Finite Group Databasebeta

← #14309052

Subgroups — #14309052

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754617C211yesmaximal, characteristic, complemented
2C2×#91121C211yesmaximal, characteristic
3C5×#5240C211yesmaximal, characteristic
4C3×#65525C211yesmaximal, characteristic
5#11745988C211yesmaximal, characteristic, complemented
6#11745975C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10600831—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C4×Dic15C2211yescharacteristic
11C2×#6710C2211yescharacteristic
12C5×#926C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91576—22no—
17#91470—21no—
18C4×#1204—22no—
19#91384—21no—
20#91121—21no—
21#91121—21no—
22C5×#834—22no—
23C3×#3969—22no—
24#70394—51nomaximal, Hall
25#65851—31no—
26C5×#410—31no—
27#65525D311yescharacteristic
28C2×#3969—31no—
29#64753—31no—
30#64752—31no—
31#64751—31no—
32C3×C5×#41—21no—
33C3×C5×#41—21no—
34C3×C5×#41—21no—
35C3×C5×#41—21no—
36C2×C3×C5×Q8Q811yescharacteristic, complemented
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4Q811yescharacteristic, complemented
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5C2311yescharacteristic
41C22×Dic15—21no—
42C2×#1204—22no—
43C4×Dic15—21no—
44C4×Dic15D411yescharacteristic
45C4×Dic15D411yescharacteristic
46#6710—21no—
47#6710—21no—
48C2×C3⋊#168—21no—
49C5×#217—21no—
50C5×#217D412yes—
51C5×C4⋊Dic3—21no—
52C5×C4⋊Dic3—21no—
53C4×C5×Dic3—21no—
54C4×C5×Dic3—21no—
55C3×C22⋊Dic5—21no—
56C3×C4⋊Dic5—21no—
57C3×C4×Dic5D412yes—
58C3×#681—21no—
59C3×#681—21no—
60C2×C3×#168—21no—
61Dic3⋊Dic5—22no—
62#6669—22no—
63#6668—22no—
64#5517—51no—
65C3×#421—51no—
66#5240D511yescharacteristic
67C2×#837—51no—
68#4556—51no—
69#4555—51no—
70#4531—51no—
71C2×C5×#41C2×D311yescharacteristic
72C5×#117—31no—
73C5×#117—31no—
74#4072C2×D311yescharacteristic
75C2×C4×Dic5—31no—
76C2×#681—31no—
77C2×#681C2×D311yescharacteristic
78#4036—61no—
79#4034—61no—
80#4024—62no—
81C4×Dic10—62no—
82C5×#99—62no—
83#3969—22no—
84#3969—61no—
85#3969—61no—
86#2031—151noSylow
87C23×C3×C5—21no—
88C23×C3×C5C2×D411yescharacteristic
89C3×C5×Q8—21no—
90C3×C5×Q8—21no—
91C3×C5×D4—21no—
92C3×C5×D4—21no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×Q811yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×Dic15—21no—
105C2×Dic15—21no—
106C2×Dic15—42no—
107C2×C5×Dic3—22no—
108C2×C5×Dic3—41no—
109C2×C3×Dic5—22no—
110#1204—44no—
111C3⋊#168—41no—
112C3×#168—41no—
113C2×C3×#41C2×D511yescharacteristic
114C2×C3×M16—51no—
115C3×#117—51no—
116#926C2×D511yescharacteristic
117#926C2×D511yescharacteristic
118C2×C4×Dic3—51no—
119C2×#208—51no—
120#890—101no—
121#888—101no—
122#878—102no—
123C4×Dic6—102no—
124C3×#104—102no—
125#837—101no—
126#837—101no—
127#834—22no—
128C5×#41—21no—
129C5×#41—21no—
130C5×#41—21no—
131C5×#41—21no—
132C2×C5×Q8Dic611yescharacteristic, complemented
133C2×C5×D4—21no—
134C2×C5×D4Dic611yescharacteristic, complemented
135C22×C4×C5—21no—
136C22×C4×C5C22×D311yescharacteristic
137C22×Dic5—61no—
138C2×Dic10—62no—
139C5×C4⋊C4—61no—
140C5×C4⋊C4—61no—
141C5×C22⋊C4—32no—
142C5×C22⋊C4—61no—
143C42×C5—61no—
144C42×C5—61no—
145C22⋊Dic5—21no—
146C4⋊Dic5—21no—
147C4⋊Dic5—62no—
148#683—62no—
149#683—62no—
150C4×Dic5—31no—
151C4×Dic5—31no—
152C4×Dic5—61no—
153C4×Dic5#6712yes—
154#681—21no—
155#681—21no—
156#681—61no—
157#681—61no—
158C2×#168—21no—
159C2×#168—61no—
160#533—151no—
161#421—51no—
162C2×#104—151no—
163#410—31no—
164#344—151no—
165#339—151no—
166H(3,ℤ/4)—151no—
167C22×C3×C5—41no—
168C22×C3×C5—41no—
169C22×C3×C5—81no—
170C22×C3×C5#12011yescharacteristic
171C22×C3×C5C22⋊Q811yescharacteristic
172C22×C3×C5#12411yescharacteristic
173C3×C4×C5—41no—
174C3×C4×C5C22⋊Q811yescharacteristic
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5#12411yescharacteristic
177C3×C4×C5#12411yescharacteristic
178Dic15—22no—
179Dic15—41no—
180C3×Dic5—42no—
181C5×Dic3—42no—
182C5×Dic3—81no—
183C3×#41—21no—
184C3×#41—21no—
185C3×#41—21no—
186C3×#41—21no—
187C2×C3×Q8Dic1011yescharacteristic, complemented
188C2×C3×D4—21no—
189C2×C3×D4Dic1011yescharacteristic, complemented
190C22×C3×C4—21no—
191C22×C3×C4C22×D511yescharacteristic
192C22×Dic3—101no—
193C2×Dic6—102no—
194C3×M16—101no—
195C3×M16—101no—
196C2×C3×C8—101no—
197C3×C4⋊C4—101no—
198C3×C22⋊C4—101no—
199C3×C42—52no—
200#217—21no—
201#217C22⋊D512yes—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—102no—
205Dic3⋊C4—102no—
206Dic3⋊C4—102no—
207C4×Dic3—21no—
208C4×Dic3—21no—
209C4×Dic3—51no—
210C4×Dic3—51no—
211C4×Dic3—101no—
212#208—101no—
213#208—101no—
214C2×C3⋊C8—101no—
215C23×C5—21no—
216C23×C5D3×D411yescharacteristic
217C5×Q8—21no—
218C5×Q8—21no—
219C5×D4—21no—
220C5×D4—21no—
221C5×D4—41no—
222C5×D4—41no—
223C2×C4×C5—21no—
224C2×C4×C5—41no—
225C2×C4×C5—41no—
226C2×C4×C5—62no—
227C2×C4×C5—121no—
228C2×C4×C5C2×Dic611yescharacteristic
229C2×C4×C5#23511yescharacteristic
230C2×C4×C5D3×D411yescharacteristic
231C2×C4×C5#23711yescharacteristic
232C2×C4×C5#23711yescharacteristic
233C2×C4×C5C2×#6711yescharacteristic
234C2×Dic5—22no—
235C2×Dic5—61no—
236C2×Dic5—61no—
237C2×Dic5—122no—
238Dic10—124no—
239#168—41no—
240#168—121no—
241C2×#41D3×D511yescharacteristic
242C2×M16—51no—
243C2×M16—151no—
244C4⋊Q8—301no—
245#125—301no—
246C22⋊Q8—302no—
247C4×Q8—302no—
248#117—31no—
249#117—31no—
250#117—51no—
251C2×C42—151no—
252#104—102no—
253#104—301no—
254#104—301no—
255#99—62no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5—81no—
260C2×C3×C5#39311yescharacteristic
261C23×C3—21no—
262C23×C3C2×C22⋊D511yescharacteristic
263C3×Q8—21no—
264C3×Q8—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—41no—
268C3×D4—41no—
269C2×C3×C4—21no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—102no—
273C2×C3×C4C2×Dic1011yescharacteristic
274C2×C3×C4#70911yescharacteristic
275C2×C3×C4D4×D511yescharacteristic
276C2×C3×C4D4×D511yescharacteristic
277C2×C3×C4#71111yescharacteristic
278C2×C3×C4#71111yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—41no—
281C2×Dic3—101no—
282C2×Dic3—101no—
283C2×Dic3—202no—
284Dic6—204no—
285C3×C8—201no—
286C3⋊C8—201no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—81no—
290C22×C5#87811yescharacteristic
291C22×C5#88511yescharacteristic
292C22×C5#93711yescharacteristic
293C4×C5—41no—
294C4×C5—122no—
295C4×C5—241no—
296C4×C5#87811yescharacteristic
297C4×C5#88511yescharacteristic
298C4×C5#92411yescharacteristic
299C4×C5#93611yescharacteristic
300Dic5—42no—
301Dic5—62no—
302Dic5—121no—
303#41—21no—
304#41—21no—
305#41—21no—
306#41—21no—
307C2×Q8—302no—
308C2×Q8#120411yescharacteristic, complemented
309C2×D4—21no—
310C2×D4#120411yescharacteristic, complemented
311C22×C4—21no—
312C22×C4—301no—
313C22×C4C2×D3×D511yescharacteristic
314M16—101no—
315M16—101no—
316M16—301no—
317M16—301no—
318C2×C8—101no—
319C2×C8—301no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—101no—
323C4⋊C4—302no—
324C4⋊C4—302no—
325C4⋊C4—302no—
326C22⋊C4—32no—
327C22⋊C4—61no—
328C22⋊C4—101no—
329C42—52no—
330C42—61no—
331C42—61no—
332C42—151no—
333C42—151no—
334C42—301no—
335C3×C5#203111yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—81no—
339C22×C3#407011yescharacteristic
340C22×C3#408211yescharacteristic
341C22×C3#408311yescharacteristic
342C3×C4—41no—
343C3×C4—202no—
344C3×C4#402411yescharacteristic
345C3×C4#402411yescharacteristic
346C3×C4#403111yescharacteristic
347C3×C4#403111yescharacteristic
348Dic3—42no—
349Dic3—81no—
350Dic3—102no—
351Dic3—201no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5—81no—
356C2×C5#494711yescharacteristic
357C23—21no—
358C23D3×C22⋊D511yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—604no—
362D4—21no—
363D4—21no—
364D4—41no—
365D4—41no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—62no—
370C2×C4—102no—
371C2×C4—121no—
372C2×C4—301no—
373C2×C4—301no—
374C2×C4—602no—
375C2×C4#677911yescharacteristic
376C2×C4#678011yescharacteristic
377C2×C4#678211yescharacteristic
378C2×C4C2×#120411yescharacteristic
379C2×C4D5×#6711yescharacteristic
380C2×C4#679011yescharacteristic
381C8—201no—
382C8—601no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—81no—
387C2×C3#6523611yescharacteristic
388C5#7039411yescharacteristic, complemented, Sylow
389C22—41no—
390C22—41no—
391C22—81no—
392C22#9152511yescharacteristic
393C22#9157511yescharacteristic
394C22#9157711yescharacteristic
395C4—41no—
396C4—122no—
397C4—202no—
398C4—241no—
399C4—302no—
400C4—601no—
401C4#9152611yescharacteristic
402C4#9153411yescharacteristic
403C4#9157411yescharacteristic
404C4#9157611yescharacteristic
405C3#1060083111yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—81no—
410C2#1174952411yescharacteristic
subgroupG/H
centerC4#91534
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
Fitting, prime radicalC2×C3×C5×#41C22
socleC2×C3×C5#393
hypercenterC2×#41D3×D5
hypocenterC3×C5#2031
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #11754617C2semidirect
5. #11745988C2semidirect
6. #11745975C2semidirect
7. #11745969C2semidirect
36. C2×C3×C5×Q8Q8semidirect
38. C2×C3×C5×D4Q8semidirect
132. C2×C5×Q8Dic6semidirect
134. C2×C5×D4Dic6semidirect
187. C2×C3×Q8Dic10semidirect
189. C2×C3×D4Dic10semidirect
308. C2×Q8#1204semidirect
310. C2×D4#1204semidirect
335. C3×C5#2031semidirect
388. C5#70394semidirect
405. C3#10600831semidirect

Maximal subgroups

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

subgroupindex
#117546172
C2×#911212
C5×#52402
C3×#655252
#117459882
#117459752
#117459692
#106008313
#703945