Finite Group Databasebeta

← #14300290

Subgroups — #14300290

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×#91914C211yesmaximal, characteristic
2C2×#91224C211yesmaximal, characteristic, complemented
3#11751822C211yesmaximal, characteristic, complemented
4C2×#91158C211yesmaximal, characteristic
5#11747642C211yesmaximal, characteristic
6C3×(C2×AGL(1,𝔽5))⋊C8C211yesmaximal, characteristic, complemented
7#11746526C211yesmaximal, characteristic, complemented
8#10603997—31nomaximal, Hall
9C2×#6773C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14C2×#6740—21no—
15C2×#6738—21no—
16#91914—22no—
17#91914—22no—
18C2×C3×C8×D5C2211yescharacteristic
19#91358C2211yescharacteristic
20#91224—22no—
21#91177—21no—
22Dic3⋊#170—21no—
23#91158—22no—
24#91105C411yescharacteristic, complemented
25C5×#816C411yescharacteristic, complemented
26#90973C411yescharacteristic, complemented
27#90969C411yescharacteristic, complemented
28#68694—51nomaximal, Hall
29C2×#4133—31no—
30C2×#4010—31no—
31C2×#3995—31no—
32#65708—31no—
33#65093—31no—
34(C2×AGL(1,𝔽5))⋊C8D311yescharacteristic, complemented
35#64875—31no—
36C22×C3×AGL(1,𝔽5)—21no—
37C2×D60C2×C411yescharacteristic
38C2×C5×Dic6C2×C411yescharacteristic
39C2×C3×C4×D5C2311yescharacteristic
40C2×#1202—21no—
41C2×#1200—21no—
42#6773—21no—
43#6773—21no—
44#6773D412yes—
45#6773Q812yes—
46C2×C3⋊#170—21no—
47C2×C3⋊#170—21no—
48#6758—21no—
49#6758—21no—
50#6758—21no—
51#6758D412yes—
52C2×C3×#170D411yescharacteristic
53C2×C3×#170D411yescharacteristic
54C3×C4×AGL(1,𝔽5)—21no—
55C3×C4×AGL(1,𝔽5)—21no—
56C3×D5⋊C8—21no—
57C3×D5⋊C8—21no—
58#6740—24no—
59#6738—24no—
60C2×C3×C5×C8C2×C411yescharacteristic
61C4⋊Dic15C2×C411yescharacteristic
62C5×C4⋊Dic3C2×C411yescharacteristic
63C2×C3×#168C2×C411yescharacteristic
64C3×C8×D5—21no—
65C3×C8×D5—21no—
66#6665—21no—
67Dic3×Dic5—21no—
68C2×#901—51no—
69#5113—51no—
70C2×#819—51no—
71#5102—51no—
72C2×#805—51no—
73C3×#336—51no—
74#4523—51no—
75C2×#713—31no—
76#4133—62no—
77#4133—62no—
78C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
79C2×#700—31no—
80C2×#700—31no—
81C2×#700—61no—
82C2×D5⋊C8—61no—
83C2×D5⋊C8C2×D311yescharacteristic
84C2×C8×D5C2×D311yescharacteristic
85#4038—31no—
86#4010—62no—
87#4004—61no—
88C4⋊#170—61no—
89#3995—62no—
90C5×#103—31no—
91#3967—31no—
92#3953—31no—
93#3941—31no—
94#1866—151noSylow
95C22×D15—21no—
96C22×C3×D5#4111yescharacteristic
97C2×C3×AGL(1,𝔽5)—22no—
98C2×C3×AGL(1,𝔽5)—41no—
99C2×C3×AGL(1,𝔽5)—41no—
100C2×C3×C4×C5C22×C411yescharacteristic
101C2×Dic15—21no—
102D60—21no—
103D60C4⋊C412yes—
104C2×C5×Dic3—21no—
105C2×C5×Dic3—21no—
106C5×Dic6—21no—
107C5×Dic6C4⋊C412yes—
108C2×C3×Dic5C2×D411yescharacteristic
109C3×C4×D5C2×D411yescharacteristic
110C3×C4×D5C2×D411yescharacteristic
111C3×C4×D5C2×Q811yescharacteristic
112C3×C4×D5#4111yescharacteristic
113#1202—22no—
114#1202—22no—
115#1200—22no—
116#1200—22no—
117C3⋊#170—22no—
118C3⋊#170—41no—
119C3×#170—21no—
120C3×#170—21no—
121C3×C5×C8—21no—
122C3×#168—21no—
123C2×#235—51no—
124C22×C3×C8—51no—
125C22×C3×C8—51no—
126C2×C3×C42—51no—
127#926—51no—
128C2×#208—51no—
129C2×#208—51no—
130#901—102no—
131#901—102no—
132C2×ΓL(1,𝔽25)—101no—
133C2×D3×C8—101no—
134#821—51no—
135#821—51no—
136#819—102no—
137#816—51no—
138#816AGL(1,𝔽5)11yescharacteristic, complemented
139Dic3⋊C8—101no—
140Dic3⋊C8—101no—
141#805—102no—
142C22×AGL(1,𝔽5)—21no—
143C2×C5×Q8—31no—
144#713—32no—
145#713—32no—
146#713—61no—
147#713—61no—
148C2×D20—31no—
149C2×D20—61no—
150C2×C4×D5—61no—
151C2×C4×D5C22×D311yescharacteristic
152C2×#170—61no—
153C2×#170—61no—
154C2×#170D1211yescharacteristic
155C2×#170D1211yescharacteristic
156C4×AGL(1,𝔽5)—21no—
157C4×AGL(1,𝔽5)—21no—
158#700—32no—
159#700—64no—
160#700—61no—
161#700—61no—
162#700—61no—
163D5⋊C8—21no—
164D5⋊C8—21no—
165D5⋊C8—64no—
166C2×C5×C8C4×D311yescharacteristic
167C5×C4⋊C4—31no—
168#685—61no—
169C4⋊Dic5—31no—
170C4×Dic5—61no—
171C2×#168C4×D311yescharacteristic
172C8×D5—21no—
173C8×D5—21no—
174C2×#131—151no—
175C2×#108—151no—
176#424—151no—
177C2×#104—151no—
178#416—151no—
179#339—151no—
180#336—51no—
181C22×C3×C5C4×D411yescharacteristic
182C2×D15—22no—
183C2×D15—41no—
184C2×D15—41no—
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#12011yescharacteristic
194C3×Dic5C22⋊Q811yescharacteristic
195C5×Dic3—22no—
196C5×Dic3—41no—
197C22×C3×C4—51no—
198C22×C3×C4—101no—
199C2×#67—101no—
200#235—52no—
201#235—52no—
202#235—101no—
203#235—101no—
204C2×D12—51no—
205C2×C4×D3—101no—
206C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
207C2×C3×C8—51no—
208C2×C3×C8—51no—
209C2×C3×C8—51no—
210C2×C3×C8—101no—
211C2×C3×C8—101no—
212C2×C3×C8—101no—
213C2×C3×C8—101no—
214C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
215C3×C42—101no—
216C3×C42—101no—
217#217—101no—
218C4⋊Dic3—51no—
219C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
220C4×Dic3—101no—
221#208—52no—
222#208—101no—
223#208—101no—
224#208—101no—
225C2×C3⋊C8—101no—
226C2×C3⋊C8—101no—
227ΓL(1,𝔽25)—104no—
228D3×C8—104no—
229C22×D5—61no—
230C22×D5#23511yescharacteristic
231C2×AGL(1,𝔽5)—22no—
232C2×AGL(1,𝔽5)—41no—
233C2×AGL(1,𝔽5)—41no—
234C5×Q8—32no—
235C5×Q8—61no—
236C2×C4×C5—61no—
237C2×C4×C5—61no—
238C2×C4×C5C2×C4×D311yescharacteristic
239C2×Dic5—61no—
240C2×Dic5C2×D1211yescharacteristic
241D20—32no—
242D20—62no—
243D20—62no—
244D20—61no—
245C4×D5—62no—
246C4×D5—62no—
247C4×D5D3×D411yescharacteristic
248C4×D5D3×D411yescharacteristic
249C4×D5#23711yescharacteristic
250C4×D5D3×Q811yescharacteristic
251#170—21no—
252#170—21no—
253#170—62no—
254#170—121no—
255C5×C8—21no—
256#168—21no—
257C2×#41—151no—
258#131—302no—
259#131—302no—
260C2×M16—151no—
261C2×M16—151no—
262C2×M16—301no—
263C22×C8—51no—
264C22×C8—51no—
265C22×C8—301no—
266#117—151no—
267C2×C42—51no—
268#108—302no—
269C4⋊C8—301no—
270C4⋊C8—301no—
271#104—302no—
272#103—31no—
273#103—151no—
274#102—151no—
275#102—151no—
276C2×C3×C5#38511yescharacteristic
277C2×C3×C5#44311yescharacteristic
278C2×C3×C5#44411yescharacteristic
279D15—42no—
280C3×D5—21no—
281C3×D5—21no—
282C23×C3—51no—
283C22×D3—101no—
284C2×C3×C4—51no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—51no—
289C2×C3×C4—102no—
290C2×C3×C4—201no—
291C2×C3×C4—201no—
292C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
293#67—102no—
294#67—102no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—101no—
298D12—52no—
299D12—101no—
300C4×D3—102no—
301C4×D3—102no—
302Dic6—21no—
303Dic6C4⋊AGL(1,𝔽5)12yes—
304C3×C8—21no—
305C3×C8—101no—
306C3×C8—101no—
307C3×C8—101no—
308C3⋊C8—102no—
309C3⋊C8—201no—
310C22×C5C4×D1211yescharacteristic
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—62no—
314C2×D5—121no—
315C2×D5—121no—
316C2×D5#88311yescharacteristic
317C2×D5#89611yescharacteristic
318AGL(1,𝔽5)—42no—
319C4×C5—62no—
320C4×C5—121no—
321C4×C5#67⋊C411yescharacteristic
322C4×C5D3×C4⋊C411yescharacteristic
323Dic5—121no—
324Dic5#88211yescharacteristic
325Dic5#89711yescharacteristic
326#41—152no—
327#41—152no—
328#41—301no—
329#41—301no—
330C2×Q8—31no—
331C2×D4—151no—
332C2×D4—301no—
333C22×C4—51no—
334C22×C4—101no—
335C22×C4—301no—
336M16—152no—
337M16—304no—
338M16—301no—
339M16—301no—
340M16—301no—
341C2×C8—51no—
342C2×C8—51no—
343C2×C8—51no—
344C2×C8—101no—
345C2×C8—101no—
346C2×C8—101no—
347C2×C8—101no—
348C2×C8—304no—
349C2×C8—301no—
350C2×C8—301no—
351C2×C8D3×AGL(1,𝔽5)11yescharacteristic
352C4⋊C4—31no—
353C4⋊C4—151no—
354C22⋊C4—301no—
355C42—101no—
356C42—101no—
357C42—301no—
358C3×C5#186611yescharacteristic, complemented, Hall
359C22×C3—51no—
360C22×C3—51no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3D4×AGL(1,𝔽5)11yescharacteristic
364C2×D3—102no—
365C2×D3—201no—
366C2×D3—201no—
367C3×C4—51no—
368C3×C4—51no—
369C3×C4—202no—
370C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
371C3×C4D4×AGL(1,𝔽5)11yescharacteristic
372Dic3—22no—
373Dic3—41no—
374Dic3—201no—
375C2×C5#466811yescharacteristic
376C2×C5#470011yescharacteristic
377C2×C5#471711yescharacteristic
378D5—21no—
379D5—21no—
380D5—122no—
381C23—51no—
382C23—301no—
383Q8—32no—
384Q8—61no—
385D4—152no—
386D4—302no—
387D4—302no—
388D4—301no—
389C2×C4—51no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—102no—
397C2×C4—201no—
398C2×C4—201no—
399C2×C4—302no—
400C2×C4—302no—
401C2×C4—301no—
402C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
403C8—21no—
404C8—101no—
405C8—101no—
406C8—101no—
407C8—302no—
408C8—601no—
409C2×C3—101no—
410C2×C3—101no—
411C2×C3#6569911yescharacteristic
412C2×C3#6573611yescharacteristic
413C2×C3#6574111yescharacteristic
414D3—202no—
415C5#6869411yescharacteristic, complemented, Sylow
416C22—51no—
417C22—51no—
418C22—101no—
419C22—101no—
420C22—302no—
421C22—601no—
422C22—601no—
423C22AGL(1,𝔽5)×D1211yescharacteristic
424C4—51no—
425C4—51no—
426C4—62no—
427C4—121no—
428C4—202no—
429C4—601no—
430C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
431C4#67⋊AGL(1,𝔽5)11yescharacteristic
432C3#1060399711yescharacteristic, complemented, Sylow
433C2—101no—
434C2—101no—
435C2—602no—
436C2#1175167911yescharacteristic
437C2#1175168411yescharacteristic
438C2#1175193411yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×D12
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×C8C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×#6773C22
cosocleC2×C3×C4×D5C23
hypercenterC2×C8D3×AGL(1,𝔽5)
hypocenterC3×C5#1866
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#91224C2semidirect
3. #11751822C2semidirect
6. C3×(C2×AGL(1,𝔽5))⋊C8C2semidirect
7. #11746526C2semidirect
24. #91105C4semidirect
25. C5×#816C4semidirect
26. #90973C4semidirect
27. #90969C4semidirect
34. (C2×AGL(1,𝔽5))⋊C8D3semidirect
138. #816AGL(1,𝔽5)semidirect
358. C3×C5#1866semidirect
415. C5#68694semidirect
432. C3#10603997semidirect

Maximal subgroups

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

subgroupindex
C2×#919142
C2×#912242
#117518222
C2×#911582
#117476422
C3×(C2×AGL(1,𝔽5))⋊C82
#117465262
#106039973
#686945