Finite Group Databasebeta

← #14288398

Subgroups — #14288398

Order 1920.

508 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65931C211yesmaximal, characteristic
2C5×#5033C211yesmaximal, characteristic, complemented
3#11748588C211yesmaximal, characteristic, complemented
4#11748370C211yesmaximal, characteristic, complemented
5#11746358C211yesmaximal, characteristic, complemented
6#11745955C211yesmaximal, characteristic, complemented
7#11745951C211yesmaximal, characteristic, complemented
8#10601960—31nomaximal, Hall
9C2×C3×#711—21no—
10C3×C5×#121C2211yescharacteristic
11C2×C5×C4⋊Dic3C2211yescharacteristic
12C3×#4085—21no—
13C3×#4085—21no—
14C3×#4076C2211yescharacteristic
15C2×C3×#683C2211yescharacteristic
16C3×#4042—21no—
17C3×#4030—21no—
18C3×#4024—21no—
19C2×#6669—21no—
20#91115C2211yescharacteristic
21#91114C2211yescharacteristic
22C5×#832—21no—
23C5×#830C2211yescharacteristic
24C5×#807—21no—
25#90994—21no—
26#90989—21no—
27#90964—21no—
28#71874—51nomaximal, Hall
29#65931D311yescharacteristic
30C5×#480—31no—
31#65269—31no—
32#65256—31no—
33#64757—31no—
34#64690—31no—
35#64689—31no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×Dic15—21no—
40C22×C5×Dic3—21no—
41C2×C3×C22⋊D5—21no—
42C2×C3×C22⋊D5—21no—
43C2×C3×C22⋊D5—21no—
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C22×C3×Dic5D411yescharacteristic
47C3×#711—42no—
48C3×#711—42no—
49C2×C3×D20D411yescharacteristic
50C2×C3×C4×D5—21no—
51C2×C3×Dic10—21no—
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C22⋊C4—21no—
54C2×C3⋊#168—21no—
55C5×C4⋊Dic3—21no—
56C5×C4⋊Dic3—21no—
57C2×C5×C3⋊C8—21no—
58C3×C22⋊Dic5—21no—
59C3×#685—21no—
60C3×#685—21no—
61C3×C4⋊Dic5D411yescharacteristic
62C3×#683—21no—
63C3×#683—21no—
64C3×#683—21no—
65#6669—42no—
66C3×#537—51no—
67#5033—51no—
68#5033D511yescharacteristic, complemented
69#4981—51no—
70#4537—51no—
71#4537—51no—
72#4528—51no—
73C2×#711—21no—
74C5×#121C2×D311yescharacteristic
75C2×C5×C4⋊C4—31no—
76#4085—21no—
77#4085—21no—
78#4076C2×D311yescharacteristic
79C2×#683—61no—
80C2×#683C2×D311yescharacteristic
81#4042—21no—
82#4030—21no—
83#4024—21no—
84C5×#107—61no—
85C5×#102—61no—
86C5×C22⋊C8—31no—
87#3963—31no—
88#3963—61no—
89C22⋊#168—31no—
90#3941—61no—
91#3939—61no—
92#1610—151noSylow
93C23×C3×C5—21no—
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—21no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×Dic15—22no—
104C2×Dic15—41no—
105C2×Dic15—41no—
106C2×C5×Dic3—41no—
107C2×C5×Dic3—41no—
108C2×C5×Dic3—41no—
109C3×C22⋊D5—42no—
110C3×C22⋊D5—41no—
111C3×C22⋊D5—41no—
112C3×C22⋊D5—81no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—42no—
118C2×C3×Dic5—41no—
119C2×C3×Dic5D812yes—
120C3×D20—41no—
121C3×C4×D5—42no—
122C3×Dic10—42no—
123C3⋊#168—41no—
124C5×C3⋊C8—41no—
125C2×C3×#41—101no—
126C3×C22⋊Q8—101no—
127C3×C22⋊Q8—101no—
128C3×#121—51no—
129C3×#121—101no—
130C3×#121—101no—
131C3×#121—101no—
132C3×#121C2×D511yescharacteristic
133C2×C3×C4⋊C4—51no—
134C2×C4⋊Dic3C2×D511yescharacteristic
135C2×Dic3⋊C4—101no—
136#832—21no—
137#832—101no—
138C3⋊#95—51no—
139C3⋊#95—101no—
140#830—51no—
141#830C2×D511yescharacteristic
142#807—21no—
143#807—101no—
144C2×C5×D4—21no—
145C2×C5×D4#6711yescharacteristic
146C22×C4×C5—61no—
147C22×C4×C5C22×D311yescharacteristic
148C2×C22⋊D5—21no—
149C2×C22⋊D5—21no—
150C2×C22⋊D5—21no—
151C22×Dic5—21no—
152C22×Dic5—61no—
153C22×Dic5#6711yescharacteristic
154C22×Dic5#6711yescharacteristic
155#711—42no—
156#711—42no—
157C2×D20#6711yescharacteristic
158C2×C4×D5—21no—
159C2×Dic10—21no—
160C2×C5×C8—61no—
161C5×C4⋊C4—61no—
162C5×C4⋊C4—61no—
163C5×C4⋊C4#6711yescharacteristic
164C5×C22⋊C4—21no—
165C22⋊Dic5—21no—
166#685—21no—
167#685—21no—
168C4⋊Dic5#6711yescharacteristic
169#683—21no—
170#683—21no—
171#683—21no—
172#683—122no—
173C2×#168—61no—
174#537—51no—
175#480—31no—
176#480—151no—
177#398—151no—
178#327—151no—
179#327—151no—
180#324—151no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5—41no—
186C22×C3×C5#12011yescharacteristic
187C22×C3×C5C2×D811yescharacteristic
188C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
189C2×C3×D5—41no—
190C2×C3×D5—41no—
191C2×C3×D5—41no—
192C3×C4×C5—21no—
193C3×C4×C5—21no—
194C3×C4×C5—41no—
195Dic15—42no—
196C3×Dic5—22no—
197C3×Dic5—41no—
198C3×Dic5—41no—
199C5×Dic3—81no—
200C3×#41—202no—
201C3×#41—202no—
202C2×C3×Q8—101no—
203C2×C3×D4—21no—
204C2×C3×D4—51no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C2×C3×D4C22⋊D511yescharacteristic
209C22×C3×C4—51no—
210C22×C3×C4—51no—
211C22×C3×C4—101no—
212C22×C3×C4—101no—
213C22×C3×C4C22×D511yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—101no—
216C3×C4⋊C4—51no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4—101no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4C22⋊D511yescharacteristic
221C3×C22⋊C4—21no—
222C3×C22⋊C4—101no—
223C3×C22⋊C4—101no—
224C3×C22⋊C4—101no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227Dic3⋊C4—202no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—101no—
230C23×C5—21no—
231C23×C5C2×#6711yescharacteristic
232C22×D5—21no—
233C5×D4—41no—
234C5×D4—41no—
235C5×D4—41no—
236C2×C4×C5—21no—
237C2×C4×C5—41no—
238C2×C4×C5—121no—
239C2×C4×C5—121no—
240C2×C4×C5—121no—
241C2×C4×C5C2×#6711yescharacteristic
242C2×C4×C5C2×#6711yescharacteristic
243C22⋊D5—42no—
244C22⋊D5—41no—
245C22⋊D5—41no—
246C22⋊D5—81no—
247C2×Dic5—21no—
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—21no—
251C2×Dic5—42no—
252C2×Dic5—41no—
253C2×Dic5—62no—
254C2×Dic5—121no—
255C2×Dic5—121no—
256C2×Dic5#21312yes—
257D20—41no—
258C4×D5—42no—
259Dic10—42no—
260C5×C8—121no—
261#168—121no—
262C2×#41—101no—
263C22⋊Q8—101no—
264C22⋊Q8—101no—
265#121—51no—
266#121—101no—
267#121—101no—
268#121—101no—
269#121D3×D511yescharacteristic
270C2×C4⋊C4—31no—
271C2×C4⋊C4—51no—
272C2×C4⋊C4—301no—
273#107—61no—
274#107—301no—
275#102—61no—
276#102—301no—
277C22⋊C8—31no—
278C22⋊C8—151no—
279#95—151no—
280#95—301no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5—81no—
284C2×C3×C5#44711yescharacteristic
285C2×C3×C5#45611yescharacteristic
286C2×C3×C5#45811yescharacteristic
287C3×D5—81no—
288C23×C3—21no—
289C23×C3—101no—
290C23×C3D4×D511yescharacteristic
291C3×Q8—202no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—201no—
296C3×D4—202no—
297C3×D4—201no—
298C3×D4—201no—
299C3×D4—401no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—52no—
303C2×C3×C4—101no—
304C2×C3×C4—101no—
305C2×C3×C4—101no—
306C2×C3×C4—101no—
307C2×C3×C4—202no—
308C2×C3×C4—202no—
309C2×C3×C4—201no—
310C2×C3×C4D4×D511yescharacteristic
311C2×C3×C4C2×C22⋊D511yescharacteristic
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—102no—
316C2×Dic3—201no—
317C2×Dic3—201no—
318C3⋊C8—41no—
319C3⋊C8—201no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—41no—
323C22×C5—41no—
324C22×C5—41no—
325C22×C5C2×#21311yescharacteristic
326C22×C5#93211yescharacteristic
327C22×C5#95311yescharacteristic
328C2×D5—41no—
329C2×D5—41no—
330C2×D5—41no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—41no—
334C4×C5—241no—
335Dic5—22no—
336Dic5—41no—
337Dic5—41no—
338Dic5—122no—
339#41—202no—
340#41—202no—
341C2×Q8—101no—
342C2×D4—21no—
343C2×D4—51no—
344C2×D4—101no—
345C2×D4—101no—
346C2×D4—101no—
347C2×D4#120111yescharacteristic
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—61no—
351C22×C4—101no—
352C22×C4—101no—
353C22×C4—301no—
354C22×C4C2×D3×D511yescharacteristic
355C2×C8—61no—
356C2×C8—301no—
357C4⋊C4—51no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—101no—
361C4⋊C4—101no—
362C4⋊C4—101no—
363C4⋊C4—602no—
364C4⋊C4#120111yescharacteristic
365C22⋊C4—21no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—101no—
369C3×C5#161011yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—201no—
376C22×C3—201no—
377C22×C3—201no—
378C22×C3D5×D811yescharacteristic
379C22×C3#405711yescharacteristic
380C22×C3#408311yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—102no—
385C3×C4—201no—
386C3×C4—201no—
387Dic3—81no—
388Dic3—202no—
389C2×C5—21no—
390C2×C5—21no—
391C2×C5—81no—
392C2×C5#521711yescharacteristic
393C2×C5#524311yescharacteristic
394C2×C5#524611yescharacteristic
395D5—81no—
396C23—21no—
397C23—101no—
398C23D5×#6711yescharacteristic
399Q8—202no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—201no—
404D4—202no—
405D4—201no—
406D4—201no—
407D4—401no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—52no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—202no—
419C2×C4—202no—
420C2×C4—201no—
421C2×C4—302no—
422C2×C4—601no—
423C2×C4—601no—
424C2×C4C2×#120111yescharacteristic
425C2×C4D5×#6711yescharacteristic
426C8—121no—
427C8—601no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—81no—
431C2×C3—401no—
432C2×C3#6534211yescharacteristic
433C2×C3#6544511yescharacteristic
434C2×C3#6549411yescharacteristic
435C5#7187411yescharacteristic, complemented, Sylow
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—201no—
442C22—201no—
443C22—201no—
444C22D5×#21311yescharacteristic
445C22#9148911yescharacteristic
446C22#9156311yescharacteristic
447C4—21no—
448C4—21no—
449C4—41no—
450C4—102no—
451C4—201no—
452C4—201no—
453C4—241no—
454C4—602no—
455C3#1060196011yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—81no—
459C2—401no—
460C2#1174902111yescharacteristic
461C2#1174918111yescharacteristic
462C2#1174946211yescharacteristic
subgroupG/H
centerC22#91563
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#121C22
socleC22×C3×C5#120
prime radicalC3×#65931C2
hypercenter#121D3×D5
hypocenterC3×C5#1610
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C5×#5033C2semidirect
3. #11748588C2semidirect
4. #11748370C2semidirect
5. #11746358C2semidirect
6. #11745955C2semidirect
7. #11745951C2semidirect
68. #5033D5semidirect
369. C3×C5#1610semidirect
435. C5#71874semidirect
455. C3#10601960semidirect

Maximal subgroups

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

subgroupindex
C3×#659312
C5×#50332
#117485882
#117483702
#117463582
#117459552
#117459512
#106019603
#718745