Finite Group Databasebeta

← #14302317

Subgroups — #14302317

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91821C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic, complemented
3C3×C5×#418C211yesmaximal, characteristic, complemented
4C2×#91117C211yesmaximal, characteristic, complemented
5#11751106C211yesmaximal, characteristic, complemented
6#11746381C211yesmaximal, characteristic, complemented
7#11746354C211yesmaximal, characteristic, complemented
8#10601122—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C2×#6716C412yescomplemented
14#91821—22no—
15#91821—22no—
16C2×#6715C412yescomplemented
17C2×#6710C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20#91796—22no—
21M16×D15—22no—
22#91790—22no—
23C3×C5×#102—22no—
24#91117—22no—
25#91089—22no—
26#70329—51nomaximal, Hall
27C2×#4078—31no—
28#66079—31no—
29C5×#418D311yescharacteristic, complemented
30C2×#3965—31no—
31#65294—31no—
32#64780—31no—
33#64753—31no—
34C24×C3×C5—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4C2×C412yes—
41C22×C3×C4×C5C2311yescharacteristic
42C2×#1220—21no—
43#6817—21no—
44#6817—21no—
45#6817D412yes—
46#6817D412yes—
47C2×D60C2×C411yescharacteristic
48C2×C4×D15—21no—
49C2×Dic30C2×C411yescharacteristic
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8D412yes—
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—21no—
56#6716—24no—
57#6715—24no—
58#6710—21no—
59#6710—21no—
60#6710—21no—
61#6710—21no—
62#6710—21no—
63#6710D412yes—
64C2×C3⋊#168—21no—
65C2×C3⋊#168—21no—
66C2×C3⋊#168C2×C412yes—
67#6705—24no—
68C8×D15—24no—
69C2×#932—51no—
70#5745—51no—
71C3×#418D511yescharacteristic, complemented
72C2×#833—51no—
73#5005—51no—
74C3⋊#338—51no—
75#4533—51no—
76C22×C5×D4C2×D311yescharacteristic
77C2×#709—31no—
78C2×C5×M16C2×D311yescharacteristic
79C5×#117C2×D311yescharacteristic
80C2×#687—32no—
81#4078—62no—
82#4078—62no—
83C2×#686—32no—
84C2×#681—31no—
85C2×#681—31no—
86C2×#681—31no—
87#4053—62no—
88D5×M16—62no—
89#4047—62no—
90C5×#102—22no—
91#3965—62no—
92#3937—62no—
93#1903—151noSylow
94C23×C3×C5—22no—
95C23×C3×C5—41no—
96C23×C3×C5—41no—
97C23×C3×C5—82no—
98C23×C3×C5#4111yescharacteristic
99C22×D15—21no—
100C3×C5×D4—22no—
101C3×C5×D4—22no—
102C3×C5×D4—41no—
103C3×C5×D4—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111#1220—22no—
112#1220—22no—
113C2×Dic15—21no—
114D60—21no—
115D60C22⋊C412yes—
116C4×D15—22no—
117C4×D15—22no—
118Dic30—21no—
119Dic30C22⋊C412yes—
120C3×C5×C8—22no—
121C3⋊#168—22no—
122C3⋊#168—22no—
123C3⋊#168—41no—
124C22×C3×D4C2×D511yescharacteristic
125C2×#235—51no—
126C2×C3×M16C2×D511yescharacteristic
127C3×#117C2×D511yescharacteristic
128C2×#214—52no—
129#932—102no—
130#932—102no—
131C2×#213—52no—
132C2×#208—51no—
133C2×#208—51no—
134C2×#208—51no—
135#907—102no—
136D3×M16—102no—
137#901—102no—
138C3×#102—22no—
139#833—102no—
140#805—102no—
141C24×C5—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4C4×D312yes—
148C22×C4×C5C22×D311yescharacteristic
149C2×C22⋊D5—61no—
150#709—32no—
151#709—32no—
152#709—61no—
153#709—61no—
154C2×D20—31no—
155C2×C4×D5—61no—
156C2×Dic10—31no—
157C5×M16—21no—
158C5×M16—21no—
159C2×C5×C8#6712yes—
160C5×C4⋊C4—21no—
161C5×C22⋊C4—21no—
162C42×C5—21no—
163#687—64no—
164#686—64no—
165#681—32no—
166#681—61no—
167#681—61no—
168#681—61no—
169#681—61no—
170#681—61no—
171C2×#168—32no—
172C2×#168—61no—
173C2×#168—61no—
174#676—64no—
175C8×D5—64no—
176C2×AGL(1,ℤ/8)—151no—
177#568—151no—
178#421—151no—
179#418D1511yescharacteristic, complemented
180C2×#100—151no—
181#344—151no—
182#338—151no—
183C22×C3×C5—41no—
184C22×C3×C5—42no—
185C22×C3×C5—82no—
186C22×C3×C5—82no—
187C22×C3×C5—82no—
188C22×C3×C5C4×D411yescharacteristic
189C22×C3×C5#12111yescharacteristic
190C22×C3×C5#12311yescharacteristic
191C2×D15—22no—
192C2×D15—81no—
193C3×C4×C5—81no—
194C3×C4×C5C2×C22⋊C411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5#12011yescharacteristic
197C3×C4×C5#12111yescharacteristic
198Dic15—22no—
199C24×C3—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C4×D512yes—
206C22×C3×C4C22×D511yescharacteristic
207C2×#67—101no—
208#235—52no—
209#235—52no—
210#235—101no—
211#235—101no—
212C2×D12—51no—
213C2×C4×D3—101no—
214C2×Dic6—51no—
215C3×M16—21no—
216C3×M16—21no—
217C2×C3×C8C22⋊D512yes—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C42—21no—
221#214—104no—
222#213—104no—
223#208—52no—
224#208—101no—
225#208—101no—
226#208—101no—
227#208—101no—
228#208—101no—
229C2×C3⋊C8—52no—
230C2×C3⋊C8—101no—
231C2×C3⋊C8—101no—
232ΓL(1,𝔽25)—104no—
233D3×C8—104no—
234C23×C5—22no—
235C23×C5—41no—
236C23×C5—41no—
237C23×C5—82no—
238C23×C5#23711yescharacteristic
239C22×D5—61no—
240C5×D4—22no—
241C5×D4—22no—
242C5×D4—41no—
243C5×D4—41no—
244C2×C4×C5—41no—
245C2×C4×C5C2×C4×D311yescharacteristic
246C2×C4×C5#23511yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5D3×D411yescharacteristic
250C2×C4×C5C2×#6711yescharacteristic
251C22⋊D5—62no—
252C22⋊D5—62no—
253C2×Dic5—61no—
254D20—32no—
255D20—61no—
256C4×D5—62no—
257C4×D5—62no—
258Dic10—32no—
259Dic10—61no—
260C5×C8—22no—
261#168—62no—
262#168—62no—
263#168—121no—
264C2×#41—151no—
265C22×D4C2×D1511yescharacteristic
266AGL(1,ℤ/8)—302no—
267AGL(1,ℤ/8)—302no—
268C2×SD16—152no—
269C2×D8—152no—
270#131—302no—
271#131—302no—
272C2×M16—151no—
273C2×M16—151no—
274C2×M16—151no—
275C2×M16—302no—
276C2×M16C2×D1511yescharacteristic
277#117C2×D1511yescharacteristic
278#104—302no—
279#102—22no—
280#100—302no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5—21no—
284C2×C3×C5—82no—
285C2×C3×C5#38611yescharacteristic
286D15—42no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—82no—
291C23×C3#71111yescharacteristic
292C22×D3—101no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—41no—
296C3×D4—41no—
297C2×C3×C4—41no—
298C2×C3×C4C2×C4×D511yescharacteristic
299C2×C3×C4#70911yescharacteristic
300C2×C3×C4D4×D511yescharacteristic
301C2×C3×C4D4×D511yescharacteristic
302C2×C3×C4D4×D511yescharacteristic
303C2×C3×C4C2×C22⋊D511yescharacteristic
304#67—102no—
305#67—102no—
306C2×Dic3—101no—
307D12—52no—
308D12—101no—
309C4×D3—102no—
310C4×D3—102no—
311Dic6—52no—
312Dic6—101no—
313C3×C8—22no—
314C3⋊C8—102no—
315C3⋊C8—102no—
316C3⋊C8—201no—
317C22×C5—41no—
318C22×C5—42no—
319C22×C5—82no—
320C22×C5—82no—
321C22×C5—82no—
322C22×C5#67⋊C411yescharacteristic
323C22×C5#88311yescharacteristic
324C22×C5#93811yescharacteristic
325C2×D5—62no—
326C2×D5—241no—
327C4×C5—81no—
328C4×C5D3×C22⋊C411yescharacteristic
329C4×C5#88411yescharacteristic
330C4×C5C4×#6711yescharacteristic
331C4×C5#93711yescharacteristic
332Dic5—62no—
333C24—21no—
334#41—152no—
335#41—152no—
336#41—301no—
337#41—301no—
338C2×Q8—151no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—151no—
345C2×D4—301no—
346C2×D4C4×D1512yes—
347C22×C4—301no—
348C22×C4C22×D1511yescharacteristic
349SD16—304no—
350D8—304no—
351M16—21no—
352M16—21no—
353M16—152no—
354M16—304no—
355M16—301no—
356M16—301no—
357M16—301no—
358M16—301no—
359M16—301no—
360C2×C8—152no—
361C2×C8—304no—
362C2×C8—301no—
363C2×C8—301no—
364C2×C8#122012yes—
365C4⋊C4—21no—
366C22⋊C4—21no—
367C42—21no—
368C3×C5#190311yescharacteristic, complemented, Hall
369C22×C3—41no—
370C22×C3—42no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3#402711yescharacteristic
375C22×C3#402911yescharacteristic
376C22×C3#408411yescharacteristic
377C2×D3—102no—
378C2×D3—401no—
379C3×C4—81no—
380C3×C4D5×C22⋊C411yescharacteristic
381C3×C4#403011yescharacteristic
382C3×C4C4×C22⋊D511yescharacteristic
383C3×C4#408311yescharacteristic
384Dic3—102no—
385C2×C5—21no—
386C2×C5—21no—
387C2×C5—21no—
388C2×C5—82no—
389C2×C5#495611yescharacteristic
390D5—122no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—82no—
395C23—301no—
396C23#681911yescharacteristic
397Q8—152no—
398Q8—301no—
399D4—22no—
400D4—22no—
401D4—41no—
402D4—41no—
403D4—152no—
404D4—302no—
405D4—302no—
406D4—301no—
407C2×C4—41no—
408C2×C4—301no—
409C2×C4—302no—
410C2×C4—302no—
411C2×C4C2×C4×D1511yescharacteristic
412C2×C4#681711yescharacteristic
413C2×C4D4×D1511yescharacteristic
414C2×C4D4×D1511yescharacteristic
415C2×C4D4×D1511yescharacteristic
416C2×C4C2×#122011yescharacteristic
417C8—22no—
418C8—302no—
419C8—302no—
420C8—601no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—82no—
425C2×C3#6524511yescharacteristic
426D3—202no—
427C5#7032911yescharacteristic, complemented, Sylow
428C22—41no—
429C22—42no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—302no—
434C22—1201no—
435C22#9177011yescharacteristic
436C22#9177211yescharacteristic
437C22#9182711yescharacteristic
438C4—81no—
439C4—302no—
440C4C22⋊C4×D1511yescharacteristic
441C4#9177311yescharacteristic
442C4C4×#122011yescharacteristic
443C4#9182611yescharacteristic
444C3#1060112211yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—21no—
448C2—82no—
449C2—602no—
450C2#1175105711yescharacteristic
subgroupG/H
centerC2#11751057
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#418C2
socleC2×C3×C5#386
prime radicalC2×#91821C2
cosocleC22×C3×C4×C5C23
hypercenter#418D15
hypocenterC3×C5#1903
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #11756504C2semidirect
3. C3×C5×#418C2semidirect
4. C2×#91117C2semidirect
5. #11751106C2semidirect
6. #11746381C2semidirect
7. #11746354C2semidirect
13. C2×#6716C4semidirect
16. C2×#6715C4semidirect
29. C5×#418D3semidirect
71. C3×#418D5semidirect
179. #418D15semidirect
368. C3×C5#1903semidirect
427. C5#70329semidirect
444. C3#10601122semidirect

Maximal subgroups

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

subgroupindex
C2×#918212
#117565042
C3×C5×#4182
C2×#911172
#117511062
#117463812
#117463542
#106011223
#703295