Finite Group Databasebeta

← #10606440

Subgroups — #10606440

Order 640.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic5C211yesmaximal, characteristic
2C2×#4070C211yesmaximal, characteristic, complemented
3C22×#683C211yesmaximal, characteristic, complemented
4C2×#4024C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic
6#65509C211yesmaximal, characteristic
7#65239C211yesmaximal, characteristic, complemented
8#65238C211yesmaximal, characteristic, complemented
9#65237C211yesmaximal, characteristic, complemented
10#65235C211yesmaximal, characteristic, complemented
11#65235C211yesmaximal, characteristic, complemented
12#64946C211yesmaximal, characteristic, complemented
13#64945C211yesmaximal, characteristic, complemented
14#64944C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C22×Dic10C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30D4×Dic5—24no—
31C2×C4⋊Dic5C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35#4070—24no—
36C2×#683—24no—
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×C4×Dic5C2211yescharacteristic
46#4024—24no—
47C5×#95C2211yescharacteristic
48C5×#95C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2449—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C2×C5×D4—22no—
59C2×C5×D4—22no—
60C2×C5×D4Q814yescomplemented
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—41no—
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×Dic5—22no—
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—22no—
76C22×Dic5—41no—
77C22×Dic5—41no—
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C2×Dic10—22no—
87C2×Dic10—22no—
88C2×Dic10—41no—
89C5×C22⋊C4—22no—
90C5×C22⋊C4—22no—
91C5×C22⋊C4—41no—
92C22⋊Dic5—22no—
93C22⋊Dic5—22no—
94C22⋊Dic5—22no—
95C22⋊Dic5—22no—
96C22⋊Dic5D414yes—
97C4⋊Dic5—22no—
98C4⋊Dic5—22no—
99C4⋊Dic5—22no—
100C4⋊Dic5—41no—
101#683—22no—
102#683—22no—
103#683—22no—
104#683—22no—
105#683—22no—
106#683—22no—
107#683—22no—
108#683D414yescomplemented
109C4×Dic5—22no—
110C2×C22⋊Q8—51no—
111C2×C22⋊Q8—51no—
112C2×C4×D4—51no—
113C22×C4⋊C4—51no—
114#399—51no—
115#398—51no—
116#396—51no—
117#395—51no—
118#389—51no—
119#387—51no—
120#386—51no—
121#386D511yescharacteristic
122#385—51no—
123#385—51no—
124#385—51no—
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×C5—21no—
135C23×C5C2×D412yes—
136C23×C5C2×Q812yes—
137C23×C5C2×Q812yes—
138C23×C5#4112yes—
139C23×C5C2411yescharacteristic
140C5×D4—28no—
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—44no—
155C2×C4×C5C2×D412yes—
156C2×C4×C5C2×Q812yes—
157C2×Dic5—28no—
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—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—44no—
178C2×Dic5—44no—
179C2×Dic5C2×D412yes—
180C2×Dic5C2×D412yes—
181C2×Dic5C2×D412yes—
182C2×Dic5C2×D412yes—
183C2×Dic5#4112yes—
184Dic10—44no—
185C22×Q8—51no—
186C22×D4C2×D511yescharacteristic
187C23×C4—51no—
188C23×C4—51no—
189C23×C4—51no—
190C23×C4C2×D511yescharacteristic
191C22⋊Q8—104no—
192C22⋊Q8—104no—
193C4×D4—104no—
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⋊C4—51no—
203C2×C4⋊C4—51no—
204C2×C4⋊C4—51no—
205C2×C4⋊C4—51no—
206C2×C4⋊C4—104no—
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—
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95C2×D511yescharacteristic
223#95C2×D511yescharacteristic
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—22no—
233C22×C5—41no—
234C22×C5—41no—
235C22×C5—41no—
236C22×C5—41no—
237C22×C5C22⋊Q814yescomplemented
238C22×C5C22×D411yescharacteristic
239C22×C5C22×D411yescharacteristic
240C22×C5C22×Q811yescharacteristic
241C22×C5C2×#4111yescharacteristic
242C22×C5C2×#4111yescharacteristic
243C22×C5H(5,𝔽2)11yescharacteristic
244C22×C52−1+411yescharacteristic
245C4×C5—22no—
246C4×C5—22no—
247C4×C5—41no—
248C4×C5—41no—
249C4×C5—41no—
250Dic5—22no—
251Dic5—22no—
252Dic5—22no—
253Dic5—22no—
254Dic5—22no—
255Dic5—41no—
256Dic5—41no—
257Dic5—41no—
258C24C22×D511yescharacteristic
259C24C22×D511yescharacteristic
260C2×Q8—102no—
261C2×Q8—102no—
262C2×Q8—201no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4Dic1014yescomplemented
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—51no—
276C22×C4—51no—
277C22×C4—102no—
278C22×C4—102no—
279C22×C4—102no—
280C22×C4—102no—
281C22×C4—102no—
282C22×C4—102no—
283C22×C4—102no—
284C22×C4—201no—
285C22×C4—201no—
286C22×C4C22×D511yescharacteristic
287C22×C4C22×D511yescharacteristic
288C22×C4C22×D511yescharacteristic
289C22×C4C22×D511yescharacteristic
290C22×C4C22×D511yescharacteristic
291C4⋊C4—54no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C4⋊C4—102no—
295C4⋊C4—102no—
296C4⋊C4—102no—
297C4⋊C4—102no—
298C4⋊C4—102no—
299C4⋊C4—102no—
300C4⋊C4—102no—
301C4⋊C4—102no—
302C4⋊C4—201no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—41no—
306C22⋊C4—54no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—102no—
310C22⋊C4—102no—
311C42—102no—
312C2×C5—24no—
313C2×C5—42no—
314C2×C5C2×C22⋊Q811yescharacteristic
315C2×C5#52511yescharacteristic
316C2×C5#53911yescharacteristic
317C2×C5D4211yescharacteristic
318C2×C5#54711yescharacteristic
319C2×C5D4×Q811yescharacteristic
320C2×C5#55411yescharacteristic
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C2×Dic1012yes—
332C23C2×Dic1012yes—
333C23D4×D512yes—
334C23#71112yes—
335C23C23×D511yescharacteristic
336Q8—204no—
337D4—28no—
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—44no—
352C2×C4—52no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—52no—
356C2×C4—52no—
357C2×C4—108no—
358C2×C4—101no—
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—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—204no—
378C2×C4—204no—
379C2×C4C2×Dic1012yes—
380C2×C4D4×D512yes—
381C5#244911yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#402414yescomplemented
396C22C22×Dic1011yescharacteristic
397C22C2×D4×D511yescharacteristic
398C22C2×D4×D511yescharacteristic
399C22C2×#71111yescharacteristic
400C22#414411yescharacteristic
401C22D5×#4111yescharacteristic
402C22#415011yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—102no—
409C4—102no—
410C4—102no—
411C4—102no—
412C4—102no—
413C4—201no—
414C4—201no—
415C4—201no—
416C2—24no—
417C2—42no—
418C2C2×#402411yescharacteristic
419C2D4×Dic1011yescharacteristic
420C2#6587311yescharacteristic
421C2#6592411yescharacteristic
422C2#6593611yescharacteristic
423C2#6594011yescharacteristic
424C2#6598911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC22×C5×D4C22
hypercenter#386D5
hypocenterC5#2449
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#4070C2semidirect
3. C22×#683C2semidirect
4. C2×#4024C2semidirect
7. #65239C2semidirect
8. #65238C2semidirect
9. #65237C2semidirect
10. #65235C2semidirect
11. #65235C2semidirect
12. #64946C2semidirect
13. #64945C2semidirect
14. #64944C2semidirect
15. #64936C2semidirect
60. C2×C5×D4Q8semidirect
108. #683D4semidirect
237. C22×C5C22⋊Q8semidirect
265. C2×D4Dic10semidirect
381. C5#2449semidirect
395. C22#4024semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40702
C22×#6832
C2×#40242
C5×#3862
#655092
#652392
#652382
#652372
#652352
#649462
#649452
#649442
#649362
#24495