Finite Group Databasebeta

← #14460170

Subgroups — #14460170

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5551C211yesmaximal, characteristic, complemented
2C3×#65897C211yesmaximal, characteristic, complemented
3D4×C3⋊#168C211yesmaximal, characteristic, complemented
4#11748545C212yesmaximal, complemented
5#11748517C212yesmaximal, complemented
6#11748404C212yesmaximal, complemented
7#11748382C212yesmaximal, complemented
8#11747059C211yesmaximal, characteristic, complemented
9C4×#6652C211yesmaximal, characteristic, complemented
10#11746923C211yesmaximal, characteristic, complemented
11#11746882C211yesmaximal, characteristic, complemented
12#10615242—31nomaximal, Hall
13C2×C5×#235C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C22×C3⋊#168C2212yes—
16C5×#938—21no—
17C5×#924C2212yes—
18C5×D3×C4⋊C4—21no—
19C5×#883—22no—
20C4×C5×D12C2211yescharacteristic
21C5×#869C2211yescharacteristic
22C3×#4084—21no—
23C3×#4076C2212yes—
24C3×#4072C2212yes—
25C3×#4038—21no—
26C3×#4031—22no—
27C3×C4×D20C2211yescharacteristic
28C3×#4015C2211yescharacteristic
29C2×#6655C2212yes—
30C2×#6652C2211yescharacteristic
31#91114C2212yes—
32C4⋊(C3⋊#168)C2211yescharacteristic
33C4×C3⋊#168C2211yescharacteristic
34#90989C2211yescharacteristic
35#90986C2212yes—
36#90972C2212yes—
37#90972C2212yes—
38#90972C2212yes—
39#90966C2211yescharacteristic
40#90966C2212yes—
41#90964C2211yescharacteristic
42#83652—51nomaximal, Hall
43C5×#547—31no—
44#65897D311yescharacteristic, complemented
45#65527—32no—
46#65522—32no—
47#65372—31no—
48#65370—31no—
49#65337—32no—
50#65334—32no—
51C4×#686—31no—
52D4×#168—31no—
53#65297—31no—
54C2×C3×C5×D4D411yescharacteristic
55C22×C3×C4×C5C2312yes—
56C2×C5×#67—22no—
57C5×#235—24no—
58C5×#235—42no—
59C2×C5×D12C2311yescharacteristic
60C2×C4×C5×D3—21no—
61C2×C4×C5×D3—22no—
62C2×C5×Dic6C2312yes—
63C2×C3×C22⋊D5—22no—
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×C3×Dic10—21no—
67C3×C5×C4⋊C4D411yescharacteristic
68C3×C5×C22⋊C4D412yes—
69C3×C42×C5C2311yescharacteristic
70C2×C3⋊#168—42no—
71C2×C3⋊#168C2311yescharacteristic
72C2×C3⋊#168C2312yes—
73C2×C3⋊#168C2311yescharacteristic
74C5×#217—22no—
75C5×(C2×D3)⋊C4—21no—
76C5×C4⋊Dic3—21no—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2311yescharacteristic
79C5×Dic3⋊C4—22no—
80C3×C22⋊Dic5—22no—
81C3×#685—21no—
82C3×#685—22no—
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5C2312yes—
85C3×C4⋊Dic5C2311yescharacteristic
86C3×C4×Dic5—22no—
87#6655—42no—
88#6652—22no—
89C3×#555—51no—
90#5551D511yescharacteristic, complemented
91#5093—51no—
92#5084—51no—
93#5041—52no—
94#5037—52no—
95#5019—51no—
96C4×#213—51no—
97D4×C3⋊C8—51no—
98#5002—52no—
99#4996—52no—
100C2×C5×#41—32no—
101C5×C4⋊Q8—31no—
102C5×#123—62no—
103C5×C22⋊Q8—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4C2×D311yescharacteristic
107C2×C5×C4⋊C4—61no—
108C2×#688—32no—
109#4084—21no—
110C2×#686—31no—
111#4076C2×D312yes—
112#4072C2×D312yes—
113C22×#168—32no—
114#4038—21no—
115#4031—22no—
116C4×D20C2×D311yescharacteristic
117#4015C2×D311yescharacteristic
118#3967—32no—
119#3967—32no—
120#3967—32no—
121#3964—32no—
122#3964—31no—
123C22⋊#168—32no—
124#3941—31no—
125#3940—32no—
126#3939—31no—
127C4⋊#168—31no—
128C4×#168—31no—
129#3303—151noSylow
130C23×C3×C5C2×D412yes—
131C22×C5×D3—21no—
132C22×C3×D5—21no—
133C3×C5×D4—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2411yescharacteristic
140C5×#67—44no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—22no—
143C5×D12—21no—
144C5×D12#4112yes—
145C4×C5×D3—44no—
146C4×C5×D3—42no—
147C5×Dic6—22no—
148C5×Dic6—22no—
149C3×C22⋊D5—82no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—22no—
152C3×D20—21no—
153C3×D20—21no—
154C3×C4×D5—42no—
155C3×Dic10—81no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3⋊#168—22no—
159C2×#235C2×D512yes—
160C3×C4⋊Q8—51no—
161C3×#124—102no—
162C3×#121—52no—
163C3×#121—101no—
164C3×C4×D4—51no—
165C3×C4×D4C2×D511yescharacteristic
166C3×#117—52no—
167C3×#117—101no—
168#938—21no—
169C2×#214—52no—
170C2×#213—51no—
171#924C2×D512yes—
172C22×C3⋊C8—52no—
173D3×C4⋊C4—21no—
174#883—22no—
175C4×D12C2×D511yescharacteristic
176#869C2×D511yescharacteristic
177#832—51no—
178#830—52no—
179#810—52no—
180#810—52no—
181#810—52no—
182#809—51no—
183#809—52no—
184#808—52no—
185#807—51no—
186C4⋊(C3⋊C8)—51no—
187C4×C3⋊C8—51no—
188C5×#41—64no—
189C5×#41—122no—
190C2×C5×Q8—32no—
191C2×C5×D4—31no—
192C2×C5×D4—62no—
193C2×C5×D4#6711yescharacteristic
194C22×C4×C5—61no—
195C22×C4×C5—62no—
196C22×C4×C5C22×D312yes—
197C2×C22⋊D5—22no—
198C2×D20C22×D311yescharacteristic
199C2×C4×D5—21no—
200C2×Dic10—21no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—61no—
203C5×C4⋊C4—62no—
204C5×C4⋊C4—61no—
205C5×C4⋊C4#6711yescharacteristic
206C5×C22⋊C4—62no—
207C5×C22⋊C4—61no—
208C5×C22⋊C4#6712yes—
209C42×C5C22×D311yescharacteristic
210C22⋊Dic5—22no—
211#688—122no—
212#686—62no—
213#685—21no—
214#685—22no—
215C4⋊Dic5—21no—
216C4⋊Dic5C22×D311yescharacteristic
217C4⋊Dic5C22×D312yes—
218C4×Dic5—22no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—32no—
222C2×#168—122no—
223#555—51no—
224#547—31no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461—151no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—22no—
235C22×C3×C5—22no—
236C22×C3×C5—22no—
237C22×C3×C5C22×D411yescharacteristic
238C2×C5×D3—22no—
239C2×C5×D3—41no—
240C2×C5×D3—41no—
241C2×C3×D5—41no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—22no—
245C3×C4×C5—21no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249C3×Dic5—41no—
250C3×Dic5—42no—
251C5×Dic3—42no—
252C5×Dic3—41no—
253C2×C3×Q8—101no—
254C2×C3×D4—51no—
255C2×C3×D4—102no—
256C2×C3×D4C22⋊D511yescharacteristic
257C22×C3×C4—101no—
258C22×C3×C4C22×D512yes—
259C2×#67—22no—
260#235—24no—
261#235—42no—
262C2×D12C22×D511yescharacteristic
263C2×C4×D3—21no—
264C2×C4×D3—22no—
265C2×Dic6C22×D512yes—
266C3×C4⋊C4—51no—
267C3×C4⋊C4—52no—
268C3×C4⋊C4—101no—
269C3×C4⋊C4C22⋊D511yescharacteristic
270C3×C22⋊C4—102no—
271C3×C22⋊C4—102no—
272C3×C22⋊C4—101no—
273C3×C22⋊C4C22⋊D512yes—
274C3×C42—102no—
275C3×C42C22×D511yescharacteristic
276#217—22no—
277#214—202no—
278#213—102no—
279(C2×D3)⋊C4—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3C22×D511yescharacteristic
283Dic3⋊C4—22no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—52no—
287C2×C3⋊C8—202no—
288C23×C5—61no—
289C23×C5C2×#6712yes—
290C22×D5—21no—
291C5×Q8—62no—
292C5×Q8—62no—
293C5×D4—22no—
294C5×D4—32no—
295C5×D4—61no—
296C5×D4—124no—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—61no—
300C2×C4×C5—62no—
301C2×C4×C5—124no—
302C2×C4×C5—122no—
303C2×C4×C5C2×#6712yes—
304C2×C4×C5C2×#6711yescharacteristic
305C2×C4×C5C2×#6711yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—82no—
308C2×Dic5—21no—
309C2×Dic5—22no—
310D20—21no—
311D20—21no—
312C4×D5—42no—
313Dic10—81no—
314#168—61no—
315#168—61no—
316#168—62no—
317C2×#41—32no—
318C2×SD16—152no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—31no—
322C4⋊Q8—51no—
323#124—102no—
324#123—62no—
325C22⋊Q8—32no—
326#121—52no—
327#121—61no—
328#121—101no—
329C4×D4—31no—
330C4×D4—51no—
331C4×D4D3×D511yescharacteristic
332#117—52no—
333#117—101no—
334C2×C4⋊C4—61no—
335#107—151no—
336#106—152no—
337C4⋊C8—151no—
338#103—152no—
339#103—152no—
340#103—152no—
341#102—151no—
342#102—151no—
343#102—152no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5—42no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350C3×D5—81no—
351C5×D3—42no—
352C23×C3—101no—
353C23×C3C2×C22⋊D512yes—
354C22×D3—21no—
355C3×Q8—401no—
356C3×D4—22no—
357C3×D4—101no—
358C3×D4—101no—
359C3×D4—402no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—101no—
363C2×C3×C4—102no—
364C2×C3×C4—202no—
365C2×C3×C4C2×C22⋊D512yes—
366C2×C3×C4C2×C22⋊D511yescharacteristic
367C2×C3×C4C2×C22⋊D511yescharacteristic
368C2×C3×C4C23×D511yescharacteristic
369#67—44no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372D12—21no—
373D12#71112yes—
374C4×D3—44no—
375C4×D3—42no—
376Dic6—22no—
377Dic6—22no—
378C3⋊C8—101no—
379C3⋊C8—101no—
380C3⋊C8—102no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—62no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5C22×#6711yescharacteristic
388C2×D5—41no—
389C2×D5—41no—
390C2×D5—41no—
391C4×C5—22no—
392C4×C5—21no—
393C4×C5—121no—
394C4×C5—122no—
395C4×C5#95012yes—
396C4×C5D3×#4111yescharacteristic
397C4×C5#100911yescharacteristic
398Dic5—41no—
399Dic5—42no—
400#41—64no—
401#41—122no—
402C2×Q8—32no—
403C2×Q8—101no—
404C2×D4—31no—
405C2×D4—51no—
406C2×D4—62no—
407C2×D4—102no—
408C2×D4#120111yescharacteristic
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—101no—
412C22×C4C2×D3×D512yes—
413SD16—602no—
414D8—302no—
415C2×C8—151no—
416C2×C8—151no—
417C2×C8—152no—
418C2×C8—602no—
419C4⋊C4—31no—
420C4⋊C4—52no—
421C4⋊C4—51no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—101no—
426C4⋊C4#120111yescharacteristic
427C22⋊C4—62no—
428C22⋊C4—61no—
429C22⋊C4—102no—
430C22⋊C4—102no—
431C22⋊C4—101no—
432C22⋊C4#120112yes—
433C42—102no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—201no—
440C22×C3—201no—
441C22×C3—201no—
442C22×C3C22×C22⋊D511yescharacteristic
443C2×D3—22no—
444C2×D3—41no—
445C2×D3—41no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—201no—
449C3×C4—202no—
450C3×C4#409612yes—
451C3×C4C2×#71111yescharacteristic
452C3×C4#414411yescharacteristic
453Dic3—42no—
454Dic3—41no—
455C2×C5—42no—
456C2×C5—122no—
457C2×C5C2×#95011yescharacteristic
458C2×C5#582911yescharacteristic
459C2×C5#583611yescharacteristic
460D5—81no—
461C23—61no—
462C23—101no—
463C23C2×#120112yes—
464Q8—62no—
465Q8—62no—
466Q8—401no—
467D4—22no—
468D4—32no—
469D4—61no—
470D4—101no—
471D4—101no—
472D4—124no—
473D4—402no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—61no—
477C2×C4—62no—
478C2×C4—101no—
479C2×C4—102no—
480C2×C4—124no—
481C2×C4—122no—
482C2×C4—202no—
483C2×C4C2×#120112yes—
484C2×C4C2×#120111yescharacteristic
485C2×C4C2×#120111yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—301no—
488C8—301no—
489C8—302no—
490C2×C3—42no—
491C2×C3—401no—
492C2×C3#6614011yescharacteristic
493C2×C3C2×#409611yescharacteristic
494C2×C3#6617111yescharacteristic
495D3—42no—
496C5#8365211yescharacteristic, complemented, Sylow
497C22—22no—
498C22—22no—
499C22—22no—
500C22—62no—
501C22—121no—
502C22—121no—
503C22—201no—
504C22—201no—
505C22—201no—
506C22C22×#120111yescharacteristic
507C4—21no—
508C4—22no—
509C4—121no—
510C4—122no—
511C4—202no—
512C4—201no—
513C4#9129712yes—
514C4D3×#71111yescharacteristic
515C4#9202711yescharacteristic
516C3#1061524211yescharacteristic, complemented, Sylow
517C2—42no—
518C2—122no—
519C2—401no—
520C2C2×#9129711yescharacteristic
521C2#1175496811yescharacteristic
522C2#1175530311yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC4×D4D3×D5
hypocenterC3×C5#3303
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C5×#5551C2semidirect
2. C3×#65897C2semidirect
3. D4×C3⋊#168C2semidirect
4. #11748545C2semidirect
5. #11748517C2semidirect
6. #11748404C2semidirect
7. #11748382C2semidirect
8. #11747059C2semidirect
9. C4×#6652C2semidirect
10. #11746923C2semidirect
11. #11746882C2semidirect
44. #65897D3semidirect
90. #5551D5semidirect
435. C3×C5#3303semidirect
496. C5#83652semidirect
516. C3#10615242semidirect

Maximal subgroups

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

subgroupindex
C5×#55512
C3×#658972
D4×C3⋊#1682
#117485452
#117485172
#117484042
#117483822
#117470592
C4×#66522
#117469232
#117468822
#106152423
#836525