Finite Group Databasebeta

← #10614316

Subgroups — #10614316

Order 640.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#120C211yesmaximal, characteristic
2C22×C22⋊Dic5C211yesmaximal, characteristic, complemented
3C2×#4081C211yesmaximal, characteristic, complemented
4C2×D4×Dic5C211yesmaximal, characteristic, complemented
5#65536C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65511C211yesmaximal, characteristic, complemented
8#65509C211yesmaximal, characteristic, complemented
9#65509C211yesmaximal, characteristic, complemented
10#65242C211yesmaximal, characteristic, complemented
11#65238C211yesmaximal, characteristic, complemented
12#65235C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#65233C211yesmaximal, characteristic, complemented
15#65231C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C22×C5×D4C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C23×Dic5C2211yescharacteristic, complemented
23C5×#120—24no—
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5—24no—
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic
37C2×C22⋊Dic5C2211yescharacteristic
38C2×C22⋊Dic5C2211yescharacteristic, complemented
39C2×C22⋊Dic5C2211yescharacteristic, complemented
40C2×C22⋊Dic5C2211yescharacteristic, complemented
41#4081—24no—
42D4×Dic5—24no—
43C2×C4⋊Dic5C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×#683C2211yescharacteristic, complemented
47C2×#683C2211yescharacteristic, complemented
48C2×C4×Dic5C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2418—51nomaximal, Sylow
56C24×C5—22no—
57C24×C5—22no—
58C24×C5—22no—
59C24×C5—22no—
60C24×C5—22no—
61C24×C5—41no—
62C24×C5—41no—
63C24×C5C2311yescharacteristic
64C24×C5C2311yescharacteristic
65C24×C5C2311yescharacteristic
66C24×C5C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4—22no—
70C2×C5×D4—22no—
71C2×C5×D4—22no—
72C2×C5×D4—22no—
73C2×C5×D4—41no—
74C2×C5×D4—41no—
75C2×C5×D4D414yes—
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C22×C4×C5C2311yescharacteristic
79C22×Dic5—22no—
80C22×Dic5—22no—
81C22×Dic5—22no—
82C22×Dic5—22no—
83C22×Dic5—22no—
84C22×Dic5—22no—
85C22×Dic5—41no—
86C22×Dic5—41no—
87C22×Dic5—41no—
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
93C22×Dic5C2311yescharacteristic
94C22×Dic5C2311yescharacteristic
95C22×Dic5C2311yescharacteristic
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—22no—
107C22⋊Dic5—22no—
108C22⋊Dic5—41no—
109C22⋊Dic5—41no—
110C22⋊Dic5—41no—
111C22⋊Dic5D414yescomplemented
112C4⋊Dic5—22no—
113#683—22no—
114#683—22no—
115#683—41no—
116#683—41no—
117C4×Dic5—22no—
118C2×#123—51no—
119C2×#120D511yescharacteristic
120C2×C4×D4—51no—
121C22×C22⋊C4—51no—
122#399—51no—
123#397—51no—
124#394—51no—
125#394—51no—
126#388—51no—
127#388—51no—
128#386—51no—
129#386—51no—
130#385—51no—
131#380—51no—
132#379—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—28no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5—21no—
146C23×C5—21no—
147C23×C5—21no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—21no—
153C23×C5—21no—
154C23×C5—21no—
155C23×C5—21no—
156C23×C5—44no—
157C23×C5—44no—
158C23×C5—44no—
159C23×C5—44no—
160C23×C5—44no—
161C23×C5—44no—
162C23×C5C2×D412yes—
163C23×C5C2×D412yes—
164C23×C5C2×D412yes—
165C23×C5#4112yes—
166C23×C5#4112yes—
167C23×C5#4112yes—
168C23×C5C2411yescharacteristic
169C5×D4—28no—
170C5×D4—44no—
171C5×D4—44no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5C2×D412yes—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—21no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—44no—
204C2×Dic5—44no—
205C2×Dic5—44no—
206C2×Dic5C2×D412yes—
207C2×Dic5C2×D412yes—
208C25C2×D511yescharacteristic
209C22×D4C2×D511yescharacteristic
210C22×D4C2×D511yescharacteristic
211C22×D4C2×D511yescharacteristic
212C23×C4—51no—
213C23×C4—51no—
214C23×C4—51no—
215#123—104no—
216#120—24no—
217C4×D4—104no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—51no—
222C2×C4⋊C4—51no—
223C2×C22⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—51no—
235C2×C22⋊C4—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C22⋊C4C2×D511yescharacteristic
240C2×C42—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95—51no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—22no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—22no—
257C22×C5—22no—
258C22×C5—22no—
259C22×C5—22no—
260C22×C5—22no—
261C22×C5—22no—
262C22×C5—416no—
263C22×C5—41no—
264C22×C5—41no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—41no—
268C22×C5—41no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5—41no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5#12314yescomplemented
276C22×C5C22×D411yescharacteristic
277C22×C5C22×D411yescharacteristic
278C22×C5C2×#4111yescharacteristic
279C22×C5C2×#4111yescharacteristic
280C22×C5C2×#4111yescharacteristic
281C22×C5H(5,𝔽2)11yescharacteristic
282C22×C5H(5,𝔽2)11yescharacteristic
283C4×C5—22no—
284C4×C5—41no—
285C4×C5—41no—
286Dic5—22no—
287Dic5—22no—
288Dic5—41no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24C22×D511yescharacteristic
302C24C22×D511yescharacteristic
303C24C22×D511yescharacteristic
304C24C22×D511yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4C22⋊D514yes—
314C22×C4—51no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—102no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—201no—
329C22×C4—201no—
330C22×C4—201no—
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—102no—
335C4⋊C4—102no—
336C4⋊C4—102no—
337C4⋊C4—201no—
338C4⋊C4—201no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—54no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—102no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C22⋊C4—201no—
353C22⋊C4—201no—
354C22⋊C4—201no—
355C42—102no—
356C2×C5—24no—
357C2×C5—42no—
358C2×C5—42no—
359C2×C5—42no—
360C2×C5C2×#12311yescharacteristic
361C2×C5#52511yescharacteristic
362C2×C5#53811yescharacteristic
363C2×C5D4211yescharacteristic
364C2×C5#54611yescharacteristic
365C2×C5#54611yescharacteristic
366C2×C5#55111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—28no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23D4×D512yes—
397C23#71112yes—
398C23#71112yes—
399C23#71112yes—
400C23C2×C22⋊D512yes—
401C23C2×C22⋊D512yes—
402C23C23×D511yescharacteristic
403D4—28no—
404D4—44no—
405D4—44no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—52no—
415C2×C4—52no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—101no—
419C2×C4—101no—
420C2×C4—101no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—204no—
439C2×C4—204no—
440C2×C4—204no—
441C2×C4C2×C22⋊D512yes—
442C5#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—416no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#408114yescomplemented
472C22C2×D4×D511yescharacteristic
473C22C2×#71111yescharacteristic
474C22C2×#71111yescharacteristic
475C22C2×#71111yescharacteristic
476C22#414411yescharacteristic
477C22#414411yescharacteristic
478C22C22×C22⋊D511yescharacteristic
479C4—22no—
480C4—41no—
481C4—41no—
482C4—102no—
483C4—102no—
484C4—201no—
485C4—201no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—42no—
494C2#6592011yescharacteristic
495C2#6592111yescharacteristic
496C2#6592511yescharacteristic
497C2#6592711yescharacteristic
498C2C2×#408111yescharacteristic
499C2D4×C22⋊D511yescharacteristic
500C2#6614011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#120C2
hypercenterC2×#120D5
hypocenterC5#2418
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊Dic5C2semidirect
3. C2×#4081C2semidirect
4. C2×D4×Dic5C2semidirect
5. #65536C2semidirect
6. #65511C2semidirect
7. #65511C2semidirect
8. #65509C2semidirect
9. #65509C2semidirect
10. #65242C2semidirect
11. #65238C2semidirect
12. #65235C2semidirect
13. #65233C2semidirect
14. #65233C2semidirect
15. #65231C2semidirect
22. C23×Dic5C22semidirect
38. C2×C22⋊Dic5C22semidirect
39. C2×C22⋊Dic5C22semidirect
40. C2×C22⋊Dic5C22semidirect
43. C2×C4⋊Dic5C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
46. C2×#683C22semidirect
47. C2×#683C22semidirect
48. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
111. C22⋊Dic5D4semidirect
275. C22×C5#123semidirect
442. C5#2418semidirect
471. C22#4081semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1202
C22×C22⋊Dic52
C2×#40812
C2×D4×Dic52
#655362
#655112
#655092
#652422
#652382
#652352
#652332
#652312
#24185