Finite Group Databasebeta

← #14300284

Subgroups — #14300284

Order 1920.

540 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91990C211yesmaximal, characteristic
2C2×C3×#4008C211yesmaximal, characteristic
3#11751825C211yesmaximal, characteristic, complemented
4C2×#91162C211yesmaximal, characteristic, complemented
5#11748961C211yesmaximal, characteristic
6#11746526C211yesmaximal, characteristic, complemented
7(C2×AGL(1,𝔽5))⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10603966—31nomaximal, Hall
9C2×C3×#711C2211yescharacteristic
10C2×#6762—21no—
11C22×C3⋊#170—21no—
12#91990—22no—
13#91990—22no—
14C2×#6758C2211yescharacteristic
15C2×D5⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
17C2×C3×#700C2211yescharacteristic
18#91358C2211yescharacteristic
19C2×D5×C3⋊C8C2211yescharacteristic
20C3×#4008—22no—
21#91177—21no—
22Dic3⋊#170—21no—
23#91162—22no—
24#91116C411yescharacteristic, complemented
25C5×#832C411yescharacteristic, complemented
26#90971C411yescharacteristic, complemented
27#90970C411yescharacteristic, complemented
28#70376—51nomaximal, Hall
29C2×#4131—31no—
30C2×#4008D311yescharacteristic
31C2×#3996—31no—
32#65708—31no—
33#65061—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35#64875—31no—
36C22×C3×AGL(1,𝔽5)—21no—
37C2×C3×C5×D4C2×C411yescharacteristic
38C2×C3×C22⋊D5—21no—
39C22×C3×Dic5—21no—
40C3×#711—21no—
41C3×#711—21no—
42C3×#711D412yes—
43C3×#711Q812yes—
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10C2×C411yescharacteristic
46#6762—24no—
47C2×C3⋊#170—24no—
48C2×C3⋊#170—21no—
49C2×C3⋊#170D411yescharacteristic
50C2×C3⋊#170D411yescharacteristic
51#6758—21no—
52#6758D412yes—
53D5⋊(C3⋊C8)—21no—
54D5⋊(C3⋊C8)—21no—
55C2×C3×#170—21no—
56C3×C4×AGL(1,𝔽5)—21no—
57C3×C4×AGL(1,𝔽5)—21no—
58C3×#700—21no—
59C3×#700—21no—
60C4⋊Dic15C2×C411yescharacteristic
61C2×C3⋊#168C2×C411yescharacteristic
62C5×C4⋊Dic3C2×C411yescharacteristic
63C2×C5×C3⋊C8C2×C411yescharacteristic
64#6665—21no—
65Dic3×Dic5—21no—
66D5×C3⋊C8—21no—
67D5×C3⋊C8—21no—
68C2×#948—51no—
69C2×C3×#104—51no—
70#5233—51no—
71C2×#822—51no—
72#5119—51no—
73#4524—51no—
74#4523—51no—
75C2×#711C2×D311yescharacteristic
76C2×#704—61no—
77C22×#170—61no—
78#4131—62no—
79#4131—62no—
80C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
81C2×#700—31no—
82C2×#700C2×D311yescharacteristic
83C2×D5⋊C8—31no—
84C2×C8×D5—31no—
85#4038—31no—
86#4008—22no—
87#4004—61no—
88C4⋊#170—61no—
89#3996—62no—
90C5×#102—31no—
91#3964—31no—
92#3948—31no—
93#3942—31no—
94#1866—151noSylow
95C23×C3×C5—21no—
96C22×C3×D5#4111yescharacteristic
97C2×C3×AGL(1,𝔽5)—22no—
98C2×C3×AGL(1,𝔽5)—41no—
99C2×C3×AGL(1,𝔽5)—41no—
100C3×C5×D4—21no—
101C3×C5×D4C4⋊C412yes—
102C2×C3×C4×C5C22×C411yescharacteristic
103C2×Dic15—21no—
104C2×C5×Dic3—21no—
105C3×C22⋊D5—22no—
106C3×C22⋊D5—22no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—22no—
109C2×C3×Dic5—22no—
110C2×C3×Dic5C2×D411yescharacteristic
111C3×C4×D5C2×D411yescharacteristic
112C3×C4×D5C2×D411yescharacteristic
113C3×C4×D5C2×Q811yescharacteristic
114C3×C4×D5#4111yescharacteristic
115C3×Dic10—21no—
116C3×Dic10C4⋊C412yes—
117C3⋊#170—21no—
118C3⋊#170—21no—
119C3⋊#170—22no—
120C3×#170—41no—
121C3⋊#168—21no—
122C5×C3⋊C8—21no—
123C2×C3×#41—51no—
124C2×C3×M16—51no—
125C2×C3×C42—51no—
126#948—102no—
127#948—102no—
128#926—51no—
129C2×#208—51no—
130C2×#208—101no—
131C22×C3⋊C8—51no—
132C22×C3⋊C8—51no—
133C22×C3⋊C8—101no—
134C3×#104—102no—
135#835—51no—
136#835—51no—
137#832—51no—
138#832AGL(1,𝔽5)11yescharacteristic, complemented
139#822—102no—
140Dic3⋊C8—101no—
141Dic3⋊C8—101no—
142C22×AGL(1,𝔽5)—21no—
143C2×C5×D4C4×D311yescharacteristic
144C2×C22⋊D5—21no—
145C22×Dic5—21no—
146#711—21no—
147#711—21no—
148#711Dic612yes—
149#711#6712yes—
150C2×C4×D5C22×D311yescharacteristic
151C2×Dic10C4×D311yescharacteristic
152#704—64no—
153C2×#170—21no—
154C2×#170—31no—
155C2×#170—31no—
156C2×#170—64no—
157C2×#170—61no—
158C4×AGL(1,𝔽5)—21no—
159C4×AGL(1,𝔽5)—21no—
160#700—21no—
161#700—21no—
162#700—32no—
163#700—61no—
164D5⋊C8—61no—
165D5⋊C8—61no—
166C2×C5×C8—31no—
167C5×C4⋊C4—31no—
168#685—61no—
169C4⋊Dic5—31no—
170C4×Dic5—61no—
171C2×#168—31no—
172C8×D5—61no—
173C8×D5—61no—
174C2×#131—151no—
175C2×#108—151no—
176#424—151no—
177C2×#104—51no—
178#416—151no—
179#339—151no—
180#336—151no—
181C22×C3×C5—22no—
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5C4×D411yescharacteristic
185C2×C3×D5—21no—
186C2×C3×D5—21no—
187C2×C3×D5C22⋊Q811yescharacteristic
188C2×C3×D5#12311yescharacteristic
189C3×AGL(1,𝔽5)—42no—
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192Dic15—41no—
193C3×Dic5—22no—
194C3×Dic5#12011yescharacteristic
195C3×Dic5C22⋊Q811yescharacteristic
196C5×Dic3—41no—
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8—51no—
202C2×C3×D4—101no—
203C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
204C22×C3×C4—51no—
205C22×C3×C4—101no—
206C22×C3×C4—101no—
207C3×M16—101no—
208C3×M16—101no—
209C2×C3×C8—101no—
210C3×C42—101no—
211C3×C42—101no—
212#217—101no—
213C4⋊Dic3—51no—
214C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
215C4×Dic3—101no—
216#208—52no—
217#208—101no—
218#208—104no—
219C2×C3⋊C8—51no—
220C2×C3⋊C8—51no—
221C2×C3⋊C8—51no—
222C2×C3⋊C8—101no—
223C2×C3⋊C8—101no—
224C2×C3⋊C8—101no—
225C2×C3⋊C8—101no—
226C2×C3⋊C8—101no—
227C2×C3⋊C8—104no—
228C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
229C23×C5—21no—
230C22×D5#23711yescharacteristic
231C2×AGL(1,𝔽5)—22no—
232C2×AGL(1,𝔽5)—41no—
233C2×AGL(1,𝔽5)—41no—
234C5×D4—21no—
235C5×D4Dic3⋊C412yes—
236C2×C4×C5—61no—
237C2×C4×C5C2×C4×D311yescharacteristic
238C22⋊D5—22no—
239C22⋊D5—22no—
240C2×Dic5—21no—
241C2×Dic5—22no—
242C2×Dic5—22no—
243C2×Dic5—61no—
244C2×Dic5D3×D411yescharacteristic
245C4×D5C2×Dic611yescharacteristic
246C4×D5D3×D411yescharacteristic
247C4×D5#23711yescharacteristic
248C4×D5C2×#6711yescharacteristic
249Dic10—21no—
250Dic10Dic3⋊C412yes—
251#170—41no—
252#170—61no—
253#170—61no—
254#170—62no—
255C5×C8—61no—
256#168—61no—
257C2×#41—51no—
258#131—302no—
259#131—302no—
260C2×M16—51no—
261C2×M16—151no—
262C2×M16—301no—
263C22×C8—151no—
264C22×C8—151no—
265C22×C8—301no—
266#117—151no—
267C2×C42—51no—
268#108—302no—
269C4⋊C8—301no—
270C4⋊C8—301no—
271#104—102no—
272#103—151no—
273#103—151no—
274#102—31no—
275#102—151no—
276C2×C3×C5—42no—
277C2×C3×C5#38511yescharacteristic
278C2×C3×C5#44311yescharacteristic
279C2×C3×C5#44411yescharacteristic
280C3×D5—21no—
281C3×D5—21no—
282C23×C3—21no—
283C23×C3—51no—
284C3×Q8—52no—
285C3×Q8—101no—
286C3×D4—21no—
287C3×D4—102no—
288C3×D4—102no—
289C3×D4C4⋊AGL(1,𝔽5)12yes—
290C2×C3×C4—51no—
291C2×C3×C4—51no—
292C2×C3×C4—51no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—102no—
296C2×C3×C4—102no—
297C2×C3×C4—101no—
298C2×C3×C4—102no—
299C2×C3×C4—201no—
300C2×C3×C4—201no—
301C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
302C2×Dic3—21no—
303C2×Dic3—101no—
304C3×C8—201no—
305C3⋊C8—21no—
306C3⋊C8—101no—
307C3⋊C8—101no—
308C3⋊C8—102no—
309C3⋊C8—101no—
310C22×C5—22no—
311C22×C5—41no—
312C22×C5—41no—
313C22×C5#67⋊C411yescharacteristic
314C2×D5—21no—
315C2×D5—21no—
316C2×D5#87811yescharacteristic
317C2×D5#93511yescharacteristic
318AGL(1,𝔽5)—42no—
319C4×C5—121no—
320C4×C5#67⋊C411yescharacteristic
321C4×C5C2×Dic3⋊C411yescharacteristic
322Dic5—22no—
323Dic5—121no—
324Dic5#87811yescharacteristic
325Dic5#93711yescharacteristic
326#41—52no—
327#41—52no—
328#41—101no—
329#41—101no—
330C2×Q8—51no—
331C2×D4—101no—
332C2×D4D3×AGL(1,𝔽5)11yescharacteristic
333C22×C4—51no—
334C22×C4—101no—
335C22×C4—101no—
336M16—101no—
337M16—101no—
338M16—152no—
339M16—301no—
340M16—304no—
341C2×C8—31no—
342C2×C8—101no—
343C2×C8—151no—
344C2×C8—151no—
345C2×C8—151no—
346C2×C8—301no—
347C2×C8—301no—
348C2×C8—301no—
349C2×C8—301no—
350C2×C8—301no—
351C2×C8—304no—
352C4⋊C4—31no—
353C4⋊C4—151no—
354C22⋊C4—301no—
355C42—101no—
356C42—101no—
357C42—301no—
358C3×C5#186611yescharacteristic, complemented, Hall
359C22×C3—22no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—51no—
363C22×C3—51no—
364C22×C3—101no—
365C22×C3—101no—
366C22×C3D4×AGL(1,𝔽5)11yescharacteristic
367C3×C4—51no—
368C3×C4—51no—
369C3×C4—102no—
370C3×C4—202no—
371C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
372C3×C4D4×AGL(1,𝔽5)11yescharacteristic
373Dic3—41no—
374Dic3—201no—
375C2×C5—42no—
376C2×C5#482311yescharacteristic
377C2×C5#482411yescharacteristic
378C2×C5#494611yescharacteristic
379D5—21no—
380D5—21no—
381C23—21no—
382C23—51no—
383Q8—52no—
384Q8—101no—
385D4—21no—
386D4—102no—
387D4—102no—
388D4Dic3⋊AGL(1,𝔽5)12yes—
389C2×C4—51no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—61no—
395C2×C4—102no—
396C2×C4—102no—
397C2×C4—101no—
398C2×C4—102no—
399C2×C4—201no—
400C2×C4—201no—
401C2×C4—301no—
402C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
403C8—61no—
404C8—201no—
405C8—301no—
406C8—301no—
407C8—302no—
408C8—301no—
409C2×C3—42no—
410C2×C3—101no—
411C2×C3—101no—
412C2×C3#6569911yescharacteristic
413C2×C3#6573211yescharacteristic
414C2×C3#6573711yescharacteristic
415C5#7037611yescharacteristic, complemented, Sylow
416C22—22no—
417C22—41no—
418C22—41no—
419C22—51no—
420C22—51no—
421C22—101no—
422C22—101no—
423C22#67⋊AGL(1,𝔽5)11yescharacteristic
424C4—51no—
425C4—51no—
426C4—102no—
427C4—121no—
428C4—202no—
429C4—601no—
430C4C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
431C4#67⋊AGL(1,𝔽5)11yescharacteristic
432C3#1060396611yescharacteristic, complemented, Sylow
433C2—42no—
434C2—101no—
435C2—101no—
436C2#1175195511yescharacteristic
437C2#1175195811yescharacteristic
438C2#1175202911yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#711C22
cosocleC2×C3×C4×D5C23
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#1866
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11751825C2semidirect
4. C2×#91162C2semidirect
6. #11746526C2semidirect
7. (C2×AGL(1,𝔽5))⋊(C3⋊C8)C2semidirect
24. #91116C4semidirect
25. C5×#832C4semidirect
26. #90971C4semidirect
27. #90970C4semidirect
138. #832AGL(1,𝔽5)semidirect
358. C3×C5#1866semidirect
415. C5#70376semidirect
432. C3#10603966semidirect

Maximal subgroups

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

subgroupindex
C2×#919902
C2×C3×#40082
#117518252
C2×#911622
#117489612
#117465262
(C2×AGL(1,𝔽5))⋊(C3⋊C8)2
#106039663
#703765