Finite Group Databasebeta

← #14282976

Subgroups — #14282976

Order 1920.

668 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65961C211yesmaximal, characteristic
2C2×#91235C211yesmaximal, characteristic, complemented
3#11752415C211yesmaximal, characteristic, complemented
4#11746797C212yesmaximal, complemented
5#11746739C212yesmaximal, complemented
6#10604499—31nomaximal, Hall
7C23×ΓL(1,𝔽16)—21no—
8C2×C3×#713C411yescharacteristic, complemented
9C22×C3×C4×D5C2211yescharacteristic
10C2×#6758—21no—
11C3×C5×C22⋊Q8C411yescharacteristic, complemented
12C3×#4092C2211yescharacteristic
13C3×#4076C411yescharacteristic, complemented
14C3×#4041C2211yescharacteristic
15C3×#4040—21no—
16C3×#4039—21no—
17C3×#4038C411yescharacteristic, complemented
18C3×#4028—21no—
19C3×#4027—21no—
20#91241—21no—
21#91235—21no—
22#91235C2212yes—
23#91232C2212yes—
24#71708—51nomaximal, Hall
25#65961D311yescharacteristic
26C2×(C2×AGL(1,𝔽5))⋊C4—31no—
27#65756—31no—
28#64926—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—24no—
32C22×ΓL(1,𝔽16)—22no—
33C22×ΓL(1,𝔽16)—42no—
34C22×ΓL(1,𝔽16)—42no—
35C2×C3×C5×Q8C2×C411yescharacteristic, complemented
36C22×C3×C4×C5C2×C411yescharacteristic
37C2×C3×C22⋊D5—21no—
38C22×C3×Dic5D411yescharacteristic
39C3×#713—42no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20C2×C411yescharacteristic, complemented
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—42no—
45C2×C3×C4×D5C2×C411yescharacteristic
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49C2×C3⋊#170—22no—
50#6758—42no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
53C3×C5×C22⋊C4—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5C2×C411yescharacteristic, complemented
59C3×#683—21no—
60C3×C4×Dic5—21no—
61C3×#525—51no—
62C2×C3⋊#95—51no—
63#5207—51no—
64#4538—52no—
65#4537—52no—
66C23×AGL(1,𝔽5)—61no—
67C2×#713Dic311yescharacteristic, complemented
68C22×C4×D5C2×D311yescharacteristic
69C2×#700—61no—
70C5×C22⋊Q8Dic311yescharacteristic, complemented
71#4092C2×D311yescharacteristic
72#4076Dic311yescharacteristic, complemented
73#4041C2×D311yescharacteristic
74#4040—21no—
75#4039—21no—
76#4038Dic311yescharacteristic, complemented
77#4028—21no—
78#4027—21no—
79C22⋊#170—61no—
80(C2×AGL(1,𝔽5))⋊C4—32no—
81(C2×AGL(1,𝔽5))⋊C4—61no—
82#4003—32no—
83#1504—151noSylow
84C23×C3×C5C22⋊C411yescharacteristic
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—21no—
88C22×C3×D5—21no—
89C22×C3×D5—21no—
90C22×C3×D5C22⋊C411yescharacteristic
91C22×C3×D5C2×D411yescharacteristic
92C2×ΓL(1,𝔽16)—24no—
93C2×ΓL(1,𝔽16)—48no—
94C2×ΓL(1,𝔽16)—42no—
95C2×ΓL(1,𝔽16)—42no—
96C3×C5×Q8—41no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C22⋊C411yescharacteristic
101C2×C3×C4×C5C22×C411yescharacteristic
102C3×C22⋊D5—42no—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—41no—
105C2×C3×Dic5C22⋊C411yescharacteristic
106C2×C3×Dic5C2×D411yescharacteristic
107C3×D20—41no—
108C3×D20—42no—
109C3×D20—41no—
110C3×D20—41no—
111C3×C4×D5—22no—
112C3×C4×D5—22no—
113C3×C4×D5—41no—
114C3×C4×D5—41no—
115C3×C4×D5—42no—
116C3⋊#170—42no—
117C2×C3×#41—51no—
118C23×C3×C4—51no—
119C23×Dic3—101no—
120C3×#123—101no—
121C3×C22⋊Q8—51no—
122C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
123C3×#121—51no—
124C3×#121—51no—
125C3×#120—101no—
126C3×C4×D4—101no—
127C3×C4×D4—101no—
128C3×#117—51no—
129C2×#208—101no—
130C3⋊#95—52no—
131C3⋊#95—101no—
132#830—52no—
133#830—101no—
134C23×D5#6711yescharacteristic
135C22×AGL(1,𝔽5)—64no—
136C22×AGL(1,𝔽5)—62no—
137C22×AGL(1,𝔽5)—122no—
138C22×AGL(1,𝔽5)—122no—
139C2×C5×Q8C2×Dic311yescharacteristic, complemented
140C22×C4×C5C2×Dic311yescharacteristic
141C2×C22⋊D5—21no—
142C22×Dic5#6711yescharacteristic
143#713—42no—
144C2×D20—21no—
145C2×D20—21no—
146C2×D20C2×Dic311yescharacteristic, complemented
147C2×C4×D5—21no—
148C2×C4×D5—42no—
149C2×C4×D5C2×Dic311yescharacteristic
150C2×C4×D5#6711yescharacteristic
151C2×C4×D5#6711yescharacteristic
152C2×C4×D5C22×D311yescharacteristic
153C2×#170—62no—
154#700—122no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4C2×Dic311yescharacteristic, complemented
157C5×C22⋊C4—21no—
158#685—21no—
159#685—21no—
160#685—21no—
161#685—21no—
162C4⋊Dic5C2×Dic311yescharacteristic, complemented
163#683—21no—
164C4×Dic5—21no—
165#525—51no—
166#407—151no—
167C2×#95—151no—
168#328—152no—
169#327—152no—
170C22×C3×C5—21no—
171C22×C3×C5—21no—
172C22×C3×C5#10412yes—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×C3×D5—24no—
175C2×C3×D5—24no—
176C2×C3×D5—21no—
177C2×C3×D5—21no—
178C2×C3×D5—21no—
179C2×C3×D5—21no—
180C2×C3×D5—41no—
181C2×C3×D5—41no—
182C2×C3×D5—41no—
183C2×C3×D5#9912yes—
184C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
185C2×C3×D5#13711yescharacteristic
186ΓL(1,𝔽16)—44no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189C3×C4×C5—41no—
190C3×C4×C5—41no—
191C3×Dic5—21no—
192C3×Dic5—21no—
193C3×Dic5—41no—
194C24×C3—51no—
195C3×#41—202no—
196C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
197C2×C3×D4—51no—
198C2×C3×D4—101no—
199C2×C3×D4—101no—
200C2×C3×D4—101no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—51no—
204C22×C3×C4—51no—
205C22×C3×C4—51no—
206C22×C3×C4—101no—
207C22×C3×C4—202no—
208C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
209C22×Dic3—102no—
210C22×Dic3—104no—
211C22×Dic3—202no—
212C22×Dic3—202no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—51no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
217C3×C22⋊C4—21no—
218C3×C22⋊C4—101no—
219C3×C22⋊C4—101no—
220C3×C22⋊C4—101no—
221C3×C22⋊C4—101no—
222C3×C42—101no—
223#208—202no—
224C2×C3⋊C8—102no—
225C23×C5#21711yescharacteristic
226C22×D5—22no—
227C22×D5—22no—
228C22×D5—21no—
229C22×D5—21no—
230C22×D5—21no—
231C22×D5#21711yescharacteristic
232C22×D5C2×#6711yescharacteristic
233C2×AGL(1,𝔽5)—64no—
234C2×AGL(1,𝔽5)—128no—
235C2×AGL(1,𝔽5)—122no—
236C2×AGL(1,𝔽5)—122no—
237C5×Q8—41no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—41no—
241C2×C4×C5#21711yescharacteristic
242C2×C4×C5C22×Dic311yescharacteristic
243C22⋊D5—42no—
244C2×Dic5—21no—
245C2×Dic5—41no—
246C2×Dic5#21711yescharacteristic
247C2×Dic5C2×#6711yescharacteristic
248D20—41no—
249D20—42no—
250D20—41no—
251D20—41no—
252C4×D5—22no—
253C4×D5—22no—
254C4×D5—41no—
255C4×D5—41no—
256C4×D5—42no—
257#170—122no—
258C2×#41—51no—
259C23×C4—51no—
260C23×C4—301no—
261C2×M16—301no—
262#123—101no—
263C22⋊Q8—51no—
264C22⋊Q8ΓL(1,𝔽16)11yescharacteristic, complemented
265#121—51no—
266#121—51no—
267#120—101no—
268C4×D4—101no—
269C4×D4—101no—
270#117—51no—
271C22⋊C8—152no—
272C22⋊C8—301no—
273#95—152no—
274#95—301no—
275C2×C3×C5—22no—
276C2×C3×C5C2×#9911yescharacteristic
277C2×C3×C5#41711yescharacteristic
278C2×C3×C5C2×#10411yescharacteristic
279C3×D5—21no—
280C3×D5—22no—
281C3×D5—21no—
282C3×D5—81no—
283C23×C3—51no—
284C23×C3—51no—
285C23×C3—101no—
286C23×C3—102no—
287C23×C3—101no—
288C23×C3—102no—
289C23×C3—101no—
290C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
291C3×Q8—41no—
292C3×D4—201no—
293C3×D4—201no—
294C3×D4—201no—
295C3×D4—202no—
296C3×D4—202no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—51no—
301C2×C3×C4—51no—
302C2×C3×C4—102no—
303C2×C3×C4—102no—
304C2×C3×C4—101no—
305C2×C3×C4—201no—
306C2×C3×C4—201no—
307C2×C3×C4—202no—
308C2×C3×C4—201no—
309C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
310C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
311C2×Dic3—104no—
312C2×Dic3—202no—
313C2×Dic3—208no—
314C2×Dic3—202no—
315C3⋊C8—202no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5#83712yes—
319C22×C5C2×#21711yescharacteristic
320C2×D5—24no—
321C2×D5—24no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—21no—
326C2×D5—41no—
327C2×D5—41no—
328C2×D5—41no—
329C2×D5#83412yes—
330C2×D5#94911yescharacteristic
331C2×D5#95111yescharacteristic
332AGL(1,𝔽5)—124no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—41no—
336C4×C5—41no—
337Dic5—21no—
338Dic5—21no—
339Dic5—41no—
340C24—51no—
341#41—202no—
342C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic, complemented
343C2×D4—51no—
344C2×D4—101no—
345C2×D4—101no—
346C2×D4—101no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—51no—
352C22×C4—101no—
353C22×C4—202no—
354C22×C4—302no—
355C22×C4—304no—
356C22×C4—602no—
357C22×C4—602no—
358C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
359M16—602no—
360C2×C8—302no—
361C4⋊C4—21no—
362C4⋊C4—51no—
363C4⋊C4—101no—
364C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic, complemented
365C22⋊C4—21no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—101no—
369C22⋊C4—101no—
370C42—101no—
371C3×C5#150411yescharacteristic, complemented, Hall
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—51no—
375C22×C3—51no—
376C22×C3—52no—
377C22×C3—104no—
378C22×C3—104no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3—201no—
384C22×C3—201no—
385C22×C3—201no—
386C22×C3#401012yes—
387C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—101no—
393C3×C4—101no—
394C3×C4—201no—
395Dic3—204no—
396C2×C5—22no—
397C2×C5C2×#83411yescharacteristic
398C2×C5#523211yescharacteristic
399C2×C5C2×#83711yescharacteristic
400D5—21no—
401D5—22no—
402D5—21no—
403D5—81no—
404C23—51no—
405C23—51no—
406C23—102no—
407C23—101no—
408C23—101no—
409C23—102no—
410C23—101no—
411C23#676311yescharacteristic
412Q8—41no—
413D4—201no—
414D4—202no—
415D4—201no—
416D4—201no—
417D4—202no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—51no—
422C2×C4—51no—
423C2×C4—102no—
424C2×C4—102no—
425C2×C4—101no—
426C2×C4—201no—
427C2×C4—201no—
428C2×C4—202no—
429C2×C4—201no—
430C2×C4—304no—
431C2×C4—602no—
432C2×C4—608no—
433C2×C4—602no—
434C2×C4#676311yescharacteristic
435C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
436C8—602no—
437C2×C3—22no—
438C2×C3—101no—
439C2×C3—101no—
440C2×C3—102no—
441C2×C3—401no—
442C2×C3D5⋊#9911yescharacteristic
443C2×C3C2×#401011yescharacteristic
444C2×C3#6579311yescharacteristic
445C5#7170811yescharacteristic, complemented, Sylow
446C22—21no—
447C22—21no—
448C22—51no—
449C22—51no—
450C22—52no—
451C22—104no—
452C22—104no—
453C22—101no—
454C22—101no—
455C22—101no—
456C22—101no—
457C22—201no—
458C22—201no—
459C22—201no—
460C22#9123912yes—
461C22C2×#676311yescharacteristic
462C4—21no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—101no—
467C4—101no—
468C4—201no—
469C4—604no—
470C3#1060449911yescharacteristic, complemented, Sylow
471C2—22no—
472C2—101no—
473C2—101no—
474C2—102no—
475C2—401no—
476C2D5⋊#83411yescharacteristic
477C2C2×#9123911yescharacteristic
478C2#1175245211yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×C22⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65961C2
cosocleC2×C3×C4×D5C23
hypercenterC22⋊Q8ΓL(1,𝔽16)
hypocenterC3×C5#1504
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#91235C2semidirect
3. #11752415C2semidirect
4. #11746797C2semidirect
5. #11746739C2semidirect
8. C2×C3×#713C4semidirect
11. C3×C5×C22⋊Q8C4semidirect
13. C3×#4076C4semidirect
17. C3×#4038C4semidirect
35. C2×C3×C5×Q8C2×C4semidirect
42. C2×C3×D20C2×C4semidirect
52. C3×C5×C4⋊C4C2×C4semidirect
58. C3×C4⋊Dic5C2×C4semidirect
67. C2×#713Dic3semidirect
70. C5×C22⋊Q8Dic3semidirect
72. #4076Dic3semidirect
76. #4038Dic3semidirect
122. C3×C22⋊Q8AGL(1,𝔽5)semidirect
139. C2×C5×Q8C2×Dic3semidirect
146. C2×D20C2×Dic3semidirect
156. C5×C4⋊C4C2×Dic3semidirect
162. C4⋊Dic5C2×Dic3semidirect
196. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
216. C3×C4⋊C4C2×AGL(1,𝔽5)semidirect
264. C22⋊Q8ΓL(1,𝔽16)semidirect
342. C2×Q8C2×ΓL(1,𝔽16)semidirect
364. C4⋊C4C2×ΓL(1,𝔽16)semidirect
371. C3×C5#1504semidirect
445. C5#71708semidirect
470. C3#10604499semidirect

Maximal subgroups

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

subgroupindex
C3×#659612
C2×#912352
#117524152
#117467972
#117467392
#106044993
#717085