Finite Group Databasebeta

← #14297839

Subgroups — #14297839

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×Dic5C211yesmaximal, characteristic
2C2×#91121C211yesmaximal, characteristic
3C2×C5×#837C211yesmaximal, characteristic
4C4×#6668C211yesmaximal, characteristic, complemented
5#11748431C211yesmaximal, characteristic, complemented
6#11745974C211yesmaximal, characteristic, complemented
7#11745973C211yesmaximal, characteristic, complemented
8#10601148—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C4×Dic15C2211yescharacteristic
11C2×#6710C2211yescharacteristic
12C2×C4×C5×Dic3C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C3×Q8×Dic5C411yescharacteristic, complemented
15C3×Q8×Dic5C411yescharacteristic, complemented
16C3×D4×Dic5—21no—
17C3×D4×Dic5—21no—
18C3×D4×Dic5C411yescharacteristic, complemented
19C3×D4×Dic5C411yescharacteristic, complemented
20C3×#4072—21no—
21C3×#4072C2211yescharacteristic
22C2×C3×C4×Dic5—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6668—21no—
25#91121—22no—
26C5×#837—22no—
27#90994—21no—
28#90982—21no—
29#90982—21no—
30#71646—51nomaximal, Hall
31#41×Dic5D311yescharacteristic
32C2×C5×#104—31no—
33C2×#3969—31no—
34#65407—31no—
35C4×#683—31no—
36#64775—31no—
37#64743—31no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41D412yes—
41C3×C5×#41Q812yes—
42C2×C3×C5×Q8C2×C411yescharacteristic
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4C2×C411yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15—21no—
48C22×C5×Dic3—21no—
49C22×C3×Dic5—21no—
50C22×C3×Dic5—21no—
51C22×C3×Dic5D411yescharacteristic
52C22×C3×Dic5D411yescharacteristic
53C4×Dic15—21no—
54C4×Dic15—21no—
55#6710—21no—
56#6710—21no—
57C2×C3⋊#168—21no—
58C4×C5×Dic3—21no—
59C4×C5×Dic3—21no—
60C5×#208—21no—
61C5×#208—21no—
62C2×C5×C3⋊C8—21no—
63C3×C22⋊Dic5—21no—
64C3×C22⋊Dic5—21no—
65C3×C22⋊Dic5—21no—
66C3×C4⋊Dic5—21no—
67C3×C4⋊Dic5—21no—
68C3×C4⋊Dic5C2×C411yescharacteristic
69C3×C4⋊Dic5C2×C411yescharacteristic
70C3×C4×Dic5—21no—
71C3×C4×Dic5—21no—
72C3×C4×Dic5—21no—
73C3×C4×Dic5C2×C411yescharacteristic
74C3×C4×Dic5C2×C411yescharacteristic
75C3×C4×Dic5D411yescharacteristic
76C3×C4×Dic5D411yescharacteristic
77#6668—24no—
78C3×C4×#41—51no—
79C2×#837—51no—
80C2×#837D511yescharacteristic
81C4×C4⋊Dic3—51no—
82#4924—51no—
83#4532—51no—
84#4532—51no—
85C2×C5×#41C2×D311yescharacteristic
86C2×C5×M16—31no—
87C2×C42×C5—31no—
88Q8×Dic5Dic311yescharacteristic, complemented
89Q8×Dic5Dic311yescharacteristic, complemented
90D4×Dic5—21no—
91D4×Dic5—21no—
92D4×Dic5Dic311yescharacteristic, complemented
93D4×Dic5Dic311yescharacteristic, complemented
94#4072—21no—
95#4072C2×D311yescharacteristic
96C2×#683—61no—
97C2×C4×Dic5—21no—
98C2×C4×Dic5—31no—
99C2×C4×Dic5C2×D311yescharacteristic
100C2×#681—31no—
101C5×#104—62no—
102#3969—62no—
103#3963—61no—
104#3946—61no—
105#3946—61no—
106#1806—151noSylow
107C23×C3×C5—21no—
108C23×C3×C5C2×D411yescharacteristic
109C3×C5×Q8—21no—
110C3×C5×Q8C4⋊C412yescomplemented
111C3×C5×D4—22no—
112C3×C5×D4—22no—
113C3×C5×D4—21no—
114C3×C5×D4C4⋊C412yescomplemented
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×Q811yescharacteristic
122C2×C3×C4×C5#4111yescharacteristic
123C2×C3×C4×C5#4111yescharacteristic
124C2×Dic15—22no—
125C2×Dic15—41no—
126C2×Dic15—41no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—41no—
129C2×C5×Dic3—41no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—21no—
132C2×C3×Dic5—21no—
133C2×C3×Dic5—41no—
134C2×C3×Dic5—42no—
135C2×C3×Dic5—42no—
136C2×C3×Dic5C22⋊C411yescharacteristic
137C2×C3×Dic5C22⋊C411yescharacteristic
138C2×C3×Dic5C22⋊C411yescharacteristic
139C2×C3×Dic5C22⋊C411yescharacteristic
140C3⋊#168—41no—
141C5×C3⋊C8—41no—
142C2×C3×#41C2×D511yescharacteristic
143C3×C4×Q8—51no—
144C3×C4×Q8—51no—
145C3×C4×D4—51no—
146C3×C4×D4—51no—
147C3×C4×D4—101no—
148C3×C4×D4—101no—
149C3×#117—51no—
150C3×#117—101no—
151C2×C3×C42—51no—
152C2×C3×C42—101no—
153C2×C4⋊Dic3—101no—
154C2×C4×Dic3—51no—
155C2×C4×Dic3C2×D511yescharacteristic
156C2×#208—51no—
157C2×#208C2×D511yescharacteristic
158#837—22no—
159#837—102no—
160C3⋊#95—101no—
161C4⋊(C3⋊C8)—101no—
162C4⋊(C3⋊C8)—101no—
163C5×#41—21no—
164C5×#41—21no—
165C5×#41Dic612yes—
166C5×#41D1212yes—
167C2×C5×Q8C2×Dic311yescharacteristic
168C2×C5×D4—21no—
169C2×C5×D4C2×Dic311yescharacteristic
170C22×C4×C5—21no—
171C22×C4×C5—61no—
172C22×C4×C5C22×D311yescharacteristic
173C22×Dic5—21no—
174C22×Dic5—21no—
175C22×Dic5—61no—
176C22×Dic5#6711yescharacteristic
177C22×Dic5#6711yescharacteristic
178C5×M16—61no—
179C5×M16—61no—
180C2×C5×C8—61no—
181C42×C5—61no—
182C42×C5—61no—
183C22⋊Dic5—21no—
184C22⋊Dic5—21no—
185C22⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5C2×Dic311yescharacteristic
189C4⋊Dic5C2×Dic311yescharacteristic
190#683—64no—
191C4×Dic5—21no—
192C4×Dic5—21no—
193C4×Dic5—21no—
194C4×Dic5—61no—
195C4×Dic5—61no—
196C4×Dic5C2×Dic311yescharacteristic
197C4×Dic5C2×Dic311yescharacteristic
198C4×Dic5#6711yescharacteristic
199C4×Dic5#6711yescharacteristic
200#681—61no—
201#681—61no—
202C2×#168—61no—
203C4×#41—51no—
204#423—151no—
205C2×#104—31no—
206C2×#104—151no—
207C4×C4⋊C4—151no—
208#339—151no—
209#339—151no—
210C22×C3×C5—21no—
211C22×C3×C5—21no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—41no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C22×C3×C5#12111yescharacteristic
217C22×C3×C5C22⋊Q811yescharacteristic
218C3×C4×C5—22no—
219C3×C4×C5C2×C4⋊C411yescharacteristic
220C3×C4×C5#11711yescharacteristic
221C3×C4×C5#12111yescharacteristic
222C3×C4×C5C22⋊Q811yescharacteristic
223Dic15—42no—
224C3×Dic5—21no—
225C3×Dic5—21no—
226C3×Dic5—21no—
227C3×Dic5—41no—
228C3×Dic5—41no—
229C3×Dic5#10412yescomplemented
230C5×Dic3—42no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41Dic1012yes—
234C3×#41C22⋊D512yes—
235C2×C3×Q8C4×D511yescharacteristic
236C2×C3×D4—21no—
237C2×C3×D4C4×D511yescharacteristic
238C22×C3×C4—21no—
239C22×C3×C4—51no—
240C22×C3×C4—51no—
241C22×C3×C4—101no—
242C22×C3×C4—101no—
243C22×C3×C4C22×D511yescharacteristic
244C22×Dic3—21no—
245C22×Dic3—101no—
246C3×C4⋊C4—51no—
247C3×C4⋊C4—51no—
248C3×C4⋊C4—101no—
249C3×C4⋊C4—101no—
250C3×C22⋊C4—101no—
251C3×C22⋊C4—101no—
252C3×C22⋊C4—101no—
253C3×C42—51no—
254C3×C42—51no—
255C3×C42—51no—
256C3×C42—51no—
257C3×C42—101no—
258C3×C42—101no—
259C3×C42—101no—
260C4⋊Dic3—104no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—101no—
264C4×Dic3—101no—
265#208—21no—
266#208—21no—
267#208—101no—
268#208—101no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—101no—
271C23×C5—21no—
272C23×C5C2×#6711yescharacteristic
273C5×Q8—21no—
274C5×Q8C4⋊Dic312yescomplemented
275C5×D4—22no—
276C5×D4—22no—
277C5×D4—21no—
278C5×D4C4⋊Dic312yescomplemented
279C2×C4×C5—21no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—62no—
283C2×C4×C5—121no—
284C2×C4×C5—121no—
285C2×C4×C5C2×Dic611yescharacteristic
286C2×C4×C5C2×D1211yescharacteristic
287C2×C4×C5#23511yescharacteristic
288C2×C4×C5#23511yescharacteristic
289C2×C4×C5C22×Dic311yescharacteristic
290C2×C4×C5C2×#6711yescharacteristic
291C2×Dic5—22no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—41no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—62no—
298C2×Dic5—121no—
299C2×Dic5—121no—
300C2×Dic5#21711yescharacteristic
301C2×Dic5#21711yescharacteristic
302C2×Dic5#21711yescharacteristic
303C2×Dic5#21711yescharacteristic
304C5×C8—121no—
305#168—121no—
306C2×#41D3×D511yescharacteristic
307C2×M16—31no—
308C2×M16—151no—
309C4×Q8—51no—
310C4×Q8—51no—
311C4×D4—51no—
312C4×D4—51no—
313C4×D4—101no—
314C4×D4—101no—
315#117—51no—
316#117—101no—
317C2×C4⋊C4—301no—
318C2×C42—31no—
319C2×C42—51no—
320C2×C42—101no—
321C2×C42—151no—
322C4⋊C8—301no—
323C4⋊C8—301no—
324#104—62no—
325#104—302no—
326#95—301no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—42no—
330C2×C3×C5#38011yescharacteristic
331C2×C3×C5C2×#10411yescharacteristic
332C2×C3×C5#42111yescharacteristic
333C23×C3—21no—
334C23×C3D4×D511yescharacteristic
335C3×Q8—21no—
336C3×Q8#68312yescomplemented
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—21no—
340C3×D4#68312yescomplemented
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—51no—
348C2×C3×C4—102no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—202no—
352C2×C3×C4—201no—
353C2×C3×C4—202no—
354C2×C3×C4C2×Dic1011yescharacteristic
355C2×C3×C4C2×C4×D511yescharacteristic
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4#71111yescharacteristic
358C2×C3×C4#71111yescharacteristic
359C2×C3×C4C2×C22⋊D511yescharacteristic
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—102no—
364C2×Dic3—201no—
365C2×Dic3—201no—
366C3⋊C8—41no—
367C3⋊C8—201no—
368C22×C5—21no—
369C22×C5—21no—
370C22×C5—22no—
371C22×C5—41no—
372C22×C5—41no—
373C22×C5#92411yescharacteristic
374C22×C5#93011yescharacteristic
375C22×C5C2×#21711yescharacteristic
376C4×C5—22no—
377C4×C5—122no—
378C4×C5#92411yescharacteristic
379C4×C5C2×C4⋊Dic311yescharacteristic
380C4×C5#92611yescharacteristic
381C4×C5#93011yescharacteristic
382Dic5—21no—
383Dic5—21no—
384Dic5—21no—
385Dic5—41no—
386Dic5—41no—
387Dic5—122no—
388Dic5#83712yescomplemented
389#41—21no—
390#41—21no—
391#41#120312yes—
392#41#120412yes—
393C2×Q8D5×Dic311yescharacteristic
394C2×D4—21no—
395C2×D4D5×Dic311yescharacteristic
396C22×C4—21no—
397C22×C4—51no—
398C22×C4—51no—
399C22×C4—61no—
400C22×C4—101no—
401C22×C4—101no—
402C22×C4—301no—
403C22×C4C2×D3×D511yescharacteristic
404M16—61no—
405M16—61no—
406M16—301no—
407M16—301no—
408C2×C8—61no—
409C2×C8—301no—
410C4⋊C4—51no—
411C4⋊C4—51no—
412C4⋊C4—101no—
413C4⋊C4—101no—
414C4⋊C4—304no—
415C22⋊C4—101no—
416C22⋊C4—101no—
417C22⋊C4—101no—
418C42—51no—
419C42—51no—
420C42—51no—
421C42—51no—
422C42—61no—
423C42—61no—
424C42—101no—
425C42—101no—
426C42—101no—
427C42—301no—
428C42—301no—
429C3×C5#180611yescharacteristic, complemented, Hall
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3#402411yescharacteristic
436C22×C3D5×C22⋊C411yescharacteristic
437C22×C3#408511yescharacteristic
438C3×C4—22no—
439C3×C4—52no—
440C3×C4—101no—
441C3×C4—101no—
442C3×C4—101no—
443C3×C4—201no—
444C3×C4—201no—
445C3×C4#402311yescharacteristic
446C3×C4#402411yescharacteristic
447C3×C4C2×#68311yescharacteristic
448C3×C4#408511yescharacteristic
449Dic3—42no—
450Dic3—202no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—42no—
454C2×C5#521211yescharacteristic
455C2×C5C2×#83711yescharacteristic
456C2×C5#523611yescharacteristic
457C23—21no—
458C23D5×#6711yescharacteristic
459Q8—21no—
460Q8#666812yescomplemented
461D4—22no—
462D4—22no—
463D4—21no—
464D4#666812yescomplemented
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—51no—
471C2×C4—51no—
472C2×C4—62no—
473C2×C4—102no—
474C2×C4—101no—
475C2×C4—101no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—202no—
479C2×C4—201no—
480C2×C4—202no—
481C2×C4—302no—
482C2×C4—601no—
483C2×C4—601no—
484C2×C4C2×D5×Dic311yescharacteristic
485C2×C4#678211yescharacteristic
486C2×C4#678211yescharacteristic
487C2×C4C2×#120311yescharacteristic
488C2×C4C2×#120411yescharacteristic
489C2×C4D5×#6711yescharacteristic
490C8—121no—
491C8—601no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—42no—
495C2×C3D5×#10411yescharacteristic
496C2×C3#6511011yescharacteristic
497C2×C3#6523111yescharacteristic
498C5#7164611yescharacteristic, complemented, Sylow
499C22—21no—
500C22—21no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22D5×#21711yescharacteristic
505C22#9156611yescharacteristic
506C22#9157411yescharacteristic
507C4—22no—
508C4—52no—
509C4—101no—
510C4—101no—
511C4—101no—
512C4—122no—
513C4—201no—
514C4—201no—
515C4—602no—
516C4#9152811yescharacteristic
517C4C2×#666811yescharacteristic
518C4#9156611yescharacteristic
519C4#9157411yescharacteristic
520C3#1060114811yescharacteristic, complemented, Sylow
521C2—21no—
522C2—21no—
523C2—42no—
524C2D5×#83711yescharacteristic
525C2#1174942711yescharacteristic
526C2#1174950611yescharacteristic
subgroupG/H
centerC2×C4D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
Fitting, prime radicalC2×C3×C5×#41C22
socleC22×C3×C5C2×C22⋊C4
cosocleC22×C3×C4×C5C23
hypercenterC2×#41D3×D5
hypocenterC3×C5#1806
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
4. C4×#6668C2semidirect
5. #11748431C2semidirect
6. #11745974C2semidirect
7. #11745973C2semidirect
14. C3×Q8×Dic5C4semidirect
15. C3×Q8×Dic5C4semidirect
18. C3×D4×Dic5C4semidirect
19. C3×D4×Dic5C4semidirect
88. Q8×Dic5Dic3semidirect
89. Q8×Dic5Dic3semidirect
92. D4×Dic5Dic3semidirect
93. D4×Dic5Dic3semidirect
110. C3×C5×Q8C4⋊C4semidirect
114. C3×C5×D4C4⋊C4semidirect
229. C3×Dic5#104semidirect
274. C5×Q8C4⋊Dic3semidirect
278. C5×D4C4⋊Dic3semidirect
336. C3×Q8#683semidirect
340. C3×D4#683semidirect
388. Dic5#837semidirect
429. C3×C5#1806semidirect
460. Q8#6668semidirect
464. D4#6668semidirect
498. C5#71646semidirect
520. C3#10601148semidirect

Maximal subgroups

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

subgroupindex
C3×#41×Dic52
C2×#911212
C2×C5×#8372
C4×#66682
#117484312
#117459742
#117459732
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