Finite Group Databasebeta

← #14459951

Subgroups — #14459951

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#547C211yesmaximal, characteristic
2#11756301C211yesmaximal, characteristic, complemented
3#11751340C212yesmaximal, complemented
4#11751334C212yesmaximal, complemented
5#11751191C211yesmaximal, characteristic, complemented
6#11751180C211yesmaximal, characteristic, complemented
7#11751152C212yesmaximal, complemented
8#11751146C212yesmaximal, complemented
9C4×#6716C211yesmaximal, characteristic, complemented
10D4×C3⋊#168C211yesmaximal, characteristic, complemented
11#11751108C211yesmaximal, characteristic, complemented
12#10615241—31nomaximal, Hall
13C2×C3×C5×#41C2212yes—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C3×C5×#123—22no—
16C3×C5×C22⋊Q8C2212yes—
17C3×C5×#121—21no—
18C3×C4×C5×D4C2211yescharacteristic
19C3×C4×C5×D4C2211yescharacteristic
20C2×C3×C5×C4⋊C4—21no—
21C2×#6718C2212yes—
22#91825—21no—
23C2×#6716C2211yescharacteristic
24#91815C2212yes—
25#91813C2212yes—
26C22×C3⋊#168C2212yes—
27#91776—21no—
28#91768—22no—
29#91758C2211yescharacteristic
30C4×Dic30C2211yescharacteristic
31#91119C2212yes—
32#91119C2212yes—
33#91119C2212yes—
34#91116C2211yescharacteristic
35#91116C2212yes—
36#91114C2212yes—
37#91094C2211yescharacteristic
38#91092C2211yescharacteristic
39#91091C2212yes—
40C4⋊(C3⋊#168)C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#83510—51nomaximal, Hall
43C5×#547D311yescharacteristic
44#65876—31no—
45#65528—32no—
46#65522—32no—
47#65379—31no—
48#65370—31no—
49#65340—32no—
50#65334—32no—
51C4×#687—31no—
52D4×#168—31no—
53#65296—31no—
54C3×C5×#41—24no—
55C3×C5×#41—42no—
56C2×C3×C5×Q8C2312yes—
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4D411yescharacteristic
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5—22no—
62C22×C3×C4×C5C2312yes—
63C2×Dic30—21no—
64C2×Dic30C2311yescharacteristic
65C3×C5×C4⋊C4—21no—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4D411yescharacteristic
69C3×C5×C4⋊C4C2311yescharacteristic
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4—21no—
72C3×C5×C22⋊C4D412yes—
73C3×C42×C5C2311yescharacteristic
74#6719—22no—
75#6719—22no—
76#6718—42no—
77#6716—22no—
78C4⋊Dic15—21no—
79C4⋊Dic15C2311yescharacteristic
80C4⋊Dic15C2312yes—
81#6712—21no—
82#6712—22no—
83C4×Dic15—21no—
84C4×Dic15—22no—
85C2×C3⋊#168—42no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2312yes—
88C2×C3⋊#168C2311yescharacteristic
89C3×#547D511yescharacteristic
90#5542—51no—
91#5239—52no—
92#5233—52no—
93#5090—51no—
94#5079—51no—
95#5051—52no—
96#5045—52no—
97C4×#214—51no—
98D4×C3⋊C8—51no—
99#5007—51no—
100C2×C5×#41C2×D312yes—
101C5×C4⋊Q8C2×D311yescharacteristic
102C5×#123—22no—
103C5×C22⋊Q8C2×D312yes—
104C5×#121—21no—
105C4×C5×D4C2×D311yescharacteristic
106C4×C5×D4C2×D311yescharacteristic
107C2×C5×C4⋊C4—21no—
108C2×#689—32no—
109#4082—61no—
110C2×#687—31no—
111#4072—32no—
112#4070—32no—
113C22×#168—32no—
114#4033—61no—
115#4025—62no—
116#4015—31no—
117C4×Dic10—31no—
118#3967—32no—
119#3967—32no—
120#3967—32no—
121#3964—32no—
122#3964—31no—
123C22⋊#168—32no—
124#3942—31no—
125#3940—31no—
126#3939—32no—
127C4⋊#168—31no—
128C4×#168—31no—
129#3302—151noSylow
130C23×C3×C5—21no—
131C23×C3×C5C2×D412yes—
132C3×C5×Q8—22no—
133C3×C5×Q8—22no—
134C3×C5×D4—22no—
135C3×C5×D4—21no—
136C3×C5×D4—44no—
137C3×C5×D4#4112yes—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—44no—
143C2×C3×C4×C5—42no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2411yescharacteristic
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—22no—
151Dic30—21no—
152Dic30—21no—
153Dic30—42no—
154C3⋊#168—21no—
155C3⋊#168—21no—
156C3⋊#168—22no—
157C2×C3×#41C2×D512yes—
158C3×C4⋊Q8C2×D511yescharacteristic
159C3×#123—22no—
160C3×C22⋊Q8C2×D512yes—
161C3×#121—21no—
162C3×C4×D4C2×D511yescharacteristic
163C3×C4×D4C2×D511yescharacteristic
164C2×C3×C4⋊C4—21no—
165C2×#216—52no—
166#936—101no—
167C2×#214—51no—
168#926—52no—
169#924—52no—
170C22×C3⋊C8—52no—
171#887—101no—
172#879—102no—
173#869—51no—
174C4×Dic6—51no—
175#835—52no—
176#835—52no—
177#835—52no—
178#832—51no—
179#832—52no—
180#830—52no—
181#810—51no—
182#808—51no—
183#807—52no—
184C4⋊(C3⋊C8)—51no—
185C4×C3⋊C8—51no—
186C5×#41—24no—
187C5×#41—42no—
188C2×C5×Q8C22×D312yes—
189C2×C5×D4—22no—
190C2×C5×D4#6711yescharacteristic
191C2×C5×D4C22×D311yescharacteristic
192C22×C4×C5—21no—
193C22×C4×C5—22no—
194C22×C4×C5C22×D312yes—
195C2×Dic10—31no—
196C2×Dic10—61no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4#6711yescharacteristic
201C5×C4⋊C4C22×D311yescharacteristic
202C5×C22⋊C4—22no—
203C5×C22⋊C4—21no—
204C5×C22⋊C4#6712yes—
205C42×C5C22×D311yescharacteristic
206C22⋊Dic5—62no—
207C22⋊Dic5—62no—
208#689—122no—
209#687—62no—
210C4⋊Dic5—32no—
211C4⋊Dic5—31no—
212C4⋊Dic5—61no—
213#683—61no—
214#683—62no—
215C4×Dic5—61no—
216C4×Dic5—62no—
217C2×#168—31no—
218C2×#168—31no—
219C2×#168—32no—
220C2×#168—122no—
221#556—151no—
222#547D1511yescharacteristic
223#487—151no—
224#485—152no—
225#474—151no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—151no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—22no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5C22×D411yescharacteristic
239C3×C4×C5—22no—
240C3×C4×C5—21no—
241C3×C4×C5—42no—
242C3×C4×C5—41no—
243C3×C4×C5#13512yes—
244C3×C4×C5C2×#4111yescharacteristic
245C3×C4×C5H(5,𝔽2)11yescharacteristic
246Dic15—41no—
247Dic15—42no—
248Dic15—41no—
249C3×#41—24no—
250C3×#41—42no—
251C2×C3×Q8C22×D512yes—
252C2×C3×D4—22no—
253C2×C3×D4C22⋊D511yescharacteristic
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C22×C3×C4—22no—
257C22×C3×C4C22×D512yes—
258C2×Dic6—51no—
259C2×Dic6—101no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—22no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4C22⋊D511yescharacteristic
264C3×C4⋊C4C22×D511yescharacteristic
265C3×C22⋊C4—22no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4C22⋊D512yes—
268C3×C42C22×D511yescharacteristic
269#217—102no—
270#217—102no—
271#216—202no—
272#214—102no—
273C4⋊Dic3—52no—
274C4⋊Dic3—51no—
275C4⋊Dic3—101no—
276Dic3⋊C4—101no—
277Dic3⋊C4—102no—
278C4×Dic3—101no—
279C4×Dic3—102no—
280C2×C3⋊C8—51no—
281C2×C3⋊C8—51no—
282C2×C3⋊C8—52no—
283C2×C3⋊C8—202no—
284C23×C5—21no—
285C23×C5C2×#6712yes—
286C5×Q8—22no—
287C5×Q8—22no—
288C5×D4—22no—
289C5×D4—21no—
290C5×D4—44no—
291C5×D4#23712yes—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—22no—
296C2×C4×C5—44no—
297C2×C4×C5—42no—
298C2×C4×C5C2×#6712yes—
299C2×C4×C5C2×#6711yescharacteristic
300C2×C4×C5C2×#6711yescharacteristic
301C2×C4×C5C23×D311yescharacteristic
302C2×Dic5—61no—
303C2×Dic5—61no—
304C2×Dic5—62no—
305Dic10—61no—
306Dic10—61no—
307Dic10—122no—
308#168—61no—
309#168—61no—
310#168—62no—
311C2×#41C2×D1512yes—
312C2×Q16—152no—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—151no—
316C4⋊Q8C2×D1511yescharacteristic
317#126—302no—
318#124—301no—
319#123—22no—
320C22⋊Q8—152no—
321C22⋊Q8C2×D1512yes—
322#121—21no—
323C4×Q8—151no—
324C4×Q8—301no—
325C4×D4C2×D1511yescharacteristic
326C4×D4C2×D1511yescharacteristic
327#117—152no—
328C2×C4⋊C4—21no—
329#107—152no—
330#106—151no—
331C4⋊C8—151no—
332#103—152no—
333#103—152no—
334#103—152no—
335#103—151no—
336#102—151no—
337#102—152no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5—42no—
341C2×C3×C5—42no—
342C2×C3×C5#54611yescharacteristic
343C2×C3×C5C2×#13511yescharacteristic
344C2×C3×C5#57811yescharacteristic
345C23×C3—21no—
346C23×C3C2×C22⋊D512yes—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×D4—22no—
350C3×D4—21no—
351C3×D4—44no—
352C3×D4#71112yes—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4C2×C22⋊D512yes—
360C2×C3×C4C2×C22⋊D511yescharacteristic
361C2×C3×C4C2×C22⋊D511yescharacteristic
362C2×C3×C4C23×D511yescharacteristic
363C2×Dic3—101no—
364C2×Dic3—101no—
365C2×Dic3—102no—
366Dic6—101no—
367Dic6—101no—
368Dic6—202no—
369C3⋊C8—101no—
370C3⋊C8—101no—
371C3⋊C8—102no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—41no—
377C22×C5—41no—
378C22×C5C22×#6711yescharacteristic
379C4×C5—22no—
380C4×C5—21no—
381C4×C5—42no—
382C4×C5—41no—
383C4×C5#95012yes—
384C4×C5C2×#23711yescharacteristic
385C4×C5#100411yescharacteristic
386Dic5—121no—
387Dic5—122no—
388Dic5—121no—
389#41—24no—
390#41—42no—
391C2×Q8—151no—
392C2×Q8—301no—
393C2×Q8C22×D1512yes—
394C2×D4—22no—
395C2×D4#122011yescharacteristic
396C2×D4C22×D1511yescharacteristic
397C22×C4—21no—
398C22×C4—22no—
399C22×C4C22×D1512yes—
400Q16—602no—
401SD16—302no—
402C2×C8—151no—
403C2×C8—151no—
404C2×C8—152no—
405C2×C8—602no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—151no—
410C4⋊C4—152no—
411C4⋊C4—301no—
412C4⋊C4—302no—
413C4⋊C4—301no—
414C4⋊C4#122011yescharacteristic
415C4⋊C4C22×D1511yescharacteristic
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—302no—
419C22⋊C4—302no—
420C22⋊C4#122012yes—
421C42—302no—
422C42—301no—
423C42C22×D1511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3C22×C22⋊D511yescharacteristic
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—42no—
435C3×C4—41no—
436C3×C4#409612yes—
437C3×C4C2×#71111yescharacteristic
438C3×C4#414411yescharacteristic
439Dic3—201no—
440Dic3—202no—
441Dic3—201no—
442C2×C5—42no—
443C2×C5—42no—
444C2×C5#580611yescharacteristic
445C2×C5C2×#95011yescharacteristic
446C2×C5#583811yescharacteristic
447C23—21no—
448C23C2×#122012yes—
449Q8—22no—
450Q8—22no—
451Q8—301no—
452Q8—301no—
453Q8—602no—
454D4—22no—
455D4—21no—
456D4—44no—
457D4#681912yes—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—301no—
465C2×C4—302no—
466C2×C4—301no—
467C2×C4C2×#122012yes—
468C2×C4C2×#122011yescharacteristic
469C2×C4C2×#122011yescharacteristic
470C2×C4C23×D1511yescharacteristic
471C8—301no—
472C8—301no—
473C8—302no—
474C2×C3—42no—
475C2×C3—42no—
476C2×C3#6614011yescharacteristic
477C2×C3C2×#409611yescharacteristic
478C2×C3#6617211yescharacteristic
479C5#8351011yescharacteristic, complemented, Sylow
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22C22×#122011yescharacteristic
487C4—22no—
488C4—21no—
489C4—42no—
490C4—41no—
491C4—601no—
492C4—602no—
493C4—601no—
494C4#9183912yes—
495C4C2×#681911yescharacteristic
496C4#9209511yescharacteristic
497C3#1061524111yescharacteristic, complemented, Sylow
498C2—42no—
499C2—42no—
500C2#1175656511yescharacteristic
501C2C2×#9183911yescharacteristic
502C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
Fitting, prime radicalC3×C5×#547C2
socleC22×C3×C5C22×D4
hypercenter#547D15
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
2. #11756301C2semidirect
3. #11751340C2semidirect
4. #11751334C2semidirect
5. #11751191C2semidirect
6. #11751180C2semidirect
7. #11751152C2semidirect
8. #11751146C2semidirect
9. C4×#6716C2semidirect
10. D4×C3⋊#168C2semidirect
11. #11751108C2semidirect
424. C3×C5#3302semidirect
479. C5#83510semidirect
497. C3#10615241semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5472
#117563012
#117513402
#117513342
#117511912
#117511802
#117511522
#117511462
C4×#67162
D4×C3⋊#1682
#117511082
#106152413
#835105