216C2×D5—41no—
217C2×D5—41no—
218C2×D5—41no—
219C2×D5—41no—
220C4×C5—22no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226Dic5—22no—
227Dic5—22no—
228Dic5—41no—
229Dic5—41no—
230Dic5—41no—
231Dic5—41no—
232Dic5—41no—
233C24—51no—
234C24C22×D511yescharacteristic
235C2×D4—201no—
236C2×D4—201no—
237C2×D4—201no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—51no—
241C22×C4—51no—
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—51no—
245C22×C4C22×D511yescharacteristic
246C22×C4C22×D511yescharacteristic
247C22×C4C22×D511yescharacteristic
248C22×C4C22×D511yescharacteristic
249C22×C4C22×D511yescharacteristic
250C22×C4C22×D511yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—102no—
255C4⋊C4—201no—
256C4⋊C4—201no—
257C4⋊C4—201no—
258C4⋊C4—201no—
259C4⋊C4—201no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—102no—
264C22⋊C4—102no—
265C22⋊C4—102no—
266C22⋊C4—102no—
267C22⋊C4—102no—
268C22⋊C4—201no—
269C22⋊C4—201no—
270C22⋊C4—201no—
271C22⋊C4—201no—
272C42—102no—
273C42—201no—
274C42—201no—
275C2×C5—81no—
276C2×C5#53911yescharacteristic
277C2×C5#54211yescharacteristic
278C2×C5#54211yescharacteristic
279C2×C5#55311yescharacteristic
280C2×C5#55311yescharacteristic
281C2×C5#55511yescharacteristic
282C2×C5#56011yescharacteristic
283D5—81no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—101no—
292C23—101no—
293C23—101no—
294C23—101no—
295C23—101no—
296C23—101no—
297C23—101no—
298C23C23×D511yescharacteristic
299D4—204no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—52no—
318C2×C4—52no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—101no—
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#71112yes—
339C5#279511yescharacteristic, complemented, Sylow
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—201no—
348C22—201no—
349C22—201no—
350C22—201no—
351C22—201no—
352C22—201no—
353C22—201no—
354C22C2×#71111yescharacteristic
355C22#414411yescharacteristic
356C22#414411yescharacteristic
357C22#414711yescharacteristic
358C22D5×#4111yescharacteristic
359C22D5×#4111yescharacteristic
360C22#414911yescharacteristic
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—102no—
368C4—102no—
369C4—201no—
370C4—201no—
371C4—201no—
372C4—201no—
373C4—201no—
374C2—81no—
375C2—401no—
376C2#6593511yescharacteristic
377C2#6595011yescharacteristic
378C2#6595411yescharacteristic
379C2#6597211yescharacteristic
380C2#6597411yescharacteristic
381C2#6597711yescharacteristic
382C2#6604311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#396C2
prime radicalC22×C22⋊D5C22
hypercenter#396D5
hypocenterC5#2795
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C5×#396C2semidirect
2. #65283C2semidirect
3. #65279C2semidirect
4. #65279C2semidirect
5. #65271C2semidirect
6. #65270C2semidirect
7. #65265C2semidirect
11. #65240C2semidirect
12. #65234C2semidirect
13. #65233C2semidirect
14. #64958C2semidirect
15. #64942C2semidirect
96. #396D5semidirect
339. C5#2795semidirect

Maximal subgroups

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

subgroupindex
C5×#3962
#652832
#652792
#652712
#652702
#652652
#652502
#652482
#652402
#652342
#652332
#649582
#649422
#27955