Finite Group Databasebeta

← #10614815

Subgroups — #10614815

Order 640.

496 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
1C2×C5×#126C211yesmaximal, characteristic, complemented
2C2×#4044C211yesmaximal, characteristic, complemented
3#65283C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65265C211yesmaximal, characteristic, complemented
6C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
7#65261C211yesmaximal, characteristic, complemented
8#65246C211yesmaximal, characteristic
9#65246C211yesmaximal, characteristic
10#65246C211yesmaximal, characteristic
11#65242C211yesmaximal, characteristic, complemented
12#65237C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15C4×#683C211yesmaximal, characteristic, complemented
16C22×C22⋊D5C2211yescharacteristic
17C5×#126—24no—
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C42×C5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4044—24no—
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2512—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⋊D5—22no—
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C5×C4⋊C4—22no—
73C5×C4⋊C4—22no—
74C5×C4⋊C4D414yescomplemented
75C5×C22⋊C4—22no—
76C5×C22⋊C4—22no—
77C5×C22⋊C4—22no—
78C42×C5—22no—
79C22⋊Dic5—22no—
80C22⋊Dic5—41no—
81C22⋊Dic5—41no—
82#685—22no—
83#685—22no—
84#685—22no—
85#685—22no—
86#685—41no—
87#685—41no—
88C4⋊Dic5—22no—
89C4⋊Dic5—22no—
90#683—22no—
91#683—22no—
92#683—22no—
93#683—22no—
94#683—41no—
95#683—41no—
96C4×Dic5—22no—
97C4×Dic5—22no—
98C4×Dic5—22no—
99C4×Dic5—22no—
100C2×#126—51no—
101C2×#126D511yescharacteristic, complemented
102#397—51no—
103#397—51no—
104#396—51no—
105#396—51no—
106#390—51no—
107#390—51no—
108#390—51no—
109#389—51no—
110#388—51no—
111#388—51no—
112#384—51no—
113C4×C4⋊C4—51no—
114C4×C4⋊C4—51no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5C2411yescharacteristic
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—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×C5C2×D412yes—
146C2×C4×C5C2×D412yes—
147C2×C4×C5C2×D412yes—
148C22⋊D5—44no—
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#4112yes—
167C2×Dic5#4112yes—
168C2×Dic5#4112yes—
169C2×Dic5#4112yes—
170C22×D4—51no—
171#126—24no—
172#126—104no—
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⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4C2×D511yescharacteristic
194C2×C22⋊C4C2×D511yescharacteristic
195C2×C22⋊C4C2×D511yescharacteristic
196C2×C42—51no—
197C2×C42—51no—
198C2×C42—51no—
199C2×C42—51no—
200C2×C42C2×D511yescharacteristic
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95—51no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5C22×D411yescharacteristic
215C22×C5C2×#4111yescharacteristic
216C22×C5C2×#4111yescharacteristic
217C22×C5C2×#4111yescharacteristic
218C22×C5C2×#4111yescharacteristic
219C22×C5H(5,𝔽2)11yescharacteristic
220C22×C52−1+411yescharacteristic
221C2×D5—41no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C4×C5—22no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—41no—
232C4×C5—41no—
233C4×C5—41no—
234Dic5—22no—
235Dic5—41no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240Dic5#12614yescomplemented
241C24—51no—
242C24C22×D511yescharacteristic
243C2×D4—102no—
244C2×D4—102no—
245C2×D4—102no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4C22×D511yescharacteristic
254C22×C4C22×D511yescharacteristic
255C22×C4C22×D511yescharacteristic
256C22×C4C22×D511yescharacteristic
257C22×C4C22×D511yescharacteristic
258C22×C4C22×D511yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—102no—
262C4⋊C4—102no—
263C4⋊C4—102no—
264C4⋊C4—102no—
265C4⋊C4—102no—
266C4⋊C4—102no—
267C4⋊C4—201no—
268C4⋊C4—201no—
269C4⋊C4C22⋊D514yescomplemented
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—102no—
274C22⋊C4—102no—
275C22⋊C4—102no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—201no—
279C22⋊C4—201no—
280C22⋊C4—201no—
281C22⋊C4—201no—
282C42—22no—
283C42—102no—
284C42—102no—
285C42—102no—
286C42—102no—
287C2×C5—81no—
288C2×C5C2×#12611yescharacteristic
289C2×C5#53211yescharacteristic
290C2×C5#54211yescharacteristic
291C2×C5#54711yescharacteristic
292C2×C5#55011yescharacteristic
293C2×C5#55311yescharacteristic
294C2×C5#55611yescharacteristic
295D5—81no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—101no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23—101no—
308C23—101no—
309C23—101no—
310C23C23×D511yescharacteristic
311D4—204no—
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—52no—
330C2×C4—52no—
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×C4C2×C22⋊D512yes—
349C2×C4C2×C22⋊D512yes—
350C2×C4C2×C22⋊D512yes—
351C5#251211yescharacteristic, complemented, Sylow
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—201no—
360C22—201no—
361C22—201no—
362C22—201no—
363C22—201no—
364C22—201no—
365C22—201no—
366C22D5×#4111yescharacteristic
367C22D5×#4111yescharacteristic
368C22D5×#4111yescharacteristic
369C22D5×#4111yescharacteristic
370C22#414911yescharacteristic
371C22#415011yescharacteristic
372C22C22×C22⋊D511yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—54no—
380C4—102no—
381C4—201no—
382C4—201no—
383C4—201no—
384C4—201no—
385C4—201no—
386C2—81no—
387C2—401no—
388C2#6603611yescharacteristic
389C2D5×#12611yescharacteristic
390C2#6604311yescharacteristic
391C2#6604411yescharacteristic
392C2#6615811yescharacteristic
393C2#6616611yescharacteristic
394C2#6616811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#126C2
prime radicalC22×C22⋊D5C22
hypercenterC2×#126D5
hypocenterC5#2512
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C5×#126C2semidirect
2. C2×#4044C2semidirect
3. #65283C2semidirect
4. #65282C2semidirect
5. #65265C2semidirect
6. C4⋊C4×Dic5C2semidirect
7. #65261C2semidirect
11. #65242C2semidirect
12. #65237C2semidirect
13. #65233C2semidirect
14. #65229C2semidirect
15. C4×#683C2semidirect
74. C5×C4⋊C4D4semidirect
101. C2×#126D5semidirect
240. Dic5#126semidirect
269. C4⋊C4C22⋊D5semidirect
351. C5#2512semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1262
C2×#40442
#652832
#652822
#652652
C4⋊C4×Dic52
#652612
#652462
#652422
#652372
#652332
#652292
C4×#6832
#25125