Finite Group Databasebeta

← #14295985

Subgroups — #14295985

Order 1920.

600 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×#91992C211yesmaximal, characteristic
2C2×C3×#4010C211yesmaximal, characteristic
3#11751782C211yesmaximal, characteristic, complemented
4C2×AGL(1,𝔽5)×C3⋊C8C211yesmaximal, characteristic, complemented
5#11749207C211yesmaximal, characteristic
6#11746532C211yesmaximal, characteristic, complemented
7#11746526C211yesmaximal, characteristic, complemented
8#10603991—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10#91992—22no—
11#91992C414yescomplemented
12C2×#6758—21no—
13C2×#6758C2211yescharacteristic
14C2×D5⋊(C3⋊C8)—21no—
15C2×D5⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
17C2×C3×#700C2211yescharacteristic
18#91358C2211yescharacteristic
19C2×D5×C3⋊C8C2211yescharacteristic
20C3×#4010C414yescomplemented
21#91169—21no—
22Dic3×#170—21no—
23AGL(1,𝔽5)×C3⋊C8—22no—
24#91119C411yescharacteristic, complemented
25C5×#835C411yescharacteristic, complemented
26#90965C411yescharacteristic, complemented
27#90964C411yescharacteristic, complemented
28#70369—51nomaximal, Hall
29C2×#4133—31no—
30C2×#4010D311yescharacteristic
31C2×C8×AGL(1,𝔽5)—31no—
32#65701—31no—
33#65093—31no—
34#64896—31no—
35#64875—31no—
36C22×C3×AGL(1,𝔽5)—21no—
37C2×C3×C5×Q8C2×C411yescharacteristic
38C3×#713—21no—
39C3×#713—21no—
40C3×#713C2×C412yes—
41C3×#713C2×C412yes—
42C2×C3×D20—21no—
43C2×C3×D20C2×C411yescharacteristic
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5C2311yescharacteristic
46C2×C3⋊#170—21no—
47C2×C3⋊#170D411yescharacteristic
48C2×C3⋊#170D411yescharacteristic
49#6758—21no—
50#6758—24no—
51#6758C2×C412yes—
52D5⋊(C3⋊C8)—24no—
53D5⋊(C3⋊C8)C2×C412yes—
54D5⋊(C3⋊C8)D412yes—
55C2×C3×#170—21no—
56C3×C4×AGL(1,𝔽5)—21no—
57C3×C4×AGL(1,𝔽5)C2×C412yes—
58C3×#700—21no—
59C3×#700C2×C412yes—
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—
71#5122—51no—
72C2×C4×C3⋊C8—51no—
73#4558—51no—
74#4523—51no—
75C2×#713C2×D311yescharacteristic
76#4133—34no—
77#4133—62no—
78C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
79C2×#700—31no—
80C2×#700—61no—
81C2×#700C2×D311yescharacteristic
82C2×D5⋊C8—31no—
83C2×D5⋊C8—61no—
84C2×C8×D5—31no—
85#4038—31no—
86#4010Dic314yescomplemented
87#4002—61no—
88C4×#170—61no—
89C8×AGL(1,𝔽5)—62no—
90C5×#103—31no—
91#3967—31no—
92#3953—31no—
93#3941—31no—
94#1764—151noSylow
95C22×C3×D5—21no—
96C22×C3×D5#4111yescharacteristic
97C2×C3×AGL(1,𝔽5)—22no—
98C2×C3×AGL(1,𝔽5)—22no—
99C2×C3×AGL(1,𝔽5)—22no—
100C3×C5×Q8—21no—
101C3×C5×Q8C4212yes—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5C22×C411yescharacteristic
104C2×Dic15—21no—
105C2×C5×Dic3—21no—
106C2×C3×Dic5C2×D411yescharacteristic
107C3×D20—22no—
108C3×D20—22no—
109C3×D20—21no—
110C3×D20C4212yes—
111C3×C4×D5—22no—
112C3×C4×D5—22no—
113C3×C4×D5C22×C411yescharacteristic
114C3×C4×D5C22×C411yescharacteristic
115C3×C4×D5C2×D411yescharacteristic
116C3×C4×D5#4111yescharacteristic
117C3⋊#170—22no—
118C3⋊#170C22⋊C412yes—
119C3⋊#170C22⋊C412yes—
120C3×#170—22no—
121C3⋊#168—21no—
122C5×C3⋊C8—21no—
123C2×C3×#41—51no—
124C2×C3×M16—51no—
125C2×C3×C42—51no—
126#948—54no—
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—54no—
135#835—51no—
136#835AGL(1,𝔽5)11yescharacteristic, complemented
137#832—51no—
138#832—51no—
139#815—101no—
140C8×Dic3—101no—
141C4×C3⋊C8—102no—
142C22×AGL(1,𝔽5)—21no—
143C2×C5×Q8C4×D311yescharacteristic
144#713—21no—
145#713—21no—
146#713C4×D312yes—
147#713C2×Dic312yes—
148C2×D20—21no—
149C2×D20C4×D311yescharacteristic
150C2×C4×D5—21no—
151C2×C4×D5C22×D311yescharacteristic
152C2×#170—21no—
153C2×#170—31no—
154C2×#170—31no—
155C2×#170—61no—
156C4×AGL(1,𝔽5)—21no—
157C4×AGL(1,𝔽5)C2×Dic312yes—
158#700—21no—
159#700—32no—
160#700—64no—
161#700—61no—
162#700C2×Dic312yes—
163D5⋊C8—32no—
164D5⋊C8—32no—
165D5⋊C8—64no—
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×#104—51no—
176#416—151no—
177#405—151no—
178C2×C4×C8—151no—
179#341—151no—
180#339—151no—
181C22×C3×C5C4×D411yescharacteristic
182C2×C3×D5—21no—
183C2×C3×D5—21no—
184C2×C3×D5—22no—
185C2×C3×D5—41no—
186C2×C3×D5—41no—
187C2×C3×D5#11711yescharacteristic
188C2×C3×D5C4×D411yescharacteristic
189C3×AGL(1,𝔽5)—24no—
190C3×C4×C5—22no—
191C3×C4×C5C2×C4211yescharacteristic
192C3×C4×C5C4×D411yescharacteristic
193Dic15—41no—
194C3×Dic5C2×C22⋊C411yescharacteristic
195C3×Dic5C4×D411yescharacteristic
196C5×Dic3—41no—
197C3×#41—52no—
198C3×#41—52no—
199C3×#41—101no—
200C3×#41—101no—
201C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
202C2×C3×D4—51no—
203C2×C3×D4—101no—
204C22×C3×C4—51no—
205C22×C3×C4—101no—
206C22×C3×C4—101no—
207C3×M16—52no—
208C3×M16—101no—
209C2×C3×C8—101no—
210C3×C42—52no—
211C3×C42—101no—
212#217—101no—
213C4⋊Dic3—51no—
214C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
215C4×Dic3—101no—
216#208—52no—
217#208—104no—
218#208—101no—
219C2×C3⋊C8—52no—
220C2×C3⋊C8—51no—
221C2×C3⋊C8—51no—
222C2×C3⋊C8—52no—
223C2×C3⋊C8—51no—
224C2×C3⋊C8—101no—
225C2×C3⋊C8—101no—
226C2×C3⋊C8—104no—
227C2×C3⋊C8—101no—
228C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
229C22×D5—21no—
230C22×D5#23711yescharacteristic
231C2×AGL(1,𝔽5)—22no—
232C2×AGL(1,𝔽5)—22no—
233C2×AGL(1,𝔽5)—22no—
234C5×Q8—21no—
235C5×Q8C4×Dic312yes—
236C2×C4×C5—21no—
237C2×C4×C5—61no—
238C2×C4×C5C2×C4×D311yescharacteristic
239C2×Dic5—61no—
240C2×Dic5D3×D411yescharacteristic
241D20—22no—
242D20—22no—
243D20—21no—
244D20C4×Dic312yes—
245C4×D5—22no—
246C4×D5—22no—
247C4×D5C2×C4×D311yescharacteristic
248C4×D5D3×D411yescharacteristic
249C4×D5#23711yescharacteristic
250C4×D5C22×Dic311yescharacteristic
251#170—22no—
252#170—32no—
253#170—32no—
254#170—62no—
255C5×C8—61no—
256#168—61no—
257C2×#41—51no—
258#131—154no—
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#104—54no—
269#103—31no—
270#103—151no—
271#102—151no—
272#102—151no—
273#97—301no—
274C4×C8—302no—
275C4×C8—301no—
276C2×C3×C5C4×C22⋊C411yescharacteristic
277C2×C3×C5#44311yescharacteristic
278C2×C3×C5#44311yescharacteristic
279C3×D5—21no—
280C3×D5—21no—
281C3×D5—42no—
282C23×C3—51no—
283C23×C3—101no—
284C3×Q8—21no—
285C3×Q8C4×AGL(1,𝔽5)12yes—
286C3×D4—52no—
287C3×D4—102no—
288C3×D4—102no—
289C3×D4—101no—
290C2×C3×C4—21no—
291C2×C3×C4—51no—
292C2×C3×C4—51no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—102no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—102no—
300C2×C3×C4—102no—
301C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
302C2×Dic3—21no—
303C2×Dic3—101no—
304C3×C8—102no—
305C3⋊C8—21no—
306C3⋊C8—52no—
307C3⋊C8—52no—
308C3⋊C8—102no—
309C3⋊C8—101no—
310C22×C5#67⋊C411yescharacteristic
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—22no—
314C2×D5—41no—
315C2×D5—41no—
316C2×D5#87711yescharacteristic
317C2×D5D4×Dic311yescharacteristic
318AGL(1,𝔽5)—24no—
319C4×C5—22no—
320C4×C5—121no—
321C4×C5#67⋊C411yescharacteristic
322C4×C5C2×C4×Dic311yescharacteristic
323Dic5—121no—
324Dic5D3×C22⋊C411yescharacteristic
325Dic5D4×Dic311yescharacteristic
326#41—52no—
327#41—52no—
328#41—101no—
329#41—101no—
330C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
331C2×D4—51no—
332C2×D4—101no—
333C22×C4—51no—
334C22×C4—101no—
335C22×C4—101no—
336M16—52no—
337M16—101no—
338M16—152no—
339M16—304no—
340M16—301no—
341C2×C8—31no—
342C2×C8—101no—
343C2×C8—151no—
344C2×C8—152no—
345C2×C8—151no—
346C2×C8—152no—
347C2×C8—151no—
348C2×C8—301no—
349C2×C8—301no—
350C2×C8—304no—
351C2×C8—301no—
352C4⋊C4—31no—
353C4⋊C4—151no—
354C22⋊C4—301no—
355C42—52no—
356C42—101no—
357C42—301no—
358C3×C5#176411yescharacteristic, complemented, Hall
359C22×C3—51no—
360C22×C3—51no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3—102no—
364C22×C3—201no—
365C22×C3—201no—
366C22×C3D4×AGL(1,𝔽5)11yescharacteristic
367C3×C4—22no—
368C3×C4—51no—
369C3×C4—51no—
370C3×C4—104no—
371C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
372C3×C4D4×AGL(1,𝔽5)11yescharacteristic
373Dic3—41no—
374Dic3—201no—
375C2×C5#482411yescharacteristic
376C2×C5#482411yescharacteristic
377C2×C5Dic3×C22⋊C411yescharacteristic
378D5—21no—
379D5—21no—
380D5—42no—
381C23—51no—
382C23—101no—
383Q8—21no—
384Q8Dic3×AGL(1,𝔽5)12yes—
385D4—52no—
386D4—102no—
387D4—102no—
388D4—101no—
389C2×C4—21no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—51no—
395C2×C4—61no—
396C2×C4—102no—
397C2×C4—102no—
398C2×C4—102no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—301no—
402C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
403C8—61no—
404C8—102no—
405C8—152no—
406C8—152no—
407C8—302no—
408C8—301no—
409C2×C3—101no—
410C2×C3—101no—
411C2×C3—202no—
412C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
413C2×C3#6573611yescharacteristic
414C2×C3#6574011yescharacteristic
415C5#7036911yescharacteristic, complemented, Sylow
416C22—51no—
417C22—51no—
418C22—101no—
419C22—101no—
420C22—102no—
421C22—201no—
422C22—201no—
423C22#67⋊AGL(1,𝔽5)11yescharacteristic
424C4—22no—
425C4—51no—
426C4—51no—
427C4—104no—
428C4—121no—
429C4—601no—
430C4C2×Dic3×AGL(1,𝔽5)11yescharacteristic
431C4#67⋊AGL(1,𝔽5)11yescharacteristic
432C3#1060399111yescharacteristic, complemented, Sylow
433C2—101no—
434C2—101no—
435C2—202no—
436C2#1175197911yescharacteristic
437C2#1175198111yescharacteristic
438C2#1175202111yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1764
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11751782C2semidirect
4. C2×AGL(1,𝔽5)×C3⋊C8C2semidirect
6. #11746532C2semidirect
7. #11746526C2semidirect
11. #91992C4semidirect
20. C3×#4010C4semidirect
24. #91119C4semidirect
25. C5×#835C4semidirect
26. #90965C4semidirect
27. #90964C4semidirect
86. #4010Dic3semidirect
136. #835AGL(1,𝔽5)semidirect
358. C3×C5#1764semidirect
415. C5#70369semidirect
432. C3#10603991semidirect

Maximal subgroups

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

subgroupindex
C2×#919922
C2×C3×#40102
#117517822
C2×AGL(1,𝔽5)×C3⋊C82
#117492072
#117465322
#117465262
#106039913
#703695