Finite Group Databasebeta

← #14471964

Subgroups — #14471964

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5722C211yesmaximal, characteristic, complemented
2#11752545C211yesmaximal, characteristic
3#11751185C211yesmaximal, characteristic, complemented
4C3×#65377C211yesmaximal, characteristic, complemented
5#11747404C212yesmaximal, complemented
6#11747403C212yesmaximal, complemented
7#11747247C212yesmaximal, complemented
8#11747221C212yesmaximal, complemented
9#11747068C211yesmaximal, characteristic, complemented
10C4×#6660C211yesmaximal, characteristic, complemented
11Dic6×#168C211yesmaximal, characteristic, complemented
12#10615434—31nomaximal, Hall
13C2×#6766—21no—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C4×D60C2211yescharacteristic
16C5×#944—22no—
17C5×#888—21no—
18C5×#888C2212yes—
19C5×#887—21no—
20C5×#887C2212yes—
21C4×C5×Dic6C2211yescharacteristic
22C4×C5×Dic6C2211yescharacteristic
23C3×#4020C2211yescharacteristic
24#91465—21no—
25#91458C2212yes—
26Dic3×D20C2212yes—
27#91409—22no—
28C2×#6660C2211yescharacteristic
29C2×#6655C2212yes—
30#91093C2212yes—
31#91092C2212yes—
32C4⋊(C3⋊#168)C2211yescharacteristic
33C3×#3941C2212yes—
34C3×#3941C2212yes—
35C3×C4×#168C2211yescharacteristic
36#90985C2211yescharacteristic
37Dic3⋊#168C2212yes—
38#90973C2211yescharacteristic
39#90969C2211yescharacteristic
40#90964C2212yes—
41Dic3×#168C2212yes—
42#81588—51nomaximal, Hall
43C5×Q82—31no—
44#65911—31no—
45#65377—32no—
46#65377D311yescharacteristic, complemented
47#65373—32no—
48#65373—31no—
49#65316—31no—
50#65316—32no—
51C4×#688—32no—
52C4×#688—31no—
53Q8×#168—31no—
54C2×C3×C5×Q8—21no—
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×C5×Dic6—22no—
58C2×C5×Dic6D411yescharacteristic
59C2×C5×Dic6C2311yescharacteristic
60C2×C3×D20—21no—
61C2×C3×D20—21no—
62C2×C3×D20C2312yes—
63C2×#1202—22no—
64C2×#1202—22no—
65C2×D5×Dic3—22no—
66#6766—44no—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4C2312yes—
69C3×C42×C5C2311yescharacteristic
70#6714—21no—
71C4⋊Dic15C2311yescharacteristic
72C2×C3⋊#168C2312yes—
73C5×C4⋊Dic3D411yescharacteristic
74C5×C4⋊Dic3C2311yescharacteristic
75C5×Dic3⋊C4—21no—
76C5×Dic3⋊C4—22no—
77C5×Dic3⋊C4—22no—
78C5×Dic3⋊C4D412yes—
79C4×C5×Dic3—21no—
80C4×C5×Dic3—22no—
81C4×C5×Dic3C2312yes—
82C2×C3×#168C2311yescharacteristic
83C2×C3×#168C2311yescharacteristic
84#6665—22no—
85#6660—22no—
86#6655—24no—
87#5722D511yescharacteristic, complemented
88#5718—51no—
89C3×#486—51no—
90#5084—51no—
91#5075—51no—
92#4762—52no—
93#4761—52no—
94#4760—52no—
95#4758—52no—
96C4×#204—51no—
97C8×Dic6—51no—
98C2×#713—61no—
99C5×C4⋊Q8—32no—
100C5×C4⋊Q8—61no—
101C5×C4⋊Q8—62no—
102C5×C4⋊Q8C2×D311yescharacteristic
103C4×C5×Q8—31no—
104C4×C5×Q8—32no—
105C4×C5×Q8—31no—
106C4×C5×Q8—61no—
107C2×#688—32no—
108C2×#688—31no—
109#4041—62no—
110#4041—61no—
111#4020—32no—
112#4020C2×D311yescharacteristic
113C4×D20—32no—
114C4×D20—31no—
115#3967—31no—
116#3941—32no—
117#3941—32no—
118#3941—31no—
119#3941C2×D312yes—
120#3941C2×D312yes—
121#3940—32no—
122#3940—31no—
123C4⋊#168—31no—
124C4⋊#168—32no—
125C4×#168—32no—
126C4×#168C2×D311yescharacteristic
127#3362—151noSylow
128C22×D15—21no—
129C22×C3×D5—22no—
130C3×C5×Q8—81no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×Dic15—21no—
136D60—21no—
137D60—21no—
138C4×D15—42no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3C2×D412yes—
142C2×C5×Dic3C2×D412yes—
143C5×Dic6—21no—
144C5×Dic6—82no—
145C5×Dic6SD1614yescomplemented
146C5×Dic6#4112yescomplemented
147C3×D20—22no—
148C3×D20—22no—
149C3×D20—42no—
150C3×D20—81no—
151#1202—44no—
152#1202—82no—
153D5×Dic3—44no—
154C3⋊#168—22no—
155C3×#168—21no—
156C3×#168—21no—
157C2×#237—101no—
158C3×C4⋊Q8C2×D511yescharacteristic
159C3×#127—51no—
160#944—22no—
161#940—52no—
162#939—102no—
163#938—101no—
164D4×Dic3—52no—
165C2×#214—52no—
166C2×#204—51no—
167#888—21no—
168#888C2×D512yes—
169#887—21no—
170#887C2×D512yes—
171C4×D12—51no—
172C4×Dic6C2×D511yescharacteristic
173C4×Dic6C2×D511yescharacteristic
174C3×#102—52no—
175C3×#102—52no—
176C3×C4×C8—51no—
177#832—52no—
178#821—51no—
179#817—51no—
180#816—51no—
181Dic3⋊C8—52no—
182C8×Dic3—52no—
183#809—52no—
184#808—52no—
185C4⋊(C3⋊C8)—51no—
186C2×C5×Q8—21no—
187C2×C5×Q8—31no—
188C2×C5×Q8—31no—
189C2×C5×Q8—62no—
190#713—124no—
191C2×D20—21no—
192C2×D20—21no—
193C2×D20—31no—
194C2×D20—62no—
195C2×D20—62no—
196C2×D20C22×D312yes—
197C2×C4×D5—62no—
198C2×C4×D5—61no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—32no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—31no—
203C5×C4⋊C4—62no—
204C5×C4⋊C4—62no—
205C5×C4⋊C4—61no—
206C5×C4⋊C4C22×D312yes—
207C42×C5—32no—
208C42×C5—62no—
209C42×C5—61no—
210C42×C5C22×D311yescharacteristic
211#688—62no—
212#688—64no—
213#685—62no—
214#685—61no—
215C4⋊Dic5—31no—
216C2×#168—32no—
217C2×#168C22×D311yescharacteristic
218C2×#168C22×D311yescharacteristic
219Q82—31no—
220#550—151no—
221#486—51no—
222#486—152no—
223#476—152no—
224#476—151no—
225#460—152no—
226#460—151no—
227C8×Q8—151no—
228C4×SD16—151no—
229C4×SD16—152no—
230C22×C3×C5C22×D411yescharacteristic
231C2×D15—41no—
232C2×D15—41no—
233C2×D15—41no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—42no—
237C3×C4×C5—21no—
238C3×C4×C5—42no—
239C3×C4×C5C2×SD1612yes—
240C3×C4×C5C2×#4111yescharacteristic
241C3×C4×C52−1+411yescharacteristic
242Dic15—41no—
243C5×Dic3—22no—
244C5×Dic3—22no—
245C5×Dic3—41no—
246C5×Dic3C2×SD1614yes—
247C2×C3×Q8—21no—
248C2×C3×D4—52no—
249C2×C3×D4—101no—
250C2×C3×D4—101no—
251C2×#67—102no—
252C2×#67—102no—
253C22×Dic3—102no—
254#237—204no—
255C2×D12—51no—
256C2×C4×D3—101no—
257C2×Dic6—22no—
258C2×Dic6C22⋊D511yescharacteristic
259C2×Dic6C22×D511yescharacteristic
260C2×C3×C8—51no—
261C2×C3×C8—51no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4C22×D512yes—
264C3×C42C22×D511yescharacteristic
265#217—102no—
266#214—104no—
267(C2×D3)⋊C4—101no—
268C4⋊Dic3—51no—
269C4⋊Dic3C22⋊D511yescharacteristic
270C4⋊Dic3C22×D511yescharacteristic
271Dic3⋊C4—21no—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4C22⋊D512yes—
275C4×Dic3—21no—
276C4×Dic3—22no—
277C4×Dic3C22×D512yes—
278C2×C3⋊C8—52no—
279#204—102no—
280C22×D5—22no—
281C22×D5—61no—
282C5×Q8—34no—
283C5×Q8—32no—
284C5×Q8—61no—
285C5×Q8—81no—
286C5×Q8—242no—
287C2×C4×C5—22no—
288C2×C4×C5—32no—
289C2×C4×C5—32no—
290C2×C4×C5—61no—
291C2×C4×C5—61no—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5D3×D411yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C2×Dic5—61no—
296D20—22no—
297D20—22no—
298D20—42no—
299D20—61no—
300D20—61no—
301D20—81no—
302D20—124no—
303D20—242no—
304C4×D5—124no—
305C4×D5—122no—
306#168—21no—
307#168—21no—
308#168—62no—
309C2×#41—301no—
310C2×SD16—151no—
311C2×SD16—152no—
312C4⋊Q8—32no—
313C4⋊Q8—61no—
314C4⋊Q8—62no—
315C4⋊Q8D3×D511yescharacteristic
316#127—51no—
317#127—152no—
318#121—301no—
319#121—302no—
320C4×Q8—31no—
321C4×Q8—32no—
322C4×Q8—31no—
323C4×Q8—61no—
324C4×D4—152no—
325C4×D4—151no—
326#106—151no—
327#106—152no—
328C4⋊C8—152no—
329C4⋊C8—151no—
330#103—151no—
331#102—52no—
332#102—52no—
333#102—151no—
334#102—152no—
335#102—152no—
336C4×C8—51no—
337C4×C8—152no—
338C2×C3×C5#54811yescharacteristic
339C2×C3×C5C22×SD1611yescharacteristic
340C2×C3×C5#57611yescharacteristic
341D15—81no—
342C3×D5—82no—
343C23×C3—102no—
344C22×D3—101no—
345C3×Q8—81no—
346C3×D4—102no—
347C3×D4—102no—
348C3×D4—202no—
349C3×D4—401no—
350C2×C3×C4—22no—
351C2×C3×C4C2×C22⋊D511yescharacteristic
352C2×C3×C4C2×C22⋊D511yescharacteristic
353C2×C3×C4C23×D511yescharacteristic
354#67—204no—
355#67—402no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—101no—
359C2×Dic3—204no—
360C2×Dic3C2×C22⋊D512yes—
361C2×Dic3C2×C22⋊D512yes—
362D12—101no—
363D12—101no—
364C4×D3—202no—
365Dic6—21no—
366Dic6—82no—
367Dic6#68814yescomplemented
368Dic6#71312yescomplemented
369C3×C8—101no—
370C3×C8—101no—
371C3⋊C8—102no—
372C22×C5C2×D3×D411yescharacteristic
373C2×D5—42no—
374C2×D5—42no—
375C2×D5—42no—
376C2×D5—121no—
377C2×D5—121no—
378C2×D5—121no—
379C4×C5—21no—
380C4×C5—34no—
381C4×C5—42no—
382C4×C5—62no—
383C4×C5—62no—
384C4×C5—121no—
385C4×C5D3×SD1612yes—
386C4×C5D3×#4111yescharacteristic
387C4×C5#101011yescharacteristic
388Dic5—121no—
389#41—604no—
390C2×Q8—21no—
391C2×Q8—31no—
392C2×Q8—31no—
393C2×Q8—62no—
394C2×D4—52no—
395C2×D4—101no—
396C2×D4—101no—
397C2×D4—151no—
398C2×D4—302no—
399C2×D4—302no—
400C22×C4—302no—
401C22×C4—301no—
402SD16—302no—
403SD16—304no—
404C2×C8—51no—
405C2×C8—51no—
406C2×C8—152no—
407C4⋊C4—21no—
408C4⋊C4—32no—
409C4⋊C4—31no—
410C4⋊C4—31no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—61no—
414C4⋊C4—151no—
415C4⋊C4C2×D3×D512yes—
416C22⋊C4—302no—
417C22⋊C4—301no—
418C42—32no—
419C42—62no—
420C42—61no—
421C42C2×D3×D511yescharacteristic
422C3×C5#336211yescharacteristic, complemented, Hall
423C22×C3—202no—
424C22×C3—202no—
425C22×C3—202no—
426C22×C3C22×C22⋊D511yescharacteristic
427C2×D3—201no—
428C2×D3—201no—
429C2×D3—201no—
430C3×C4—21no—
431C3×C4—42no—
432C3×C4C2×#68812yes—
433C3×C4C2×#71311yescharacteristic
434C3×C4#414711yescharacteristic
435Dic3—22no—
436Dic3—22no—
437Dic3—41no—
438Dic3—201no—
439Dic3C2×#68814yes—
440C2×C5#567711yescharacteristic
441C2×C5#575711yescharacteristic
442C2×C5C2×D3×SD1611yescharacteristic
443D5—82no—
444D5—241no—
445C23—102no—
446C23—301no—
447Q8—34no—
448Q8—32no—
449Q8—61no—
450Q8—81no—
451Q8—242no—
452D4—102no—
453D4—102no—
454D4—202no—
455D4—301no—
456D4—301no—
457D4—401no—
458D4—604no—
459D4—1202no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—301no—
466C2×C4—604no—
467C2×C4—602no—
468C2×C4D3×C22⋊D511yescharacteristic
469C2×C4D3×C22⋊D511yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—101no—
472C8—101no—
473C8—302no—
474C2×C3—402no—
475C2×C3C22×#68811yescharacteristic
476C2×C3#6615111yescharacteristic
477C2×C3#6616911yescharacteristic
478D3—401no—
479C5#8158811yescharacteristic, complemented, Sylow
480C22—202no—
481C22—202no—
482C22—202no—
483C22—601no—
484C22—601no—
485C22—601no—
486C22C2×D3×C22⋊D511yescharacteristic
487C4—21no—
488C4—34no—
489C4—42no—
490C4—62no—
491C4—62no—
492C4—121no—
493C4—601no—
494C4D3×#68812yes—
495C4#9202811yescharacteristic
496C4D3×#71311yescharacteristic
497C3#1061543411yescharacteristic, complemented, Sylow
498C2—402no—
499C2—1201no—
500C2#1175469711yescharacteristic
501C2C2×D3×#68811yescharacteristic
502C2#1175523011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radical#11752545C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3362
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5722C2semidirect
3. #11751185C2semidirect
4. C3×#65377C2semidirect
5. #11747404C2semidirect
6. #11747403C2semidirect
7. #11747247C2semidirect
8. #11747221C2semidirect
9. #11747068C2semidirect
10. C4×#6660C2semidirect
11. Dic6×#168C2semidirect
46. #65377D3semidirect
87. #5722D5semidirect
145. C5×Dic6SD16semidirect
146. C5×Dic6#41semidirect
367. Dic6#688semidirect
368. Dic6#713semidirect
422. C3×C5#3362semidirect
479. C5#81588semidirect
497. C3#10615434semidirect

Maximal subgroups

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

subgroupindex
C5×#57222
#117525452
#117511852
C3×#653772
#117474042
#117474032
#117472472
#117472212
#117470682
C4×#66602
Dic6×#1682
#106154343
#815885