Finite Group Databasebeta

← #10614092

Subgroups — #10614092

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×C4×C5×D4C211yesmaximal, characteristic
2C2×#4024C211yesmaximal, characteristic, complemented
3#65509C211yesmaximal, characteristic
4#65509C211yesmaximal, characteristic
5#65501C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65499C211yesmaximal, characteristic, complemented
8C2×#3963C211yesmaximal, characteristic, complemented
9#65275C211yesmaximal, characteristic, complemented
10#65273C211yesmaximal, characteristic, complemented
11#65239C211yesmaximal, characteristic, complemented
12#65237C211yesmaximal, characteristic, complemented
13#65236C211yesmaximal, characteristic, complemented
14#65222C211yesmaximal, characteristic, complemented
15#65218C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C22×Dic10C2211yescharacteristic
22C4×C5×D4—24no—
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C42×C5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40#4024—24no—
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2450—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C2×C5×D4—22no—
57C2×C5×D4—22no—
58C2×C5×D4Q814yescomplemented
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—41no—
64C22×C4×C5—41no—
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—41no—
74C22×Dic5—41no—
75C22×Dic5—41no—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C2×Dic10—22no—
85C2×Dic10—22no—
86C2×Dic10—41no—
87C5×C4⋊C4—22no—
88C5×C22⋊C4—22no—
89C5×C22⋊C4D414yes—
90C42×C5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—22no—
94C22⋊Dic5—41no—
95C22⋊Dic5—41no—
96C22⋊Dic5—41no—
97C4⋊Dic5—22no—
98C4⋊Dic5—41no—
99C4⋊Dic5—41no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683—41no—
104C2×C22⋊Q8—51no—
105C2×C4×D4D511yescharacteristic
106#400—51no—
107#398—51no—
108#396—51no—
109#393—51no—
110#391—51no—
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×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5C2×D412yes—
130C23×C5C2×Q812yes—
131C23×C5C2×Q812yes—
132C23×C5#4112yes—
133C23×C5C2411yescharacteristic
134C5×D4—28no—
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—44no—
147C2×C4×C5—44no—
148C2×C4×C5C2×D412yes—
149C2×C4×C5C2×D412yes—
150C2×C4×C5C2×Q812yes—
151C2×C4×C5#4112yes—
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—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—44no—
176C2×Dic5—44no—
177C2×Dic5#4112yes—
178Dic10—44no—
179C22×Q8—51no—
180C22×D4C2×D511yescharacteristic
181C23×C4—51no—
182C23×C4—51no—
183C23×C4C2×D511yescharacteristic
184C23×C4C2×D511yescharacteristic
185C22⋊Q8—104no—
186C4×D4—24no—
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⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
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#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—22no—
221C22×C5—22no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5C22⋊Q814yes—
230C22×C5C22×D411yescharacteristic
231C22×C5C22×Q811yescharacteristic
232C22×C5C2×#4111yescharacteristic
233C22×C5C2×#4111yescharacteristic
234C22×C5C2×#4111yescharacteristic
235C22×C5H(5,𝔽2)11yescharacteristic
236C22×C5H(5,𝔽2)11yescharacteristic
237C4×C5—22no—
238C4×C5—22no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242Dic5—22no—
243Dic5—41no—
244Dic5—41no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248Dic5—41no—
249Dic5—41no—
250C24C22×D511yescharacteristic
251C24C22×D511yescharacteristic
252C2×Q8—102no—
253C2×Q8—102no—
254C2×Q8—201no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4Dic1014yescomplemented
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—102no—
273C22×C4—102no—
274C22×C4—102no—
275C22×C4—201no—
276C22×C4—201no—
277C22×C4—201no—
278C22×C4C22×D511yescharacteristic
279C22×C4C22×D511yescharacteristic
280C22×C4C22×D511yescharacteristic
281C22×C4C22×D511yescharacteristic
282C22×C4C22×D511yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—201no—
289C4⋊C4—201no—
290C4⋊C4—201no—
291C22⋊C4—22no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C22⋊C4C22⋊D514yes—
299C42—22no—
300C2×C5—24no—
301C2×C5—42no—
302C2×C5C2×C22⋊Q811yescharacteristic
303C2×C5#52911yescharacteristic
304C2×C5#53811yescharacteristic
305C2×C5#54611yescharacteristic
306C2×C5#54611yescharacteristic
307C2×C5#55411yescharacteristic
308C2×C5#55411yescharacteristic
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C2×Dic1012yes—
320C23C2×Dic1012yes—
321C23#70912yes—
322C23C2×C22⋊D512yes—
323C23C23×D511yescharacteristic
324Q8—204no—
325D4—28no—
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—44no—
338C2×C4—44no—
339C2×C4—52no—
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×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—101no—
363C2×C4—204no—
364C2×C4—204no—
365C2×C4C2×Dic1012yes—
366C2×C4#71112yes—
367C2×C4C2×C22⋊D512yes—
368C2×C4C2×C22⋊D512yes—
369C5#245011yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#407014yes—
384C22C22×Dic1011yescharacteristic
385C22C2×#70911yescharacteristic
386C22C2×#71111yescharacteristic
387C22#414411yescharacteristic
388C22D5×#4111yescharacteristic
389C22#414911yescharacteristic
390C22C22×C22⋊D511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—102no—
397C4—201no—
398C4—201no—
399C4—201no—
400C4—201no—
401C4—201no—
402C4—201no—
403C4—201no—
404C2—24no—
405C2—42no—
406C2#6587211yescharacteristic
407C2#6587311yescharacteristic
408C2#6587711yescharacteristic
409C2#6589411yescharacteristic
410C2C2×#407011yescharacteristic
411C2#6614011yescharacteristic
412C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×D4C2
prime radicalC22×C5×D4C22
hypercenterC2×C4×D4D5
hypocenterC5#2450
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#4024C2semidirect
5. #65501C2semidirect
6. #65499C2semidirect
7. #65499C2semidirect
8. C2×#3963C2semidirect
9. #65275C2semidirect
10. #65273C2semidirect
11. #65239C2semidirect
12. #65237C2semidirect
13. #65236C2semidirect
14. #65222C2semidirect
15. #65218C2semidirect
58. C2×C5×D4Q8semidirect
257. C2×D4Dic10semidirect
369. C5#2450semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×D42
C2×#40242
#655092
#655012
#654992
C2×#39632
#652752
#652732
#652392
#652372
#652362
#652222
#652182
#24505