Finite Group Databasebeta

← #10610073

Subgroups — #10610073

Order 640.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic
2C2×#4038C211yesmaximal, characteristic
3C5×#400C211yesmaximal, characteristic, complemented
4#65273C211yesmaximal, characteristic, complemented
5#65262C211yesmaximal, characteristic, complemented
6#64957C211yesmaximal, characteristic
7#64957C211yesmaximal, characteristic
8#64956C211yesmaximal, characteristic
9#64953C211yesmaximal, characteristic
10D5×#95C211yesmaximal, characteristic
11#64946C211yesmaximal, characteristic, complemented
12#64945C211yesmaximal, characteristic, complemented
13#64942C211yesmaximal, characteristic, complemented
14#64941C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4042—24no—
38#4038—24no—
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2438—51nomaximal, Sylow
54C23×D5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C2×C4×D5—22no—
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5—22no—
73C2×C4×D5—22no—
74C2×C4×D5—22no—
75C2×C4×D5—41no—
76C2×C4×D5—41no—
77C2×C4×D5Q814yescomplemented
78C2×Dic10—22no—
79C2×Dic10—22no—
80C2×Dic10—41no—
81C5×C4⋊C4—22no—
82C5×C4⋊C4—22no—
83#685—22no—
84#685—22no—
85#685—22no—
86#685D414yes—
87C4⋊Dic5—22no—
88C4⋊Dic5—22no—
89C4⋊Dic5—22no—
90#683—22no—
91#683—22no—
92#683—22no—
93#683—41no—
94#683—41no—
95#683—41no—
96C4×Dic5—22no—
97C4×Dic5—22no—
98C2×C22⋊Q8—51no—
99C2×#117—51no—
100#400D511yescharacteristic, complemented
101#398—51no—
102#398—51no—
103#395—51no—
104#391—51no—
105#389—51no—
106#387—51no—
107#387—51no—
108#385—51no—
109#385—51no—
110#381—51no—
111#380—51no—
112C2×#95—51no—
113C23×C5C2411yescharacteristic
114C22×D5—21no—
115C22×D5—21no—
116C22×D5—21no—
117C22×D5—21no—
118C22×D5C2×D412yes—
119C22×D5C2×Q812yes—
120C22×D5#4112yes—
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×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×C5C2×D412yes—
141C2×C4×C5C2×Q812yes—
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—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5C2×D412yes—
160C2×Dic5C2×Q812yes—
161C2×Dic5#4112yes—
162C2×Dic5#4112yes—
163C4×D5—28no—
164C4×D5—44no—
165C4×D5—44no—
166Dic10—44no—
167C22×Q8—51no—
168C23×C4—51no—
169C23×C4—51no—
170C23×C4—51no—
171C22⋊Q8—104no—
172#117—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⋊C4—51no—
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×C42—51no—
189C2×C42—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#95—51no—
199#95C2×D511yescharacteristic
200#95C2×D511yescharacteristic
201#95C2×D511yescharacteristic
202#95C2×D511yescharacteristic
203#95C2×D511yescharacteristic
204C22×C5C22×D411yescharacteristic
205C22×C5C22×Q811yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5C2×#4111yescharacteristic
208C22×C5C2×#4111yescharacteristic
209C22×C5H(5,𝔽2)11yescharacteristic
210C22×C52−1+411yescharacteristic
211C2×D5—22no—
212C2×D5—22no—
213C2×D5—22no—
214C2×D5—22no—
215C2×D5—22no—
216C2×D5—22no—
217C2×D5C22⋊Q814yes—
218C4×C5—22no—
219C4×C5—22no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5—41no—
224C4×C5—41no—
225Dic5—22no—
226Dic5—22no—
227Dic5—22no—
228Dic5—22no—
229Dic5—41no—
230Dic5—41no—
231Dic5—41no—
232C24—51no—
233C2×Q8—102no—
234C2×Q8—102no—
235C2×Q8—201no—
236C22×C4—51no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—54no—
241C22×C4—51no—
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—102no—
245C22×C4—102no—
246C22×C4—102no—
247C22×C4—102no—
248C22×C4—102no—
249C22×C4—102no—
250C22×C4—201no—
251C22×C4—201no—
252C22×C4C22×D511yescharacteristic
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⋊C4—201no—
270C22⋊C4—54no—
271C22⋊C4—102no—
272C22⋊C4—102no—
273C22⋊C4—102no—
274C42—102no—
275C42—102no—
276C2×C5C2×C22⋊Q811yescharacteristic
277C2×C5#53311yescharacteristic
278C2×C5#54411yescharacteristic
279C2×C5#54611yescharacteristic
280C2×C5#54711yescharacteristic
281C2×C5#55111yescharacteristic
282C2×C5#55211yescharacteristic
283D5—24no—
284C23—52no—
285C23—52no—
286C23—52no—
287C23—101no—
288C23—101no—
289C23—101no—
290C23—101no—
291C23C23×D511yescharacteristic
292Q8—204no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—52no—
313C2×C4—52no—
314C2×C4—52no—
315C2×C4—52no—
316C2×C4—108no—
317C2×C4—101no—
318C2×C4—101no—
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—204no—
335C2×C4—204no—
336C2×C4D4×D512yes—
337C2×C4Q8×D512yes—
338C5#243811yescharacteristic, complemented, Sylow
339C22—54no—
340C22—102no—
341C22—102no—
342C22—102no—
343C22—102no—
344C22—102no—
345C22—102no—
346C22C2×D4×D511yescharacteristic
347C22C2×Q8×D511yescharacteristic
348C22D5×#4111yescharacteristic
349C22D5×#4111yescharacteristic
350C22D5×#4111yescharacteristic
351C22#414911yescharacteristic
352C22#415011yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—102no—
361C4—102no—
362C4—102no—
363C4—102no—
364C4—201no—
365C4—201no—
366C4—201no—
367C2—104no—
368C2D5×C22⋊Q811yescharacteristic
369C2#6598911yescharacteristic
370C2#6601011yescharacteristic
371C2#6602211yescharacteristic
372C2#6602311yescharacteristic
373C2#6603911yescharacteristic
374C2#6604211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC23×D5C23
hypercenter#400D5
hypocenterC5#2438
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C5×#400C2semidirect
4. #65273C2semidirect
5. #65262C2semidirect
11. #64946C2semidirect
12. #64945C2semidirect
13. #64942C2semidirect
14. #64941C2semidirect
15. #64935C2semidirect
77. C2×C4×D5Q8semidirect
100. #400D5semidirect
338. C5#2438semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40382
C5×#4002
#652732
#652622
#649572
#649562
#649532
D5×#952
#649462
#649452
#649422
#649412
#649352
#24385