Finite Group Databasebeta

← #10614386

Subgroups — #10614386

Order 640.

674 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×#121C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic, complemented
3C2×D4×Dic5C211yesmaximal, characteristic, complemented
4C22×#683C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65509C211yesmaximal, characteristic, complemented
7#65509C211yesmaximal, characteristic, complemented
8#65509C211yesmaximal, characteristic, complemented
9#65499C211yesmaximal, characteristic, complemented
10#65272C211yesmaximal, characteristic, complemented
11#65267C211yesmaximal, characteristic, complemented
12#65241C211yesmaximal, characteristic, complemented
13#65238C211yesmaximal, characteristic, complemented
14#65235C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C23×C4×C5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C23×Dic5C2211yescharacteristic
23C5×#121—24no—
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36#4081—24no—
37D4×Dic5—24no—
38C2×C4⋊Dic5C2211yescharacteristic
39C2×#683—24no—
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×#683C2211yescharacteristic
46C2×C4×Dic5C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2443—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C24×C5C2311yescharacteristic
59C2×C5×D4—22no—
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—22no—
66C2×C5×D4—41no—
67C2×C5×D4D414yes—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—41no—
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×Dic5—22no—
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—22no—
81C22×Dic5—22no—
82C22×Dic5—41no—
83C22×Dic5—41no—
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C5×C4⋊C4—22no—
93C5×C22⋊C4—22no—
94C5×C22⋊C4—22no—
95C22⋊Dic5—22no—
96C22⋊Dic5—22no—
97C22⋊Dic5—22no—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—41no—
104C4⋊Dic5—22no—
105#683—22no—
106#683—22no—
107#683—22no—
108#683—22no—
109#683—41no—
110#683D414yescomplemented
111C4×Dic5—22no—
112C2×#123—51no—
113C2×#121D511yescharacteristic
114C2×C4×D4—51no—
115C22×C4⋊C4—51no—
116#399—51no—
117#398—51no—
118#397—51no—
119#394—51no—
120#388—51no—
121#387—51no—
122#386—51no—
123#386—51no—
124#386—51no—
125#385—51no—
126#385—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×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5C2×D412yes—
145C23×C5C2×D412yes—
146C23×C5#4112yes—
147C23×C5#4112yes—
148C23×C5C2411yescharacteristic
149C5×D4—28no—
150C5×D4—44no—
151C5×D4—44no—
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—44no—
163C2×C4×C5C2×D412yes—
164C2×C4×C5C2×D412yes—
165C2×Dic5—28no—
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—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—21no—
187C2×Dic5—44no—
188C2×Dic5—44no—
189C2×Dic5C2×D412yes—
190C2×Dic5C2×D412yes—
191C2×Dic5#4112yes—
192C22×D4C2×D511yescharacteristic
193C22×D4C2×D511yescharacteristic
194C22×D4C2×D511yescharacteristic
195C23×C4—51no—
196C23×C4—51no—
197C23×C4—51no—
198C23×C4C2×D511yescharacteristic
199#123—104no—
200#121—24no—
201C4×D4—104no—
202C2×C4⋊C4—51no—
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⋊C4—104no—
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⋊C4—51no—
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—22no—
238C22×C5—22no—
239C22×C5—22no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—41no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5#12314yescomplemented
252C22×C5C22×D411yescharacteristic
253C22×C5C22×D411yescharacteristic
254C22×C5C2×#4111yescharacteristic
255C22×C5C2×#4111yescharacteristic
256C22×C5C2×#4111yescharacteristic
257C22×C5H(5,𝔽2)11yescharacteristic
258C22×C52−1+411yescharacteristic
259C4×C5—22no—
260C4×C5—22no—
261C4×C5—41no—
262C4×C5—41no—
263Dic5—22no—
264Dic5—22no—
265Dic5—22no—
266Dic5—41no—
267Dic5—41no—
268Dic5—41no—
269Dic5—41no—
270Dic5—41no—
271C24C22×D511yescharacteristic
272C24C22×D511yescharacteristic
273C24C22×D511yescharacteristic
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4C22⋊D514yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—51no—
287C22×C4—51no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—51no—
294C22×C4—102no—
295C22×C4—102no—
296C22×C4—102no—
297C22×C4—102no—
298C22×C4—102no—
299C22×C4—102no—
300C22×C4—102no—
301C22×C4—201no—
302C22×C4—201no—
303C22×C4C22×D511yescharacteristic
304C22×C4C22×D511yescharacteristic
305C22×C4C22×D511yescharacteristic
306C22×C4C22×D511yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—54no—
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—102no—
325C22⋊C4—201no—
326C42—102no—
327C2×C5—24no—
328C2×C5—42no—
329C2×C5—81no—
330C2×C5C2×#12311yescharacteristic
331C2×C5#52511yescharacteristic
332C2×C5#53911yescharacteristic
333C2×C5D4211yescharacteristic
334C2×C5#54711yescharacteristic
335C2×C5#54811yescharacteristic
336C2×C5#55311yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23#71112yes—
355C23#71112yes—
356C23C2×C22⋊D512yes—
357C23C2×C22⋊D512yes—
358C23C23×D511yescharacteristic
359D4—28no—
360D4—44no—
361D4—44no—
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—44no—
373C2×C4—52no—
374C2×C4—52no—
375C2×C4—52no—
376C2×C4—108no—
377C2×C4—101no—
378C2×C4—101no—
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—101no—
398C2×C4—204no—
399C2×C4—204no—
400C2×C4D4×D512yes—
401C2×C4C2×C22⋊D512yes—
402C5#244311yescharacteristic, complemented, Sylow
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#408114yescomplemented
424C22C2×D4×D511yescharacteristic
425C22C2×#71111yescharacteristic
426C22C2×#71111yescharacteristic
427C22#414411yescharacteristic
428C22D5×#4111yescharacteristic
429C22#415011yescharacteristic
430C22C22×C22⋊D511yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—102no—
436C4—102no—
437C4—102no—
438C4—201no—
439C4—201no—
440C4—201no—
441C4—201no—
442C4—201no—
443C2—24no—
444C2—42no—
445C2—81no—
446C2#6593211yescharacteristic
447C2#6593511yescharacteristic
448C2#6593611yescharacteristic
449C2#6594011yescharacteristic
450C2C2×#408111yescharacteristic
451C2D4×C22⋊D511yescharacteristic
452C2#6615811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2443
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4081C2semidirect
3. C2×D4×Dic5C2semidirect
4. C22×#683C2semidirect
5. #65511C2semidirect
6. #65509C2semidirect
7. #65509C2semidirect
8. #65509C2semidirect
9. #65499C2semidirect
10. #65272C2semidirect
11. #65267C2semidirect
12. #65241C2semidirect
13. #65238C2semidirect
14. #65235C2semidirect
15. #65233C2semidirect
110. #683D4semidirect
251. C22×C5#123semidirect
402. C5#2443semidirect
423. C22#4081semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40812
C2×D4×Dic52
C22×#6832
#655112
#655092
#654992
#652722
#652672
#652412
#652382
#652352
#652332
#24435