Finite Group Databasebeta

← #10614713

Subgroups — #10614713

Order 640.

562 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×C5×#124C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3C2×#4081C211yesmaximal, characteristic
4C2×#4024C211yesmaximal, characteristic, complemented
5#65518C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic
7#65241C211yesmaximal, characteristic, complemented
8#65240C211yesmaximal, characteristic, complemented
9#65238C211yesmaximal, characteristic, complemented
10#65237C211yesmaximal, characteristic, complemented
11#65236C211yesmaximal, characteristic, complemented
12#65235C211yesmaximal, characteristic, complemented
13#65234C211yesmaximal, characteristic, complemented
14#65222C211yesmaximal, characteristic, complemented
15#65217C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C5×#124—24no—
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C42×C5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33#4090—24no—
34#4081—24no—
35C2×C4⋊Dic5C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4024—24no—
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2674—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C2×C5×Q8—22no—
59C2×C5×Q8—22no—
60C2×C5×Q8—41no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—41no—
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×Dic5—22no—
70C22×Dic5—22no—
71C22×Dic5—41no—
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C2×Dic10—22no—
81C2×Dic10—22no—
82C2×Dic10—41no—
83C5×C22⋊C4—22no—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4D414yes—
87C42×C5—22no—
88C22⋊Dic5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—22no—
94C22⋊Dic5—41no—
95C4⋊Dic5—22no—
96C4⋊Dic5—22no—
97#683—22no—
98#683—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683—41no—
104#683D414yes—
105C4×Dic5—22no—
106C2×C4⋊Q8—51no—
107C2×#124D511yescharacteristic
108C2×#123—51no—
109C2×C22⋊Q8—51no—
110#399—51no—
111#397—51no—
112#396—51no—
113#395—51no—
114#394—51no—
115#393—51no—
116#389—51no—
117#388—51no—
118#388—51no—
119#387—51no—
120#385—51no—
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×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5C2×D412yes—
135C23×C5C2411yescharacteristic
136C5×Q8—44no—
137C5×D4—44no—
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×C5C2×D412yes—
151C2×C4×C5C2×D412yes—
152C2×C4×C5C2×D412yes—
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—44no—
175C2×Dic5C2×D412yes—
176C2×Dic5C2×D412yes—
177C2×Dic5#4112yes—
178Dic10—44no—
179C22×Q8—51no—
180C22×Q8C2×D511yescharacteristic
181C22×D4C2×D511yescharacteristic
182C23×C4—51no—
183C4⋊Q8—104no—
184#124—24no—
185#123—104no—
186C22⋊Q8—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⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊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×C22⋊C4C2×D511yescharacteristic
205C2×C22⋊C4C2×D511yescharacteristic
206C2×C22⋊C4C2×D511yescharacteristic
207C2×C22⋊C4C2×D511yescharacteristic
208C2×C42—51no—
209C2×C42C2×D511yescharacteristic
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217#95—51no—
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×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5C22×D411yescharacteristic
233C22×C5C22×D411yescharacteristic
234C22×C5C2×#4111yescharacteristic
235C22×C5H(5,𝔽2)11yescharacteristic
236C22×C52−1+411yescharacteristic
237C22×C52−1+411yescharacteristic
238C22×C52−1+411yescharacteristic
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—22no—
242C4×C5—41no—
243C4×C5—41no—
244Dic5—22no—
245Dic5—22no—
246Dic5—22no—
247Dic5—41no—
248Dic5—41no—
249Dic5—41no—
250Dic5—41no—
251Dic5—41no—
252C24C22×D511yescharacteristic
253C24C22×D511yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—102no—
258C2×Q8—102no—
259C2×Q8—201no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—102no—
272C22×C4—102no—
273C22×C4—201no—
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C4⋊C4—54no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—201no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—102no—
295C22⋊C4—102no—
296C22⋊C4—102no—
297C22⋊C4—102no—
298C22⋊C4—201no—
299C22⋊C4C22⋊D514yes—
300C42—22no—
301C42—102no—
302C2×C5—42no—
303C2×C5—81no—
304C2×C5#53611yescharacteristic
305C2×C5#53611yescharacteristic
306C2×C5#54211yescharacteristic
307C2×C5D4211yescharacteristic
308C2×C5#54711yescharacteristic
309C2×C5#54811yescharacteristic
310C2×C5#56311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23C2×C22⋊D512yes—
325C23C23×D511yescharacteristic
326Q8—44no—
327Q8—204no—
328D4—44no—
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—52no—
342C2×C4—52no—
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—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—204no—
366C2×C4D4×D512yes—
367C2×C4C2×C22⋊D512yes—
368C2×C4C2×C22⋊D512yes—
369C5#267411yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22C2×D4×D511yescharacteristic
385C22#414411yescharacteristic
386C22D5×#4111yescharacteristic
387C22#415011yescharacteristic
388C22#415011yescharacteristic
389C22#415011yescharacteristic
390C22C22×C22⋊D511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—102no—
397C4—102no—
398C4—102no—
399C4—201no—
400C4—201no—
401C4—201no—
402C4—201no—
403C4—201no—
404C2—42no—
405C2—81no—
406C2#6600311yescharacteristic
407C2#6600611yescharacteristic
408C2#6600911yescharacteristic
409C2#6601111yescharacteristic
410C2D4×C22⋊D511yescharacteristic
411C2#6615811yescharacteristic
412C2#6615911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#124C2
prime radicalC22×C5×D4C22
hypercenterC2×#124D5
hypocenterC5#2674
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
4. C2×#4024C2semidirect
5. #65518C2semidirect
7. #65241C2semidirect
8. #65240C2semidirect
9. #65238C2semidirect
10. #65237C2semidirect
11. #65236C2semidirect
12. #65235C2semidirect
13. #65234C2semidirect
14. #65222C2semidirect
15. #65217C2semidirect
369. C5#2674semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1242
C2×#40902
C2×#40812
C2×#40242
#655182
#655112
#652412
#652402
#652382
#652372
#652362
#652352
#652342
#652222
#652172
#26745