Finite Group Databasebeta

← #14309032

Subgroups — #14309032

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#11756276C211yesmaximal, characteristic, complemented
2C3×C5×#410C211yesmaximal, characteristic
3#11751337C211yesmaximal, characteristic
4C2×#91121C211yesmaximal, characteristic
5#11746354C211yesmaximal, characteristic, complemented
6#11746353C211yesmaximal, characteristic, complemented
7#11746352C211yesmaximal, characteristic, complemented
8#10600831—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×C4×Dic15C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15C2×#6710C2211yescharacteristic
16#91779—21no—
17#91777—21no—
18#91767—22no—
19C4×Dic30—22no—
20C3×C5×#99—22no—
21#91121—21no—
22#91121—21no—
23#91121—22no—
24#70038—51nomaximal, Hall
25#65851—31no—
26C5×#410D311yescharacteristic
27#65525—31no—
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×Dic30—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C22⋊C4—21no—
46C3×C5×C22⋊C4D412yes—
47C3×C42×C5—21no—
48C3×C42×C5—21no—
49#6719—21no—
50C4⋊Dic15—22no—
51C4⋊Dic15—21no—
52#6712—22no—
53#6712—22no—
54C4×Dic15—21no—
55C4×Dic15D412yes—
56C4×Dic15D411yescharacteristic
57C4×Dic15D411yescharacteristic
58#6710—21no—
59#6710—21no—
60#6710—21no—
61#6710—21no—
62C2×C3⋊#168—21no—
63C2×C3⋊#168—21no—
64#5517—51no—
65C3×#410D511yescharacteristic
66#5236—51no—
67C2×#837—51no—
68#4533—51no—
69#4532—51no—
70#4531—51no—
71C2×C5×#41C2×D311yescharacteristic
72C5×#117C2×D311yescharacteristic
73C5×#117C2×D311yescharacteristic
74#4072—31no—
75C2×C4×Dic5—31no—
76C2×#681—31no—
77C2×#681—31no—
78#4036—61no—
79#4034—61no—
80#4024—62no—
81C4×Dic10—62no—
82C5×#99—22no—
83#3969—61no—
84#3969—61no—
85#3969—62no—
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—22no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×Q811yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×Dic15—21no—
107C2×Dic15—21no—
108C2×Dic15—22no—
109C2×Dic15—42no—
110Dic30—44no—
111C3⋊#168—41no—
112C3⋊#168—41no—
113C2×C3×#41C2×D511yescharacteristic
114C3×#117C2×D511yescharacteristic
115C3×#117C2×D511yescharacteristic
116#926—51no—
117C2×C4×Dic3—51no—
118C2×#208—51no—
119C2×#208—51no—
120#890—101no—
121#888—101no—
122#878—102no—
123C4×Dic6—102no—
124C3×#99—22no—
125#837—101no—
126#837—101no—
127#837—102no—
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—21no—
140C5×C4⋊C4—21no—
141C5×C22⋊C4—21no—
142C5×C22⋊C4#6712yes—
143C42×C5—21no—
144C42×C5—21no—
145C22⋊Dic5—61no—
146C4⋊Dic5—62no—
147C4⋊Dic5—61no—
148#683—62no—
149#683—62no—
150C4×Dic5—32no—
151C4×Dic5—31no—
152C4×Dic5—31no—
153C4×Dic5—61no—
154#681—61no—
155#681—61no—
156#681—61no—
157#681—61no—
158C2×#168—61no—
159C2×#168—61no—
160#533—151no—
161#421—151no—
162C2×#104—151no—
163#410D1511yescharacteristic
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—42no—
174C3×C4×C5—41no—
175C3×C4×C5—81no—
176C3×C4×C5C22⋊Q811yescharacteristic
177C3×C4×C5C22⋊Q811yescharacteristic
178C3×C4×C5#12411yescharacteristic
179C3×C4×C5#12411yescharacteristic
180Dic15—22no—
181Dic15—42no—
182Dic15—41no—
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×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4C22⋊D512yes—
198C3×C42—21no—
199C3×C42—21no—
200#217—101no—
201C4⋊Dic3—102no—
202C4⋊Dic3—101no—
203Dic3⋊C4—102no—
204Dic3⋊C4—102no—
205C4×Dic3—52no—
206C4×Dic3—51no—
207C4×Dic3—51no—
208C4×Dic3—101no—
209#208—101no—
210#208—101no—
211#208—101no—
212#208—101no—
213C2×C3⋊C8—101no—
214C2×C3⋊C8—101no—
215C23×C5—21no—
216C23×C5C2×#6711yescharacteristic
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—22no—
225C2×C4×C5—41no—
226C2×C4×C5—41no—
227C2×C4×C5—41no—
228C2×C4×C5C2×Dic611yescharacteristic
229C2×C4×C5#23511yescharacteristic
230C2×C4×C5D3×D411yescharacteristic
231C2×C4×C5D3×D411yescharacteristic
232C2×C4×C5#23711yescharacteristic
233C2×C4×C5#23711yescharacteristic
234C2×Dic5—61no—
235C2×Dic5—61no—
236C2×Dic5—62no—
237C2×Dic5—122no—
238Dic10—124no—
239#168—121no—
240#168—121no—
241C2×#41C2×D1511yescharacteristic
242C2×M16—151no—
243C2×M16—151no—
244C4⋊Q8—301no—
245#125—301no—
246C22⋊Q8—302no—
247C4×Q8—302no—
248#117—151no—
249#117C2×D1511yescharacteristic
250#117C2×D1511yescharacteristic
251C2×C42—151no—
252#104—301no—
253#104—301no—
254#104—302no—
255#99—22no—
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—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4C2×Dic1011yescharacteristic
275C2×C3×C4#70911yescharacteristic
276C2×C3×C4D4×D511yescharacteristic
277C2×C3×C4D4×D511yescharacteristic
278C2×C3×C4#71111yescharacteristic
279C2×C3×C4#71111yescharacteristic
280C2×Dic3—101no—
281C2×Dic3—101no—
282C2×Dic3—102no—
283C2×Dic3—202no—
284Dic6—204no—
285C3⋊C8—201no—
286C3⋊C8—201no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—81no—
290C22×C5#92411yescharacteristic
291C22×C5#93611yescharacteristic
292C22×C5#93711yescharacteristic
293C4×C5—42no—
294C4×C5—41no—
295C4×C5—81no—
296C4×C5#87811yescharacteristic
297C4×C5#87811yescharacteristic
298C4×C5#88511yescharacteristic
299C4×C5#88511yescharacteristic
300Dic5—62no—
301Dic5—122no—
302Dic5—121no—
303#41—21no—
304#41—21no—
305#41—21no—
306#41—21no—
307C2×Q8—302no—
308C2×Q8Dic3011yescharacteristic, complemented
309C2×D4—21no—
310C2×D4Dic3011yescharacteristic, complemented
311C22×C4—21no—
312C22×C4—301no—
313C22×C4C22×D1511yescharacteristic
314M16—301no—
315M16—301no—
316M16—301no—
317M16—301no—
318C2×C8—301no—
319C2×C8—301no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—302no—
323C4⋊C4—302no—
324C4⋊C4—302no—
325C4⋊C4—301no—
326C22⋊C4—21no—
327C22⋊C4—301no—
328C22⋊C4#122012yes—
329C42—21no—
330C42—21no—
331C42—151no—
332C42—151no—
333C42—152no—
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—42no—
343C3×C4—41no—
344C3×C4—81no—
345C3×C4#402411yescharacteristic
346C3×C4#402411yescharacteristic
347C3×C4#403111yescharacteristic
348C3×C4#403111yescharacteristic
349Dic3—102no—
350Dic3—202no—
351Dic3—201no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5—81no—
356C2×C5#494711yescharacteristic
357C23—21no—
358C23C2×#122011yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—604no—
362D4—21no—
363D4—21no—
364D4—41no—
365D4—41no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—301no—
372C2×C4—301no—
373C2×C4—302no—
374C2×C4—602no—
375C2×C4C2×Dic3011yescharacteristic
376C2×C4#681711yescharacteristic
377C2×C4D4×D1511yescharacteristic
378C2×C4D4×D1511yescharacteristic
379C2×C4#681911yescharacteristic
380C2×C4#681911yescharacteristic
381C8—601no—
382C8—601no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—81no—
387C2×C3#6523611yescharacteristic
388C5#7003811yescharacteristic, complemented, Sylow
389C22—41no—
390C22—41no—
391C22—81no—
392C22#9181311yescharacteristic
393C22#9182511yescharacteristic
394C22#9182611yescharacteristic
395C4—42no—
396C4—41no—
397C4—81no—
398C4—302no—
399C4—601no—
400C4—602no—
401C4#9176711yescharacteristic
402C4#9176711yescharacteristic
403C4#9177411yescharacteristic
404C4#9177411yescharacteristic
405C3#1060083111yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—81no—
410C2#1175104811yescharacteristic
subgroupG/H
centerC4#91774
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC3×C5×#410C2
socleC2×C3×C5#393
prime radicalC2×C3×C5×#41C22
hypercenter#410D15
hypocenterC3×C5#2031
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #11756276C2semidirect
5. #11746354C2semidirect
6. #11746353C2semidirect
7. #11746352C2semidirect
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×Q8Dic30semidirect
310. C2×D4Dic30semidirect
335. C3×C5#2031semidirect
388. C5#70038semidirect
405. C3#10600831semidirect

Maximal subgroups

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

subgroupindex
#117562762
C3×C5×#4102
#117513372
C2×#911212
#117463542
#117463532
#117463522
#106008313
#700385