Finite Group Databasebeta

← #14302951

Subgroups — #14302951

Order 1920.

628 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×#65850C211yesmaximal, characteristic, complemented
2#11754255C211yesmaximal, characteristic
3C5×#5005C211yesmaximal, characteristic
4#11748425C211yesmaximal, characteristic
5C2×#90961C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7#11745970C211yesmaximal, characteristic, complemented
8#10601204—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C3×Dic10C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16C3×#4033—22no—
17C3×#4023—22no—
18C3×C4×Dic10—22no—
19#91539—22no—
20#91385C412yescomplemented
21#91370C412yescomplemented
22#91369—22no—
23C5×#805—22no—
24#90979—22no—
25#90961—22no—
26#71663—51nomaximal, Hall
27#66159—31no—
28#65850D311yescharacteristic, complemented
29C5×#421—31no—
30#65525—31no—
31C2×#3956—31no—
32#64777—31no—
33#64751—31no—
34C22×C3×C4×C5C2311yescharacteristic
35C2×C5×#67—21no—
36C5×#235—21no—
37C5×#235—21no—
38C5×#235—21no—
39C5×#235—21no—
40C2×C5×D12C2×C411yescharacteristic
41C2×C4×C5×D3—21no—
42C2×C5×Dic6C2×C411yescharacteristic
43C22×C3×Dic5D412yes—
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10C2×C412yes—
49C2×C3×Dic10D412yes—
50C3×C5×C4⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C42×C5D412yes—
53#6719—21no—
54C4⋊Dic15—21no—
55C4×Dic15C2×C412yes—
56#6710—21no—
57#6710—21no—
58C2×C3⋊#168—21no—
59C5×#208—21no—
60C5×#208—21no—
61C2×C5×C3⋊C8—21no—
62C3×C22⋊Dic5—21no—
63C3×C4⋊Dic5—21no—
64C3×#683—24no—
65C3×C4×Dic5—24no—
66C3×C4×Dic5D412yes—
67#6669—24no—
68(C2×D3)⋊Dic5—24no—
69C3×#519—51no—
70#5816—51no—
71C2×#836—51no—
72#5005—51no—
73#5005D511yescharacteristic
74#4531—51no—
75#4531—51no—
76C2×C5×#41—31no—
77C22×Dic10C2×D311yescharacteristic
78C2×C5×M16—31no—
79C5×#117C2×D311yescharacteristic
80#4090—32no—
81#4082—32no—
82#4081—62no—
83#4072—31no—
84#4072C2×D311yescharacteristic
85#4070—62no—
86C2×#681—31no—
87#4033—22no—
88#4023—22no—
89C4×Dic10—22no—
90C5×#104—62no—
91#3969—62no—
92#3956—62no—
93#1912—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—21no—
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—22no—
104C5×#67—41no—
105C5×#67—41no—
106C2×C5×Dic3—21no—
107C5×D12—21no—
108C5×D12—21no—
109C4×C5×D3—41no—
110C4×C5×D3—41no—
111C5×Dic6—21no—
112C5×Dic6—21no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5—22no—
115C2×C3×Dic5—22no—
116C2×C3×Dic5C22⋊C414yes—
117C3×Dic10—22no—
118C3×Dic10—22no—
119C3×Dic10—22no—
120C3×Dic10—22no—
121C3⋊#168—41no—
122C5×C3⋊C8—41no—
123C22×C3×Q8—51no—
124C2×#235C2×D511yescharacteristic
125C3×C4×Q8—102no—
126C3×C4×Q8—102no—
127C3×#117—51no—
128C3×#117—102no—
129C3×#117C2×D511yescharacteristic
130#947—52no—
131#946—102no—
132#944—52no—
133#929—102no—
134#926—51no—
135C2×#208—51no—
136C2×#208C2×D511yescharacteristic
137#836—102no—
138#805—22no—
139#805—102no—
140C5×#41—61no—
141C5×#41—61no—
142C5×#41—61no—
143C5×#41—61no—
144C2×C5×Q8—31no—
145C2×C5×D4—31no—
146C2×C5×D4—61no—
147C22×C4×C5—61no—
148C22×C4×C5C22×D311yescharacteristic
149C22×Dic5#6712yes—
150C2×Dic10—21no—
151C2×Dic10—21no—
152C2×Dic10—21no—
153C2×Dic10—21no—
154C2×Dic10C4×D312yes—
155C2×Dic10D1212yes—
156C5×M16—61no—
157C5×M16—61no—
158C2×C5×C8—61no—
159C5×C4⋊C4—21no—
160C5×C22⋊C4—21no—
161C42×C5#6712yes—
162C22⋊Dic5—21no—
163C22⋊Dic5—61no—
164C22⋊Dic5—64no—
165C4⋊Dic5—21no—
166C4⋊Dic5—61no—
167#683—24no—
168#683—64no—
169C4×Dic5—24no—
170C4×Dic5—32no—
171C4×Dic5#6712yes—
172#681—61no—
173#681—61no—
174C2×#168—61no—
175#536—151no—
176#519—51no—
177#421—31no—
178#421—151no—
179C2×#101—151no—
180H(3,ℤ/4)—151no—
181H(3,ℤ/4)—151no—
182C22×C3×C5—41no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5#12311yescharacteristic
186C2×C5×D3—42no—
187C2×C5×D3—41no—
188C3×C4×C5—42no—
189C3×C4×C5C4×D411yescharacteristic
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5#12111yescharacteristic
192C3×C4×C5#12111yescharacteristic
193Dic15—42no—
194C3×Dic5—24no—
195C3×Dic5—42no—
196C5×Dic3—41no—
197C2×C3×Q8—52no—
198C2×C3×Q8—52no—
199C2×C3×Q8—101no—
200C2×C3×Q8—101no—
201C2×C3×Q8—101no—
202C2×C3×Q8—101no—
203C22×C3×C4—52no—
204C22×C3×C4C22×D511yescharacteristic
205C2×#67—21no—
206#235—21no—
207#235—21no—
208#235—21no—
209#235—21no—
210C2×D12C4×D511yescharacteristic
211C2×C4×D3—21no—
212C2×Dic6C4×D511yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4—104no—
215C3×C4⋊C4—101no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—101no—
218C3×C42—52no—
219C3×C42—104no—
220C3×C42C22⋊D512yes—
221#217—101no—
222(C2×D3)⋊C4—104no—
223C4⋊Dic3—101no—
224Dic3⋊C4—104no—
225C4×Dic3—52no—
226#208—21no—
227#208—21no—
228#208—101no—
229#208—101no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—101no—
232C23×C5—61no—
233C23×C5C2×#6711yescharacteristic
234C5×Q8—61no—
235C5×Q8—61no—
236C5×D4—61no—
237C5×D4—61no—
238C5×D4—121no—
239C5×D4—121no—
240C2×C4×C5—22no—
241C2×C4×C5—61no—
242C2×C4×C5—121no—
243C2×C4×C5—121no—
244C2×C4×C5C2×C4×D311yescharacteristic
245C2×C4×C5C2×D1211yescharacteristic
246C2×C4×C5#23511yescharacteristic
247C2×C4×C5#23511yescharacteristic
248C2×C4×C5C2×#6711yescharacteristic
249C2×C4×C5C2×#6711yescharacteristic
250C2×Dic5—22no—
251C2×Dic5—22no—
252C2×Dic5—22no—
253C2×Dic5—62no—
254C2×Dic5(C2×D3)⋊C414yes—
255Dic10—22no—
256Dic10—22no—
257Dic10—22no—
258Dic10—22no—
259C5×C8—121no—
260#168—121no—
261C2×#41—31no—
262C22×Q8—51no—
263C2×M16—31no—
264C2×M16—151no—
265C4⋊Q8—152no—
266#124—152no—
267#123—302no—
268C22⋊Q8—302no—
269C4×Q8—102no—
270C4×Q8—102no—
271#117—51no—
272#117—102no—
273#117—151no—
274#117D3×D511yescharacteristic
275#104—62no—
276#104—302no—
277#101—302no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38611yescharacteristic
282C5×D3—81no—
283C23×C3D4×D511yescharacteristic
284C22×D3—21no—
285C3×Q8—102no—
286C3×Q8—102no—
287C3×Q8—102no—
288C3×Q8—102no—
289C2×C3×C4—22no—
290C2×C3×C4—54no—
291C2×C3×C4—102no—
292C2×C3×C4—102no—
293C2×C3×C4—102no—
294C2×C3×C4C2×C4×D511yescharacteristic
295C2×C3×C4#70911yescharacteristic
296C2×C3×C4D4×D511yescharacteristic
297C2×C3×C4D4×D511yescharacteristic
298C2×C3×C4#71111yescharacteristic
299C2×C3×C4C2×C22⋊D511yescharacteristic
300#67—41no—
301#67—41no—
302C2×Dic3—21no—
303C2×Dic3—102no—
304D12—21no—
305D12—21no—
306C4×D3—41no—
307C4×D3—41no—
308Dic6—21no—
309Dic6—21no—
310C3⋊C8—41no—
311C3⋊C8—201no—
312C22×C5—41no—
313C22×C5—122no—
314C22×C5—121no—
315C22×C5C2×(C2×D3)⋊C411yescharacteristic
316C22×C5#92911yescharacteristic
317C22×C5#95311yescharacteristic
318C4×C5—42no—
319C4×C5—121no—
320C4×C5C4×#6711yescharacteristic
321C4×C5C4×#6711yescharacteristic
322C4×C5#93011yescharacteristic
323C4×C5#93011yescharacteristic
324Dic5—24no—
325Dic5—42no—
326Dic5—122no—
327#41—61no—
328#41—61no—
329#41—61no—
330#41—61no—
331C2×Q8—31no—
332C2×Q8—52no—
333C2×Q8—52no—
334C2×Q8—101no—
335C2×Q8—101no—
336C2×Q8—101no—
337C2×Q8—101no—
338C2×D4—31no—
339C2×D4—61no—
340C22×C4—52no—
341C22×C4—61no—
342C22×C4C2×D3×D511yescharacteristic
343M16—61no—
344M16—61no—
345M16—301no—
346M16—301no—
347C2×C8—61no—
348C2×C8—301no—
349C4⋊C4—21no—
350C4⋊C4—104no—
351C4⋊C4—101no—
352C4⋊C4—304no—
353C4⋊C4—301no—
354C22⋊C4—21no—
355C22⋊C4—101no—
356C22⋊C4—304no—
357C22⋊C4—301no—
358C42—52no—
359C42—104no—
360C42—152no—
361C42#120112yes—
362C3×C5#191211yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3D5×C22⋊C411yescharacteristic
365C22×C3#402911yescharacteristic
366C22×C3#408311yescharacteristic
367C2×D3—42no—
368C2×D3—41no—
369C3×C4—42no—
370C3×C4—104no—
371C3×C4—202no—
372C3×C4#402711yescharacteristic
373C3×C4#403011yescharacteristic
374C3×C4C4×C22⋊D511yescharacteristic
375C3×C4#408411yescharacteristic
376Dic3—41no—
377Dic3—202no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5—241no—
382C2×C5#521311yescharacteristic
383C23—61no—
384C23D5×#6711yescharacteristic
385Q8—61no—
386Q8—61no—
387Q8—102no—
388Q8—102no—
389Q8—102no—
390Q8—102no—
391D4—61no—
392D4—61no—
393D4—121no—
394D4—121no—
395C2×C4—22no—
396C2×C4—54no—
397C2×C4—61no—
398C2×C4—102no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—302no—
404C2×C4#677111yescharacteristic
405C2×C4#677211yescharacteristic
406C2×C4C4×D3×D511yescharacteristic
407C2×C4D5×D1211yescharacteristic
408C2×C4C2×#120111yescharacteristic
409C2×C4D5×#6711yescharacteristic
410C8—121no—
411C8—601no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3#6524511yescharacteristic
416D3—81no—
417C5#7166311yescharacteristic, complemented, Sylow
418C22—41no—
419C22—122no—
420C22—121no—
421C22#9144711yescharacteristic
422C22D5×(C2×D3)⋊C411yescharacteristic
423C22#9156311yescharacteristic
424C4—42no—
425C4—104no—
426C4—121no—
427C4—202no—
428C4—602no—
429C4C4×#120111yescharacteristic
430C4#9144011yescharacteristic
431C4#9144811yescharacteristic
432C4#9144911yescharacteristic
433C3#1060120411yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—241no—
438C2#1174890511yescharacteristic
subgroupG/H
centerC2#11748905
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1912
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
1. C3×#65850C2semidirect
5. C2×#90961C2semidirect
6. #11746352C2semidirect
7. #11745970C2semidirect
20. #91385C4semidirect
21. #91370C4semidirect
28. #65850D3semidirect
362. C3×C5#1912semidirect
417. C5#71663semidirect
433. C3#10601204semidirect

Maximal subgroups

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

subgroupindex
C3×#658502
#117542552
C5×#50052
#117484252
C2×#909612
#117463522
#117459702
#106012043
#716635