Finite Group Databasebeta

← #10609045

Subgroups — #10609045

Order 640.

620 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic, complemented
2C2×#4033C211yesmaximal, characteristic, complemented
3C2×#4027C211yesmaximal, characteristic
4C2×#4024C211yesmaximal, characteristic, complemented
5C5×#396C211yesmaximal, characteristic, complemented
6#65283C211yesmaximal, characteristic, complemented
7#65283C211yesmaximal, characteristic, complemented
8#65265C211yesmaximal, characteristic, complemented
9#65261C211yesmaximal, characteristic, complemented
10#65246C211yesmaximal, characteristic
11#65246C211yesmaximal, characteristic
12#65237C211yesmaximal, characteristic, complemented
13#65231C211yesmaximal, characteristic, complemented
14#64956C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C22×C22⋊D5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×C4×Dic5C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46C2×C4×Dic5C2211yescharacteristic
47#4042—24no—
48#4033—24no—
49#4027—24no—
50#4024—24no—
51C5×#95C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2538—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—22no—
65C2×C22⋊D5—22no—
66C2×C22⋊D5Q814yescomplemented
67C22×Dic5—22no—
68C22×Dic5—22no—
69C22×Dic5—22no—
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5—22no—
80C2×Dic10—22no—
81C2×Dic10—22no—
82C2×Dic10D414yescomplemented
83C5×C4⋊C4—22no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4—41no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4—41no—
89C22⋊Dic5—22no—
90C22⋊Dic5—41no—
91#685—22no—
92#685—22no—
93#685—22no—
94#685—41no—
95#685—41no—
96C4⋊Dic5—22no—
97C4⋊Dic5—41no—
98#683—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—22no—
103#683—22no—
104#683—22no—
105#683—22no—
106#683—22no—
107C4×Dic5—22no—
108C4×Dic5—22no—
109C4×Dic5D414yes—
110C2×C22⋊Q8—51no—
111C2×C22⋊Q8—51no—
112C2×C4×Q8—51no—
113C2×C4×D4—51no—
114#396—51no—
115#396—51no—
116#396—51no—
117#396D511yescharacteristic, complemented
118#390—51no—
119#390—51no—
120#389—51no—
121#389—51no—
122#384—51no—
123#380—51no—
124#380—51no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5C2×Q812yes—
132C23×C5C2411yescharacteristic
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5C2×Q812yes—
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×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5C2×D412yes—
157C2×C4×C5C2×D412yes—
158C22⋊D5—28no—
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—44no—
174C2×Dic5C2×D412yes—
175C2×Dic5C2×D412yes—
176C2×Dic5C2×D412yes—
177C2×Dic5C2×D412yes—
178C2×Dic5C2×Q812yes—
179C2×Dic5#4112yes—
180C2×Dic5#4112yes—
181C4×D5—44no—
182Dic10—28no—
183C22×Q8—51no—
184C22×D4—51no—
185C23×C4—51no—
186C23×C4—51no—
187C22⋊Q8—104no—
188C22⋊Q8—104no—
189C4×Q8—104no—
190C4×D4—104no—
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×C4⋊C4—51no—
198C2×C4⋊C4—51no—
199C2×C4⋊C4—51no—
200C2×C4⋊C4—51no—
201C2×C4⋊C4—51no—
202C2×C4⋊C4C2×D511yescharacteristic
203C2×C4⋊C4C2×D511yescharacteristic
204C2×C4⋊C4C2×D511yescharacteristic
205C2×C22⋊C4—51no—
206C2×C22⋊C4—51no—
207C2×C22⋊C4—51no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4—51no—
210C2×C22⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4C2×D511yescharacteristic
213C2×C22⋊C4C2×D511yescharacteristic
214C2×C22⋊C4C2×D511yescharacteristic
215C2×C42—51no—
216C2×C42—51no—
217C2×C42—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95C2×D511yescharacteristic
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5C22×D411yescharacteristic
230C22×C5C22×D411yescharacteristic
231C22×C5C22×Q811yescharacteristic
232C22×C5C2×#4111yescharacteristic
233C22×C5C2×#4111yescharacteristic
234C22×C5H(5,𝔽2)11yescharacteristic
235C22×C52−1+411yescharacteristic
236C2×D5—22no—
237C2×D5—22no—
238C2×D5—22no—
239C2×D5—41no—
240C2×D5—41no—
241C2×D5—41no—
242C2×D5—41no—
243C4×C5—22no—
244C4×C5—22no—
245C4×C5—41no—
246C4×C5—41no—
247C4×C5—41no—
248C4×C5—41no—
249Dic5—22no—
250Dic5—41no—
251Dic5—41no—
252Dic5—41no—
253Dic5—41no—
254Dic5#12114yes—
255Dic5C22⋊Q814yescomplemented
256C24—51no—
257C24C22×D511yescharacteristic
258C2×Q8—54no—
259C2×Q8—102no—
260C2×Q8—102no—
261C2×D4—54no—
262C2×D4—102no—
263C2×D4—102no—
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—102no—
274C22×C4—102no—
275C22×C4—102no—
276C22×C4—102no—
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C22×C4C22×D511yescharacteristic
280C22×C4C22×D511yescharacteristic
281C22×C4C22×D511yescharacteristic
282C22×C4C22×D511yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—102no—
287C4⋊C4—102no—
288C4⋊C4—102no—
289C4⋊C4—102no—
290C4⋊C4—102no—
291C4⋊C4—102no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C4⋊C4—102no—
295C4⋊C4—102no—
296C4⋊C4—201no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—102no—
301C22⋊C4—102no—
302C22⋊C4—102no—
303C22⋊C4—102no—
304C22⋊C4—201no—
305C22⋊C4—201no—
306C22⋊C4—201no—
307C42—54no—
308C42—102no—
309C42—102no—
310C2×C5—42no—
311C2×C5C2×#12111yescharacteristic
312C2×C5C2×C22⋊Q811yescharacteristic
313C2×C5#53711yescharacteristic
314C2×C5D4×Q811yescharacteristic
315C2×C5#55011yescharacteristic
316C2×C5#55411yescharacteristic
317C2×C5#55611yescharacteristic
318D5—42no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—52no—
326C23—101no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23—101no—
331C23—101no—
332C23Q8×D512yes—
333C23C23×D511yescharacteristic
334Q8—108no—
335D4—108no—
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—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—52no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—52no—
356C2×C4—52no—
357C2×C4—52no—
358C2×C4—52no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—204no—
374C2×C4—204no—
375C2×C4D4×D512yes—
376C2×C4D4×D512yes—
377C5#253811yescharacteristic, complemented, Sylow
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—102no—
386C22—102no—
387C22—102no—
388C22—201no—
389C22—201no—
390C22—201no—
391C22—201no—
392C22C2×D4×D511yescharacteristic
393C22C2×D4×D511yescharacteristic
394C22#414411yescharacteristic
395C22C2×Q8×D511yescharacteristic
396C22D5×#4111yescharacteristic
397C22D5×#4111yescharacteristic
398C22#415011yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—54no—
406C4—54no—
407C4—102no—
408C4—201no—
409C4—201no—
410C4—201no—
411C4—201no—
412C2—42no—
413C2—202no—
414C2#6593111yescharacteristic
415C2#6593311yescharacteristic
416C2D5×#12111yescharacteristic
417C2D5×C22⋊Q811yescharacteristic
418C2#6596611yescharacteristic
419C2#6597011yescharacteristic
420C2#6603611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#396C2
prime radicalC22×C22⋊D5C22
hypercenter#396D5
hypocenterC5#2538
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
2. C2×#4033C2semidirect
4. C2×#4024C2semidirect
5. C5×#396C2semidirect
6. #65283C2semidirect
7. #65283C2semidirect
8. #65265C2semidirect
9. #65261C2semidirect
12. #65237C2semidirect
13. #65231C2semidirect
14. #64956C2semidirect
15. #64936C2semidirect
66. C2×C22⋊D5Q8semidirect
82. C2×Dic10D4semidirect
117. #396D5semidirect
255. Dic5C22⋊Q8semidirect
377. C5#2538semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40332
C2×#40272
C2×#40242
C5×#3962
#652832
#652652
#652612
#652462
#652372
#652312
#649562
#649362
#25385