Finite Group Databasebeta

← #14309035

Subgroups — #14309035

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
2#11751240C211yesmaximal, characteristic
3C2×#91109C211yesmaximal, characteristic
4#11751107C211yesmaximal, characteristic
5C3×C5×#344C211yesmaximal, characteristic, complemented
6#11746378C211yesmaximal, characteristic, complemented
7#11746376C211yesmaximal, characteristic, complemented
8#10599835—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13#91815C2211yescharacteristic
14#91815C2211yescharacteristic
15C2×C4×Dic15C2211yescharacteristic
16#91779—21no—
17#91777—21no—
18#91767—22no—
19C4×Dic30—22no—
20#91109—21no—
21#91109—21no—
22#91109—22no—
23#91090—22no—
24#69042—51nomaximal, Hall
25#65851—31no—
26#65428—31no—
27C2×#3957—31no—
28#65295—31no—
29C5×#344D311yescharacteristic, complemented
30#64777—31no—
31#64775—31no—
32C22×C3×C4×C5C2311yescharacteristic
33C2×#1220—21no—
34C22×Dic15—21no—
35#6817—21no—
36#6817—21no—
37#6817—21no—
38#6817—21no—
39C2×D60Q811yescharacteristic, complemented
40C2×C4×D15—21no—
41C2×Dic30—22no—
42C2×Dic30Q811yescharacteristic, complemented
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C3×C5×M16—21no—
46C3×C5×M16—21no—
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C22⋊C4D412yes—
51C3×C42×C5—21no—
52#6719—21no—
53#6719—21no—
54C4⋊Dic15—22no—
55C4⋊Dic15—21no—
56C4⋊Dic15—21no—
57#6712—22no—
58#6712—22no—
59C4×Dic15—21no—
60C4×Dic15—21no—
61C4×Dic15D412yes—
62C4×Dic15D411yescharacteristic
63C4×Dic15D411yescharacteristic
64#5517—51no—
65#5139—51no—
66C2×#825—51no—
67#5006—51no—
68C3×#344D511yescharacteristic, complemented
69#4557—51no—
70#4555—51no—
71C2×#709—31no—
72C2×C5×M16C2×D311yescharacteristic
73C2×C5×M16C2×D311yescharacteristic
74C5×#117C2×D311yescharacteristic
75#4072—31no—
76#4072—31no—
77C2×C4×Dic5—31no—
78#4036—61no—
79#4034—61no—
80#4024—62no—
81C4×Dic10—62no—
82#3957—61no—
83#3957—61no—
84#3957—62no—
85#3938—62no—
86#2031—151noSylow
87C23×C3×C5C2×D411yescharacteristic
88C22×D15—21no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×Q811yescharacteristic
93C2×C3×C4×C5#4111yescharacteristic
94C2×C3×C4×C5#4111yescharacteristic
95C2×C3×C4×C5#4111yescharacteristic
96#1220—41no—
97#1220—41no—
98C2×Dic15—21no—
99C2×Dic15—21no—
100C2×Dic15—22no—
101C2×Dic15—21no—
102C2×Dic15—42no—
103C2×Dic15—41no—
104D60—21no—
105D60—21no—
106C4×D15—41no—
107C4×D15—41no—
108Dic30—21no—
109Dic30—21no—
110Dic30—44no—
111C3×C5×C8—41no—
112C3×C5×C8—41no—
113C2×#235—51no—
114C2×C3×M16C2×D511yescharacteristic
115C2×C3×M16C2×D511yescharacteristic
116C3×#117C2×D511yescharacteristic
117#926—51no—
118#926—51no—
119C2×C4×Dic3—51no—
120#890—101no—
121#888—101no—
122#878—102no—
123C4×Dic6—102no—
124#825—101no—
125#825—101no—
126#825—102no—
127#806—102no—
128C22×C4×C5C22×D311yescharacteristic
129C2×C22⋊D5—61no—
130C22×Dic5—61no—
131#709—61no—
132#709—61no—
133#709—61no—
134#709—61no—
135C2×D20—31no—
136C2×C4×D5—61no—
137C2×Dic10—31no—
138C2×Dic10—62no—
139C5×M16—21no—
140C5×M16—21no—
141C5×M16—21no—
142C5×M16—21no—
143C2×C5×C8—21no—
144C2×C5×C8—21no—
145C5×C4⋊C4—21no—
146C5×C22⋊C4D1212yes—
147C42×C5—21no—
148C22⋊Dic5—61no—
149C22⋊Dic5—61no—
150C4⋊Dic5—62no—
151C4⋊Dic5—61no—
152C4⋊Dic5—61no—
153#683—62no—
154#683—62no—
155C4×Dic5—32no—
156C4×Dic5—31no—
157C4×Dic5—31no—
158C4×Dic5—61no—
159C4×Dic5—61no—
160#533—151no—
161#421—151no—
162C2×#104—151no—
163#410—151no—
164#344D1511yescharacteristic, complemented
165#339—151no—
166H(3,ℤ/4)—151no—
167C22×C3×C5—41no—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q811yescharacteristic
170C22×C3×C5#12411yescharacteristic
171C2×D15—41no—
172C2×D15—81no—
173C3×C4×C5—42no—
174C3×C4×C5C22⋊Q811yescharacteristic
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5#12411yescharacteristic
177C3×C4×C5#12411yescharacteristic
178Dic15—22no—
179Dic15—42no—
180Dic15—41no—
181Dic15—41no—
182Dic15—81no—
183C22×C3×C4C22×D511yescharacteristic
184C2×#67—101no—
185C22×Dic3—101no—
186#235—101no—
187#235—101no—
188#235—101no—
189#235—101no—
190C2×D12—51no—
191C2×C4×D3—101no—
192C2×Dic6—51no—
193C2×Dic6—102no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C3×C4⋊C4—21no—
201C3×C22⋊C4D2012yes—
202C3×C42—21no—
203#217—101no—
204#217—101no—
205C4⋊Dic3—102no—
206C4⋊Dic3—101no—
207C4⋊Dic3—101no—
208Dic3⋊C4—102no—
209Dic3⋊C4—102no—
210C4×Dic3—52no—
211C4×Dic3—51no—
212C4×Dic3—51no—
213C4×Dic3—101no—
214C4×Dic3—101no—
215C23×C5C2×D1211yescharacteristic
216C22×D5—61no—
217C2×C4×C5—22no—
218C2×C4×C5#23511yescharacteristic
219C2×C4×C5#23511yescharacteristic
220C2×C4×C5D3×D411yescharacteristic
221C2×C4×C5D3×D411yescharacteristic
222C2×C4×C5#23711yescharacteristic
223C2×C4×C5D3×Q811yescharacteristic
224C22⋊D5—121no—
225C22⋊D5—121no—
226C2×Dic5—61no—
227C2×Dic5—61no—
228C2×Dic5—62no—
229C2×Dic5—61no—
230C2×Dic5—122no—
231C2×Dic5—121no—
232D20—61no—
233D20—61no—
234C4×D5—121no—
235C4×D5—121no—
236Dic10—61no—
237Dic10—61no—
238Dic10—124no—
239C5×C8—41no—
240C5×C8—41no—
241C2×#41—151no—
242C2×M16C2×D1511yescharacteristic
243C2×M16C2×D1511yescharacteristic
244C4⋊Q8—301no—
245#125—301no—
246C22⋊Q8—302no—
247C4×Q8—302no—
248#117—151no—
249#117—151no—
250#117C2×D1511yescharacteristic
251C2×C42—151no—
252#104—301no—
253#104—301no—
254#104—302no—
255#99—302no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5#39311yescharacteristic
260D15—81no—
261C23×C3C2×D2011yescharacteristic
262C22×D3—101no—
263C2×C3×C4—22no—
264C2×C3×C4#70911yescharacteristic
265C2×C3×C4#70911yescharacteristic
266C2×C3×C4D4×D511yescharacteristic
267C2×C3×C4D4×D511yescharacteristic
268C2×C3×C4#71111yescharacteristic
269C2×C3×C4Q8×D511yescharacteristic
270#67—201no—
271#67—201no—
272C2×Dic3—101no—
273C2×Dic3—101no—
274C2×Dic3—102no—
275C2×Dic3—101no—
276C2×Dic3—202no—
277C2×Dic3—201no—
278D12—101no—
279D12—101no—
280C4×D3—201no—
281C4×D3—201no—
282Dic6—101no—
283Dic6—101no—
284Dic6—204no—
285C3×C8—41no—
286C3×C8—41no—
287C22×C5—41no—
288C22×C5#87511yescharacteristic
289C22×C5#88211yescharacteristic
290C22×C5#89711yescharacteristic
291C2×D5—121no—
292C2×D5—241no—
293C4×C5—42no—
294C4×C5#88511yescharacteristic
295C4×C5#88511yescharacteristic
296C4×C5#89611yescharacteristic
297C4×C5#89611yescharacteristic
298Dic5—62no—
299Dic5—122no—
300Dic5—121no—
301Dic5—121no—
302Dic5—241no—
303#41—301no—
304#41—301no—
305#41—301no—
306#41—301no—
307C2×Q8—151no—
308C2×Q8—302no—
309C2×D4—151no—
310C2×D4—301no—
311C22×C4—301no—
312C22×C4—301no—
313C22×C4C22×D1511yescharacteristic
314M16—21no—
315M16—21no—
316M16—21no—
317M16—21no—
318C2×C8—21no—
319C2×C8—21no—
320C4⋊C4—21no—
321C4⋊C4—302no—
322C4⋊C4—302no—
323C4⋊C4—302no—
324C4⋊C4—301no—
325C4⋊C4—301no—
326C22⋊C4—301no—
327C22⋊C4—301no—
328C22⋊C4D6012yes—
329C42—21no—
330C42—151no—
331C42—151no—
332C42—152no—
333C42—301no—
334C42—301no—
335C3×C5#203111yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3#402111yescharacteristic
338C22×C3#402811yescharacteristic
339C22×C3#404311yescharacteristic
340C2×D3—201no—
341C2×D3—401no—
342C3×C4—42no—
343C3×C4#403111yescharacteristic
344C3×C4#403111yescharacteristic
345C3×C4#404211yescharacteristic
346C3×C4#404211yescharacteristic
347Dic3—102no—
348Dic3—202no—
349Dic3—201no—
350Dic3—201no—
351Dic3—401no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5#467311yescharacteristic
356D5—241no—
357C23—301no—
358C23C2×D6011yescharacteristic
359Q8—301no—
360Q8—301no—
361Q8—604no—
362D4—301no—
363D4—301no—
364D4—601no—
365D4—601no—
366C2×C4—22no—
367C2×C4—301no—
368C2×C4—301no—
369C2×C4—302no—
370C2×C4—301no—
371C2×C4—602no—
372C2×C4—601no—
373C2×C4—601no—
374C2×C4—601no—
375C2×C4#681711yescharacteristic
376C2×C4#681711yescharacteristic
377C2×C4D4×D1511yescharacteristic
378C2×C4D4×D1511yescharacteristic
379C2×C4#681911yescharacteristic
380C2×C4Q8×D1511yescharacteristic
381C8—41no—
382C8—41no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#6496211yescharacteristic
387D3—401no—
388C5#6904211yescharacteristic, complemented, Sylow
389C22—41no—
390C22—601no—
391C22—1201no—
392C22#9176411yescharacteristic
393C22#9177111yescharacteristic
394C22#9178611yescharacteristic
395C4—42no—
396C4—302no—
397C4—601no—
398C4—602no—
399C4—601no—
400C4—1201no—
401C4#9177411yescharacteristic
402C4#9177411yescharacteristic
403C4#9178511yescharacteristic
404C4#9178511yescharacteristic
405C3#1059983511yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—1201no—
410C2#1175077411yescharacteristic
subgroupG/H
centerC4#91774
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC3×C5×#344C2
socleC2×C3×C5#393
prime radicalC2×#6817C22
hypercenter#344D15
hypocenterC3×C5#2031
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #11756276C2semidirect
5. C3×C5×#344C2semidirect
6. #11746378C2semidirect
7. #11746376C2semidirect
29. C5×#344D3semidirect
39. C2×D60Q8semidirect
42. C2×Dic30Q8semidirect
68. C3×#344D5semidirect
164. #344D15semidirect
335. C3×C5#2031semidirect
388. C5#69042semidirect
405. C3#10599835semidirect

Maximal subgroups

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

subgroupindex
#117562762
#117512402
C2×#911092
#117511072
C3×C5×#3442
#117463782
#117463762
#105998353
#690425