Finite Group Databasebeta

← #14471544

Subgroups — #14471544

Order 1920.

740 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
1Dic6×D20C211yesmaximal, characteristic
2#11752490C211yesmaximal, characteristic, complemented
3#11750782C211yesmaximal, characteristic, complemented
4C5×C8×Dic6C211yesmaximal, characteristic, complemented
5C3×C4×#677C211yesmaximal, characteristic
6#11748028C212yesmaximal, complemented
7#11747867C212yesmaximal, complemented
8#11747753C212yesmaximal
9#11747739C212yesmaximal
10#11747062C211yesmaximal, characteristic
11#11746943C211yesmaximal, characteristic, complemented
12#10611656—31nomaximal, Hall
13C2×D5×Dic6—21no—
14#91758C2211yescharacteristic
15#91758C2211yescharacteristic
16C4×C5×Dic6C2211yescharacteristic
17C2×C3×#677C2211yescharacteristic
18C3×C4×D20C2211yescharacteristic
19C3×C4×Dic10C2211yescharacteristic
20#91469C2212yes—
21Dic3×D20C2212yes—
22#91422—22no—
23#91389C2212yes—
24#91388—21no—
25#91360—21no—
26#91343—22no—
27#91340C2212yes—
28Dic5×Dic6—21no—
29C2×#6657C2212yes—
30C3×C4×C5×C8C2211yescharacteristic
31#91100C2212yes—
32#91100C2212yes—
33C5×Dic3⋊C8C2212yes—
34C5×C8×Dic3C2212yes—
35C5×C4⋊(C3⋊C8)C2211yescharacteristic
36C3×#3953C2211yescharacteristic
37C3×#3949C2211yescharacteristic
38C3×#3948C2211yescharacteristic
39#90987C2212yes—
40#90974C2212yes—
41#90971C2212yes—
42#79619—51nomaximal, Hall
43Q8×D20—31no—
44#65904—31no—
45C5×C8×Q8—31no—
46#65140—31no—
47#65140—32no—
48#65130—32no—
49#65130—31no—
50C4×#677—32no—
51C4×#677D311yescharacteristic
52#64970—31no—
53#64970—32no—
54C2×Dic30—21no—
55C2×Dic30C2312yes—
56C2×C5×Dic6D411yescharacteristic
57C2×C3×D20C2311yescharacteristic
58C2×C3×C4×D5—21no—
59C2×C3×Dic10C2311yescharacteristic
60C2×#1204—22no—
61C2×D5×Dic3—22no—
62D5×Dic6—44no—
63C2×C3×C5×C8C2311yescharacteristic
64C2×C3×C5×C8C2311yescharacteristic
65C3×C42×C5C2311yescharacteristic
66C4⋊Dic15—21no—
67C4⋊Dic15—21no—
68C4⋊Dic15—21no—
69C4⋊Dic15C2312yes—
70C5×C4⋊Dic3D411yescharacteristic
71C5×Dic3⋊C4D412yes—
72C4×C5×Dic3C2312yes—
73C2×C5×C3⋊C8C2312yes—
74C3×#685—21no—
75C3×C4⋊Dic5C2311yescharacteristic
76C3×C4⋊Dic5C2311yescharacteristic
77C3×#683—21no—
78C3×C4×Dic5—21no—
79C3×#677—22no—
80Dic3⋊Dic5—22no—
81Dic3⋊Dic5—22no—
82Dic3⋊Dic5—22no—
83#6669—22no—
84#6665—22no—
85Dic3×Dic5—22no—
86#6657—24no—
87#5567—51no—
88D4×Dic6—51no—
89C3×C4×SD16—51no—
90#5162—52no—
91#5020—51no—
92#5019—51no—
93#4854—52no—
94#4792—52no—
95#4758—52no—
96#4681—51no—
97C8×Dic6D511yescharacteristic, complemented
98C2×Q8×D5—61no—
99C4×C5×Q8—31no—
100Q8×Dic5—61no—
101C2×#677—32no—
102C2×#677C2×D311yescharacteristic
103#4043—62no—
104#4043—61no—
105#4036—61no—
106#4036—62no—
107C4×D20—32no—
108C4×D20C2×D311yescharacteristic
109#4015—31no—
110#4015—32no—
111#4015—32no—
112#4015—31no—
113C4×Dic10—32no—
114C4×Dic10C2×D311yescharacteristic
115C5×C4⋊C8—32no—
116C5×C4⋊C8—31no—
117C4×C5×C8—32no—
118C4×C5×C8C2×D311yescharacteristic
119#3953C2×D311yescharacteristic
120#3949—32no—
121#3949C2×D311yescharacteristic
122#3948—32no—
123#3948—32no—
124#3948—32no—
125#3948—32no—
126#3948C2×D311yescharacteristic
127#3359—151noSylow
128C22×C3×D5—21no—
129C2×C3×C4×C5C2×D411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2411yescharacteristic
132C2×Dic15—22no—
133C2×Dic15—22no—
134Dic30—22no—
135Dic30—22no—
136Dic30—42no—
137C2×C5×Dic3C2×D412yes—
138C2×C5×Dic3C2×D412yes—
139C5×Dic6SD1614yescomplemented
140C2×C3×Dic5—21no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C3×D20—21no—
144C3×D20#4112yes—
145C3×C4×D5—42no—
146C3×Dic10—21no—
147C3×Dic10—21no—
148#1204—44no—
149D5×Dic3—44no—
150C3×C5×C8—21no—
151C3×C5×C8—21no—
152C5×C3⋊C8—22no—
153C22×Dic6—101no—
154C2×C3×SD16—51no—
155C3×C4×Q8—51no—
156C3×C4×D4—51no—
157#944—52no—
158D4×Dic3—52no—
159C2×#214—52no—
160#924—101no—
161#889—102no—
162#888—52no—
163#887—52no—
164#878—102no—
165#870—101no—
166#869—51no—
167#869—51no—
168C4×Dic6—101no—
169C4×Dic6C2×D511yescharacteristic
170C3×#106—51no—
171C3×#103—51no—
172C3×#102—51no—
173C3×C4×C8C2×D511yescharacteristic
174#835—52no—
175#816—52no—
176#816—52no—
177Dic3⋊C8C2×D512yes—
178C8×Dic3C2×D512yes—
179#810—52no—
180#808—52no—
181C4⋊(C3⋊C8)C2×D511yescharacteristic
182C2×C5×Q8—31no—
183Q8×D5—124no—
184C2×D20C22×D311yescharacteristic
185C2×C4×D5—21no—
186C2×C4×D5—62no—
187C2×Dic10—32no—
188C2×Dic10—62no—
189C2×Dic10—61no—
190C2×Dic10C22×D311yescharacteristic
191C2×C5×C8—32no—
192C2×C5×C8C22×D311yescharacteristic
193C2×C5×C8C22×D311yescharacteristic
194C5×C4⋊C4—32no—
195C5×C4⋊C4—31no—
196C42×C5—32no—
197C42×C5C22×D311yescharacteristic
198#685—21no—
199#685—62no—
200C4⋊Dic5—32no—
201C4⋊Dic5—62no—
202C4⋊Dic5—62no—
203C4⋊Dic5—62no—
204C4⋊Dic5—61no—
205C4⋊Dic5—61no—
206C4⋊Dic5—61no—
207C4⋊Dic5C22×D311yescharacteristic
208C4⋊Dic5C22×D311yescharacteristic
209#683—21no—
210#683—62no—
211C4×Dic5—21no—
212C4×Dic5—62no—
213#677—22no—
214#677—64no—
215#557—151no—
216D4×Q8—151no—
217#487—152no—
218#487—151no—
219#475—152no—
220#475—151no—
221#461—151no—
222#461—152no—
223C8×Q8—31no—
224C4×SD16—51no—
225C4×SD16—152no—
226C22×C3×C5C22×D411yescharacteristic
227C2×C3×D5—22no—
228C2×C3×D5—41no—
229C2×C3×D5—41no—
230C3×C4×C5—21no—
231C3×C4×C5C2×SD1612yes—
232C3×C4×C5C2×#4111yescharacteristic
233C3×C4×C52−1+411yescharacteristic
234Dic15—42no—
235Dic15—42no—
236C3×Dic5—41no—
237C3×Dic5—41no—
238C3×Dic5—41no—
239C5×Dic3—22no—
240C5×Dic3C2×SD1614yes—
241C2×C3×Q8—51no—
242C2×C3×D4—51no—
243C22×C3×C4—101no—
244C22×Dic3—102no—
245C2×Dic6—52no—
246C2×Dic6—101no—
247C2×Dic6—102no—
248C2×Dic6—204no—
249C2×Dic6D2011yescharacteristic
250C3×SD16—102no—
251C2×C3×C8C22×D511yescharacteristic
252C2×C3×C8C22×D511yescharacteristic
253C3×C4⋊C4—51no—
254C3×C4⋊C4—51no—
255C3×C4⋊C4—101no—
256C3×C22⋊C4—101no—
257C3×C42—101no—
258C3×C42C22×D511yescharacteristic
259#217—102no—
260#214—104no—
261C4⋊Dic3—52no—
262C4⋊Dic3—101no—
263C4⋊Dic3—101no—
264C4⋊Dic3—101no—
265C4⋊Dic3D2011yescharacteristic
266Dic3⋊C4—102no—
267Dic3⋊C4—102no—
268Dic3⋊C4—102no—
269Dic3⋊C4—102no—
270Dic3⋊C4D2012yes—
271C4×Dic3—102no—
272C4×Dic3C22×D512yes—
273C2×C3⋊C8C22×D512yes—
274C22×D5—21no—
275C5×Q8—34no—
276C2×C4×C5—32no—
277C2×C4×C5—32no—
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5D3×D411yescharacteristic
280C2×C4×C5C23×D311yescharacteristic
281C2×Dic5—21no—
282C2×Dic5—21no—
283C2×Dic5—21no—
284C2×Dic5—62no—
285C2×Dic5—62no—
286D20—21no—
287D20#23512yes—
288C4×D5—42no—
289C4×D5—124no—
290Dic10—21no—
291Dic10—21no—
292Dic10—62no—
293Dic10—62no—
294Dic10—124no—
295Dic10—122no—
296C5×C8—21no—
297C5×C8—21no—
298C5×C8—62no—
299C22×Q8—301no—
300C2×SD16—51no—
301C2×SD16—152no—
302C4⋊Q8—152no—
303C4⋊Q8—152no—
304C4⋊Q8—151no—
305C4⋊Q8—151no—
306#125—301no—
307#125—302no—
308C22⋊Q8—302no—
309C22⋊Q8—301no—
310C4×Q8—31no—
311C4×Q8—51no—
312C4×Q8—152no—
313C4×Q8—301no—
314C4×D4—51no—
315C4×D4—152no—
316#106—51no—
317#106—152no—
318C4⋊C8—32no—
319C4⋊C8—31no—
320#103—51no—
321#103—152no—
322#103—152no—
323#103—152no—
324#103—152no—
325#102—51no—
326C4×C8—32no—
327C4×C8D3×D511yescharacteristic
328C2×C3×C5#54811yescharacteristic
329C2×C3×C5C22×SD1611yescharacteristic
330C2×C3×C5#57811yescharacteristic
331C3×D5—42no—
332C23×C3—101no—
333C3×Q8—101no—
334C3×Q8—101no—
335C3×D4—52no—
336C3×D4—101no—
337C2×C3×C4—101no—
338C2×C3×C4—101no—
339C2×C3×C4—101no—
340C2×C3×C4—202no—
341C2×C3×C4C2×D2011yescharacteristic
342C2×C3×C4C2×D2011yescharacteristic
343C2×C3×C4C23×D511yescharacteristic
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—204no—
347C2×Dic3C2×D2012yes—
348C2×Dic3C2×D2012yes—
349Dic6—102no—
350Dic6—102no—
351Dic6—202no—
352Dic6—204no—
353Dic6#67714yescomplemented
354C3×C8—21no—
355C3×C8—21no—
356C3⋊C8—22no—
357C22×C5C2×D3×D411yescharacteristic
358C2×D5—22no—
359C2×D5—41no—
360C2×D5—41no—
361C4×C5—21no—
362C4×C5—34no—
363C4×C5—62no—
364C4×C5D3×SD1612yes—
365C4×C5C2×#23511yescharacteristic
366C4×C5#101011yescharacteristic
367Dic5—41no—
368Dic5—41no—
369Dic5—41no—
370Dic5—122no—
371Dic5—122no—
372C2×Q8—31no—
373C2×Q8—51no—
374C2×Q8—152no—
375C2×Q8—302no—
376C2×Q8—301no—
377C2×Q8—604no—
378C2×D4—51no—
379C22×C4—101no—
380C22×C4—302no—
381SD16—102no—
382SD16—304no—
383C2×C8—32no—
384C2×C8C2×D3×D511yescharacteristic
385C2×C8C2×D3×D511yescharacteristic
386C4⋊C4—32no—
387C4⋊C4—31no—
388C4⋊C4—51no—
389C4⋊C4—51no—
390C4⋊C4—101no—
391C4⋊C4—152no—
392C4⋊C4—302no—
393C4⋊C4—302no—
394C4⋊C4—302no—
395C4⋊C4—302no—
396C4⋊C4—301no—
397C4⋊C4—301no—
398C4⋊C4—301no—
399C22⋊C4—101no—
400C22⋊C4—302no—
401C42—32no—
402C42—101no—
403C42—302no—
404C42C2×D3×D511yescharacteristic
405C3×C5#335911yescharacteristic, complemented, Hall
406C22×C3—102no—
407C22×C3—201no—
408C22×C3—201no—
409C22×C3C22×D2011yescharacteristic
410C3×C4—21no—
411C3×C4—201no—
412C3×C4—201no—
413C3×C4—201no—
414C3×C4C2×#67712yes—
415C3×C4#414711yescharacteristic
416C3×C4D5×#4111yescharacteristic
417Dic3—22no—
418Dic3—202no—
419Dic3—202no—
420Dic3C2×#67714yes—
421C2×C5#555211yescharacteristic
422C2×C5C2×D3×SD1611yescharacteristic
423C2×C5#577111yescharacteristic
424D5—42no—
425C23—101no—
426Q8—34no—
427Q8—101no—
428Q8—101no—
429Q8—302no—
430Q8—302no—
431Q8—604no—
432Q8—602no—
433D4—52no—
434D4—101no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4—202no—
441C2×C4—302no—
442C2×C4—302no—
443C2×C4—604no—
444C2×C4D3×D2011yescharacteristic
445C2×C4D3×D2011yescharacteristic
446C2×C4C22×D3×D511yescharacteristic
447C8—21no—
448C8—21no—
449C8—62no—
450C2×C3—202no—
451C2×C3#6591011yescharacteristic
452C2×C3C22×#67711yescharacteristic
453C2×C3#6608711yescharacteristic
454C5#7961911yescharacteristic, complemented, Sylow
455C22—102no—
456C22—201no—
457C22—201no—
458C22C2×D3×D2011yescharacteristic
459C4—21no—
460C4—34no—
461C4—62no—
462C4—201no—
463C4—201no—
464C4—201no—
465C4—602no—
466C4—602no—
467C4D3×#67712yes—
468C4#9200011yescharacteristic
469C4D5×#23511yescharacteristic
470C3#1061165611yescharacteristic, complemented, Sylow
471C2—202no—
472C2#1175254111yescharacteristic
473C2C2×D3×#67711yescharacteristic
474C2#1175400311yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×D20C23
hypercenterC4×C8D3×D5
hypocenterC3×C5#3359
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #11752490C2semidirect
3. #11750782C2semidirect
4. C5×C8×Dic6C2semidirect
6. #11748028C2semidirect
7. #11747867C2semidirect
11. #11746943C2semidirect
97. C8×Dic6D5semidirect
139. C5×Dic6SD16semidirect
353. Dic6#677semidirect
405. C3×C5#3359semidirect
454. C5#79619semidirect
470. C3#10611656semidirect

Maximal subgroups

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

subgroupindex
Dic6×D202
#117524902
#117507822
C5×C8×Dic62
C3×C4×#6772
#117480282
#117478672
#117477532
#117477392
#117470622
#117469432
#106116563
#796195