Finite Group Databasebeta

← #10608295

Subgroups — #10608295

Order 640.

536 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4092C211yesmaximal, characteristic, complemented
2C2×#4042C211yesmaximal, characteristic, complemented
3C5×#391C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65281C211yesmaximal, characteristic, complemented
6#65228C211yesmaximal, characteristic
7#65222C211yesmaximal, characteristic, complemented
8#64965C211yesmaximal, characteristic, complemented
9#64962C211yesmaximal, characteristic, complemented
10#64957C211yesmaximal, characteristic, complemented
11#64956C211yesmaximal, characteristic, complemented
12#64954C211yesmaximal, characteristic
13#64954C211yesmaximal, characteristic
14#64946C211yesmaximal, characteristic, complemented
15#64945C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C22×D20C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C42×C5C2211yescharacteristic
23#4092—24no—
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
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
42#4042—24no—
43C5×#95C2211yescharacteristic
44C5×#95C2211yescharacteristic
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2692—51nomaximal, Sylow
54C23×D5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C2×C5×Q8—22no—
57C2×C5×Q8—22no—
58C2×C5×Q8—41no—
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C2×D20—41no—
73C2×D20—41no—
74C2×D20Q814yescomplemented
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—41no—
79C2×C4×D5—41no—
80C2×C4×D5—41no—
81C2×Dic10—22no—
82C2×Dic10—41no—
83C2×Dic10—41no—
84C5×C4⋊C4—22no—
85C42×C5—41no—
86#685—22no—
87#685—22no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—41no—
92#685—41no—
93#685—41no—
94#685—41no—
95#685—41no—
96C4⋊Dic5—22no—
97C4⋊Dic5—41no—
98#683—22no—
99#683—22no—
100#683—41no—
101#683—41no—
102#683—41no—
103#683D414yescomplemented
104C2×C22⋊Q8—51no—
105C2×C22⋊Q8—51no—
106#399—51no—
107#398—51no—
108#396—51no—
109#395—51no—
110#393—51no—
111#393—51no—
112#391D511yescharacteristic, complemented
113#390—51no—
114#387—51no—
115#386—51no—
116#386—51no—
117#385—51no—
118#380—51no—
119C23×C5C2411yescharacteristic
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5C2×Q812yes—
133C22×D5C2×Q812yes—
134C5×Q8—44no—
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—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5C2×D412yes—
154C2×C4×C5C2×Q812yes—
155C2×C4×C5#4112yes—
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—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5C2×D412yes—
173C2×Dic5C2×D412yes—
174D20—44no—
175C4×D5—44no—
176C4×D5—44no—
177Dic10—44no—
178C22×Q8—51no—
179C22×Q8C2×D511yescharacteristic
180C22×D4—51no—
181C23×C4—51no—
182C23×C4—51no—
183C22⋊Q8—104no—
184C22⋊Q8—104no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4—51no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C42C2×D511yescharacteristic
205#95—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95C2×D511yescharacteristic
212#95C2×D511yescharacteristic
213#95C2×D511yescharacteristic
214#95C2×D511yescharacteristic
215C22×C5C22×D411yescharacteristic
216C22×C5C22×Q811yescharacteristic
217C22×C5C2×#4111yescharacteristic
218C22×C5H(5,𝔽2)11yescharacteristic
219C22×C5H(5,𝔽2)11yescharacteristic
220C22×C5H(5,𝔽2)11yescharacteristic
221C22×C52−1+411yescharacteristic
222C2×D5—22no—
223C2×D5—22no—
224C2×D5—22no—
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—41no—
231C2×D5—41no—
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C2×D5—41no—
236C4×C5—22no—
237C4×C5—22no—
238C4×C5—22no—
239C4×C5—41no—
240C4×C5—41no—
241C4×C5—41no—
242C4×C5—41no—
243Dic5—22no—
244Dic5—22no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248Dic5—41no—
249C24—51no—
250C24—51no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—102no—
255C2×Q8—201no—
256C2×Q8—201no—
257C2×D4—54no—
258C2×D4—201no—
259C2×D4—201no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—102no—
267C22×C4—102no—
268C22×C4—102no—
269C22×C4—201no—
270C22×C4—201no—
271C22×C4—201no—
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—54no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—201no—
285C4⋊C4—201no—
286C4⋊C4—201no—
287C4⋊C4—201no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C42—41no—
299C2×C5#53511yescharacteristic
300C2×C5#54211yescharacteristic
301C2×C5#54311yescharacteristic
302C2×C5#54611yescharacteristic
303C2×C5D4×Q811yescharacteristic
304C2×C5#55411yescharacteristic
305C2×C5#56211yescharacteristic
306D5—42no—
307D5—42no—
308C23—52no—
309C23—52no—
310C23—101no—
311C23—101no—
312C23—101no—
313C23—101no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23—101no—
322C23C23×D511yescharacteristic
323Q8—44no—
324Q8—204no—
325D4—204no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—52no—
345C2×C4—52no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—204no—
363C2×C4—204no—
364C2×C4#70912yes—
365C2×C4D4×D512yes—
366C2×C4Q8×D512yes—
367C5#269211yescharacteristic, complemented, Sylow
368C22—102no—
369C22—102no—
370C22—102no—
371C22—102no—
372C22—102no—
373C22—102no—
374C22—201no—
375C22—201no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22—201no—
381C22—201no—
382C22C2×#70911yescharacteristic
383C22C2×D4×D511yescharacteristic
384C22#414411yescharacteristic
385C22C2×Q8×D511yescharacteristic
386C22#414711yescharacteristic
387C22#414911yescharacteristic
388C22#414911yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—102no—
397C4—102no—
398C4—201no—
399C4—201no—
400C4—201no—
401C4—201no—
402C2—202no—
403C2—202no—
404C2#6582411yescharacteristic
405C2#6591311yescharacteristic
406C2#6591611yescharacteristic
407C2#6596811yescharacteristic
408C2#6598111yescharacteristic
409C2#6600811yescharacteristic
410C2#6606111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC22×D20C22
hypercenter#391D5
hypocenterC5#2692
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
2. C2×#4042C2semidirect
3. C5×#391C2semidirect
4. #65520C2semidirect
5. #65281C2semidirect
7. #65222C2semidirect
8. #64965C2semidirect
9. #64962C2semidirect
10. #64957C2semidirect
11. #64956C2semidirect
14. #64946C2semidirect
15. #64945C2semidirect
74. C2×D20Q8semidirect
103. #683D4semidirect
112. #391D5semidirect
367. C5#2692semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×#40422
C5×#3912
#655202
#652812
#652282
#652222
#649652
#649622
#649572
#649562
#649542
#649462
#649452
#26925