Finite Group Databasebeta

← #14287878

Subgroups — #14287878

Order 1920.

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

normal subgroups only

This group has 532 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×#65957C211yesmaximal, characteristic, complemented
2#11751147C211yesmaximal, characteristic
3#11748769C211yesmaximal, characteristic
4#11748551C211yesmaximal, characteristic
5C5×#4538C211yesmaximal, characteristic, complemented
6#11745961C211yesmaximal, characteristic, complemented
7#11745951C211yesmaximal, characteristic, complemented
8#10599118—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C3×C5×C22⋊Q8C2211yescharacteristic
11C2×#6717—21no—
12C3×#4090—21no—
13C3×#4070C2211yescharacteristic
14C2×C3×#683C2211yescharacteristic
15C3×#4035—21no—
16C3×#4035—21no—
17C3×#4034—21no—
18C3×#4024—21no—
19C3×#4023—21no—
20C2×#6667—21no—
21C2×#6667—21no—
22C2×#6658—21no—
23#91114C2211yescharacteristic
24#91093—21no—
25C5×C3⋊#95C2211yescharacteristic
26C5×#830C2211yescharacteristic
27#90968—21no—
28#71902—51nomaximal, Hall
29#65957D311yescharacteristic, complemented
30#65335—31no—
31#65020—31no—
32#64961—31no—
33C5×#328—31no—
34#64689—31no—
35#64672—31no—
36C23×D15—21no—
37C2×C3×C5×Q8D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C2×D60—21no—
40C2×D60—21no—
41C2×D60—42no—
42C22×C5×Dic3—21no—
43C22×C3×Dic5D411yescharacteristic
44C22×C3×Dic5D411yescharacteristic
45C2×C3×Dic10—21no—
46C2×C3×Dic10D411yescharacteristic
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C22⋊C4—21no—
50#6717—42no—
51C2×C3⋊#168—21no—
52C2×C5×C3⋊C8—21no—
53C3×C22⋊Dic5—21no—
54C3×C4⋊Dic5—21no—
55C3×C4⋊Dic5D411yescharacteristic
56C3×#683—21no—
57C3×#683—21no—
58C3×#683—21no—
59C3×#683—21no—
60C3×#683—21no—
61C3×#683—21no—
62C3×#683—21no—
63C3×C4×Dic5—21no—
64C3×C4×Dic5—21no—
65#6667—42no—
66#6667—42no—
67#6658—42no—
68C3×#544—51no—
69#5046—51no—
70#5046—51no—
71#4991—51no—
72#4538—51no—
73#4538D511yescharacteristic, complemented
74#4528—51no—
75C22×D20—31no—
76C5×C22⋊Q8C2×D311yescharacteristic
77#4090—21no—
78C2×#688—61no—
79C2×#685—61no—
80C2×#685—61no—
81#4070C2×D311yescharacteristic
82C2×#683C2×D311yescharacteristic
83C2×#677—61no—
84#4035—21no—
85#4035—21no—
86#4034—21no—
87#4024—21no—
88#4023—21no—
89C5×C22⋊C8—31no—
90C5×#95—31no—
91C22⋊#168—31no—
92#3953—61no—
93#3941—61no—
94#1599—151noSylow
95C23×C3×C5C2×D411yescharacteristic
96C22×D15—22no—
97C22×D15—41no—
98C22×D15—41no—
99C22×D15—41no—
100C22×D15—41no—
101C22×D15—41no—
102C22×D15—41no—
103C3×C5×Q8—41no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109D60—22no—
110D60—22no—
111D60—41no—
112D60—41no—
113C2×C5×Dic3—41no—
114C2×C5×Dic3—41no—
115C2×C5×Dic3—41no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—41no—
122C2×C3×Dic5SD1612yescomplemented
123C3×Dic10—41no—
124C3×Dic10—81no—
125C3⋊#168—41no—
126C5×C3⋊C8—41no—
127C22×D12—51no—
128C3×C4⋊Q8—101no—
129C3×C4⋊Q8—101no—
130C3×#125—101no—
131C3×#125—101no—
132C3×C22⋊Q8—51no—
133C3×C22⋊Q8—101no—
134C3×C22⋊Q8C2×D511yescharacteristic
135C3×#117—101no—
136C2×C3×C4⋊C4—51no—
137C2×#215—101no—
138C2×#215—101no—
139C2×(C2×D3)⋊C4—101no—
140C2×(C2×D3)⋊C4—101no—
141C3⋊#95C2×D511yescharacteristic
142#830—51no—
143#830C2×D511yescharacteristic
144#809—101no—
145#809—101no—
146C23×D5—61no—
147C2×C5×Q8#6711yescharacteristic
148C22×C4×C5—61no—
149C22×C4×C5C22×D311yescharacteristic
150C22×Dic5#6711yescharacteristic
151C22×Dic5#6711yescharacteristic
152C2×D20—61no—
153C2×D20—61no—
154C2×D20—122no—
155C2×Dic10—21no—
156C2×Dic10#6711yescharacteristic
157C2×C5×C8—61no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4#6711yescharacteristic
160C5×C22⋊C4—21no—
161C22⋊Dic5—21no—
162#688—122no—
163#685—122no—
164#685—122no—
165C4⋊Dic5—21no—
166C4⋊Dic5#6711yescharacteristic
167#683—21no—
168#683—21no—
169#683—21no—
170#683—21no—
171#683—21no—
172#683—21no—
173#683—21no—
174C4×Dic5—21no—
175C4×Dic5—21no—
176C2×#168—61no—
177#677—122no—
178#544—51no—
179#450—151no—
180#450—151no—
181#394—151no—
182#328—31no—
183#328—151no—
184#324—151no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5#12011yescharacteristic
188C22×C3×C5C2×SD1611yescharacteristic
189C22×C3×C5#13711yescharacteristic
190C2×D15—42no—
191C2×D15—42no—
192C2×D15—42no—
193C2×D15—81no—
194C2×D15—81no—
195C2×D15—81no—
196C2×D15—81no—
197C3×C4×C5—21no—
198C3×C4×C5—21no—
199C3×C4×C5—41no—
200C3×C4×C5—41no—
201C3×Dic5—22no—
202C3×Dic5—41no—
203C3×Dic5—41no—
204C3×Dic5—41no—
205C5×Dic3—81no—
206C23×D3—101no—
207C2×C3×Q8—51no—
208C2×C3×Q8—101no—
209C2×C3×Q8D2011yescharacteristic
210C22×C3×C4—51no—
211C22×C3×C4—51no—
212C22×C3×C4C22×D511yescharacteristic
213C22×Dic3—21no—
214C2×D12—101no—
215C2×D12—101no—
216C2×D12—202no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—51no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4—101no—
222C3×C4⋊C4—101no—
223C3×C4⋊C4—101no—
224C3×C4⋊C4—101no—
225C3×C4⋊C4—101no—
226C3×C4⋊C4—101no—
227C3×C4⋊C4D2011yescharacteristic
228C3×C22⋊C4—21no—
229C3×C22⋊C4—101no—
230C3×C42—101no—
231C3×C42—101no—
232#215—202no—
233#215—202no—
234(C2×D3)⋊C4—202no—
235(C2×D3)⋊C4—202no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—101no—
238C23×C5C2×#6711yescharacteristic
239C22×D5—62no—
240C22×D5—121no—
241C22×D5—121no—
242C22×D5—121no—
243C22×D5—121no—
244C22×D5—121no—
245C22×D5—121no—
246C5×Q8—41no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—41no—
250C2×C4×C5—121no—
251C2×C4×C5—121no—
252C2×C4×C5—121no—
253C2×C4×C5C2×#6711yescharacteristic
254C2×C4×C5C2×#6711yescharacteristic
255C2×Dic5—21no—
256C2×Dic5—21no—
257C2×Dic5—21no—
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5—41no—
261C2×Dic5#21512yescomplemented
262D20—62no—
263D20—62no—
264D20—121no—
265D20—121no—
266Dic10—41no—
267Dic10—81no—
268C5×C8—121no—
269#168—121no—
270C22×D4—151no—
271C2×SD16—301no—
272C2×SD16—301no—
273C4⋊Q8—101no—
274C4⋊Q8—101no—
275#125—101no—
276#125—101no—
277C22⋊Q8—51no—
278C22⋊Q8—101no—
279C22⋊Q8D3×D511yescharacteristic
280#117—101no—
281C2×C4⋊C4—51no—
282C2×C22⋊C4—301no—
283C2×C22⋊C4—301no—
284#102—301no—
285#102—301no—
286C22⋊C8—31no—
287C22⋊C8—151no—
288#95—31no—
289C2×C3×C5—21no—
290C2×C3×C5—21no—
291C2×C3×C5#44811yescharacteristic
292C2×C3×C5#45411yescharacteristic
293C2×C3×C5B(4,𝔽2)11yescharacteristic
294D15—82no—
295C23×C3D4×D511yescharacteristic
296C22×D3—102no—
297C22×D3—201no—
298C22×D3—201no—
299C22×D3—201no—
300C22×D3—201no—
301C22×D3—201no—
302C22×D3—201no—
303C3×Q8—41no—
304C3×Q8—201no—
305C3×Q8—401no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—41no—
309C2×C3×C4—52no—
310C2×C3×C4—101no—
311C2×C3×C4—101no—
312C2×C3×C4—101no—
313C2×C3×C4—101no—
314C2×C3×C4—101no—
315C2×C3×C4—201no—
316C2×C3×C4C2×D2011yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321D12—102no—
322D12—102no—
323D12—201no—
324D12—201no—
325C3⋊C8—41no—
326C3⋊C8—201no—
327C22×C5—21no—
328C22×C5—21no—
329C22×C5C2×#21511yescharacteristic
330C22×C5#94211yescharacteristic
331C22×C5#95311yescharacteristic
332C2×D5—122no—
333C2×D5—122no—
334C2×D5—122no—
335C2×D5—241no—
336C2×D5—241no—
337C2×D5—241no—
338C2×D5—241no—
339C4×C5—21no—
340C4×C5—21no—
341C4×C5—41no—
342C4×C5—41no—
343C4×C5—241no—
344Dic5—22no—
345Dic5—41no—
346Dic5—41no—
347Dic5—41no—
348C24—301no—
349C2×Q8—51no—
350C2×Q8—101no—
351C2×Q8#120211yescharacteristic
352C2×D4—301no—
353C2×D4—301no—
354C2×D4—602no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—61no—
358C22×C4C2×D3×D511yescharacteristic
359SD16—602no—
360SD16—602no—
361C2×C8—61no—
362C2×C8—301no—
363C4⋊C4—21no—
364C4⋊C4—51no—
365C4⋊C4—101no—
366C4⋊C4—101no—
367C4⋊C4—101no—
368C4⋊C4—101no—
369C4⋊C4—101no—
370C4⋊C4—101no—
371C4⋊C4—101no—
372C4⋊C4—101no—
373C4⋊C4#120211yescharacteristic
374C22⋊C4—21no—
375C22⋊C4—101no—
376C22⋊C4—602no—
377C22⋊C4—602no—
378C42—101no—
379C42—101no—
380C3×C5#159911yescharacteristic, complemented, Hall
381C22×C3—21no—
382C22×C3—21no—
383C22×C3#402811yescharacteristic
384C22×C3D5×SD1611yescharacteristic
385C22×C3#406411yescharacteristic
386C2×D3—202no—
387C2×D3—202no—
388C2×D3—202no—
389C2×D3—401no—
390C2×D3—401no—
391C2×D3—401no—
392C2×D3—401no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—102no—
398C3×C4—201no—
399C3×C4—201no—
400C3×C4—201no—
401Dic3—81no—
402C2×C5—21no—
403C2×C5—21no—
404C2×C5#522711yescharacteristic
405C2×C5#524411yescharacteristic
406C2×C5#524511yescharacteristic
407D5—242no—
408C23—302no—
409C23—601no—
410C23—601no—
411C23—601no—
412C23—601no—
413C23—601no—
414C23—601no—
415C23D5×#6711yescharacteristic
416Q8—41no—
417Q8—201no—
418Q8—401no—
419D4—302no—
420D4—302no—
421D4—601no—
422D4—601no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—41no—
426C2×C4—52no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—201no—
436C2×C4C2×#120211yescharacteristic
437C2×C4D5×#6711yescharacteristic
438C8—121no—
439C8—601no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3#6503011yescharacteristic
443C2×C3#6509511yescharacteristic
444C2×C3#6511611yescharacteristic
445D3—402no—
446C5#7190211yescharacteristic, complemented, Sylow
447C22—21no—
448C22—21no—
449C22—602no—
450C22—602no—
451C22—602no—
452C22—1201no—
453C22—1201no—
454C22—1201no—
455C22—1201no—
456C22D5×#21511yescharacteristic
457C22#9151511yescharacteristic
458C22#9156411yescharacteristic
459C4—21no—
460C4—21no—
461C4—41no—
462C4—41no—
463C4—102no—
464C4—201no—
465C4—201no—
466C4—201no—
467C4—241no—
468C3#1059911811yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—1202no—
472C2#1174917611yescharacteristic
473C2#1174926811yescharacteristic
474C2#1174945711yescharacteristic
subgroupG/H
centerC22#91564
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC3×C5×C22⋊Q8C22
socleC22×C3×C5#120
prime radicalC22×D60C22
hypercenterC22⋊Q8D3×D5
hypocenterC3×C5#1599
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#65957C2semidirect
5. C5×#4538C2semidirect
6. #11745961C2semidirect
7. #11745951C2semidirect
29. #65957D3semidirect
73. #4538D5semidirect
122. C2×C3×Dic5SD16semidirect
261. C2×Dic5#215semidirect
380. C3×C5#1599semidirect
446. C5#71902semidirect
468. C3#10599118semidirect

Maximal subgroups

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

subgroupindex
C3×#659572
#117511472
#117487692
#117485512
C5×#45382
#117459612
#117459512
#105991183
#719025