Finite Group Databasebeta

← #14307433

Subgroups — #14307433

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753568C211yesmaximal, characteristic
2C4×Dic3×AGL(1,𝔽5)C211yesmaximal, characteristic
3#11751824C211yesmaximal, characteristic
4#11746746C211yesmaximal, characteristic
5C3×#64875C211yesmaximal, characteristic
6#11746526C211yesmaximal, characteristic
7#11746524C211yesmaximal, characteristic
8#10604665—31nomaximal, Hall
9C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×#700C2211yescharacteristic
13C2×Dic3×AGL(1,𝔽5)—21no—
14C2×Dic3×AGL(1,𝔽5)—21no—
15#91779C411yescharacteristic, complemented
16C5×#888C411yescharacteristic, complemented
17C3×#4038C2211yescharacteristic
18#91469C411yescharacteristic, complemented
19#91422—21no—
20#91421—21no—
21C4×D5×Dic3C2211yescharacteristic
22#91384C411yescharacteristic, complemented
23#91358C2211yescharacteristic
24Dic5×Dic6—21no—
25#91323—21no—
26Dic3⋊#170—21no—
27Dic3⋊#170—21no—
28#68731—51nomaximal, Hall
29#66062—31no—
30C42×AGL(1,𝔽5)—31no—
31#65681—31no—
32#64875—31no—
33#64875—31no—
34#64875—31no—
35#64875D311yescharacteristic
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C5×Dic6—21no—
39C2×C3×C4×D5C2311yescharacteristic
40C2×#1204—21no—
41C2×D5×Dic3D411yescharacteristic
42C2×D5×Dic3D411yescharacteristic
43C2×C3⋊#170—21no—
44C4×ΓL(1,𝔽16)—21no—
45C4×ΓL(1,𝔽16)Q812yescomplemented
46#6758—21no—
47#6758—21no—
48C2×C3×#170—21no—
49C3×C4×AGL(1,𝔽5)—21no—
50C3×C4×AGL(1,𝔽5)Q812yescomplemented
51C3×#700—21no—
52C3×#700—21no—
53Dic3×AGL(1,𝔽5)—24no—
54Dic3×AGL(1,𝔽5)—24no—
55C3×C5×C4⋊C4C2×C411yescharacteristic
56C4⋊Dic15—21no—
57C4⋊Dic15C2×C411yescharacteristic
58#6712—21no—
59C4×Dic15C2×C411yescharacteristic
60C5×C4⋊Dic3C2×C411yescharacteristic
61C5×Dic3⋊C4—21no—
62C4×C5×Dic3C2×C411yescharacteristic
63C3×#685—21no—
64C3×C4⋊Dic5C2×C411yescharacteristic
65C3×C4×Dic5—21no—
66Dic3⋊Dic5—21no—
67Dic3⋊Dic5—21no—
68#6668—21no—
69#6665—21no—
70Dic3×Dic5—21no—
71Dic3×Dic5D411yescharacteristic
72Dic3×Dic5D411yescharacteristic
73#5517—51no—
74#5118—51no—
75C42×Dic3—51no—
76C3×#339—51no—
77#4555—51no—
78#4532—51no—
79#4523—51no—
80C2×C4×AGL(1,𝔽5)—31no—
81C2×C4×AGL(1,𝔽5)—61no—
82C2×C4×AGL(1,𝔽5)—61no—
83C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
84C2×#700—31no—
85C2×#700C2×D311yescharacteristic
86C5×C4⋊Q8—31no—
87#4092—61no—
88Q8×Dic5—61no—
89#4043—61no—
90#4038—31no—
91#4038C2×D311yescharacteristic
92#4036—31no—
93#4036—31no—
94#4033—61no—
95C42×D5—31no—
96#4015—31no—
97C4⋊#170—61no—
98C4⋊#170—61no—
99#1994—151noSylow
100C22×C3×D5C2×D411yescharacteristic
101C2×ΓL(1,𝔽16)—22no—
102C2×ΓL(1,𝔽16)—22no—
103C2×ΓL(1,𝔽16)—22no—
104C2×C3×AGL(1,𝔽5)—22no—
105C2×C3×AGL(1,𝔽5)—22no—
106C2×C3×AGL(1,𝔽5)—22no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×Dic15—21no—
110C2×Dic15C22⋊C411yescharacteristic
111C2×Dic15C22⋊C411yescharacteristic
112C2×C5×Dic3—21no—
113C2×C5×Dic3C22⋊C411yescharacteristic
114C2×C5×Dic3C22⋊C411yescharacteristic
115C5×Dic6—42no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5C2×D411yescharacteristic
118C3×C4×D5C2×Q811yescharacteristic
119C3×C4×D5C2×Q811yescharacteristic
120C3×C4×D5#4111yescharacteristic
121C3×C4×D5#4111yescharacteristic
122#1204—42no—
123D5×Dic3—22no—
124D5×Dic3—22no—
125C3⋊#170—41no—
126C3×#170—41no—
127C2×C3×M16—51no—
128C3×#117—51no—
129C2×C3×C42—51no—
130#926—51no—
131C2×C4×Dic3—51no—
132C2×C4×Dic3—51no—
133C2×C4×Dic3—101no—
134C2×C4×Dic3—101no—
135C2×#208—51no—
136#890—51no—
137#890—51no—
138#888—51no—
139#888AGL(1,𝔽5)11yescharacteristic, complemented
140#878—101no—
141#878—101no—
142C4×Dic6—101no—
143C4×Dic6—101no—
144Dic3⋊C8—101no—
145Dic3⋊C8—101no—
146C22×AGL(1,𝔽5)—21no—
147C22×AGL(1,𝔽5)—61no—
148C2×C5×Q8—61no—
149C2×C4×D5—31no—
150C2×C4×D5—31no—
151C2×C4×D5C22×D311yescharacteristic
152C2×Dic10—61no—
153C2×#170—21no—
154C2×#170—61no—
155C4×AGL(1,𝔽5)—21no—
156C4×AGL(1,𝔽5)—32no—
157C4×AGL(1,𝔽5)—64no—
158C4×AGL(1,𝔽5)—64no—
159C4×AGL(1,𝔽5)—61no—
160C4×AGL(1,𝔽5)Dic612yescomplemented
161#700—21no—
162#700—21no—
163#700—61no—
164#700—61no—
165C5×C4⋊C4—31no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4C4×D311yescharacteristic
168C42×C5—31no—
169#685—21no—
170#685—61no—
171C4⋊Dic5—31no—
172C4⋊Dic5—61no—
173C4⋊Dic5—61no—
174C4⋊Dic5—61no—
175C4⋊Dic5C4×D311yescharacteristic
176#683—61no—
177#683—61no—
178C4×Dic5—21no—
179C4×Dic5—31no—
180C4×Dic5—31no—
181C4×Dic5—31no—
182C4×Dic5—61no—
183#533—151no—
184#423—151no—
185C43—151no—
186#339—51no—
187#339—151no—
188#339—151no—
189#339—151no—
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C2×C3×D5—21no—
192C2×C3×D5—21no—
193C2×C3×D5C22⋊Q811yescharacteristic
194C2×C3×D5C22⋊Q811yescharacteristic
195ΓL(1,𝔽16)—24no—
196C3×AGL(1,𝔽5)—24no—
197C3×C4×C5—41no—
198C3×C4×C5C4×Q811yescharacteristic
199C3×C4×C5C4×Q811yescharacteristic
200Dic15—21no—
201Dic15—21no—
202Dic15—41no—
203C3×Dic5—41no—
204C3×Dic5#12411yescharacteristic
205C3×Dic5C4⋊Q811yescharacteristic
206C5×Dic3—41no—
207C5×Dic3#10412yes—
208C5×Dic3#10412yes—
209C22×C3×C4—51no—
210C22×C3×C4—101no—
211C22×Dic3—51no—
212C22×Dic3—51no—
213C22×Dic3—101no—
214C2×Dic6—21no—
215C2×Dic6—101no—
216C3×M16—101no—
217C3×M16—101no—
218C2×C3×C8—101no—
219C3×C4⋊C4—51no—
220C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
221C3×C22⋊C4—101no—
222C3×C42—52no—
223C3×C42—101no—
224C3×C42—101no—
225#217—101no—
226C4⋊Dic3—51no—
227C4⋊Dic3—101no—
228C4⋊Dic3—101no—
229C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
230Dic3⋊C4—21no—
231Dic3⋊C4—101no—
232Dic3⋊C4—101no—
233Dic3⋊C4—101no—
234C4×Dic3—51no—
235C4×Dic3—51no—
236C4×Dic3—51no—
237C4×Dic3—52no—
238C4×Dic3—101no—
239C4×Dic3—101no—
240C4×Dic3—104no—
241C4×Dic3—104no—
242C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
243#208—101no—
244#208—101no—
245C2×C3⋊C8—101no—
246C22×D5D3×D411yescharacteristic
247C2×AGL(1,𝔽5)—22no—
248C2×AGL(1,𝔽5)—22no—
249C2×AGL(1,𝔽5)—22no—
250C2×AGL(1,𝔽5)—62no—
251C2×AGL(1,𝔽5)—62no—
252C2×AGL(1,𝔽5)—62no—
253C5×Q8—122no—
254C2×C4×C5—21no—
255C2×C4×C5—31no—
256C2×C4×C5—31no—
257C2×C4×C5—61no—
258C2×C4×C5C2×C4×D311yescharacteristic
259C2×Dic5—21no—
260C2×Dic5—31no—
261C2×Dic5—31no—
262C2×Dic5—61no—
263C2×Dic5D3×D411yescharacteristic
264C4×D5—62no—
265C4×D5—62no—
266C4×D5C2×Dic611yescharacteristic
267C4×D5#23511yescharacteristic
268C4×D5#23711yescharacteristic
269C4×D5D3×Q811yescharacteristic
270Dic10—122no—
271#170—41no—
272#170—121no—
273C2×M16—51no—
274C2×M16—151no—
275C4⋊Q8—31no—
276C4⋊Q8—151no—
277#125—151no—
278#125—151no—
279C22⋊Q8—301no—
280C22⋊Q8—301no—
281C4×Q8—301no—
282C4×Q8—301no—
283#117—51no—
284#117—151no—
285C2×C42—51no—
286C2×C42—151no—
287C2×C42—151no—
288C2×C42—301no—
289C2×C42—301no—
290C4⋊C8—301no—
291C4⋊C8—301no—
292C2×C3×C5#39111yescharacteristic
293C2×C3×C5C2×#10411yescharacteristic
294C2×C3×C5C2×#10411yescharacteristic
295C3×D5—21no—
296C3×D5—21no—
297C23×C3—51no—
298C2×C3×C4—21no—
299C2×C3×C4—51no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—51no—
303C2×C3×C4—51no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—102no—
307C2×C3×C4—101no—
308C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
309C2×Dic3—21no—
310C2×Dic3—51no—
311C2×Dic3—51no—
312C2×Dic3—102no—
313C2×Dic3—102no—
314C2×Dic3—102no—
315C2×Dic3—102no—
316C2×Dic3—102no—
317C2×Dic3—101no—
318C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
319C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
320Dic6—42no—
321Dic6—202no—
322C3×C8—201no—
323C3⋊C8—201no—
324C22×C5D3×C22⋊C411yescharacteristic
325C2×D5—21no—
326C2×D5—21no—
327C2×D5#87811yescharacteristic
328C2×D5#89611yescharacteristic
329AGL(1,𝔽5)—24no—
330AGL(1,𝔽5)—64no—
331C4×C5—32no—
332C4×C5—32no—
333C4×C5—41no—
334C4×C5—121no—
335C4×C5C4×Dic611yescharacteristic
336C4×C5#88711yescharacteristic
337Dic5—41no—
338Dic5—61no—
339Dic5—61no—
340Dic5—121no—
341Dic5#88511yescharacteristic
342Dic5#88811yescharacteristic
343C2×Q8—61no—
344C2×Q8—301no—
345C22×C4—51no—
346C22×C4—101no—
347C22×C4—151no—
348C22×C4—151no—
349C22×C4—301no—
350M16—101no—
351M16—101no—
352M16—301no—
353M16—301no—
354C2×C8—101no—
355C2×C8—301no—
356C4⋊C4—31no—
357C4⋊C4—51no—
358C4⋊C4—61no—
359C4⋊C4—151no—
360C4⋊C4—301no—
361C4⋊C4—301no—
362C4⋊C4—301no—
363C4⋊C4—301no—
364C4⋊C4—301no—
365C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
366C22⋊C4—101no—
367C22⋊C4—301no—
368C42—31no—
369C42—52no—
370C42—101no—
371C42—101no—
372C42—151no—
373C42—151no—
374C42—152no—
375C42—151no—
376C42—301no—
377C42—301no—
378C42—304no—
379C42—304no—
380C3×C5#199411yescharacteristic, complemented, Hall
381C22×C3—51no—
382C22×C3—51no—
383C22×C3—101no—
384C22×C3—101no—
385C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
386C3×C4—41no—
387C3×C4—51no—
388C3×C4—51no—
389C3×C4—104no—
390C3×C4—201no—
391C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
392C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
393Dic3—41no—
394Dic3—104no—
395Dic3—101no—
396Dic3—101no—
397Dic3—201no—
398Dic3#400812yes—
399Dic3#401012yes—
400C2×C5#464611yescharacteristic
401C2×C5D3×#10411yescharacteristic
402C2×C5D3×#10411yescharacteristic
403D5—21no—
404D5—21no—
405C23—51no—
406Q8—122no—
407Q8—602no—
408C2×C4—21no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—51no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—51no—
416C2×C4—61no—
417C2×C4—102no—
418C2×C4—102no—
419C2×C4—102no—
420C2×C4—101no—
421C2×C4—151no—
422C2×C4—151no—
423C2×C4—302no—
424C2×C4—302no—
425C2×C4—302no—
426C2×C4—302no—
427C2×C4—302no—
428C2×C4—301no—
429C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
430C8—201no—
431C8—601no—
432C2×C3—101no—
433C2×C3—101no—
434C2×C3C2×#400811yescharacteristic
435C2×C3C2×#401011yescharacteristic
436C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
437C5#6873111yescharacteristic, complemented, Sylow
438C22—51no—
439C22—51no—
440C22—101no—
441C22—101no—
442C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
443C4—32no—
444C4—32no—
445C4—41no—
446C4—51no—
447C4—51no—
448C4—104no—
449C4—121no—
450C4—201no—
451C4—304no—
452C4—301no—
453C4—301no—
454C4—601no—
455C4AGL(1,𝔽5)×Dic611yescharacteristic
456C4ΓL(1,𝔽16)⋊Q811yescharacteristic
457C3#1060466511yescharacteristic, complemented, Sylow
458C2—101no—
459C2—101no—
460C2#1175184211yescharacteristic
461C2D3×#400811yescharacteristic
462C2D3×#401011yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1994
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
15. #91779C4semidirect
16. C5×#888C4semidirect
18. #91469C4semidirect
22. #91384C4semidirect
45. C4×ΓL(1,𝔽16)Q8semidirect
50. C3×C4×AGL(1,𝔽5)Q8semidirect
139. #888AGL(1,𝔽5)semidirect
160. C4×AGL(1,𝔽5)Dic6semidirect
380. C3×C5#1994semidirect
437. C5#68731semidirect
457. C3#10604665semidirect

Maximal subgroups

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

subgroupindex
#117535682
C4×Dic3×AGL(1,𝔽5)2
#117518242
#117467462
C3×#648752
#117465262
#117465242
#106046653
#687315