Finite Group Databasebeta

← #14460233

Subgroups — #14460233

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
1C3×#66060C211yesmaximal, characteristic, complemented
2#11752718C211yesmaximal, characteristic, complemented
3#11751180C211yesmaximal, characteristic, complemented
4C5×#5083C211yesmaximal, characteristic, complemented
5#11749365C212yesmaximal, complemented
6#11749270C212yesmaximal, complemented
7#11749210C212yesmaximal, complemented
8#11749207C212yesmaximal, complemented
9C4×#6652C211yesmaximal, characteristic, complemented
10#11746996C211yesmaximal, characteristic, complemented
11#11746881C211yesmaximal, characteristic, complemented
12#10612182—31nomaximal, Hall
13C2×C3×#713C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15#91758C2211yescharacteristic
16C2×C5×#215C2212yes—
17C4×C5×D12C2211yescharacteristic
18C3×#4092C2212yes—
19C3×#4041—21no—
20C3×#4040—22no—
21C3×D5×C4⋊C4—21no—
22C3×C4×D20C2211yescharacteristic
23C3×C4×D20C2211yescharacteristic
24#91462—22no—
25#91456—21no—
26#91449C2212yes—
27#91358C2212yes—
28#91338—21no—
29C2×#6652C2211yescharacteristic
30C2×D5×C3⋊C8C2212yes—
31#91119C2212yes—
32#91119C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C5×#835C2212yes—
35C5×C4⋊(C3⋊C8)C2211yescharacteristic
36#90989C2211yescharacteristic
37#90987C2212yes—
38#90966C2211yescharacteristic
39#90965C2212yes—
40#90964C2211yescharacteristic
41#90955C2212yes—
42#83511—51nomaximal, Hall
43#66060D311yescharacteristic, complemented
44#65897—31no—
45C5×#461—31no—
46#65461—32no—
47#65460—32no—
48#65370—31no—
49C4×#686—31no—
50#65131—31no—
51#65123—31no—
52#65104—32no—
53#65093—32no—
54C2×C3×C5×Q8C2312yes—
55C2×Dic30—21no—
56C2×C5×D12C2311yescharacteristic
57C2×C4×C5×D3—21no—
58C3×#713—24no—
59C3×#713—42no—
60C2×C3×D20—22no—
61C2×C3×D20D411yescharacteristic
62C2×C3×D20C2311yescharacteristic
63C2×C3×C4×D5—21no—
64C2×C3×C4×D5—22no—
65C2×C3×C4×D5C2312yes—
66C2×#1201—22no—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70C4⋊Dic15—21no—
71C4⋊Dic15C2312yes—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×#215—42no—
75C5×(C2×D3)⋊C4—21no—
76C5×C4⋊Dic3C2311yescharacteristic
77C2×C5×C3⋊C8C2312yes—
78C3×#685—22no—
79C3×#685—21no—
80C3×#685D412yes—
81C3×C4⋊Dic5D411yescharacteristic
82C3×C4⋊Dic5C2311yescharacteristic
83C3×#683—22no—
84(C2×D3)⋊Dic5—22no—
85#6665—22no—
86Dic3×Dic5—22no—
87#6652—22no—
88D5×C3⋊C8—42no—
89C3×#547—51no—
90#5563—51no—
91#5238—52no—
92#5233—52no—
93#5083D511yescharacteristic, complemented
94#5079—51no—
95#5048—52no—
96#5045—52no—
97C4×#213—51no—
98D4×C3⋊C8—51no—
99#5008—51no—
100C2×#713C2×D312yes—
101C2×C5×SD16—32no—
102C5×C4⋊Q8C2×D311yescharacteristic
103C4×C5×D4—31no—
104#4092C2×D312yes—
105#4084—32no—
106C2×#686—31no—
107#4076—61no—
108#4072—61no—
109C2×C8×D5—32no—
110#4041—21no—
111#4040—22no—
112#4038—32no—
113D5×C4⋊C4—21no—
114#4031—62no—
115C4×D20C2×D311yescharacteristic
116C4×D20C2×D311yescharacteristic
117#4015—31no—
118C5×C4⋊C8—31no—
119C5×#103—32no—
120#3967—32no—
121#3967—32no—
122#3964—31no—
123#3953—32no—
124#3952—32no—
125#3949—32no—
126#3941—31no—
127#3939—31no—
128C4×#168—31no—
129#3303—151noSylow
130C22×C5×D3—21no—
131C22×C3×D5—21no—
132C22×C3×D5C2×D412yes—
133C3×C5×Q8—22no—
134C3×C5×Q8—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139C2×Dic15—22no—
140Dic30—81no—
141C2×C5×Dic3—21no—
142C5×D12—21no—
143C5×D12—21no—
144C4×C5×D3—42no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5C2×D412yes—
147C3×D20—22no—
148C3×D20—21no—
149C3×D20—44no—
150C3×D20#4112yes—
151C3×C4×D5—22no—
152C3×C4×D5—22no—
153C3×C4×D5—44no—
154C3×C4×D5—42no—
155#1201—82no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C5×C3⋊C8—22no—
159C2×C3×#41—52no—
160C3×C4⋊Q8C2×D511yescharacteristic
161C3×#123—102no—
162C3×C22⋊Q8—52no—
163C3×#121—101no—
164C3×C4×D4—51no—
165C3×C4×D4—51no—
166C2×C3×C4⋊C4—101no—
167C2×#215C2×D512yes—
168#938—101no—
169C2×#213—51no—
170#930—52no—
171#926—52no—
172C22×C3⋊C8—52no—
173#892—101no—
174#885—102no—
175C4×D12C2×D511yescharacteristic
176#869—51no—
177#835—52no—
178#835—52no—
179#835C2×D512yes—
180#832—51no—
181#832—52no—
182#830—52no—
183#809—51no—
184#808—52no—
185#807—51no—
186C4⋊(C3⋊C8)C2×D511yescharacteristic
187C4×C3⋊C8—51no—
188C2×C5×Q8C22×D312yes—
189C2×C5×D4—31no—
190C22×C4×C5—61no—
191C2×C22⋊D5—62no—
192#713—24no—
193#713—42no—
194C2×D20—22no—
195C2×D20#6711yescharacteristic
196C2×D20C22×D311yescharacteristic
197C2×C4×D5—21no—
198C2×C4×D5—22no—
199C2×C4×D5C22×D312yes—
200C2×Dic10—61no—
201C5×SD16—122no—
202C2×C5×C8—32no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—31no—
206C5×C22⋊C4—61no—
207C42×C5C22×D311yescharacteristic
208C22⋊Dic5—62no—
209#686—62no—
210#685—22no—
211#685—21no—
212#685—62no—
213#685#6712yes—
214C4⋊Dic5—32no—
215C4⋊Dic5—61no—
216C4⋊Dic5#6711yescharacteristic
217C4⋊Dic5C22×D311yescharacteristic
218#683—22no—
219C4×Dic5—62no—
220C2×#168—31no—
221C2×#168—31no—
222C8×D5—122no—
223#555—151no—
224#547—51no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461—31no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5C22×D411yescharacteristic
235C2×C5×D3—41no—
236C2×C5×D3—41no—
237C2×C5×D3—41no—
238C2×C3×D5—22no—
239C2×C3×D5—22no—
240C2×C3×D5—22no—
241C2×C3×D5—22no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—21no—
245C3×C4×C5—42no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—42no—
250C3×Dic5—22no—
251C3×Dic5—41no—
252C5×Dic3—41no—
253C3×#41—104no—
254C3×#41—202no—
255C2×C3×Q8C22×D512yes—
256C2×C3×D4—51no—
257C2×C3×D4—51no—
258C2×C3×D4—102no—
259C22×C3×C4—52no—
260C22×C3×C4—102no—
261C22×C3×C4—101no—
262C2×#67—102no—
263C2×D12C22×D511yescharacteristic
264C2×C4×D3—21no—
265C2×Dic6—101no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—51no—
269C3×C4⋊C4—51no—
270C3×C4⋊C4—102no—
271C3×C22⋊C4—52no—
272C3×C22⋊C4—102no—
273C3×C22⋊C4—101no—
274C3×C42C22×D511yescharacteristic
275#217—102no—
276#215—42no—
277#213—102no—
278(C2×D3)⋊C4—21no—
279(C2×D3)⋊C4—102no—
280C4⋊Dic3—52no—
281C4⋊Dic3—101no—
282C4⋊Dic3C22×D511yescharacteristic
283C4×Dic3—102no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—202no—
287C2×C3⋊C8C22×D512yes—
288C23×C5—61no—
289C22×D5—21no—
290C22×D5C2×#6712yes—
291C5×Q8—22no—
292C5×Q8—22no—
293C5×D4—61no—
294C5×D4—61no—
295C2×C4×C5—22no—
296C2×C4×C5—61no—
297C2×C4×C5—122no—
298C2×C4×C5C2×#6711yescharacteristic
299C2×C4×C5C2×#6711yescharacteristic
300C2×C4×C5C23×D311yescharacteristic
301C22⋊D5—242no—
302C2×Dic5—21no—
303C2×Dic5—62no—
304C2×Dic5C2×#6712yes—
305D20—22no—
306D20—21no—
307D20—44no—
308D20#23712yes—
309C4×D5—22no—
310C4×D5—22no—
311C4×D5—44no—
312C4×D5—42no—
313Dic10—241no—
314C5×C8—62no—
315#168—61no—
316#168—61no—
317C2×#41—52no—
318C2×SD16—32no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—151no—
322C4⋊Q8D3×D511yescharacteristic
323#124—302no—
324#123—102no—
325C22⋊Q8—52no—
326#121—101no—
327#121—152no—
328#121—301no—
329C4×D4—31no—
330C4×D4—51no—
331C4×D4—51no—
332#117—152no—
333#117—301no—
334C2×C4⋊C4—101no—
335#107—151no—
336#106—152no—
337C4⋊C8—31no—
338#103—32no—
339#103—152no—
340#103—152no—
341#102—151no—
342#102—151no—
343#102—152no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5#54611yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5#57711yescharacteristic
349C3×D5—42no—
350C3×D5—42no—
351C5×D3—81no—
352C23×C3—52no—
353C23×C3—101no—
354C22×D3—21no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×D4—52no—
358C3×D4—101no—
359C3×D4—102no—
360C3×D4—204no—
361C2×C3×C4—22no—
362C2×C3×C4—52no—
363C2×C3×C4—102no—
364C2×C3×C4—102no—
365C2×C3×C4—101no—
366C2×C3×C4—204no—
367C2×C3×C4—202no—
368C2×C3×C4D4×D511yescharacteristic
369C2×C3×C4D4×D511yescharacteristic
370C2×C3×C4C23×D511yescharacteristic
371#67—402no—
372C2×Dic3—21no—
373C2×Dic3—102no—
374D12—21no—
375D12—21no—
376C4×D3—42no—
377Dic6—401no—
378C3⋊C8—22no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—121no—
382C22×C5—121no—
383C22×C5—121no—
384C22×C5C22×#6711yescharacteristic
385C2×D5—22no—
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—22no—
389C2×D5—41no—
390C2×D5—41no—
391C4×C5—21no—
392C4×C5—42no—
393C4×C5—121no—
394C4×C5#95012yes—
395C4×C5C2×#23711yescharacteristic
396C4×C5#100411yescharacteristic
397Dic5—22no—
398Dic5—41no—
399Dic5—122no—
400#41—104no—
401#41—202no—
402C2×Q8—301no—
403C2×Q8C2×D3×D512yes—
404C2×D4—31no—
405C2×D4—51no—
406C2×D4—51no—
407C2×D4—102no—
408C2×D4—302no—
409C22×C4—52no—
410C22×C4—61no—
411C22×C4—102no—
412C22×C4—101no—
413SD16—122no—
414D8—302no—
415C2×C8—32no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—31no—
422C4⋊C4—51no—
423C4⋊C4—51no—
424C4⋊C4—102no—
425C4⋊C4—152no—
426C4⋊C4—301no—
427C22⋊C4—52no—
428C22⋊C4—61no—
429C22⋊C4—102no—
430C22⋊C4—101no—
431C22⋊C4—302no—
432C22⋊C4—302no—
433C42—302no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—102no—
437C22×C3—102no—
438C22×C3—102no—
439C22×C3—102no—
440C22×C3—201no—
441C22×C3—201no—
442C22×C3C2×D4×D511yescharacteristic
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—41no—
446C3×C4—21no—
447C3×C4—42no—
448C3×C4—102no—
449C3×C4—201no—
450C3×C4#406212yes—
451C3×C4#414411yescharacteristic
452C3×C4D5×#4111yescharacteristic
453Dic3—41no—
454Dic3—202no—
455C2×C5—241no—
456C2×C5#580611yescharacteristic
457C2×C5C2×#95011yescharacteristic
458C2×C5#583711yescharacteristic
459D5—42no—
460D5—42no—
461C23—52no—
462C23—61no—
463C23—101no—
464Q8—22no—
465Q8—22no—
466Q8—1201no—
467D4—52no—
468D4—61no—
469D4—61no—
470D4—101no—
471D4—102no—
472D4—204no—
473D4—1202no—
474C2×C4—22no—
475C2×C4—52no—
476C2×C4—61no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—101no—
480C2×C4—122no—
481C2×C4—204no—
482C2×C4—202no—
483C2×C4—302no—
484C2×C4D5×#6711yescharacteristic
485C2×C4D5×#6711yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—62no—
488C8—301no—
489C8—301no—
490C2×C3—202no—
491C2×C3—202no—
492C2×C3#6594611yescharacteristic
493C2×C3C2×#406211yescharacteristic
494C2×C3#6611211yescharacteristic
495D3—81no—
496C5#8351111yescharacteristic, complemented, Sylow
497C22—102no—
498C22—102no—
499C22—102no—
500C22—102no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—201no—
505C22—201no—
506C22C2×D5×#6711yescharacteristic
507C4—21no—
508C4—42no—
509C4—102no—
510C4—121no—
511C4—201no—
512C4—602no—
513C4#9151412yes—
514C4D5×#23711yescharacteristic
515C4#9202511yescharacteristic
516C3#1061218211yescharacteristic, complemented, Sylow
517C2—202no—
518C2—202no—
519C2—241no—
520C2#1175493011yescharacteristic
521C2#1175508611yescharacteristic
522C2C2×#9151411yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3303
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. C3×#66060C2semidirect
2. #11752718C2semidirect
3. #11751180C2semidirect
4. C5×#5083C2semidirect
5. #11749365C2semidirect
6. #11749270C2semidirect
7. #11749210C2semidirect
8. #11749207C2semidirect
9. C4×#6652C2semidirect
10. #11746996C2semidirect
11. #11746881C2semidirect
43. #66060D3semidirect
93. #5083D5semidirect
435. C3×C5#3303semidirect
496. C5#83511semidirect
516. C3#10612182semidirect

Maximal subgroups

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

subgroupindex
C3×#660602
#117527182
#117511802
C5×#50832
#117493652
#117492702
#117492102
#117492072
C4×#66522
#117469962
#117468812
#106121823
#835115