Finite Group Databasebeta

← #14295136

Subgroups — #14295136

Order 1920.

672 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
1C2×#91989C211yesmaximal, characteristic
2C3×D5⋊(C22⋊C8)C211yesmaximal, characteristic
3C2×#91150C211yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽16)⋊C8C211yesmaximal, characteristic, complemented
5D5×#830C211yesmaximal, characteristic
6#11746517C211yesmaximal, characteristic, complemented
7#11746517C211yesmaximal, characteristic, complemented
8#10603909—31nomaximal, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6763C412yes—
11#91989—22no—
12#91989—22no—
13C2×C4⋊ΓL(1,𝔽16)C412yes—
14C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
15C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
16C2×C3×D5⋊C8C2211yescharacteristic
17C2×C3×D5⋊C8C2211yescharacteristic
18C2×D5×C3⋊C8C2211yescharacteristic
19C2×D5×C3⋊C8C2211yescharacteristic
20C3×C22⋊#170C412yescomplemented
21C3×#4003C412yescomplemented
22#91150—22no—
23ΓL(1,𝔽16)⋊C8—22no—
24#91114C411yescharacteristic, complemented
25C5×#830C411yescharacteristic, complemented
26#90955C411yescharacteristic, complemented
27#90955C411yescharacteristic, complemented
28#69907—51nomaximal, Hall
29C2×#4130—31no—
30D5⋊(C22⋊C8)D311yescharacteristic
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)—24no—
51C4×ΓL(1,𝔽16)—21no—
52C4×ΓL(1,𝔽16)—21no—
53C4×ΓL(1,𝔽16)D412yes—
54C4×ΓL(1,𝔽16)D412yes—
55C2×C3×#170C2×C412yes—
56C2×C3×#170C2×C412yes—
57C3×D5⋊C8—21no—
58C3×D5⋊C8—21no—
59C3×D5⋊C8—21no—
60C3×D5⋊C8—21no—
61C2×C3⋊#168C2×C411yescharacteristic
62C2×C3⋊#168C2×C411yescharacteristic
63C2×C5×C3⋊C8C2×C411yescharacteristic
64C2×C5×C3⋊C8C2×C411yescharacteristic
65D5×C3⋊C8—21no—
66D5×C3⋊C8—21no—
67D5×C3⋊C8—21no—
68D5×C3⋊C8—21no—
69C2×#926—51no—
70C2×C3×C22⋊C8—51no—
71C2×#830—51no—
72C2×#815—51no—
73C2×C8×Dic3—51no—
74#4550—51no—
75#4550—51no—
76C22×C4×D5C2×D311yescharacteristic
77C2×C22⋊AGL(1,𝔽5)—32no—
78#4130—62no—
79#4130—62no—
80C2×C4⋊AGL(1,𝔽5)—32no—
81C2×C4×AGL(1,𝔽5)—31no—
82C2×C4×AGL(1,𝔽5)—31no—
83C2×D5⋊C8C2×D311yescharacteristic
84C2×D5⋊C8C2×D311yescharacteristic
85C2×C8×D5—31no—
86C2×C8×D5—31no—
87C22⋊#170Dic312yescomplemented
88#4003Dic312yescomplemented
89#3992—62no—
90C8×AGL(1,𝔽5)—62no—
91C5×C22⋊C8—31no—
92C22⋊#168—31no—
93#3952—31no—
94#3952—31no—
95#1750—151noSylow
96C23×C3×C5C4211yescharacteristic
97C22×C3×D5—21no—
98C22×C3×D5—21no—
99C22×C3×D5—22no—
100C22×C3×D5—22no—
101C22×C3×D5C4211yescharacteristic
102C22×C3×D5C22×C411yescharacteristic
103C2×ΓL(1,𝔽16)—22no—
104C2×ΓL(1,𝔽16)—22no—
105C2×ΓL(1,𝔽16)—22no—
106C2×ΓL(1,𝔽16)—22no—
107C2×ΓL(1,𝔽16)—22no—
108C2×ΓL(1,𝔽16)C22⋊C414yes—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5C4211yescharacteristic
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5C4211yescharacteristic
116C2×C3×Dic5C22×C411yescharacteristic
117C3×C4×D5—21no—
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5C2×D411yescharacteristic
124C3×C4×D5C2×D411yescharacteristic
125C3×C4×D5#4111yescharacteristic
126C3×C4×D5#4111yescharacteristic
127C3×#170—22no—
128C3×#170—22no—
129C3⋊#168—21no—
130C3⋊#168—21no—
131C5×C3⋊C8—21no—
132C5×C3⋊C8—21no—
133C23×C3×C4—51no—
134C22×C3×C8—51no—
135C22×C3×C8—51no—
136C2×#217—52no—
137#926—102no—
138#926—102no—
139C2×C4⋊Dic3—52no—
140C2×C4×Dic3—51no—
141C2×C4×Dic3—51no—
142C22×C3⋊C8—51no—
143C22×C3⋊C8—51no—
144C3×C22⋊C8—52no—
145C3×C22⋊C8—52no—
146#830—51no—
147#830—51no—
148#830—51no—
149#830AGL(1,𝔽5)11yescharacteristic, complemented
150#815—102no—
151C8×Dic3—102no—
152C23×D5C4×D311yescharacteristic
153C22×AGL(1,𝔽5)—32no—
154C22×AGL(1,𝔽5)—32no—
155C22×C4×C5C4×D311yescharacteristic
156C22×Dic5C2×Dic311yescharacteristic
157C2×C4×D5—21no—
158C2×C4×D5—21no—
159C2×C4×D5—21no—
160C2×C4×D5—21no—
161C2×C4×D5C4×D311yescharacteristic
162C2×C4×D5C4×D311yescharacteristic
163C2×C4×D5C2×Dic311yescharacteristic
164C2×C4×D5C22×D311yescharacteristic
165C22⋊AGL(1,𝔽5)—64no—
166C2×#170C2×Dic312yes—
167C2×#170C2×Dic312yes—
168C4⋊AGL(1,𝔽5)—64no—
169C4×AGL(1,𝔽5)—32no—
170C4×AGL(1,𝔽5)—32no—
171C4×AGL(1,𝔽5)—61no—
172C4×AGL(1,𝔽5)—61no—
173D5⋊C8—21no—
174D5⋊C8—21no—
175D5⋊C8—21no—
176D5⋊C8—21no—
177C2×C5×C8—31no—
178C2×C5×C8—31no—
179C2×#168—31no—
180C2×#168—31no—
181C8×D5—61no—
182C8×D5—61no—
183C8×D5—61no—
184C8×D5—61no—
185C2×#117—151no—
186C2×C22⋊C8—51no—
187C2×C22⋊C8—151no—
188C2×#97—151no—
189C2×C4×C8—151no—
190#336—151no—
191#336—151no—
192C22×C3×C5—21no—
193C22×C3×C5—21no—
194C22×C3×C5—21no—
195C22×C3×C5C2×C4211yescharacteristic
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—21no—
199C2×C3×D5—42no—
200C2×C3×D5—42no—
201C2×C3×D5C2×C22⋊C411yescharacteristic
202C2×C3×D5#11711yescharacteristic
203C2×C3×D5#13112yes—
204C2×C3×D5#13112yes—
205ΓL(1,𝔽16)—24no—
206ΓL(1,𝔽16)—42no—
207C3×C4×C5—21no—
208C3×C4×C5C4×D411yescharacteristic
209C3×C4×C5C4×D411yescharacteristic
210C3×Dic5—21no—
211C3×Dic5C4×D411yescharacteristic
212C3×Dic5C4×D411yescharacteristic
213C24×C3—51no—
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—51no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—101no—
220C22×C3×C4—101no—
221C22×C3×C4—101no—
222C22×C3×C4—101no—
223C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
224C22×Dic3—52no—
225C22×Dic3—52no—
226C2×C3×C8—52no—
227C2×C3×C8—52no—
228C2×C3×C8—101no—
229C2×C3×C8—101no—
230C2×C3×C8—101no—
231C2×C3×C8—101no—
232#217—104no—
233C4⋊Dic3—104no—
234C4×Dic3—52no—
235C4×Dic3—52no—
236C4×Dic3—101no—
237C4×Dic3—101no—
238C2×C3⋊C8—51no—
239C2×C3⋊C8—51no—
240C2×C3⋊C8—101no—
241C2×C3⋊C8—101no—
242C2×C3⋊C8—101no—
243C2×C3⋊C8—101no—
244C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
245C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
246C23×C5C4×Dic311yescharacteristic
247C22×D5—21no—
248C22×D5—21no—
249C22×D5—22no—
250C22×D5—22no—
251C22×D5C4×Dic311yescharacteristic
252C22×D5C2×C4×D311yescharacteristic
253C2×AGL(1,𝔽5)—34no—
254C2×AGL(1,𝔽5)—62no—
255C2×AGL(1,𝔽5)—62no—
256C2×AGL(1,𝔽5)—62no—
257C2×AGL(1,𝔽5)—62no—
258C2×AGL(1,𝔽5)—62no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5C4×Dic311yescharacteristic
262C2×C4×C5C2×C4×D311yescharacteristic
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5C4×Dic311yescharacteristic
266C2×Dic5C22×Dic311yescharacteristic
267C4×D5—21no—
268C4×D5—21no—
269C4×D5—21no—
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—21no—
273C4×D5D3×D411yescharacteristic
274C4×D5D3×D411yescharacteristic
275C4×D5#23711yescharacteristic
276C4×D5#23711yescharacteristic
277#170—22no—
278#170—22no—
279C5×C8—61no—
280C5×C8—61no—
281#168—61no—
282#168—61no—
283C23×C4—51no—
284C22×C8—51no—
285C22×C8—51no—
286C22×C8—151no—
287C22×C8—151no—
288#117—302no—
289#117—302no—
290C2×C4⋊C4—152no—
291C2×C22⋊C4—152no—
292C2×C42—151no—
293C2×C42—151no—
294C22⋊C8—31no—
295C22⋊C8—52no—
296C22⋊C8—52no—
297C22⋊C8—151no—
298C22⋊C8—151no—
299C22⋊C8—151no—
300#97—302no—
301C4×C8—302no—
302C2×C3×C5—41no—
303C2×C3×C5C4×C22⋊C411yescharacteristic
304C2×C3×C5#40511yescharacteristic
305C2×C3×C5#40511yescharacteristic
306C3×D5—41no—
307C3×D5#40812yes—
308C3×D5#43312yes—
309C23×C3—51no—
310C23×C3—51no—
311C23×C3—101no—
312C23×C3—101no—
313C23×C3—102no—
314C23×C3—102no—
315C23×C3C4×AGL(1,𝔽5)11yescharacteristic
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—51no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—51no—
324C2×C3×C4—101no—
325C2×C3×C4—101no—
326C2×C3×C4—101no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—101no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
333C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
334C2×Dic3—54no—
335C2×Dic3—102no—
336C2×Dic3—102no—
337C2×Dic3—102no—
338C2×Dic3—102no—
339C2×Dic3—102no—
340C3×C8—102no—
341C3×C8—102no—
342C3⋊C8—21no—
343C3⋊C8—21no—
344C3⋊C8—101no—
345C3⋊C8—101no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—21no—
349C22×C5C2×C4×Dic311yescharacteristic
350C2×D5—21no—
351C2×D5—21no—
352C2×D5—21no—
353C2×D5—42no—
354C2×D5—42no—
355C2×D5#87711yescharacteristic
356C2×D5D3×C22⋊C411yescharacteristic
357C2×D5#90112yes—
358C2×D5#90712yes—
359AGL(1,𝔽5)—64no—
360AGL(1,𝔽5)—122no—
361C4×C5—21no—
362C4×C5#67⋊C411yescharacteristic
363C4×C5#67⋊C411yescharacteristic
364Dic5—21no—
365Dic5D4×Dic311yescharacteristic
366Dic5D4×Dic311yescharacteristic
367C24—51no—
368C22×C4—51no—
369C22×C4—51no—
370C22×C4—51no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—101no—
374C22×C4—101no—
375C22×C4—101no—
376C22×C4—101no—
377C22×C4—152no—
378C22×C4—152no—
379C22×C4D3×AGL(1,𝔽5)11yescharacteristic
380C2×C8—31no—
381C2×C8—31no—
382C2×C8—52no—
383C2×C8—52no—
384C2×C8—101no—
385C2×C8—101no—
386C2×C8—101no—
387C2×C8—101no—
388C2×C8—151no—
389C2×C8—151no—
390C2×C8—301no—
391C2×C8—301no—
392C2×C8—301no—
393C2×C8—301no—
394C4⋊C4—304no—
395C22⋊C4—304no—
396C42—152no—
397C42—152no—
398C42—301no—
399C42—301no—
400C3×C5#175011yescharacteristic, complemented, Hall
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—51no—
405C22×C3—51no—
406C22×C3—52no—
407C22×C3—52no—
408C22×C3—101no—
409C22×C3—101no—
410C22×C3—101no—
411C22×C3—202no—
412C22×C3—202no—
413C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
414C3×C4—21no—
415C3×C4—51no—
416C3×C4—51no—
417C3×C4—101no—
418C3×C4D4×AGL(1,𝔽5)11yescharacteristic
419C3×C4D4×AGL(1,𝔽5)11yescharacteristic
420Dic3—104no—
421Dic3—202no—
422C2×C5—41no—
423C2×C5Dic3×C22⋊C411yescharacteristic
424C2×C5#510111yescharacteristic
425C2×C5#512211yescharacteristic
426D5—41no—
427D5#472012yes—
428D5#472712yes—
429C23—51no—
430C23—51no—
431C23—101no—
432C23—101no—
433C23—102no—
434C23—102no—
435C23Dic3×AGL(1,𝔽5)11yescharacteristic
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—101no—
447C2×C4—101no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—101no—
451C2×C4—101no—
452C2×C4—154no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4—302no—
456C2×C4—302no—
457C2×C4—302no—
458C2×C4Dic3×AGL(1,𝔽5)11yescharacteristic
459C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
460C8—61no—
461C8—61no—
462C8—102no—
463C8—102no—
464C8—301no—
465C8—301no—
466C2×C3—41no—
467C2×C3—52no—
468C2×C3—52no—
469C2×C3—201no—
470C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
471C2×C3#6571811yescharacteristic
472C2×C3#6572811yescharacteristic
473C5#6990711yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—21no—
477C22—52no—
478C22—52no—
479C22—51no—
480C22—51no—
481C22—101no—
482C22—101no—
483C22—101no—
484C22—202no—
485C22—202no—
486C22C2×Dic3×AGL(1,𝔽5)11yescharacteristic
487C4—21no—
488C4—51no—
489C4—51no—
490C4—101no—
491C4—304no—
492C4—602no—
493C4#67⋊AGL(1,𝔽5)11yescharacteristic
494C4#67⋊AGL(1,𝔽5)11yescharacteristic
495C3#1060390911yescharacteristic, complemented, Sylow
496C2—41no—
497C2—52no—
498C2—52no—
499C2—201no—
500C2#1175169411yescharacteristic
501C2#1175169611yescharacteristic
502C2#1175202111yescharacteristic
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#1750
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
20. C3×C22⋊#170C4semidirect
21. C3×#4003C4semidirect
24. #91114C4semidirect
25. C5×#830C4semidirect
26. #90955C4semidirect
27. #90955C4semidirect
87. C22⋊#170Dic3semidirect
88. #4003Dic3semidirect
149. #830AGL(1,𝔽5)semidirect
400. C3×C5#1750semidirect
473. C5#69907semidirect
495. C3#10603909semidirect

Maximal subgroups

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

subgroupindex
C2×#919892
C3×D5⋊(C22⋊C8)2
C2×#911502
C2×ΓL(1,𝔽16)⋊C82
D5×#8302
#117465172
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