Finite Group Databasebeta

← #14309026

Subgroups — #14309026

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756229C211yesmaximal, characteristic, complemented
2C2×C3×C5×#104C211yesmaximal, characteristic
3#11751341C211yesmaximal, characteristic
4#11751337C211yesmaximal, characteristic
5#11746380C211yesmaximal, characteristic, complemented
6#11746378C211yesmaximal, characteristic, complemented
7#11746344C211yesmaximal, characteristic, complemented
8#10601161—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12#91815C2211yescharacteristic
13#91815C2211yescharacteristic
14#91815C2211yescharacteristic
15#91813—22no—
16C2×#6710C2211yescharacteristic
17#91759—21no—
18#91758—21no—
19C4×Dic30—22no—
20C3×C5×#104—21no—
21C3×C5×#104—21no—
22#91121—22no—
23#91118—22no—
24#70368—51nomaximal, Hall
25#65804—31no—
26C2×C5×#104D311yescharacteristic
27#65529—31no—
28#65525—31no—
29#64779—31no—
30#64777—31no—
31#64743—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×C5—21no—
41C22×C3×C4×C5C2311yescharacteristic
42C2×Dic30—22no—
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C2×C3×C5×C8—21no—
46C3×C42×C5—21no—
47C3×C42×C5D411yescharacteristic
48C3×C42×C5D411yescharacteristic
49#6719—21no—
50#6719—21no—
51#6719D412yes—
52C4⋊Dic15—22no—
53C4⋊Dic15—21no—
54C4⋊Dic15—21no—
55C4⋊Dic15—21no—
56#6712—24no—
57C4×Dic15—21no—
58C4×Dic15—21no—
59C4×Dic15D412yes—
60#6710—21no—
61#6710—21no—
62C2×C3⋊#168—21no—
63#5470—51no—
64C2×C3×#104D511yescharacteristic
65#5240—51no—
66#5236—51no—
67#4559—51no—
68#4557—51no—
69#4523—51no—
70C2×C5×#41C2×D311yescharacteristic
71C2×C5×M16C2×D311yescharacteristic
72C2×C42×C5C2×D311yescharacteristic
73#4072—31no—
74#4072—31no—
75#4072—31no—
76#4070—62no—
77C2×#681—31no—
78#4016—61no—
79#4015—61no—
80C4×Dic10—62no—
81C5×#104—21no—
82C5×#104—21no—
83#3969—62no—
84#3966—62no—
85#2031—151noSylow
86C23×C3×C5—21no—
87C23×C3×C5C2×D411yescharacteristic
88C3×C5×Q8—21no—
89C3×C5×Q8—21no—
90C3×C5×D4—21no—
91C3×C5×D4—21no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—42no—
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—22no—
107C2×Dic15—22no—
108C2×Dic15—41no—
109Dic30—44no—
110C3×C5×C8—41no—
111C3⋊#168—41no—
112C2×C3×#41C2×D511yescharacteristic
113C2×C3×M16C2×D511yescharacteristic
114C2×C3×C42C2×D511yescharacteristic
115#926—51no—
116#926—51no—
117#926—51no—
118#924—102no—
119C2×#208—51no—
120#870—101no—
121#869—101no—
122C4×Dic6—102no—
123C3×#104—21no—
124C3×#104—21no—
125#837—102no—
126#834—102no—
127C5×#41—21no—
128C5×#41—21no—
129C5×#41—21no—
130C5×#41—21no—
131C2×C5×Q8Dic611yescharacteristic, complemented
132C2×C5×D4—21no—
133C2×C5×D4Dic611yescharacteristic, complemented
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5C22×D311yescharacteristic
137C2×Dic10—62no—
138C5×M16—21no—
139C5×M16—21no—
140C2×C5×C8—21no—
141C42×C5—21no—
142C42×C5#6711yescharacteristic
143C42×C5#6711yescharacteristic
144C22⋊Dic5—32no—
145C22⋊Dic5—61no—
146C22⋊Dic5—61no—
147C4⋊Dic5—62no—
148C4⋊Dic5—61no—
149C4⋊Dic5—61no—
150C4⋊Dic5—61no—
151#683—64no—
152C4×Dic5—32no—
153C4×Dic5—61no—
154C4×Dic5—61no—
155#681—61no—
156#681—61no—
157C2×#168—61no—
158#533—151no—
159#421—151no—
160C2×#104D1511yescharacteristic
161#410—151no—
162#344—151no—
163#339—151no—
164H(3,ℤ/4)—151no—
165C22×C3×C5—41no—
166C22×C3×C5—41no—
167C22×C3×C5—81no—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q811yescharacteristic
170C22×C3×C5#12411yescharacteristic
171C3×C4×C5—22no—
172C3×C4×C5—41no—
173C3×C4×C5—41no—
174C3×C4×C5C22⋊Q811yescharacteristic
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5#12411yescharacteristic
177C3×C4×C5#12411yescharacteristic
178Dic15—42no—
179Dic15—42no—
180Dic15—81no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—21no—
185C2×C3×Q8Dic1011yescharacteristic, complemented
186C2×C3×D4—21no—
187C2×C3×D4Dic1011yescharacteristic, complemented
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C22×D511yescharacteristic
191C2×Dic6—102no—
192C3×M16—21no—
193C3×M16—21no—
194C2×C3×C8—21no—
195C3×C42—21no—
196C3×C42C22⋊D511yescharacteristic
197C3×C42C22⋊D511yescharacteristic
198#217—52no—
199#217—101no—
200#217—101no—
201C4⋊Dic3—102no—
202C4⋊Dic3—101no—
203C4⋊Dic3—101no—
204C4⋊Dic3—101no—
205Dic3⋊C4—104no—
206C4×Dic3—52no—
207C4×Dic3—101no—
208C4×Dic3—101no—
209#208—101no—
210#208—101no—
211C2×C3⋊C8—101no—
212C23×C5—21no—
213C23×C5D3×D411yescharacteristic
214C5×Q8—21no—
215C5×Q8—21no—
216C5×D4—21no—
217C5×D4—21no—
218C5×D4—41no—
219C5×D4—41no—
220C2×C4×C5—21no—
221C2×C4×C5—21no—
222C2×C4×C5—21no—
223C2×C4×C5—42no—
224C2×C4×C5—41no—
225C2×C4×C5—41no—
226C2×C4×C5C2×Dic611yescharacteristic
227C2×C4×C5#23511yescharacteristic
228C2×C4×C5D3×D411yescharacteristic
229C2×C4×C5#23711yescharacteristic
230C2×C4×C5#23711yescharacteristic
231C2×C4×C5C2×#6711yescharacteristic
232C2×Dic5—62no—
233C2×Dic5—62no—
234C2×Dic5—121no—
235Dic10—124no—
236C5×C8—41no—
237#168—121no—
238C2×#41C2×D1511yescharacteristic
239C2×M16—151no—
240C2×M16C2×D1511yescharacteristic
241C4⋊Q8—301no—
242#125—301no—
243C22⋊Q8—302no—
244C4×Q8—302no—
245#117—151no—
246#117—151no—
247#117—151no—
248C2×C42C2×D1511yescharacteristic
249#104—21no—
250#104—21no—
251#104—302no—
252#99—302no—
253C2×C3×C5—21no—
254C2×C3×C5—21no—
255C2×C3×C5—21no—
256C2×C3×C5—81no—
257C2×C3×C5#39311yescharacteristic
258C23×C3—21no—
259C23×C3D4×D511yescharacteristic
260C3×Q8—21no—
261C3×Q8—21no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—41no—
265C3×D4—41no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4C2×Dic1011yescharacteristic
273C2×C3×C4#70911yescharacteristic
274C2×C3×C4D4×D511yescharacteristic
275C2×C3×C4#71111yescharacteristic
276C2×C3×C4#71111yescharacteristic
277C2×C3×C4C2×C22⋊D511yescharacteristic
278C2×Dic3—102no—
279C2×Dic3—102no—
280C2×Dic3—201no—
281Dic6—204no—
282C3×C8—41no—
283C3⋊C8—201no—
284C22×C5—41no—
285C22×C5—41no—
286C22×C5—81no—
287C22×C5#87811yescharacteristic
288C22×C5#88511yescharacteristic
289C22×C5#93711yescharacteristic
290C4×C5—22no—
291C4×C5—41no—
292C4×C5—41no—
293C4×C5#87811yescharacteristic
294C4×C5#88511yescharacteristic
295C4×C5#92411yescharacteristic
296C4×C5#93611yescharacteristic
297Dic5—122no—
298Dic5—122no—
299Dic5—241no—
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304C2×Q8—302no—
305C2×Q8Dic3011yescharacteristic, complemented
306C2×D4—21no—
307C2×D4Dic3011yescharacteristic, complemented
308C22×C4—21no—
309C22×C4—21no—
310C22×C4C22×D1511yescharacteristic
311M16—21no—
312M16—21no—
313M16—301no—
314M16—301no—
315C2×C8—21no—
316C2×C8—301no—
317C4⋊C4—304no—
318C4⋊C4—302no—
319C4⋊C4—301no—
320C4⋊C4—301no—
321C4⋊C4—301no—
322C22⋊C4—152no—
323C22⋊C4—301no—
324C22⋊C4—301no—
325C42—21no—
326C42—152no—
327C42—301no—
328C42—301no—
329C42#122011yescharacteristic
330C42#122011yescharacteristic
331C3×C5#203111yescharacteristic, complemented, Hall
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—81no—
335C22×C3#402411yescharacteristic
336C22×C3#403111yescharacteristic
337C22×C3#408311yescharacteristic
338C3×C4—22no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4#402411yescharacteristic
342C3×C4#403111yescharacteristic
343C3×C4#407011yescharacteristic
344C3×C4#408211yescharacteristic
345Dic3—202no—
346Dic3—202no—
347Dic3—401no—
348C2×C5—21no—
349C2×C5—21no—
350C2×C5—21no—
351C2×C5—81no—
352C2×C5#494711yescharacteristic
353C23—21no—
354C23D4×D1511yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—604no—
358D4—21no—
359D4—21no—
360D4—41no—
361D4—41no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—302no—
369C2×C4—302no—
370C2×C4—601no—
371C2×C4C2×Dic3011yescharacteristic
372C2×C4#681711yescharacteristic
373C2×C4D4×D1511yescharacteristic
374C2×C4#681911yescharacteristic
375C2×C4#681911yescharacteristic
376C2×C4C2×#122011yescharacteristic
377C8—41no—
378C8—601no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—81no—
383C2×C3#6523611yescharacteristic
384C5#7036811yescharacteristic, complemented, Sylow
385C22—41no—
386C22—41no—
387C22—81no—
388C22#9176711yescharacteristic
389C22#9177411yescharacteristic
390C22#9182611yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—602no—
395C4—602no—
396C4—1201no—
397C4#9176711yescharacteristic
398C4#9177411yescharacteristic
399C4#9181311yescharacteristic
400C4#9182511yescharacteristic
401C3#1060116111yescharacteristic, complemented, Sylow
402C2—21no—
403C2—21no—
404C2—21no—
405C2—81no—
406C2#1175104811yescharacteristic
subgroupG/H
centerC4#91774
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC2×C3×C5×#104C2
socleC2×C3×C5#393
prime radicalC2×C3×C5×#41C22
hypercenterC2×#104D15
hypocenterC3×C5#2031
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #11756229C2semidirect
5. #11746380C2semidirect
6. #11746378C2semidirect
7. #11746344C2semidirect
36. C2×C3×C5×Q8Q8semidirect
38. C2×C3×C5×D4Q8semidirect
131. C2×C5×Q8Dic6semidirect
133. C2×C5×D4Dic6semidirect
185. C2×C3×Q8Dic10semidirect
187. C2×C3×D4Dic10semidirect
305. C2×Q8Dic30semidirect
307. C2×D4Dic30semidirect
331. C3×C5#2031semidirect
384. C5#70368semidirect
401. C3#10601161semidirect

Maximal subgroups

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

subgroupindex
#117562292
C2×C3×C5×#1042
#117513412
#117513372
#117463802
#117463782
#117463442
#106011613
#703685