Finite Group Databasebeta

← #14302813

Subgroups — #14302813

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65860C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic
3#11748624C211yesmaximal, characteristic
4#11748400C211yesmaximal, characteristic, complemented
5#11748399C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7C5×#4531C211yesmaximal, characteristic, complemented
8#10599505—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C3×#4043—22no—
17C3×#4032—22no—
18C3×#4021C412yescomplemented
19C3×#4015C412yescomplemented
20#91284—22no—
21#91283—22no—
22D5×#208—22no—
23#90995—22no—
24#90971—22no—
25#90970—22no—
26#71865—51nomaximal, Hall
27#66079—31no—
28#65860D311yescharacteristic
29#65409—31no—
30#65295—31no—
31#65293—31no—
32C5×H(3,ℤ/4)—31no—
33#64751—31no—
34C22×C3×C4×C5C2311yescharacteristic
35C2×C3×C22⋊D5D412yes—
36C22×C3×Dic5—21no—
37C3×#709—21no—
38C3×#709—21no—
39C3×#709—21no—
40C3×#709—21no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5D412yes—
43C2×C3×Dic10—21no—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C412yes—
49C2×C3×Dic10C2×C411yescharacteristic
50C3×C5×C4⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C42×C5C2×C412yes—
53#6719—21no—
54C4⋊Dic15—21no—
55C4×Dic15—21no—
56#6710—21no—
57#6710—21no—
58C2×C3⋊#168D412yes—
59C5×#217—21no—
60C5×C4⋊Dic3—21no—
61C4×C5×Dic3—21no—
62C5×#208—21no—
63C5×#208—21no—
64C2×C5×C3⋊C8D412yes—
65C3×#685—24no—
66C3×C4⋊Dic5—24no—
67#6649—24no—
68D5×C3⋊C8—24no—
69C3×#536—51no—
70#5819—51no—
71#5240—51no—
72#5225—51no—
73#5225—51no—
74#4531—51no—
75#4531D511yescharacteristic, complemented
76C2×#709C2×D311yescharacteristic
77C22×Dic10C2×D311yescharacteristic
78C2×C5×M16—31no—
79C5×#117—31no—
80C5×#117C2×D311yescharacteristic
81#4072—31no—
82C2×#681—31no—
83#4053—62no—
84D5×M16—62no—
85#4047—62no—
86#4043—22no—
87#4032—22no—
88#4021Dic312yescomplemented
89#4015Dic312yescomplemented
90#3948—62no—
91#3942—62no—
92#3938—62no—
93#1909—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5C22⋊C412yes—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×Dic15—41no—
104C2×C5×Dic3—41no—
105C3×C22⋊D5—22no—
106C3×C22⋊D5—22no—
107C2×C3×Dic5—22no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5—41no—
110C2×C3×Dic5C22⋊C412yes—
111C3×D20—21no—
112C3×D20—21no—
113C3×C4×D5—22no—
114C3×C4×D5—22no—
115C3×Dic10—21no—
116C3×Dic10—21no—
117C3×Dic10—22no—
118C3×Dic10—22no—
119C3×Dic10—41no—
120C3×Dic10—41no—
121C3⋊#168—22no—
122C5×C3⋊C8—22no—
123C2×C3×#41—51no—
124C22×C3×Q8—51no—
125C3×C4⋊Q8—52no—
126C3×#124—52no—
127C3×#123—102no—
128C3×C22⋊Q8—102no—
129C3×#117C2×D511yescharacteristic
130#948—102no—
131#948—102no—
132#926—51no—
133#926C2×D511yescharacteristic
134C2×#208—51no—
135C2×#208—102no—
136C2×#208C2×D511yescharacteristic
137#835—102no—
138#835—102no—
139#834—102no—
140C22×C4×C5C22×D311yescharacteristic
141C2×C22⋊D5#6712yes—
142C22×Dic5—21no—
143#709—21no—
144#709—21no—
145#709—21no—
146#709—21no—
147C2×D20C2×Dic311yescharacteristic
148C2×C4×D5#6712yes—
149C2×Dic10—21no—
150C2×Dic10—21no—
151C2×Dic10—21no—
152C2×Dic10—21no—
153C2×Dic10—21no—
154C2×Dic10C2×Dic312yes—
155C2×Dic10C2×Dic311yescharacteristic
156C5×M16—61no—
157C5×M16—61no—
158C2×C5×C8—32no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—61no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—61no—
163C42×C5—61no—
164C42×C5C2×Dic312yes—
165C22⋊Dic5—61no—
166#685—24no—
167C4⋊Dic5—24no—
168C4⋊Dic5—61no—
169C4×Dic5—61no—
170#681—61no—
171#681—61no—
172C2×#168—32no—
173#676—64no—
174C8×D5—64no—
175#568—151no—
176#536—51no—
177#419—151no—
178#419—151no—
179#410—151no—
180H(3,ℤ/4)—31no—
181H(3,ℤ/4)—151no—
182C22×C3×C5—41no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5#12111yescharacteristic
185C22×C3×C5#12111yescharacteristic
186C2×C3×D5—22no—
187C2×C3×D5—42no—
188C3×C4×C5—42no—
189C3×C4×C5C4×D411yescharacteristic
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5#12011yescharacteristic
192C3×C4×C5#12311yescharacteristic
193Dic15—81no—
194C3×Dic5—22no—
195C3×Dic5—42no—
196C5×Dic3—81no—
197C3×#41—101no—
198C3×#41—101no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—51no—
202C2×C3×Q8—52no—
203C2×C3×Q8—101no—
204C2×C3×Q8—101no—
205C2×C3×Q8—101no—
206C2×C3×Q8—101no—
207C2×C3×Q8—101no—
208C2×C3×D4—52no—
209C2×C3×D4—51no—
210C22×C3×C4—52no—
211C22×C3×C4—101no—
212C22×C3×C4C22×D511yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4—104no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—104no—
217C3×C42C4×D512yes—
218#217—21no—
219#217—101no—
220C4⋊Dic3—21no—
221C4⋊Dic3—101no—
222C4×Dic3—21no—
223C4×Dic3—101no—
224#208—21no—
225#208—21no—
226#208—104no—
227#208—101no—
228#208—101no—
229C2×C3⋊C8—52no—
230C2×C3⋊C8—104no—
231C2×C3⋊C8D2012yes—
232C23×C5C2×#6711yescharacteristic
233C22×D5#21712yes—
234C2×C4×C5—22no—
235C2×C4×C5—121no—
236C2×C4×C5D3×D411yescharacteristic
237C2×C4×C5D3×D411yescharacteristic
238C2×C4×C5#23711yescharacteristic
239C2×C4×C5#23711yescharacteristic
240C2×C4×C5C22×Dic311yescharacteristic
241C2×C4×C5C2×#6711yescharacteristic
242C22⋊D5—22no—
243C22⋊D5—22no—
244C2×Dic5—22no—
245C2×Dic5—41no—
246C2×Dic5—41no—
247C2×Dic5—121no—
248C2×Dic5#21712yes—
249D20—21no—
250D20—21no—
251C4×D5—22no—
252C4×D5—22no—
253Dic10—21no—
254Dic10—21no—
255Dic10—22no—
256Dic10—22no—
257Dic10—41no—
258Dic10—41no—
259C5×C8—62no—
260#168—62no—
261C2×#41—51no—
262C22×Q8—51no—
263#131—302no—
264#131—302no—
265C2×M16—31no—
266C2×M16—151no—
267C2×M16—302no—
268C4⋊Q8—52no—
269#124—52no—
270#123—102no—
271C22⋊Q8—102no—
272#117—31no—
273#117—151no—
274#117D3×D511yescharacteristic
275#103—302no—
276#103—302no—
277#99—302no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38611yescharacteristic
282C3×D5—42no—
283C23×C3—52no—
284C23×C3D4×D511yescharacteristic
285C3×Q8—101no—
286C3×Q8—101no—
287C3×Q8—102no—
288C3×Q8—102no—
289C3×Q8—201no—
290C3×Q8—201no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4—101no—
294C3×D4—101no—
295C2×C3×C4—22no—
296C2×C3×C4—52no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—102no—
300C2×C3×C4—201no—
301C2×C3×C4—201no—
302C2×C3×C4C2×C4×D511yescharacteristic
303C2×C3×C4C2×D2011yescharacteristic
304C2×C3×C4#70911yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4#71311yescharacteristic
308C2×Dic3—41no—
309C2×Dic3—201no—
310C3⋊C8—22no—
311C3⋊C8—102no—
312C22×C5—41no—
313C22×C5#93911yescharacteristic
314C22×C5#93911yescharacteristic
315C22×C5C2×#21711yescharacteristic
316C2×D5—22no—
317C2×D5—42no—
318C4×C5—42no—
319C4×C5—241no—
320C4×C5D4×Dic311yescharacteristic
321C4×C5D4×Dic311yescharacteristic
322C4×C5#93511yescharacteristic
323C4×C5#93711yescharacteristic
324Dic5—22no—
325Dic5—42no—
326Dic5—241no—
327#41—101no—
328#41—101no—
329#41—101no—
330#41—101no—
331C2×Q8—51no—
332C2×Q8—52no—
333C2×Q8—101no—
334C2×Q8—101no—
335C2×Q8—101no—
336C2×Q8—101no—
337C2×Q8—101no—
338C2×D4—52no—
339C2×D4—51no—
340C22×C4—52no—
341C22×C4—101no—
342C22×C4C2×D3×D511yescharacteristic
343M16—61no—
344M16—61no—
345M16—304no—
346M16—301no—
347M16—301no—
348C2×C8—32no—
349C2×C8—152no—
350C2×C8—304no—
351C4⋊C4—21no—
352C4⋊C4—61no—
353C4⋊C4—104no—
354C4⋊C4—301no—
355C22⋊C4—21no—
356C22⋊C4—61no—
357C22⋊C4—104no—
358C22⋊C4—301no—
359C42—61no—
360C42—301no—
361C42D5×Dic312yes—
362C3×C5#190911yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3—102no—
365C22×C3—202no—
366C22×C3D5×C22⋊C411yescharacteristic
367C22×C3#403011yescharacteristic
368C22×C3#404111yescharacteristic
369C3×C4—42no—
370C3×C4—102no—
371C3×C4—202no—
372C3×C4C4×D2011yescharacteristic
373C3×C4#402811yescharacteristic
374C3×C4#403911yescharacteristic
375C3×C4#404011yescharacteristic
376Dic3—81no—
377Dic3—401no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5#522011yescharacteristic
382D5—42no—
383C23—52no—
384C23D5×#6711yescharacteristic
385Q8—101no—
386Q8—101no—
387Q8—102no—
388Q8—102no—
389Q8—201no—
390Q8—201no—
391D4—102no—
392D4—102no—
393D4—101no—
394D4—101no—
395C2×C4—22no—
396C2×C4—52no—
397C2×C4—102no—
398C2×C4—102no—
399C2×C4—102no—
400C2×C4—121no—
401C2×C4—201no—
402C2×C4—201no—
403C2×C4—601no—
404C2×C4#676511yescharacteristic
405C2×C4#676611yescharacteristic
406C2×C4D3×D2011yescharacteristic
407C2×C4#677711yescharacteristic
408C2×C4C2×D5×Dic311yescharacteristic
409C2×C4D5×#6711yescharacteristic
410C8—62no—
411C8—302no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—202no—
416C2×C3#6495411yescharacteristic
417C5#7186511yescharacteristic, complemented, Sylow
418C22—41no—
419C22—102no—
420C22—202no—
421C22#9140811yescharacteristic
422C22#9140911yescharacteristic
423C22D5×#21711yescharacteristic
424C4—42no—
425C4—102no—
426C4—202no—
427C4—241no—
428C4—1201no—
429C4#9140711yescharacteristic
430C4Dic3×D2011yescharacteristic
431C4#9143411yescharacteristic
432C4#9147411yescharacteristic
433C3#1059950511yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—202no—
438C2#1174889611yescharacteristic
subgroupG/H
centerC2#11748896
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1909
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #11748400C2semidirect
5. #11748399C2semidirect
6. #11746352C2semidirect
7. C5×#4531C2semidirect
18. C3×#4021C4semidirect
19. C3×#4015C4semidirect
75. #4531D5semidirect
88. #4021Dic3semidirect
89. #4015Dic3semidirect
362. C3×C5#1909semidirect
417. C5#71865semidirect
433. C3#10599505semidirect

Maximal subgroups

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

subgroupindex
C3×#658602
#117541202
#117486242
#117484002
#117483992
#117463522
C5×#45312
#105995053
#718655