Finite Group Databasebeta

← #10608113

Subgroups — #10608113

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×#4024C211yesmaximal, characteristic, complemented
2C2×#4024C211yesmaximal, characteristic, complemented
3C5×#390C211yesmaximal, characteristic
4#65509C211yesmaximal, characteristic
5#65509C211yesmaximal, characteristic
6#65273C211yesmaximal, characteristic, complemented
7#65268C211yesmaximal, characteristic, complemented
8#65238C211yesmaximal, characteristic, complemented
9#65237C211yesmaximal, characteristic, complemented
10#65236C211yesmaximal, characteristic, complemented
11#65236C211yesmaximal, characteristic, complemented
12#65235C211yesmaximal, characteristic, complemented
13#65231C211yesmaximal, characteristic, complemented
14#65222C211yesmaximal, characteristic, complemented
15#65218C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊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×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41#4024—24no—
42#4024—24no—
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2692—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C2×C5×D4—41no—
57C2×C5×D4—41no—
58C2×C5×D4Q814yescomplemented
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×Dic5—22no—
65C22×Dic5—22no—
66C22×Dic5—22no—
67C22×Dic5—41no—
68C22×Dic5—41no—
69C22×Dic5—41no—
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C2×Dic10—22no—
79C2×Dic10—22no—
80C2×Dic10—22no—
81C2×Dic10—41no—
82C2×Dic10—41no—
83C2×Dic10—41no—
84C5×C4⋊C4—41no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C42×C5—41no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—41no—
94C22⋊Dic5—41no—
95C22⋊Dic5—41no—
96C4⋊Dic5—22no—
97C4⋊Dic5—22no—
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#391—51no—
113#390D511yescharacteristic
114#387—51no—
115#386—51no—
116#386—51no—
117#385—51no—
118#380—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×C5—21no—
130C23×C5—21no—
131C23×C5C2×Q812yes—
132C23×C5C2×Q812yes—
133C23×C5C2411yescharacteristic
134C5×D4—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×C5C2×D412yes—
149C2×C4×C5C2×Q812yes—
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—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—44no—
172C2×Dic5—44no—
173C2×Dic5C2×D412yes—
174C2×Dic5C2×D412yes—
175C2×Dic5#4112yes—
176Dic10—44no—
177Dic10—44no—
178C22×Q8—51no—
179C22×Q8—51no—
180C22×D4C2×D511yescharacteristic
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⋊C4C2×D511yescharacteristic
201C2×C22⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
203C2×C22⋊C4C2×D511yescharacteristic
204C2×C42C2×D511yescharacteristic
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—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—22no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5—41no—
224C22×C5—41no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5C22×D411yescharacteristic
230C22×C5C22×Q811yescharacteristic
231C22×C5C2×#4111yescharacteristic
232C22×C5H(5,𝔽2)11yescharacteristic
233C22×C5H(5,𝔽2)11yescharacteristic
234C22×C5H(5,𝔽2)11yescharacteristic
235C22×C52−1+411yescharacteristic
236C4×C5—22no—
237C4×C5—22no—
238C4×C5—41no—
239C4×C5—41no—
240C4×C5—41no—
241Dic5—22no—
242Dic5—22no—
243Dic5—22no—
244Dic5—41no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248Dic5—41no—
249C24C22×D511yescharacteristic
250C24C22×D511yescharacteristic
251C2×Q8—102no—
252C2×Q8—102no—
253C2×Q8—102no—
254C2×Q8—201no—
255C2×Q8—201no—
256C2×Q8—201no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4Dic1014yescomplemented
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—102no—
269C22×C4—102no—
270C22×C4—102no—
271C22×C4—201no—
272C22×C4—201no—
273C22×C4—201no—
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—54no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—201no—
286C4⋊C4—201no—
287C4⋊C4—201no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C42—41no—
299C2×C5—42no—
300C2×C5—42no—
301C2×C5#53511yescharacteristic
302C2×C5#54211yescharacteristic
303C2×C5#54311yescharacteristic
304C2×C5#54611yescharacteristic
305C2×C5D4×Q811yescharacteristic
306C2×C5#55411yescharacteristic
307C2×C5#56211yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×Dic1012yes—
321C23C2×Dic1012yes—
322C23C23×D511yescharacteristic
323Q8—204no—
324Q8—204no—
325D4—44no—
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—52no—
340C2×C4—52no—
341C2×C4—52no—
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×C4D4×D512yes—
367C5#269211yescharacteristic, complemented, Sylow
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22C22×Dic1011yescharacteristic
383C22C2×D4×D511yescharacteristic
384C22#414411yescharacteristic
385C22#414411yescharacteristic
386C22#414411yescharacteristic
387C22D5×#4111yescharacteristic
388C22#415011yescharacteristic
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—102no—
395C4—102no—
396C4—102no—
397C4—201no—
398C4—201no—
399C4—201no—
400C4—201no—
401C4—201no—
402C2—42no—
403C2—42no—
404C2#6581711yescharacteristic
405C2D4×Dic1011yescharacteristic
406C2#6587311yescharacteristic
407C2#6594411yescharacteristic
408C2#6595511yescharacteristic
409C2#6600311yescharacteristic
410C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC22×C5×D4C22
hypercenter#390D5
hypocenterC5#2692
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#4024C2semidirect
2. C2×#4024C2semidirect
6. #65273C2semidirect
7. #65268C2semidirect
8. #65238C2semidirect
9. #65237C2semidirect
10. #65236C2semidirect
11. #65236C2semidirect
12. #65235C2semidirect
13. #65231C2semidirect
14. #65222C2semidirect
15. #65218C2semidirect
58. C2×C5×D4Q8semidirect
103. #683D4semidirect
259. C2×D4Dic10semidirect
367. C5#2692semidirect

Maximal subgroups

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

subgroupindex
C2×#40242
C5×#3902
#655092
#652732
#652682
#652382
#652372
#652362
#652352
#652312
#652222
#652182
#26925