Finite Group Databasebeta

← #14302798

Subgroups — #14302798

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×#66079C211yesmaximal, characteristic
2#11754251C211yesmaximal, characteristic
3C5×#5120C211yesmaximal, characteristic, complemented
4#11748624C211yesmaximal, characteristic
5#11748409C211yesmaximal, characteristic, complemented
6#11746378C211yesmaximal, characteristic, complemented
7#11745971C211yesmaximal, characteristic, complemented
8#10601143—31nomaximal, Hall
9C22×C5×Dic6C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13#91815C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16C3×#4053—22no—
17C3×D5×M16—22no—
18C3×#4047—22no—
19#91536—22no—
20#91385C412yescomplemented
21#91368C412yescomplemented
22#91359—22no—
23C5×#816—22no—
24#90995—22no—
25#90973—22no—
26#72024—51nomaximal, Hall
27#66150—31no—
28#66079D311yescharacteristic
29C5×#419—31no—
30#65514—31no—
31#65295—31no—
32#64777—31no—
33#64751—31no—
34C22×C3×C4×C5C2311yescharacteristic
35C22×C5×Dic3—21no—
36C2×C5×Dic6—21no—
37C2×C5×Dic6—21no—
38C2×C5×Dic6—21no—
39C2×C5×Dic6—21no—
40C2×C5×Dic6—21no—
41C2×C5×Dic6C2×C412yes—
42C2×C3×C22⋊D5D412yes—
43C3×#709—21no—
44C3×#709—21no—
45C3×#709—21no—
46C3×#709—21no—
47C2×C3×D20C2×C411yescharacteristic
48C2×C3×C4×D5D412yes—
49C2×C3×Dic10C2×C411yescharacteristic
50C3×C5×M16—21no—
51C3×C5×M16—21no—
52C2×C3×C5×C8D412yes—
53#6719—21no—
54#6719—21no—
55C4⋊Dic15—21no—
56C4⋊Dic15—21no—
57C4×Dic15—21no—
58C4×Dic15C2×C412yes—
59C5×#217—21no—
60C5×C4⋊Dic3—21no—
61C4×C5×Dic3—21no—
62C3×#681—21no—
63C3×#681—21no—
64C2×C3×#168D412yes—
65C3×#676—24no—
66C3×C8×D5—24no—
67#6669—24no—
68#6665—24no—
69C3×#568—51no—
70#5825—51no—
71#5240—51no—
72#5120—51no—
73#5120D511yescharacteristic, complemented
74#4557—51no—
75#4557—51no—
76C22×C5×Q8—31no—
77C2×#709C2×D311yescharacteristic
78C2×C5×M16C2×D311yescharacteristic
79C5×#117—31no—
80#4093—32no—
81#4092—62no—
82#4090—32no—
83#4075—62no—
84#4072—31no—
85#4072—31no—
86C2×#681C2×D311yescharacteristic
87#4053—22no—
88D5×M16—22no—
89#4047—22no—
90C5×#103—62no—
91#3967—62no—
92#3938—62no—
93#1909—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5C22⋊C412yes—
96C2×C3×C4×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×Dic15—22no—
103C2×Dic15—41no—
104C2×C5×Dic3—22no—
105C2×C5×Dic3—41no—
106C2×C5×Dic3—41no—
107C2×C5×Dic3—41no—
108C5×Dic6—22no—
109C5×Dic6—22no—
110C5×Dic6—41no—
111C5×Dic6—41no—
112C3×C22⋊D5—22no—
113C3×C22⋊D5—22no—
114C2×C3×Dic5C22⋊C412yes—
115C3×D20—21no—
116C3×D20—21no—
117C3×C4×D5—22no—
118C3×C4×D5—22no—
119C3×Dic10—21no—
120C3×Dic10—21no—
121C3×C5×C8—22no—
122C3×#168—22no—
123C2×C3×#41—51no—
124C22×Dic6C2×D511yescharacteristic
125C3×#131—102no—
126C3×#131—102no—
127C2×C3×M16—51no—
128C2×C3×M16—102no—
129C2×C3×M16C2×D511yescharacteristic
130#944—52no—
131#936—52no—
132#935—102no—
133#926—51no—
134#926—51no—
135#926C2×D511yescharacteristic
136#924—102no—
137#834—102no—
138#816—22no—
139#816—102no—
140C2×C5×Q8—32no—
141C2×C5×Q8—61no—
142C2×C5×Q8—61no—
143C2×C5×Q8—61no—
144C2×C5×Q8—61no—
145C2×C5×Q8—61no—
146C22×C4×C5—61no—
147C22×C4×C5C22×D311yescharacteristic
148C2×C22⋊D5D1212yes—
149#709—21no—
150#709—21no—
151#709—21no—
152#709—21no—
153C2×D20C4×D311yescharacteristic
154C2×C4×D5#6712yes—
155C2×Dic10C4×D311yescharacteristic
156C5×M16—21no—
157C5×M16—21no—
158C2×C5×C8#6712yes—
159C5×C4⋊C4—61no—
160C5×C22⋊C4—61no—
161C42×C5—61no—
162C22⋊Dic5—61no—
163C22⋊Dic5—61no—
164#685—64no—
165C4⋊Dic5—61no—
166C4⋊Dic5—61no—
167#683—64no—
168C4×Dic5—32no—
169C4×Dic5—61no—
170#681—21no—
171#681—21no—
172C2×#168#6712yes—
173#676—24no—
174C8×D5—24no—
175#568—51no—
176#536—151no—
177#419—31no—
178#419—151no—
179#410—151no—
180H(3,ℤ/4)—151no—
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×C5C4×D411yescharacteristic
189C3×C4×C5C4×D411yescharacteristic
190C3×C4×C5#12011yescharacteristic
191C3×C4×C5#12311yescharacteristic
192Dic15—42no—
193Dic15—81no—
194C3×Dic5—22no—
195C5×Dic3—42no—
196C5×Dic3—81no—
197C3×#41—101no—
198C3×#41—101no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—51no—
202C2×C3×D4—52no—
203C2×C3×D4—51no—
204C22×C3×C4—52no—
205C22×C3×C4C22×D511yescharacteristic
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C4×D512yes—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—104no—
216C3×M16—101no—
217C3×M16—101no—
218C2×C3×C8—52no—
219C2×C3×C8—104no—
220C2×C3×C8C22⋊D512yes—
221#217—21no—
222#217—104no—
223#217—101no—
224#217—101no—
225C4⋊Dic3—21no—
226C4⋊Dic3—101no—
227C4⋊Dic3—101no—
228Dic3⋊C4—104no—
229C4×Dic3—21no—
230C4×Dic3—52no—
231C4×Dic3—101no—
232C23×C5C2×D1211yescharacteristic
233C22×D5(C2×D3)⋊C412yes—
234C5×Q8—62no—
235C5×Q8—62no—
236C5×Q8—121no—
237C5×Q8—121no—
238C2×C4×C5—62no—
239C2×C4×C5—121no—
240C2×C4×C5—121no—
241C2×C4×C5—121no—
242C2×C4×C5C2×C4×D311yescharacteristic
243C2×C4×C5#23511yescharacteristic
244C2×C4×C5#23511yescharacteristic
245C2×C4×C5C2×#6711yescharacteristic
246C2×C4×C5C2×#6711yescharacteristic
247C2×C4×C5C2×#6711yescharacteristic
248C22⋊D5—22no—
249C22⋊D5—22no—
250C2×Dic5—62no—
251C2×Dic5—121no—
252C2×Dic5(C2×D3)⋊C412yes—
253D20—21no—
254D20—21no—
255C4×D5—22no—
256C4×D5—22no—
257Dic10—21no—
258Dic10—21no—
259C5×C8—22no—
260#168—22no—
261C2×#41—51no—
262C22×Q8—31no—
263#131—102no—
264#131—102no—
265C2×M16—51no—
266C2×M16—102no—
267C2×M16D3×D511yescharacteristic
268C4⋊Q8—152no—
269#124—152no—
270#123—302no—
271C22⋊Q8—302no—
272#117—31no—
273#117—151no—
274#117—151no—
275#103—62no—
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×D4—102no—
288C3×D4—102no—
289C3×D4—101no—
290C3×D4—101no—
291C2×C3×C4—52no—
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
300C2×Dic3—22no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—102no—
305C2×Dic3—201no—
306Dic6—22no—
307Dic6—22no—
308Dic6—41no—
309Dic6—41no—
310C3×C8—22no—
311C3×C8—102no—
312C22×C5—41no—
313C22×C5C2×(C2×D3)⋊C411yescharacteristic
314C22×C5#93011yescharacteristic
315C22×C5#93011yescharacteristic
316C2×D5—22no—
317C2×D5—42no—
318C4×C5—122no—
319C4×C5—241no—
320C4×C5C4×#6711yescharacteristic
321C4×C5C4×#6711yescharacteristic
322C4×C5#92911yescharacteristic
323C4×C5#95311yescharacteristic
324Dic5—22no—
325Dic5—122no—
326Dic5—241no—
327#41—101no—
328#41—101no—
329#41—101no—
330#41—101no—
331C2×Q8—32no—
332C2×Q8—51no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×D4—52no—
339C2×D4—51no—
340C22×C4—52no—
341C22×C4—61no—
342C22×C4C2×D3×D511yescharacteristic
343M16—21no—
344M16—21no—
345M16—104no—
346M16—101no—
347M16—101no—
348C2×C8—52no—
349C2×C8—104no—
350C2×C8#120112yes—
351C4⋊C4—61no—
352C4⋊C4—304no—
353C4⋊C4—301no—
354C4⋊C4—301no—
355C22⋊C4—61no—
356C22⋊C4—304no—
357C22⋊C4—301no—
358C22⋊C4—301no—
359C42—61no—
360C42—152no—
361C42—301no—
362C3×C5#190911yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3—102no—
365C22×C3—202no—
366C22×C3D5×C22⋊C411yescharacteristic
367C22×C3#403011yescharacteristic
368C22×C3#408411yescharacteristic
369C3×C4—102no—
370C3×C4#402711yescharacteristic
371C3×C4#402911yescharacteristic
372C3×C4C4×C22⋊D511yescharacteristic
373C3×C4#408311yescharacteristic
374Dic3—42no—
375Dic3—81no—
376Dic3—202no—
377Dic3—401no—
378C2×C5—21no—
379C2×C5—21no—
380C2×C5—21no—
381C2×C5#521311yescharacteristic
382D5—42no—
383C23—52no—
384C23D5×D1211yescharacteristic
385Q8—62no—
386Q8—62no—
387Q8—101no—
388Q8—101no—
389Q8—121no—
390Q8—121no—
391D4—102no—
392D4—102no—
393D4—101no—
394D4—101no—
395C2×C4—52no—
396C2×C4—62no—
397C2×C4—102no—
398C2×C4—102no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—302no—
403C2×C4—601no—
404C2×C4#677111yescharacteristic
405C2×C4#677211yescharacteristic
406C2×C4C4×D3×D511yescharacteristic
407C2×C4C2×#120111yescharacteristic
408C2×C4D5×#6711yescharacteristic
409C2×C4D5×#6711yescharacteristic
410C8—22no—
411C8—102no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—202no—
416C2×C3#6524511yescharacteristic
417C5#7202411yescharacteristic, complemented, Sylow
418C22—41no—
419C22—102no—
420C22—202no—
421C22#9144811yescharacteristic
422C22#9144911yescharacteristic
423C22D5×(C2×D3)⋊C411yescharacteristic
424C4—102no—
425C4—122no—
426C4—241no—
427C4—602no—
428C4—1201no—
429C4C4×#120111yescharacteristic
430C4#9144011yescharacteristic
431C4#9144711yescharacteristic
432C4#9156311yescharacteristic
433C3#1060114311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—21no—
437C2—202no—
438C2#1174890511yescharacteristic
subgroupG/H
centerC2#11748905
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1909
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C5×#5120C2semidirect
5. #11748409C2semidirect
6. #11746378C2semidirect
7. #11745971C2semidirect
20. #91385C4semidirect
21. #91368C4semidirect
73. #5120D5semidirect
362. C3×C5#1909semidirect
417. C5#72024semidirect
433. C3#10601143semidirect

Maximal subgroups

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

subgroupindex
C3×#660792
#117542512
C5×#51202
#117486242
#117484092
#117463782
#117459712
#106011433
#720245