Finite Group Databasebeta

← #10614088

Subgroups — #10614088

Order 640.

610 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×#4085C211yesmaximal, characteristic
3#65509C211yesmaximal, characteristic, complemented
4#65502C211yesmaximal, characteristic, complemented
5#65502C211yesmaximal, characteristic, complemented
6#65500C211yesmaximal, characteristic, complemented
7C2×#3963C211yesmaximal, characteristic, complemented
8#65282C211yesmaximal, characteristic, complemented
9#65271C211yesmaximal, characteristic, complemented
10#65248C211yesmaximal, characteristic, complemented
11#65247C211yesmaximal, characteristic, complemented
12#65241C211yesmaximal, characteristic, complemented
13#65235C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15#65222C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C23×Dic5C2211yescharacteristic
23C4×C5×D4—24no—
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C42×C5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32#4085—24no—
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2444—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C23×D5C2311yescharacteristic
57C2×C5×D4—22no—
58C2×C5×D4—22no—
59C2×C5×D4D414yes—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—41no—
65C22×C4×C5—41no—
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C2×C22⋊D5—22no—
72C2×C22⋊D5—22no—
73C2×C22⋊D5—22no—
74C2×C22⋊D5—22no—
75C2×C22⋊D5—22no—
76C2×C22⋊D5—41no—
77C22×Dic5—22no—
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—41no—
81C22×Dic5—41no—
82C22×Dic5—41no—
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C5×C4⋊C4—22no—
91C5×C22⋊C4—22no—
92C5×C22⋊C4—22no—
93C42×C5—22no—
94C22⋊Dic5—22no—
95C22⋊Dic5—22no—
96C22⋊Dic5—41no—
97C22⋊Dic5—41no—
98#685—22no—
99#685—22no—
100#685—41no—
101#685—41no—
102#685—41no—
103#683—22no—
104#683—22no—
105#683—41no—
106C2×#121—51no—
107C2×C4×D4D511yescharacteristic, complemented
108#400—51no—
109#397—51no—
110#397—51no—
111#394—51no—
112#388—51no—
113#387—51no—
114#386—51no—
115#386—51no—
116#386—51no—
117#386—51no—
118#385—51no—
119#381—51no—
120C2×#95—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×C5C2×D412yes—
132C23×C5C2×D412yes—
133C23×C5#4112yes—
134C23×C5#4112yes—
135C23×C5C2411yescharacteristic
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5—21no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—21no—
143C5×D4—28no—
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×C5—21no—
157C2×C4×C5—44no—
158C2×C4×C5—44no—
159C2×C4×C5C2×D412yes—
160C2×C4×C5#4112yes—
161C22⋊D5—44no—
162C22⋊D5—44no—
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—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—44no—
184C2×Dic5—44no—
185C2×Dic5#4112yes—
186C22×D4—51no—
187C22×D4—51no—
188C22×D4C2×D511yescharacteristic
189C23×C4—51no—
190C23×C4—51no—
191C23×C4C2×D511yescharacteristic
192C23×C4C2×D511yescharacteristic
193#121—104no—
194C4×D4—24no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4—51no—
197C2×C4⋊C4—51no—
198C2×C4⋊C4C2×D511yescharacteristic
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C22⋊C4—51no—
206C2×C22⋊C4—51no—
207C2×C22⋊C4—51no—
208C2×C22⋊C4C2×D511yescharacteristic
209C2×C22⋊C4C2×D511yescharacteristic
210C2×C42C2×D511yescharacteristic
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95—51no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5—22no—
230C22×C5—22no—
231C22×C5—22no—
232C22×C5—41no—
233C22×C5—41no—
234C22×C5—41no—
235C22×C5—41no—
236C22×C5#12314yes—
237C22×C5C22×D411yescharacteristic
238C22×C5C2×#4111yescharacteristic
239C22×C5C2×#4111yescharacteristic
240C22×C5C2×#4111yescharacteristic
241C22×C5C2×#4111yescharacteristic
242C22×C5H(5,𝔽2)11yescharacteristic
243C22×C5H(5,𝔽2)11yescharacteristic
244C2×D5—41no—
245C2×D5—41no—
246C2×D5—41no—
247C2×D5—41no—
248C2×D5—41no—
249C2×D5—41no—
250C2×D5—41no—
251C4×C5—22no—
252C4×C5—22no—
253C4×C5—41no—
254C4×C5—41no—
255C4×C5—41no—
256Dic5—22no—
257Dic5—41no—
258Dic5—41no—
259Dic5—41no—
260Dic5—41no—
261Dic5—41no—
262Dic5—41no—
263C24—51no—
264C24C22×D511yescharacteristic
265C24C22×D511yescharacteristic
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—102no—
269C2×D4—102no—
270C2×D4—102no—
271C2×D4—102no—
272C2×D4—102no—
273C2×D4—201no—
274C2×D4C22⋊D514yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—51no—
286C22×C4—51no—
287C22×C4—51no—
288C22×C4—102no—
289C22×C4—102no—
290C22×C4—102no—
291C22×C4—201no—
292C22×C4—201no—
293C22×C4—201no—
294C22×C4C22×D511yescharacteristic
295C22×C4C22×D511yescharacteristic
296C22×C4C22×D511yescharacteristic
297C22×C4C22×D511yescharacteristic
298C22×C4C22×D511yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—102no—
301C4⋊C4—102no—
302C4⋊C4—201no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—102no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—201no—
310C22⋊C4—201no—
311C22⋊C4—201no—
312C22⋊C4—201no—
313C22⋊C4—201no—
314C42—22no—
315C2×C5—24no—
316C2×C5—42no—
317C2×C5C2×#12311yescharacteristic
318C2×C5#52911yescharacteristic
319C2×C5#53811yescharacteristic
320C2×C5#54611yescharacteristic
321C2×C5#54611yescharacteristic
322C2×C5#55311yescharacteristic
323C2×C5#55311yescharacteristic
324D5—81no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—101no—
336C23—101no—
337C23—101no—
338C23—101no—
339C23—101no—
340C23—101no—
341C23—101no—
342C23#70912yes—
343C23#70912yes—
344C23C2×C22⋊D512yes—
345C23C2×C22⋊D512yes—
346C23C23×D511yescharacteristic
347D4—28no—
348D4—204no—
349D4—204no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—52no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
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—204no—
387C2×C4—204no—
388C2×C4#71112yes—
389C2×C4C2×C22⋊D512yes—
390C5#244411yescharacteristic, complemented, Sylow
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22—201no—
408C22—201no—
409C22—201no—
410C22—201no—
411C22#407514yes—
412C22C2×#70911yescharacteristic
413C22C2×#70911yescharacteristic
414C22C2×#71111yescharacteristic
415C22#414411yescharacteristic
416C22D5×#4111yescharacteristic
417C22#414911yescharacteristic
418C22C22×C22⋊D511yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—102no—
425C4—201no—
426C4—201no—
427C4—201no—
428C4—201no—
429C4—201no—
430C4—201no—
431C2—24no—
432C2—42no—
433C2—401no—
434C2#6587211yescharacteristic
435C2#6587411yescharacteristic
436C2#6589411yescharacteristic
437C2#6589911yescharacteristic
438C2C2×#407511yescharacteristic
439C2#6614011yescharacteristic
440C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×D4C2
prime radicalC2×#4085C2
hypercenterC2×C4×D4D5
hypocenterC5#2444
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×D4C2semidirect
3. #65509C2semidirect
4. #65502C2semidirect
5. #65502C2semidirect
6. #65500C2semidirect
7. C2×#3963C2semidirect
8. #65282C2semidirect
9. #65271C2semidirect
10. #65248C2semidirect
11. #65247C2semidirect
12. #65241C2semidirect
13. #65235C2semidirect
14. #65229C2semidirect
15. #65222C2semidirect
107. C2×C4×D4D5semidirect
390. C5#2444semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×D42
C2×#40852
#655092
#655022
#655002
C2×#39632
#652822
#652712
#652482
#652472
#652412
#652352
#652292
#652222
#24445