Finite Group Databasebeta

← #14300240

Subgroups — #14300240

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×#91992C211yesmaximal, characteristic
2C2×#91239C211yesmaximal, characteristic
3C2×#91225C211yesmaximal, characteristic, complemented
4#11752367C211yesmaximal, characteristic, complemented
5C3×#65093C211yesmaximal, characteristic
6#11746763C211yesmaximal, characteristic, complemented
7#11746746C211yesmaximal, characteristic, complemented
8#10603998—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10#91992—22no—
11#91992—22no—
12C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
13C2×#6758—21no—
14C2×#6758C2211yescharacteristic
15C2×#6758C2211yescharacteristic
16C2×D5⋊(C3⋊C8)—21no—
17C2×D5⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C8×D5C2211yescharacteristic
19C3×#4038C2211yescharacteristic
20#91239—22no—
21#91233—21no—
22C4⋊(C3⋊#170)—21no—
23#91225—22no—
24C3×C5×#103C411yescharacteristic, complemented
25C3×#3967C411yescharacteristic, complemented
26C3×#3953C411yescharacteristic, complemented
27C3×#3941C411yescharacteristic, complemented
28#70292—51nomaximal, Hall
29C2×#4133—31no—
30C2×#4010—31no—
31C2×#3996—31no—
32#65708—31no—
33#65093D311yescharacteristic
34(C2×AGL(1,𝔽5))⋊C8—31no—
35#64875—31no—
36C22×ΓL(1,𝔽16)—21no—
37C2×C3×C5×Q8C2×C411yescharacteristic
38C3×#713—21no—
39C3×#713—21no—
40C3×#713D412yes—
41C3×#713Q812yes—
42C2×C3×D20—21no—
43C2×C3×D20C2×C411yescharacteristic
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5C2311yescharacteristic
46C2×C3⋊#170—21no—
47C2×C3⋊#170—21no—
48C2×C3⋊#170D411yescharacteristic
49C2×C3⋊#170D411yescharacteristic
50C4×ΓL(1,𝔽16)—21no—
51C4×ΓL(1,𝔽16)—21no—
52#6758—24no—
53#6758—21no—
54#6758—21no—
55#6758—21no—
56#6758D412yes—
57D5⋊(C3⋊C8)—24no—
58D5⋊(C3⋊C8)—21no—
59D5⋊(C3⋊C8)—21no—
60C2×C3×C5×C8C2×C411yescharacteristic
61C3×C5×C4⋊C4C2×C411yescharacteristic
62C3×#685—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×C4×Dic5—21no—
65C2×C3×#168C2×C411yescharacteristic
66C3×C8×D5—21no—
67C3×C8×D5—21no—
68C2×#948—51no—
69C3×#416—51no—
70C2×#837—51no—
71C2×#819—51no—
72#4999—51no—
73#4550—51no—
74#4532—51no—
75C2×#713C2×D311yescharacteristic
76#4133—62no—
77#4133—62no—
78C2×C4×AGL(1,𝔽5)—31no—
79C2×#700—31no—
80C2×#700—31no—
81C2×#700—61no—
82C2×D5⋊C8—31no—
83C2×D5⋊C8—61no—
84C2×C8×D5C2×D311yescharacteristic
85#4038C2×D311yescharacteristic
86#4010—62no—
87#4004—61no—
88C4⋊#170—61no—
89#3996—62no—
90C5×#103Dic311yescharacteristic, complemented
91#3967Dic311yescharacteristic, complemented
92#3953Dic311yescharacteristic, complemented
93#3941Dic311yescharacteristic, complemented
94#1866—151noSylow
95C22×C3×D5—21no—
96C22×C3×D5#4111yescharacteristic
97C2×ΓL(1,𝔽16)—22no—
98C2×ΓL(1,𝔽16)—41no—
99C2×ΓL(1,𝔽16)—41no—
100C3×C5×Q8—21no—
101C3×C5×Q8C4⋊C412yes—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×Dic5—21no—
106C2×C3×Dic5C2×D411yescharacteristic
107C3×D20—22no—
108C3×D20—22no—
109C3×D20—21no—
110C3×D20C4⋊C412yes—
111C3×C4×D5—22no—
112C3×C4×D5—22no—
113C3×C4×D5C2×D411yescharacteristic
114C3×C4×D5C2×D411yescharacteristic
115C3×C4×D5C2×Q811yescharacteristic
116C3×C4×D5#4111yescharacteristic
117C3⋊#170—21no—
118C3⋊#170—21no—
119C3⋊#170—22no—
120C3⋊#170—41no—
121C3×C5×C8—21no—
122C3×#168—21no—
123C2×C3×#41—51no—
124C22×C3×C8—51no—
125C3×#117—51no—
126#948—102no—
127#948—102no—
128C2×C4×Dic3—51no—
129C2×#208—51no—
130C2×#208—51no—
131C2×#208—101no—
132C22×C3⋊C8—51no—
133C22×C3⋊C8—101no—
134C3×#103—51no—
135C3×#103AGL(1,𝔽5)11yescharacteristic, complemented
136C3×#102—51no—
137C3×#102—51no—
138#837—102no—
139#819—102no—
140C4⋊(C3⋊C8)—101no—
141C4⋊(C3⋊C8)—101no—
142C22×AGL(1,𝔽5)—61no—
143C2×C5×Q8C2×Dic311yescharacteristic
144#713—21no—
145#713—21no—
146#713Dic612yes—
147#713D1212yes—
148C2×D20—21no—
149C2×D20C2×Dic311yescharacteristic
150C2×C4×D5—21no—
151C2×C4×D5C22×D311yescharacteristic
152C2×#170—31no—
153C2×#170—31no—
154C2×#170—61no—
155C2×#170—61no—
156C4×AGL(1,𝔽5)—61no—
157C4×AGL(1,𝔽5)—61no—
158#700—32no—
159#700—64no—
160#700—61no—
161#700—61no—
162#700—61no—
163D5⋊C8—64no—
164D5⋊C8—61no—
165D5⋊C8—61no—
166C2×C5×C8C2×Dic311yescharacteristic
167C5×C4⋊C4C2×Dic311yescharacteristic
168#685—21no—
169C4⋊Dic5C2×Dic311yescharacteristic
170C4×Dic5—21no—
171C2×#168C2×Dic311yescharacteristic
172C8×D5—21no—
173C8×D5—21no—
174C2×#131—151no—
175C2×#108—151no—
176#424—151no—
177C2×#104—151no—
178#416—51no—
179#339—151no—
180#336—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×D5C22⋊Q811yescharacteristic
188C2×C3×D5#12311yescharacteristic
189ΓL(1,𝔽16)—42no—
190C3×C4×C5—22no—
191C3×C4×C5—41no—
192C3×C4×C5C2×C4⋊C411yescharacteristic
193C3×C4×C5C4×D411yescharacteristic
194C3×Dic5—41no—
195C3×Dic5#12011yescharacteristic
196C3×Dic5C22⋊Q811yescharacteristic
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×Dic3—101no—
207C2×C3×C8—51no—
208C2×C3×C8—101no—
209C2×C3×C8—101no—
210C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
211C3×C4⋊C4—51no—
212C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
213C3×C22⋊C4—101no—
214C3×C42—101no—
215C4×Dic3—101no—
216C4×Dic3—101no—
217#208—52no—
218#208—101no—
219#208—104no—
220#208—101no—
221#208—101no—
222C2×C3⋊C8—51no—
223C2×C3⋊C8—51no—
224C2×C3⋊C8—101no—
225C2×C3⋊C8—101no—
226C2×C3⋊C8—101no—
227C2×C3⋊C8—104no—
228C2×C3⋊C8—101no—
229C22×D5—21no—
230C22×D5#23711yescharacteristic
231C2×AGL(1,𝔽5)—62no—
232C2×AGL(1,𝔽5)—121no—
233C2×AGL(1,𝔽5)—121no—
234C5×Q8—21no—
235C5×Q8C4⋊Dic312yes—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5C22×Dic311yescharacteristic
239C2×Dic5—21no—
240C2×Dic5D3×D411yescharacteristic
241D20—22no—
242D20—22no—
243D20—21no—
244D20C4⋊Dic312yes—
245C4×D5—22no—
246C4×D5—22no—
247C4×D5C2×Dic611yescharacteristic
248C4×D5C2×D1211yescharacteristic
249C4×D5D3×D411yescharacteristic
250C4×D5#23711yescharacteristic
251#170—61no—
252#170—62no—
253#170—61no—
254#170—121no—
255C5×C8—21no—
256#168—21no—
257C2×#41—51no—
258#131—302no—
259#131—302no—
260C2×M16—151no—
261C2×M16—151no—
262C2×M16—301no—
263C22×C8—51no—
264C22×C8—151no—
265C22×C8—301no—
266#117—51no—
267C2×C42—151no—
268#108—302no—
269C4⋊C8—301no—
270C4⋊C8—301no—
271#104—302no—
272#103—51no—
273#103ΓL(1,𝔽16)11yescharacteristic, complemented
274#102—51no—
275#102—51no—
276C2×C3×C5#38511yescharacteristic
277C2×C3×C5#44311yescharacteristic
278C2×C3×C5#44411yescharacteristic
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—21no—
292C2×C3×C4—51no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—51no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—101no—
300C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
301C2×Dic3—102no—
302C2×Dic3—201no—
303C2×Dic3—201no—
304C3×C8—21no—
305C3×C8—101no—
306C3⋊C8—101no—
307C3⋊C8—101no—
308C3⋊C8—102no—
309C3⋊C8—201no—
310C22×C5D4×Dic311yescharacteristic
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—22no—
314C2×D5—41no—
315C2×D5—41no—
316C2×D5#87811yescharacteristic
317C2×D5#88611yescharacteristic
318AGL(1,𝔽5)—122no—
319C4×C5—22no—
320C4×C5—41no—
321C4×C5C2×C4⋊Dic311yescharacteristic
322C4×C5D4×Dic311yescharacteristic
323Dic5—41no—
324Dic5#87811yescharacteristic
325Dic5#88211yescharacteristic
326#41—52no—
327#41—52no—
328#41—101no—
329#41—101no—
330C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
331C2×D4—51no—
332C2×D4—101no—
333C22×C4—51no—
334C22×C4—101no—
335C22×C4—301no—
336M16—152no—
337M16—301no—
338M16—304no—
339M16—301no—
340M16—301no—
341C2×C8—51no—
342C2×C8—101no—
343C2×C8—101no—
344C2×C8—151no—
345C2×C8—151no—
346C2×C8—301no—
347C2×C8—301no—
348C2×C8—301no—
349C2×C8—304no—
350C2×C8—301no—
351C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
352C4⋊C4—51no—
353C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
354C22⋊C4—101no—
355C42—101no—
356C42—301no—
357C42—301no—
358C3×C5#186611yescharacteristic, 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—41no—
369C3×C4—51no—
370C3×C4—51no—
371C3×C4—201no—
372C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
373C3×C4D4×AGL(1,𝔽5)11yescharacteristic
374Dic3—202no—
375C2×C5#495011yescharacteristic
376C2×C5#519611yescharacteristic
377C2×C5#519711yescharacteristic
378D5—21no—
379D5—21no—
380D5—42no—
381C23—51no—
382C23—101no—
383Q8—21no—
384Q8C4⋊ΓL(1,𝔽16)12yes—
385D4—52no—
386D4—102no—
387D4—102no—
388D4—101no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—51no—
395C2×C4—51no—
396C2×C4—102no—
397C2×C4—102no—
398C2×C4—101no—
399C2×C4—302no—
400C2×C4—601no—
401C2×C4—601no—
402C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
403C8—21no—
404C8—101no—
405C8—301no—
406C8—301no—
407C8—302no—
408C8—601no—
409C2×C3—101no—
410C2×C3—101no—
411C2×C3—202no—
412C2×C3#6569911yescharacteristic
413C2×C3#6573711yescharacteristic
414C2×C3#6574011yescharacteristic
415C5#7029211yescharacteristic, complemented, Sylow
416C22—51no—
417C22—51no—
418C22—101no—
419C22—101no—
420C22—102no—
421C22—201no—
422C22—201no—
423C22D4×ΓL(1,𝔽16)11yescharacteristic
424C4—22no—
425C4—41no—
426C4—51no—
427C4—51no—
428C4—201no—
429C4—602no—
430C4C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
431C4D4×ΓL(1,𝔽16)11yescharacteristic
432C3#1060399811yescharacteristic, complemented, Sylow
433C2—101no—
434C2—101no—
435C2—202no—
436C2#1175235811yescharacteristic
437C2#1175239611yescharacteristic
438C2#1175239911yescharacteristic
subgroupG/H
centerC22D4×ΓL(1,𝔽16)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#103C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenter#103ΓL(1,𝔽16)
hypocenterC3×C5#1866
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. C2×#91225C2semidirect
4. #11752367C2semidirect
6. #11746763C2semidirect
7. #11746746C2semidirect
24. C3×C5×#103C4semidirect
25. C3×#3967C4semidirect
26. C3×#3953C4semidirect
27. C3×#3941C4semidirect
90. C5×#103Dic3semidirect
91. #3967Dic3semidirect
92. #3953Dic3semidirect
93. #3941Dic3semidirect
135. C3×#103AGL(1,𝔽5)semidirect
273. #103ΓL(1,𝔽16)semidirect
358. C3×C5#1866semidirect
415. C5#70292semidirect
432. C3#10603998semidirect

Maximal subgroups

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

subgroupindex
C2×#919922
C2×#912392
C2×#912252
#117523672
C3×#650932
#117467632
#117467462
#106039983
#702925