Finite Group Databasebeta

← #10608359

Subgroups — #10608359

Order 640.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4092C211yesmaximal, characteristic, complemented
2C2×C4×D20C211yesmaximal, characteristic
3C5×#391C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65283C211yesmaximal, characteristic, complemented
6#65278C211yesmaximal, characteristic, complemented
7#65222C211yesmaximal, characteristic, complemented
8#64957C211yesmaximal, characteristic, complemented
9#64957C211yesmaximal, characteristic, complemented
10#64956C211yesmaximal, characteristic, complemented
11#64954C211yesmaximal, characteristic
12#64954C211yesmaximal, characteristic
13D5×#95C211yesmaximal, characteristic, complemented
14#64950C211yesmaximal, characteristic, complemented
15#64949C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C22×D20C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C42×C5C2211yescharacteristic
24#4092—24no—
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C4×D20—24no—
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44C5×#95C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2450—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—22no—
73C2×D20—22no—
74C2×D20Q814yescomplemented
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5—22no—
80C2×C4×D5—22no—
81C2×C4×D5—22no—
82C2×C4×D5—41no—
83C2×C4×D5—41no—
84C2×C4×D5—41no—
85C2×C4×D5—41no—
86C2×C4×D5—41no—
87C5×C4⋊C4—41no—
88C42×C5—22no—
89#685—22no—
90#685—22no—
91#685—22no—
92#685—22no—
93#685—41no—
94#685—41no—
95#685—41no—
96#685D414yes—
97C4⋊Dic5—22no—
98C4⋊Dic5—22no—
99C4⋊Dic5—41no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683—41no—
104C2×C22⋊Q8—51no—
105C2×C4×D4—51no—
106#400—51no—
107#398—51no—
108#396—51no—
109#393—51no—
110#391D511yescharacteristic, complemented
111#387—51no—
112#386—51no—
113#386—51no—
114#385—51no—
115#385—51no—
116#385—51no—
117#380—51no—
118C2×#95—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×D5C2×D412yes—
131C22×D5C2×Q812yes—
132C22×D5C2×Q812yes—
133C22×D5#4112yes—
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×C5C2×D412yes—
153C2×C4×C5C2×Q812yes—
154C2×C4×C5#4112yes—
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×Dic5—21no—
173C2×Dic5C2×D412yes—
174D20—28no—
175C4×D5—44no—
176C4×D5—44no—
177C4×D5—44no—
178C4×D5—44no—
179C22×Q8C2×D511yescharacteristic
180C22×D4—51no—
181C23×C4—51no—
182C23×C4—51no—
183C23×C4—51no—
184C23×C4—51no—
185C22⋊Q8—104no—
186C4×D4—104no—
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⋊C4—51no—
194C2×C4⋊C4C2×D511yescharacteristic
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×C42C2×D511yescharacteristic
204#95—51no—
205#95—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95C2×D511yescharacteristic
213#95C2×D511yescharacteristic
214#95C2×D511yescharacteristic
215#95C2×D511yescharacteristic
216C22×C5C22×D411yescharacteristic
217C22×C5C22×Q811yescharacteristic
218C22×C5C2×#4111yescharacteristic
219C22×C5C2×#4111yescharacteristic
220C22×C5C2×#4111yescharacteristic
221C22×C5H(5,𝔽2)11yescharacteristic
222C22×C5H(5,𝔽2)11yescharacteristic
223C2×D5—22no—
224C2×D5—22no—
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—22no—
231C2×D5—22no—
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C2×D5—41no—
236C2×D5C22⋊Q814yes—
237C4×C5—22no—
238C4×C5—22no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242C4×C5—41no—
243C4×C5—41no—
244Dic5—22no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248Dic5—41no—
249Dic5—41no—
250C24—51no—
251C24—51no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×D4—54no—
256C2×D4—102no—
257C2×D4—102no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—102no—
265C22×C4—102no—
266C22×C4—102no—
267C22×C4—102no—
268C22×C4—102no—
269C22×C4—102no—
270C22×C4—102no—
271C22×C4—201no—
272C22×C4—201no—
273C22×C4—201no—
274C22×C4—201no—
275C22×C4—201no—
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C22×C4C22×D511yescharacteristic
280C22×C4C22×D511yescharacteristic
281C22×C4C22×D511yescharacteristic
282C22×C4C22×D511yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—102no—
289C4⋊C4—201no—
290C4⋊C4—201no—
291C22⋊C4—54no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—102no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C22⋊C4—201no—
299C42—22no—
300C2×C5C2×C22⋊Q811yescharacteristic
301C2×C5#52911yescharacteristic
302C2×C5#53811yescharacteristic
303C2×C5#54611yescharacteristic
304C2×C5#54611yescharacteristic
305C2×C5#55411yescharacteristic
306C2×C5#55411yescharacteristic
307D5—24no—
308D5—42no—
309C23—52no—
310C23—52no—
311C23—52no—
312C23—52no—
313C23—101no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23—101no—
322C23—101no—
323C23C23×D511yescharacteristic
324Q8—44no—
325D4—108no—
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—52no—
344C2×C4—101no—
345C2×C4—101no—
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—204no—
362C2×C4—204no—
363C2×C4—204no—
364C2×C4—204no—
365C2×C4#70912yes—
366C2×C4D4×D512yes—
367C2×C4Q8×D512yes—
368C2×C4#71312yes—
369C5#245011yescharacteristic, complemented, Sylow
370C22—54no—
371C22—102no—
372C22—102no—
373C22—102no—
374C22—102no—
375C22—102no—
376C22—102no—
377C22—102no—
378C22—102no—
379C22—102no—
380C22—201no—
381C22—201no—
382C22—201no—
383C22—201no—
384C22C2×#70911yescharacteristic
385C22C2×D4×D511yescharacteristic
386C22#414411yescharacteristic
387C22C2×Q8×D511yescharacteristic
388C22C2×#71311yescharacteristic
389C22D5×#4111yescharacteristic
390C22#414911yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—102no—
399C4—201no—
400C4—201no—
401C4—201no—
402C4—201no—
403C4—201no—
404C2—104no—
405C2—202no—
406C2#6582411yescharacteristic
407C2#6591311yescharacteristic
408C2#6591411yescharacteristic
409C2D5×C22⋊Q811yescharacteristic
410C2#6596411yescharacteristic
411C2#6601211yescharacteristic
412C2#6606411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC22×D20C22
hypercenter#391D5
hypocenterC5#2450
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
3. C5×#391C2semidirect
4. #65520C2semidirect
5. #65283C2semidirect
6. #65278C2semidirect
7. #65222C2semidirect
8. #64957C2semidirect
9. #64957C2semidirect
10. #64956C2semidirect
13. D5×#95C2semidirect
14. #64950C2semidirect
15. #64949C2semidirect
74. C2×D20Q8semidirect
110. #391D5semidirect
369. C5#2450semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×C4×D202
C5×#3912
#655202
#652832
#652782
#652222
#649572
#649562
#649542
D5×#952
#649502
#649492
#24505