Finite Group Databasebeta

← #14301199

Subgroups — #14301199

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754138C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic, complemented
3C5×#5110C211yesmaximal, characteristic, complemented
4C3×#65402C211yesmaximal, characteristic, complemented
5#11748425C211yesmaximal, characteristic, complemented
6#11746373C211yesmaximal, characteristic, complemented
7#11745992C211yesmaximal, characteristic, complemented
8#10600805—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16#91309—21no—
17#91307C412yescomplemented
18#91299C412yescomplemented
19#91298—21no—
20#91297—22no—
21#91284—22no—
22#91283—22no—
23D5×#208—22no—
24C5×(C2×D3)⋊C8—22no—
25C3×#3953—21no—
26C3×#3948—21no—
27#90979—22no—
28#70293—51nomaximal, Hall
29#66156—31no—
30#66079—31no—
31C5×#408—31no—
32#65525—31no—
33#65402D311yescharacteristic, complemented
34#64776—31no—
35#64772—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×C5×#67—21no—
38C5×#235—21no—
39C5×#235—21no—
40C5×#235—21no—
41C5×#235—21no—
42C2×C5×D12C2×C411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C5×Dic6C2×C411yescharacteristic
45C2×C3×C22⋊D5—21no—
46C3×#709—21no—
47C3×#709—21no—
48C3×#709C2×C412yes—
49C3×#709D412yes—
50C2×C3×D20D411yescharacteristic
51C2×C3×C4×D5—21no—
52C2×C3×Dic10D411yescharacteristic
53C2×C3×C5×C8—21no—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56#6710—21no—
57#6710—21no—
58#6710C2×C412yes—
59#6710D412yes—
60C2×C3⋊#168—21no—
61C2×C3⋊#168—21no—
62C5×#208—21no—
63C5×#208—21no—
64C2×C5×C3⋊C8D412yes—
65C3×C22⋊Dic5—21no—
66C3×C4⋊Dic5—21no—
67C3×C4×Dic5—21no—
68#6661—22no—
69#6656—22no—
70#6655—22no—
71#6652—22no—
72#6649—24no—
73D5×C3⋊C8—24no—
74#5822—51no—
75#5819—51no—
76C3×#416—51no—
77#5110D511yescharacteristic, complemented
78#5005—51no—
79#4552—51no—
80#4533—51no—
81C2×C5×#41—31no—
82C2×#709C2×D311yescharacteristic
83C2×C5×M16—31no—
84C22×C5×C8C2×D311yescharacteristic
85#4097—61no—
86#4096—62no—
87#4095—61no—
88#4088—32no—
89#4078—32no—
90#4072C2×D311yescharacteristic
91C2×#681—31no—
92C2×#681—31no—
93#4053—62no—
94D5×M16—62no—
95#4047—62no—
96C5×C22⋊C8—62no—
97#3969—62no—
98#3953—21no—
99#3948—21no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×C5×D3—21no—
103C22×C3×D5—21no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C5×#67—41no—
111C5×#67—41no—
112C2×C5×Dic3—21no—
113C5×D12—21no—
114C5×D12—21no—
115C4×C5×D3—41no—
116C4×C5×D3—41no—
117C5×Dic6—21no—
118C5×Dic6—21no—
119C3×C22⋊D5—22no—
120C3×C22⋊D5—22no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—41no—
123C3×D20—21no—
124C3×D20C22⋊C412yes—
125C3×C4×D5—22no—
126C3×C4×D5—22no—
127C3×Dic10—21no—
128C3×Dic10C22⋊C412yes—
129C3×C5×C8—41no—
130C3⋊#168—22no—
131C3⋊#168—22no—
132C5×C3⋊C8—22no—
133C2×C3×#41—51no—
134C2×#235C2×D511yescharacteristic
135C22×C3×C8C2×D511yescharacteristic
136C3×#117—51no—
137#951—52no—
138#950—102no—
139#949—52no—
140#948—102no—
141#948—102no—
142#942—101no—
143#932—101no—
144C2×#208—51no—
145C2×#208—51no—
146C2×#208—102no—
147C2×#208C2×D511yescharacteristic
148C3×#103—101no—
149C3×#102—101no—
150(C2×D3)⋊C8—22no—
151#805—102no—
152C5×#41—61no—
153C5×#41—61no—
154C5×#41—61no—
155C5×#41—61no—
156C2×C5×Q8—31no—
157C2×C5×D4—31no—
158C2×C5×D4—61no—
159C22×C4×C5—61no—
160C22×C4×C5C22×D311yescharacteristic
161C2×C22⋊D5—21no—
162#709—21no—
163#709—21no—
164#709C4×D312yes—
165#709D1212yes—
166C2×D20#6711yescharacteristic
167C2×C4×D5—21no—
168C2×Dic10#6711yescharacteristic
169C5×M16—61no—
170C5×M16—61no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—32no—
175C22⋊Dic5—21no—
176#689—62no—
177#688—62no—
178#687—62no—
179#686—62no—
180C4⋊Dic5—21no—
181C4×Dic5—21no—
182#681—32no—
183#681—32no—
184#681—61no—
185#681—61no—
186C2×#168—61no—
187C2×#168—61no—
188#676—64no—
189C8×D5—64no—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—51no—
194#408—31no—
195#344—151no—
196#341—151no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×C5×D3—42no—
202C2×C5×D3—41no—
203C2×C3×D5—22no—
204C2×C3×D5—81no—
205C3×C4×C5C2×C22⋊C411yescharacteristic
206C3×C4×C5C4×D411yescharacteristic
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12111yescharacteristic
209C3×Dic5—22no—
210C3×Dic5—81no—
211C5×Dic3—41no—
212C3×#41—52no—
213C3×#41—52no—
214C3×#41—101no—
215C3×#41—101no—
216C2×C3×Q8—51no—
217C2×C3×D4—51no—
218C2×C3×D4—101no—
219C22×C3×C4—101no—
220C22×C3×C4C22×D511yescharacteristic
221C2×#67—21no—
222#235—21no—
223#235—21no—
224#235—21no—
225#235—21no—
226C2×D12C4×D511yescharacteristic
227C2×C4×D3—21no—
228C2×Dic6C4×D511yescharacteristic
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C3×C4⋊C4—101no—
233C3×C22⋊C4—101no—
234C3×C42—101no—
235#216—102no—
236#215—102no—
237#214—102no—
238#213—102no—
239#208—21no—
240#208—21no—
241#208—52no—
242#208—52no—
243#208—104no—
244#208—101no—
245#208—101no—
246C2×C3⋊C8—104no—
247C2×C3⋊C8—101no—
248C2×C3⋊C8—101no—
249C2×C3⋊C8C22⋊D512yes—
250C23×C5—61no—
251C23×C5#23711yescharacteristic
252C22×D5—21no—
253C5×Q8—61no—
254C5×Q8—61no—
255C5×D4—61no—
256C5×D4—61no—
257C5×D4—121no—
258C5×D4—121no—
259C2×C4×C5—61no—
260C2×C4×C5—121no—
261C2×C4×C5—121no—
262C2×C4×C5C2×C4×D311yescharacteristic
263C2×C4×C5C2×D1211yescharacteristic
264C2×C4×C5D3×D411yescharacteristic
265C2×C4×C5D3×D411yescharacteristic
266C2×C4×C5#23711yescharacteristic
267C2×C4×C5C2×#6711yescharacteristic
268C22⋊D5—22no—
269C22⋊D5—22no—
270C2×Dic5—21no—
271C2×Dic5—41no—
272D20—21no—
273D20(C2×D3)⋊C412yes—
274C4×D5—22no—
275C4×D5—22no—
276Dic10—21no—
277Dic10(C2×D3)⋊C412yes—
278C5×C8—41no—
279C5×C8—62no—
280#168—62no—
281#168—62no—
282C2×#41—31no—
283C2×#41—51no—
284#137—152no—
285#137—301no—
286AGL(1,ℤ/8)—152no—
287AGL(1,ℤ/8)—301no—
288#135—302no—
289#131—302no—
290#131—302no—
291C2×M16—31no—
292C2×M16—151no—
293C2×M16—151no—
294C2×M16—302no—
295C22×C8D3×D511yescharacteristic
296#117—51no—
297#104—302no—
298#103—101no—
299#102—101no—
300C22⋊C8—62no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5#38611yescharacteristic
305C3×D5—42no—
306C5×D3—81no—
307C23×C3—101no—
308C23×C3#70911yescharacteristic
309C22×D3—21no—
310C3×Q8—52no—
311C3×Q8—101no—
312C3×D4—52no—
313C3×D4—102no—
314C3×D4—102no—
315C3×D4—101no—
316C2×C3×C4—101no—
317C2×C3×C4—102no—
318C2×C3×C4—102no—
319C2×C3×C4—201no—
320C2×C3×C4C2×C4×D511yescharacteristic
321C2×C3×C4D4×D511yescharacteristic
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4D4×D511yescharacteristic
324C2×C3×C4#71111yescharacteristic
325C2×C3×C4C2×C22⋊D511yescharacteristic
326#67—41no—
327#67—41no—
328C2×Dic3—21no—
329D12—21no—
330D12—21no—
331C4×D3—41no—
332C4×D3—41no—
333Dic6—21no—
334Dic6—21no—
335C3×C8—41no—
336C3⋊C8—22no—
337C3⋊C8—102no—
338C3⋊C8—102no—
339C22×C5—41no—
340C22×C5—122no—
341C22×C5—121no—
342C22×C5#67⋊C411yescharacteristic
343C22×C5#88611yescharacteristic
344C22×C5#93911yescharacteristic
345C2×D5—22no—
346C2×D5—81no—
347C4×C5—121no—
348C4×C5#67⋊C411yescharacteristic
349C4×C5#88211yescharacteristic
350C4×C5C2×(C2×D3)⋊C411yescharacteristic
351C4×C5#93911yescharacteristic
352Dic5—22no—
353Dic5—81no—
354#41—52no—
355#41—52no—
356#41—61no—
357#41—61no—
358#41—61no—
359#41—61no—
360#41—101no—
361#41—101no—
362C2×Q8—31no—
363C2×Q8—51no—
364C2×D4—31no—
365C2×D4—51no—
366C2×D4—61no—
367C2×D4—101no—
368C22×C4—61no—
369C22×C4—101no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—61no—
376M16—61no—
377M16—152no—
378M16—152no—
379M16—304no—
380M16—301no—
381M16—301no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—32no—
386C2×C8—304no—
387C2×C8—301no—
388C2×C8—301no—
389C4⋊C4—101no—
390C22⋊C4—101no—
391C42—101no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—102no—
395C22×C3—401no—
396C22×C3#402911yescharacteristic
397C22×C3#403011yescharacteristic
398C22×C3C4×C22⋊D511yescharacteristic
399C2×D3—42no—
400C2×D3—41no—
401C3×C4—102no—
402C3×C4—401no—
403C3×C4D5×C22⋊C411yescharacteristic
404C3×C4#402711yescharacteristic
405C3×C4#408311yescharacteristic
406C3×C4#408411yescharacteristic
407Dic3—41no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—241no—
412C2×C5#495811yescharacteristic
413D5—42no—
414C23—61no—
415C23—101no—
416C23#676511yescharacteristic
417Q8—52no—
418Q8—61no—
419Q8—61no—
420Q8—101no—
421D4—52no—
422D4—61no—
423D4—61no—
424D4—102no—
425D4—102no—
426D4—101no—
427D4—121no—
428D4—121no—
429C2×C4—61no—
430C2×C4—101no—
431C2×C4—102no—
432C2×C4—102no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—201no—
436C2×C4C4×D3×D511yescharacteristic
437C2×C4D5×D1211yescharacteristic
438C2×C4#677711yescharacteristic
439C2×C4#678011yescharacteristic
440C2×C4D5×#6711yescharacteristic
441C2×C4D3×C22⋊D511yescharacteristic
442C8—41no—
443C8—62no—
444C8—302no—
445C8—302no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—202no—
450C2×C3#6524511yescharacteristic
451D3—81no—
452C5#7029311yescharacteristic, complemented, Sylow
453C22—41no—
454C22—102no—
455C22—122no—
456C22—121no—
457C22—401no—
458C22#9140811yescharacteristic
459C22#9141411yescharacteristic
460C22#9144111yescharacteristic
461C4—102no—
462C4—121no—
463C4—401no—
464C4#9144211yescharacteristic
465C4D5×(C2×D3)⋊C411yescharacteristic
466C4#9147511yescharacteristic
467C4#9154911yescharacteristic
468C3#1060080511yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—202no—
473C2—241no—
474C2#1174890611yescharacteristic
subgroupG/H
centerC4#91475
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×C8C22
socleC2×C3×C5#386
prime radical#11754138C2
cosocleC22×C3×C4×C5C23
hypercenterC22×C8D3×D5
hypocenterC3×C5#1881
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. #11754120C2semidirect
3. C5×#5110C2semidirect
4. C3×#65402C2semidirect
5. #11748425C2semidirect
6. #11746373C2semidirect
7. #11745992C2semidirect
17. #91307C4semidirect
18. #91299C4semidirect
33. #65402D3semidirect
77. #5110D5semidirect
392. C3×C5#1881semidirect
452. C5#70293semidirect
468. C3#10600805semidirect

Maximal subgroups

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

subgroupindex
#117541382
#117541202
C5×#51102
C3×#654022
#117484252
#117463732
#117459922
#106008053
#702935