Finite Group Databasebeta

← #14295357

Subgroups — #14295357

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5⋊M16C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic
3C2×#91150C211yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽16)⋊C8C211yesmaximal, characteristic, complemented
5D5×#830C211yesmaximal, characteristic
6#11746517C211yesmaximal, characteristic, complemented
7#11746517C211yesmaximal, characteristic, complemented
8#10603915—31nomaximal, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763C412yes—
11C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
12C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
13C2×C3×#704C412yescomplemented
14C3×D5⋊M16—22no—
15C3×D5⋊M16—22no—
16C2×C3×#700C412yescomplemented
17C2×C3×D5⋊C8C2211yescharacteristic
18C2×C3×D5⋊C8C2211yescharacteristic
19C2×D5×C3⋊C8C2211yescharacteristic
20C2×D5×C3⋊C8C2211yescharacteristic
21#91235C412yes—
22#91150—22no—
23ΓL(1,𝔽16)⋊C8—22no—
24#91114C411yescharacteristic, complemented
25C5×#830C411yescharacteristic, complemented
26#90955C411yescharacteristic, complemented
27#90955C411yescharacteristic, complemented
28#71983—51nomaximal, Hall
29C2×D5⋊M16D311yescharacteristic
30C4×C22⋊AGL(1,𝔽5)—31no—
31C2×#3992—31no—
32C2×C8×AGL(1,𝔽5)—31no—
33D5×C22⋊C8—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35(C2×AGL(1,𝔽5))⋊C8—31no—
36C23×C3×D5C2×C411yescharacteristic
37C22×ΓL(1,𝔽16)C2×C412yes—
38C22×ΓL(1,𝔽16)C2×C412yes—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5C2×C411yescharacteristic
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5C2×C411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49#6763—24no—
50C4×ΓL(1,𝔽16)—21no—
51C4×ΓL(1,𝔽16)—21no—
52C4×ΓL(1,𝔽16)—21no—
53C4×ΓL(1,𝔽16)—21no—
54C3×#704—24no—
55C2×C3×#170C2×C412yes—
56C2×C3×#170C2×C412yes—
57C3×#700—24no—
58C3×D5⋊C8—21no—
59C3×D5⋊C8—21no—
60C3×D5⋊C8D412yes—
61C3×D5⋊C8D412yes—
62C2×C3⋊#168C2×C411yescharacteristic
63C2×C3⋊#168C2×C411yescharacteristic
64C2×C5×C3⋊C8C2×C411yescharacteristic
65C2×C5×C3⋊C8C2×C411yescharacteristic
66D5×C3⋊C8—21no—
67D5×C3⋊C8—21no—
68D5×C3⋊C8—21no—
69D5×C3⋊C8—21no—
70C22×C3×M16—51no—
71C4×#217—51no—
72C2×#830—51no—
73C2×#815—51no—
74C2×C8×Dic3—51no—
75#4550—51no—
76#4550—51no—
77C22×C4×D5C2×D311yescharacteristic
78C2×C22⋊AGL(1,𝔽5)—32no—
79C2×#704Dic312yescomplemented
80C2×C4×AGL(1,𝔽5)—31no—
81C2×C4×AGL(1,𝔽5)—31no—
82D5⋊M16—22no—
83D5⋊M16—22no—
84C2×#700Dic312yescomplemented
85C2×D5⋊C8C2×D311yescharacteristic
86C2×D5⋊C8C2×D311yescharacteristic
87C2×C8×D5—31no—
88C2×C8×D5—31no—
89(C2×AGL(1,𝔽5))⋊C4—32no—
90#3992—62no—
91C8×AGL(1,𝔽5)—62no—
92C5×C22⋊C8—31no—
93C22⋊#168—31no—
94#3952—31no—
95#3952—31no—
96#1753—151noSylow
97C23×C3×C5C4211yescharacteristic
98C22×C3×D5—21no—
99C22×C3×D5—21no—
100C22×C3×D5—22no—
101C22×C3×D5—22no—
102C22×C3×D5C4211yescharacteristic
103C22×C3×D5C22×C411yescharacteristic
104C2×ΓL(1,𝔽16)—22no—
105C2×ΓL(1,𝔽16)—22no—
106C2×ΓL(1,𝔽16)—22no—
107C2×ΓL(1,𝔽16)—22no—
108C2×ΓL(1,𝔽16)—22no—
109C2×ΓL(1,𝔽16)M1614yescomplemented
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5C4211yescharacteristic
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×Dic5—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5C4211yescharacteristic
117C2×C3×Dic5C22×C411yescharacteristic
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5—21no—
124C3×C4×D5C2×D411yescharacteristic
125C3×C4×D5C2×D411yescharacteristic
126C3×C4×D5#4111yescharacteristic
127C3×C4×D5#4111yescharacteristic
128C3×#170—22no—
129C3×#170C22⋊C414yescomplemented
130C3⋊#168—21no—
131C3⋊#168—21no—
132C5×C3⋊C8—21no—
133C5×C3⋊C8—21no—
134C23×C3×C4—51no—
135C2×C3×M16—52no—
136C2×C3×M16—52no—
137C2×C3×M16—102no—
138C2×C3×M16—102no—
139C22×C3×C8—51no—
140C22×C3×C8—51no—
141C2×#217—52no—
142C2×C4×Dic3—51no—
143C2×C4×Dic3—51no—
144C22×C3⋊C8—51no—
145C22×C3⋊C8—51no—
146C3⋊#95—52no—
147#830—51no—
148#830—51no—
149#830—51no—
150#830AGL(1,𝔽5)11yescharacteristic, complemented
151#815—102no—
152C8×Dic3—102no—
153C23×D5C4×D311yescharacteristic
154C22×AGL(1,𝔽5)—32no—
155C22×AGL(1,𝔽5)—32no—
156C22×C4×C5C4×D311yescharacteristic
157C22×Dic5C2×Dic311yescharacteristic
158C2×C4×D5—21no—
159C2×C4×D5—21no—
160C2×C4×D5—21no—
161C2×C4×D5—21no—
162C2×C4×D5C4×D311yescharacteristic
163C2×C4×D5C4×D311yescharacteristic
164C2×C4×D5C2×Dic311yescharacteristic
165C2×C4×D5C22×D311yescharacteristic
166C22⋊AGL(1,𝔽5)—64no—
167#704—24no—
168C2×#170C2×Dic312yes—
169C2×#170C2×Dic312yes—
170C4×AGL(1,𝔽5)—61no—
171C4×AGL(1,𝔽5)—61no—
172C4×AGL(1,𝔽5)—61no—
173C4×AGL(1,𝔽5)—61no—
174#700—24no—
175D5⋊C8—21no—
176D5⋊C8—21no—
177D5⋊C8#6712yes—
178D5⋊C8#6712yes—
179C2×C5×C8—31no—
180C2×C5×C8—31no—
181C2×#168—31no—
182C2×#168—31no—
183C8×D5—61no—
184C8×D5—61no—
185C8×D5—61no—
186C8×D5—61no—
187C22×M16—51no—
188C2×C22⋊C8—151no—
189C2×#97—151no—
190C2×C4×C8—151no—
191C4×C22⋊C4—151no—
192#336—151no—
193#336—151no—
194C22×C3×C5—21no—
195C22×C3×C5—21no—
196C22×C3×C5—21no—
197C22×C3×C5C2×C4211yescharacteristic
198C2×C3×D5—21no—
199C2×C3×D5—21no—
200C2×C3×D5—21no—
201C2×C3×D5—42no—
202C2×C3×D5—42no—
203C2×C3×D5C4×D411yescharacteristic
204C2×C3×D5C4×D411yescharacteristic
205C2×C3×D5C2×M1612yes—
206C2×C3×D5#13112yes—
207ΓL(1,𝔽16)—24no—
208ΓL(1,𝔽16)—42no—
209C3×C4×C5—21no—
210C3×C4×C5C4×D411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×Dic5—21no—
213C3×Dic5C2×C22⋊C411yescharacteristic
214C3×Dic5#11711yescharacteristic
215C24×C3—51no—
216C22×C3×C4—51no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—101no—
222C22×C3×C4—101no—
223C22×C3×C4—101no—
224C22×C3×C4—101no—
225C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
226C22×Dic3—52no—
227C22×Dic3—52no—
228C3×M16—104no—
229C3×M16—104no—
230C2×C3×C8—52no—
231C2×C3×C8—52no—
232C2×C3×C8—52no—
233C2×C3×C8—52no—
234C2×C3×C8—101no—
235C2×C3×C8—101no—
236#217—104no—
237C4×Dic3—101no—
238C4×Dic3—101no—
239C4×Dic3—101no—
240C4×Dic3—101no—
241C2×C3⋊C8—51no—
242C2×C3⋊C8—51no—
243C2×C3⋊C8—101no—
244C2×C3⋊C8—101no—
245C2×C3⋊C8—101no—
246C2×C3⋊C8—101no—
247C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
248C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
249C23×C5C4×Dic311yescharacteristic
250C22×D5—22no—
251C22×D5—22no—
252C22×D5—21no—
253C22×D5—21no—
254C22×D5C4×Dic311yescharacteristic
255C22×D5C2×C4×D311yescharacteristic
256C2×AGL(1,𝔽5)—34no—
257C2×AGL(1,𝔽5)—62no—
258C2×AGL(1,𝔽5)—62no—
259C2×AGL(1,𝔽5)—62no—
260C2×AGL(1,𝔽5)—62no—
261C2×AGL(1,𝔽5)—62no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5C4×Dic311yescharacteristic
265C2×C4×C5C2×C4×D311yescharacteristic
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5C4×Dic311yescharacteristic
269C2×Dic5C22×Dic311yescharacteristic
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—21no—
273C4×D5—21no—
274C4×D5—21no—
275C4×D5—21no—
276C4×D5#23511yescharacteristic
277C4×D5#23511yescharacteristic
278C4×D5C2×#6711yescharacteristic
279C4×D5C2×#6711yescharacteristic
280#170—22no—
281#170#21714yescomplemented
282C5×C8—61no—
283C5×C8—61no—
284#168—61no—
285#168—61no—
286C23×C4—51no—
287C2×M16—52no—
288C2×M16—52no—
289C2×M16—102no—
290C2×M16—102no—
291C22×C8—51no—
292C22×C8—51no—
293C22×C8—151no—
294C22×C8—151no—
295C2×C22⋊C4—152no—
296C2×C42—151no—
297C2×C42—151no—
298C22⋊C8—31no—
299C22⋊C8—151no—
300C22⋊C8—151no—
301C22⋊C8—151no—
302#97—302no—
303C4×C8—302no—
304#95—152no—
305C2×C3×C5—41no—
306C2×C3×C5C4×C22⋊C411yescharacteristic
307C2×C3×C5C4×M1611yescharacteristic
308C2×C3×C5#40511yescharacteristic
309C3×D5—41no—
310C3×D5#43512yes—
311C3×D5#43612yes—
312C23×C3—51no—
313C23×C3—51no—
314C23×C3—102no—
315C23×C3—101no—
316C23×C3—101no—
317C23×C3—102no—
318C23×C3C4×AGL(1,𝔽5)11yescharacteristic
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—51no—
324C2×C3×C4—51no—
325C2×C3×C4—51no—
326C2×C3×C4—51no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—101no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—101no—
333C2×C3×C4—101no—
334C2×C3×C4—101no—
335C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
336C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
337C2×Dic3—54no—
338C2×Dic3—102no—
339C2×Dic3—102no—
340C2×Dic3—102no—
341C2×Dic3—102no—
342C2×Dic3—102no—
343C3×C8—54no—
344C3×C8—102no—
345C3⋊C8—21no—
346C3⋊C8—21no—
347C3⋊C8—101no—
348C3⋊C8—101no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5C2×C4×Dic311yescharacteristic
353C2×D5—21no—
354C2×D5—21no—
355C2×D5—21no—
356C2×D5—42no—
357C2×D5—42no—
358C2×D5D3×M1612yes—
359C2×D5#90712yes—
360C2×D5C4×#6711yescharacteristic
361C2×D5C4×#6711yescharacteristic
362AGL(1,𝔽5)—64no—
363AGL(1,𝔽5)—122no—
364C4×C5—21no—
365C4×C5C4×#6711yescharacteristic
366C4×C5C4×#6711yescharacteristic
367Dic5—21no—
368Dic5#92611yescharacteristic
369Dic5C2×#21711yescharacteristic
370C24—51no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—101no—
377C22×C4—101no—
378C22×C4—101no—
379C22×C4—101no—
380C22×C4—152no—
381C22×C4—152no—
382C22×C4D3×AGL(1,𝔽5)11yescharacteristic
383M16—104no—
384M16—104no—
385C2×C8—31no—
386C2×C8—31no—
387C2×C8—52no—
388C2×C8—52no—
389C2×C8—52no—
390C2×C8—52no—
391C2×C8—101no—
392C2×C8—101no—
393C2×C8—151no—
394C2×C8—151no—
395C2×C8—301no—
396C2×C8—301no—
397C2×C8—301no—
398C2×C8—301no—
399C22⋊C4—304no—
400C42—301no—
401C42—301no—
402C42—301no—
403C42—301no—
404C3×C5#175311yescharacteristic, complemented, Hall
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—51no—
409C22×C3—51no—
410C22×C3—52no—
411C22×C3—52no—
412C22×C3—101no—
413C22×C3—101no—
414C22×C3—101no—
415C22×C3—202no—
416C22×C3—202no—
417C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
418C3×C4—21no—
419C3×C4—51no—
420C3×C4—51no—
421C3×C4—101no—
422C3×C4D4×AGL(1,𝔽5)11yescharacteristic
423C3×C4D4×AGL(1,𝔽5)11yescharacteristic
424Dic3—104no—
425Dic3—202no—
426C2×C5—41no—
427C2×C5Dic3×M1611yescharacteristic
428C2×C5#512211yescharacteristic
429C2×C5C4×#21711yescharacteristic
430D5—41no—
431D5#512712yes—
432D5#512812yes—
433C23—51no—
434C23—51no—
435C23—102no—
436C23—101no—
437C23—101no—
438C23—102no—
439C23Dic3×AGL(1,𝔽5)11yescharacteristic
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—101no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—154no—
457C2×C4—302no—
458C2×C4—302no—
459C2×C4—302no—
460C2×C4—302no—
461C2×C4—302no—
462C2×C4Dic3×AGL(1,𝔽5)11yescharacteristic
463C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
464C8—54no—
465C8—61no—
466C8—61no—
467C8—102no—
468C8—301no—
469C8—301no—
470C2×C3—41no—
471C2×C3—52no—
472C2×C3—52no—
473C2×C3—201no—
474C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
475C2×C3#6571811yescharacteristic
476C2×C3M16×AGL(1,𝔽5)11yescharacteristic
477C5#7198311yescharacteristic, complemented, Sylow
478C22—21no—
479C22—21no—
480C22—21no—
481C22—52no—
482C22—52no—
483C22—51no—
484C22—51no—
485C22—101no—
486C22—101no—
487C22—101no—
488C22—202no—
489C22—202no—
490C22C2×Dic3×AGL(1,𝔽5)11yescharacteristic
491C4—21no—
492C4—51no—
493C4—51no—
494C4—101no—
495C4—304no—
496C4—602no—
497C4AGL(1,𝔽5)×#6711yescharacteristic
498C4AGL(1,𝔽5)×#6711yescharacteristic
499C3#1060391511yescharacteristic, complemented, Sylow
500C2—41no—
501C2—52no—
502C2—52no—
503C2—201no—
504C2#1175169611yescharacteristic
505C2#1175169811yescharacteristic
506C2AGL(1,𝔽5)×#21711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C2×C42
FittingC22×C3×C4×C5C2×C4
prime radicalC23×C3×D5C2×C4
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1753
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#91150C2semidirect
4. C2×ΓL(1,𝔽16)⋊C8C2semidirect
6. #11746517C2semidirect
7. #11746517C2semidirect
13. C2×C3×#704C4semidirect
16. C2×C3×#700C4semidirect
24. #91114C4semidirect
25. C5×#830C4semidirect
26. #90955C4semidirect
27. #90955C4semidirect
79. C2×#704Dic3semidirect
84. C2×#700Dic3semidirect
109. C2×ΓL(1,𝔽16)M16semidirect
129. C3×#170C22⋊C4semidirect
150. #830AGL(1,𝔽5)semidirect
281. #170#217semidirect
404. C3×C5#1753semidirect
477. C5#71983semidirect
499. C3#10603915semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D5⋊M162
C4×#67632
C2×#911502
C2×ΓL(1,𝔽16)⋊C82
D5×#8302
#117465172
#106039153
#719835