Finite Group Databasebeta

← #10614577

Subgroups — #10614577

Order 640.

666 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#123C211yesmaximal, characteristic, complemented
2C2×#4085C211yesmaximal, characteristic
3C2×#4075C211yesmaximal, characteristic, complemented
4C2×#4027C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic, complemented
6C2×#3963C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65276C211yesmaximal, characteristic, complemented
9#65269C211yesmaximal, characteristic, complemented
10#65262C211yesmaximal, characteristic, complemented
11#65250C211yesmaximal, characteristic, complemented
12#65245C211yesmaximal, characteristic, complemented
13#65239C211yesmaximal, characteristic, complemented
14#65237C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C5×#123—24no—
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C4⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C5×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33#4085—24no—
34#4075—24no—
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×C4⋊Dic5C2211yescharacteristic
40C2×C4⋊Dic5C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4027—24no—
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2425—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C23×D5C2311yescharacteristic
59C2×C5×D4—22no—
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5—22no—
73C2×C22⋊D5—22no—
74C2×C22⋊D5—41no—
75C2×C22⋊D5D414yes—
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—41no—
81C22×Dic5—41no—
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C2×C4×D5—22no—
90C2×C4×D5—22no—
91C2×C4×D5—41no—
92C5×C4⋊C4—22no—
93C5×C4⋊C4—22no—
94C5×C22⋊C4—22no—
95C5×C22⋊C4—22no—
96C5×C22⋊C4D414yescomplemented
97C22⋊Dic5—22no—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—41no—
101#685—22no—
102#685—22no—
103#685—22no—
104#685—22no—
105C4⋊Dic5—22no—
106C4⋊Dic5—41no—
107#683—22no—
108#683—22no—
109#683—22no—
110#683—22no—
111C4×Dic5—22no—
112C2×#123—51no—
113C2×#123D511yescharacteristic, complemented
114C2×#121—51no—
115C2×C4×D4—51no—
116#398—51no—
117#397—51no—
118#396—51no—
119#394—51no—
120#389—51no—
121#388—51no—
122#386—51no—
123#386—51no—
124#385—51no—
125#380—51no—
126C2×#95—51no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5C2×D412yes—
139C23×C5C2×D412yes—
140C23×C5#4112yes—
141C23×C5C2411yescharacteristic
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5C2×D412yes—
149C5×D4—44no—
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×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—44no—
164C2×C4×C5C2×D412yes—
165C2×C4×C5C2×D412yes—
166C22⋊D5—28no—
167C22⋊D5—44no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—44no—
187C2×Dic5—44no—
188C2×Dic5C2×D412yes—
189C2×Dic5#4112yes—
190C2×Dic5#4112yes—
191C4×D5—44no—
192C22×D4—51no—
193C22×D4—51no—
194C22×D4C2×D511yescharacteristic
195C23×C4—51no—
196C23×C4—51no—
197C23×C4—51no—
198C23×C4C2×D511yescharacteristic
199#123—24no—
200#123—104no—
201#121—104no—
202C4×D4—104no—
203C2×C4⋊C4—51no—
204C2×C4⋊C4—51no—
205C2×C4⋊C4—51no—
206C2×C4⋊C4—51no—
207C2×C4⋊C4—51no—
208C2×C4⋊C4—51no—
209C2×C4⋊C4C2×D511yescharacteristic
210C2×C4⋊C4C2×D511yescharacteristic
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4—51no—
215C2×C22⋊C4—51no—
216C2×C22⋊C4—51no—
217C2×C22⋊C4—51no—
218C2×C22⋊C4—51no—
219C2×C22⋊C4C2×D511yescharacteristic
220C2×C22⋊C4C2×D511yescharacteristic
221C2×C22⋊C4C2×D511yescharacteristic
222C2×C42—51no—
223#95—51no—
224#95—51no—
225#95—51no—
226#95—51no—
227#95—51no—
228#95—51no—
229#95—51no—
230#95—51no—
231C22×C5—22no—
232C22×C5—22no—
233C22×C5—22no—
234C22×C5—22no—
235C22×C5—22no—
236C22×C5—22no—
237C22×C5—41no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—41no—
243C22×C5—41no—
244C22×C5#12114yes—
245C22×C5C22×D411yescharacteristic
246C22×C5C22×D411yescharacteristic
247C22×C5C2×#4111yescharacteristic
248C22×C5C2×#4111yescharacteristic
249C22×C5C2×#4111yescharacteristic
250C22×C5H(5,𝔽2)11yescharacteristic
251C22×C52−1+411yescharacteristic
252C2×D5—22no—
253C2×D5—22no—
254C2×D5—22no—
255C2×D5—41no—
256C2×D5—41no—
257C2×D5—41no—
258C2×D5—41no—
259C4×C5—22no—
260C4×C5—22no—
261C4×C5—41no—
262C4×C5—41no—
263C4×C5—41no—
264Dic5—22no—
265Dic5—22no—
266Dic5—22no—
267Dic5—41no—
268Dic5—41no—
269Dic5—41no—
270Dic5—41no—
271C24—51no—
272C24C22×D511yescharacteristic
273C24C22×D511yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—54no—
278C2×D4—102no—
279C2×D4—102no—
280C2×D4—102no—
281C2×D4—102no—
282C2×D4—201no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—51no—
287C22×C4—51no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—102no—
294C22×C4—102no—
295C22×C4—102no—
296C22×C4—102no—
297C22×C4—102no—
298C22×C4—102no—
299C22×C4—201no—
300C22×C4—201no—
301C22×C4—201no—
302C22×C4C22×D511yescharacteristic
303C22×C4C22×D511yescharacteristic
304C22×C4C22×D511yescharacteristic
305C22×C4C22×D511yescharacteristic
306C22×C4C22×D511yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—102no—
310C4⋊C4—102no—
311C4⋊C4—102no—
312C4⋊C4—102no—
313C4⋊C4—102no—
314C4⋊C4—201no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—102no—
318C22⋊C4—102no—
319C22⋊C4—102no—
320C22⋊C4—102no—
321C22⋊C4—102no—
322C22⋊C4—102no—
323C22⋊C4—102no—
324C22⋊C4—201no—
325C22⋊C4C22⋊D514yescomplemented
326C42—102no—
327C2×C5—24no—
328C2×C5—81no—
329C2×C5C2×#12111yescharacteristic
330C2×C5#52911yescharacteristic
331C2×C5#53911yescharacteristic
332C2×C5#54611yescharacteristic
333C2×C5#54611yescharacteristic
334C2×C5#54711yescharacteristic
335C2×C5#54711yescharacteristic
336D5—42no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—52no—
349C23—101no—
350C23—101no—
351C23—101no—
352C23—101no—
353C23—101no—
354C23—101no—
355C23D4×D512yes—
356C23#71112yes—
357C23C2×C22⋊D512yes—
358C23C23×D511yescharacteristic
359D4—44no—
360D4—108no—
361D4—204no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—52no—
377C2×C4—52no—
378C2×C4—52no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—101no—
383C2×C4—101no—
384C2×C4—101no—
385C2×C4—101no—
386C2×C4—101no—
387C2×C4—101no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—101no—
395C2×C4—101no—
396C2×C4—101no—
397C2×C4—204no—
398C2×C4—204no—
399C2×C4—204no—
400C2×C4C2×C22⋊D512yes—
401C2×C4C2×C22⋊D512yes—
402C5#242511yescharacteristic, complemented, Sylow
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—102no—
417C22—102no—
418C22—102no—
419C22—201no—
420C22—201no—
421C22—201no—
422C22—201no—
423C22#408414yes—
424C22C2×D4×D511yescharacteristic
425C22C2×#71111yescharacteristic
426C22D5×#4111yescharacteristic
427C22D5×#4111yescharacteristic
428C22#414911yescharacteristic
429C22#415011yescharacteristic
430C22C22×C22⋊D511yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—102no—
437C4—102no—
438C4—102no—
439C4—201no—
440C4—201no—
441C4—201no—
442C4—201no—
443C2—24no—
444C2—81no—
445C2—202no—
446C2#6598311yescharacteristic
447C2#6598811yescharacteristic
448C2#6598911yescharacteristic
449C2#6599911yescharacteristic
450C2C2×#408411yescharacteristic
451C2#6615811yescharacteristic
452C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC2×#4085C2
hypercenterC2×#123D5
hypocenterC5#2425
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C5×#123C2semidirect
3. C2×#4075C2semidirect
4. C2×#4027C2semidirect
5. #65509C2semidirect
6. C2×#3963C2semidirect
7. #65282C2semidirect
8. #65276C2semidirect
9. #65269C2semidirect
10. #65262C2semidirect
11. #65250C2semidirect
12. #65245C2semidirect
13. #65239C2semidirect
14. #65237C2semidirect
15. #65233C2semidirect
96. C5×C22⋊C4D4semidirect
113. C2×#123D5semidirect
325. C22⋊C4C22⋊D5semidirect
402. C5#2425semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40852
C2×#40752
C2×#40272
#655092
C2×#39632
#652822
#652762
#652692
#652622
#652502
#652452
#652392
#652372
#652332
#24255