Finite Group Databasebeta

← #14295679

Subgroups — #14295679

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#713C211yesmaximal, characteristic
2C2×#91239C211yesmaximal, characteristic
3C2×C3×#4010C211yesmaximal, characteristic
4Dic3×#700C211yesmaximal, characteristic, complemented
5#11751748C211yesmaximal, characteristic, complemented
6#11746526C211yesmaximal, characteristic, complemented
7#11746524C211yesmaximal, characteristic, complemented
8#10604450—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14Q8×Dic15C411yescharacteristic, complemented
15C5×Q8×Dic3C411yescharacteristic, complemented
16#91434—21no—
17#91434C411yescharacteristic, complemented
18Dic3×D20—21no—
19Dic3×D20C411yescharacteristic, complemented
20C4×D5×Dic3—21no—
21C4×D5×Dic3C2211yescharacteristic
22#91358—21no—
23#91358C2211yescharacteristic
24#91239C414yescomplemented
25C3×#4010C414yescomplemented
26#91169—21no—
27Dic3×#170—21no—
28(C2×AGL(1,𝔽5))⋊Dic3—21no—
29(C2×AGL(1,𝔽5))⋊Dic3—21no—
30#70354—51nomaximal, Hall
31C4×#713—31no—
32C2×#4010—31no—
33C2×#4010D311yescharacteristic
34#65686—31no—
35C4×#700—31no—
36#64875—31no—
37#64875—31no—
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C3×#713—21no—
42C3×#713—21no—
43C3×#713C2×C412yes—
44C3×#713C2×C412yes—
45C2×C3×D20—21no—
46C2×C3×D20C2×C411yescharacteristic
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5C2311yescharacteristic
49C2×D5×Dic3—21no—
50C2×D5×Dic3—21no—
51C2×D5×Dic3D411yescharacteristic
52C2×D5×Dic3D411yescharacteristic
53C2×C3⋊#170—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)C2×C412yescomplemented
56#6758—21no—
57#6758C2×C412yescomplemented
58C2×C3×#170—21no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)C2×C412yescomplemented
61C3×#700—21no—
62C3×#700C2×C412yescomplemented
63C4⋊Dic15—21no—
64C4⋊Dic15C2×C411yescharacteristic
65C4×Dic15—21no—
66C4×Dic15C2×C411yescharacteristic
67C5×C4⋊Dic3—21no—
68C5×C4⋊Dic3C2×C411yescharacteristic
69C4×C5×Dic3—21no—
70C4×C5×Dic3C2×C411yescharacteristic
71#6665—21no—
72#6665—21no—
73#6665—21no—
74Dic3×Dic5—21no—
75Dic3×Dic5D411yescharacteristic
76Dic3×Dic5D411yescharacteristic
77Dic3×#41—51no—
78C2×C3×#104—51no—
79C2×#837—51no—
80Dic3×M16—51no—
81#4932—51no—
82#4555—51no—
83#4523—51no—
84C2×#713C2×D311yescharacteristic
85C2×C4×AGL(1,𝔽5)—31no—
86C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
87C2×#700—31no—
88C2×#700C2×D311yescharacteristic
89C4×C5×Q8—31no—
90Q8×Dic5—31no—
91#4039—31no—
92#4039—61no—
93#4038—31no—
94#4038—61no—
95C4×D20—31no—
96C4×D20—61no—
97C42×D5—31no—
98C42×D5—61no—
99#4010—34no—
100#4010Dic314yescomplemented
101(C2×AGL(1,𝔽5))⋊C4—61no—
102(C2×AGL(1,𝔽5))⋊C4—61no—
103#4002—61no—
104C4×#170—61no—
105#1759—151noSylow
106C22×C3×D5—21no—
107C22×C3×D5C2×D411yescharacteristic
108C2×ΓL(1,𝔽16)—22no—
109C2×ΓL(1,𝔽16)—41no—
110C2×ΓL(1,𝔽16)—41no—
111C2×C3×AGL(1,𝔽5)—22no—
112C2×C3×AGL(1,𝔽5)—41no—
113C2×C3×AGL(1,𝔽5)—41no—
114C3×C5×Q8—21no—
115C3×C5×Q8C4212yes—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5C22×C411yescharacteristic
118C2×Dic15—21no—
119C2×Dic15C22⋊C411yescharacteristic
120C2×Dic15C22⋊C411yescharacteristic
121C2×C5×Dic3—21no—
122C2×C5×Dic3C22⋊C411yescharacteristic
123C2×C5×Dic3C22⋊C411yescharacteristic
124C2×C3×Dic5C2×D411yescharacteristic
125C3×D20—22no—
126C3×D20—22no—
127C3×D20—21no—
128C3×D20C4212yes—
129C3×C4×D5—22no—
130C3×C4×D5—22no—
131C3×C4×D5C22×C411yescharacteristic
132C3×C4×D5C22×C411yescharacteristic
133C3×C4×D5#4111yescharacteristic
134C3×C4×D5#4111yescharacteristic
135D5×Dic3—41no—
136D5×Dic3—41no—
137D5×Dic3—42no—
138D5×Dic3—42no—
139C3⋊#170—22no—
140C3×#170—22no—
141C2×C3×#41—51no—
142C2×C3×M16—51no—
143C2×C3×C42—51no—
144Q8×Dic3—51no—
145Q8×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
146D4×Dic3—51no—
147D4×Dic3—51no—
148D4×Dic3—101no—
149D4×Dic3—101no—
150#926—51no—
151#926—101no—
152C2×C4×Dic3—51no—
153C2×C4×Dic3—51no—
154C2×C4×Dic3—101no—
155C2×#208—51no—
156C3×#104—54no—
157#837—54no—
158C3⋊#95—101no—
159C3⋊#95—101no—
160#815—101no—
161C8×Dic3—101no—
162C22×AGL(1,𝔽5)—21no—
163C22×AGL(1,𝔽5)—61no—
164C2×C5×Q8C4×D311yescharacteristic
165#713—21no—
166#713—21no—
167#713C4×D312yes—
168#713C2×Dic312yes—
169C2×D20—21no—
170C2×D20C4×D311yescharacteristic
171C2×C4×D5—21no—
172C2×C4×D5—31no—
173C2×C4×D5—31no—
174C2×C4×D5—61no—
175C2×C4×D5—61no—
176C2×C4×D5C22×D311yescharacteristic
177C2×#170—21no—
178C2×#170—61no—
179C4×AGL(1,𝔽5)—21no—
180C4×AGL(1,𝔽5)—32no—
181C4×AGL(1,𝔽5)—61no—
182C4×AGL(1,𝔽5)C2×Dic312yescomplemented
183#700—21no—
184#700—32no—
185#700—61no—
186#700C2×Dic312yescomplemented
187C5×C4⋊C4—31no—
188C5×C4⋊C4—61no—
189C42×C5—31no—
190C42×C5—61no—
191#685—61no—
192#685—61no—
193#685—61no—
194C4⋊Dic5—31no—
195C4⋊Dic5—61no—
196C4×Dic5—31no—
197C4×Dic5—31no—
198C4×Dic5—31no—
199C4×Dic5—61no—
200C4×Dic5—61no—
201C4×#41—151no—
202C2×#104—51no—
203C2×#104—151no—
204C4×M16—151no—
205#376—151no—
206#339—151no—
207#339—151no—
208C22×C3×C5C2×C22⋊C411yescharacteristic
209C2×C3×D5—21no—
210C2×C3×D5—21no—
211C2×C3×D5—22no—
212C2×C3×D5—41no—
213C2×C3×D5—41no—
214C2×C3×D5C4×D411yescharacteristic
215C2×C3×D5C4×D411yescharacteristic
216ΓL(1,𝔽16)—42no—
217C3×AGL(1,𝔽5)—42no—
218C3×C4×C5—22no—
219C3×C4×C5C2×C4211yescharacteristic
220C3×C4×C5#11711yescharacteristic
221Dic15—21no—
222Dic15—21no—
223Dic15—41no—
224C3×Dic5C4×D411yescharacteristic
225C3×Dic5C4×D411yescharacteristic
226C5×Dic3—21no—
227C5×Dic3—21no—
228C5×Dic3—41no—
229C3×#41—52no—
230C3×#41—52no—
231C3×#41—101no—
232C3×#41—101no—
233C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
234C2×C3×D4—51no—
235C2×C3×D4—101no—
236C22×C3×C4—51no—
237C22×C3×C4—101no—
238C22×C3×C4—101no—
239C22×Dic3—51no—
240C22×Dic3—51no—
241C22×Dic3—101no—
242C22×Dic3—101no—
243C22×Dic3—101no—
244C3×M16—52no—
245C3×M16—101no—
246C2×C3×C8—101no—
247C3×C42—52no—
248C3×C42—101no—
249#217—101no—
250#217—101no—
251#217—101no—
252C4⋊Dic3—21no—
253C4⋊Dic3—51no—
254C4⋊Dic3—101no—
255C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
256C4×Dic3—21no—
257C4×Dic3—51no—
258C4×Dic3—51no—
259C4×Dic3—51no—
260C4×Dic3—52no—
261C4×Dic3—101no—
262C4×Dic3—101no—
263C4×Dic3—101no—
264C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
265#208—52no—
266#208—101no—
267C2×C3⋊C8—101no—
268C22×D5—21no—
269C22×D5D3×D411yescharacteristic
270C2×AGL(1,𝔽5)—22no—
271C2×AGL(1,𝔽5)—41no—
272C2×AGL(1,𝔽5)—41no—
273C2×AGL(1,𝔽5)—62no—
274C2×AGL(1,𝔽5)—121no—
275C2×AGL(1,𝔽5)—121no—
276C5×Q8—21no—
277C5×Q8C4×Dic312yes—
278C2×C4×C5—21no—
279C2×C4×C5—31no—
280C2×C4×C5—31no—
281C2×C4×C5—61no—
282C2×C4×C5C2×C4×D311yescharacteristic
283C2×Dic5—31no—
284C2×Dic5—31no—
285C2×Dic5—61no—
286C2×Dic5D3×D411yescharacteristic
287D20—22no—
288D20—22no—
289D20—21no—
290D20C4×Dic312yes—
291C4×D5—22no—
292C4×D5—22no—
293C4×D5—121no—
294C4×D5—121no—
295C4×D5—122no—
296C4×D5—122no—
297C4×D5C2×C4×D311yescharacteristic
298C4×D5#23711yescharacteristic
299C4×D5#23711yescharacteristic
300C4×D5C22×Dic311yescharacteristic
301#170—22no—
302#170—62no—
303C2×#41—51no—
304C2×M16—51no—
305C2×M16—151no—
306C4×Q8—31no—
307C4×Q8—151no—
308C4×D4—151no—
309C4×D4—151no—
310C4×D4—301no—
311C4×D4—301no—
312#117—151no—
313#117—301no—
314C2×C42—51no—
315C2×C42—151no—
316C2×C42—151no—
317C2×C42—301no—
318#104—54no—
319#104—154no—
320#97—301no—
321C4×C8—301no—
322#95—301no—
323#95—301no—
324C2×C3×C5C4×C22⋊C411yescharacteristic
325C2×C3×C5#42111yescharacteristic
326C2×C3×C5#42111yescharacteristic
327C3×D5—21no—
328C3×D5—21no—
329C3×D5—42no—
330C23×C3—51no—
331C23×C3—101no—
332C3×Q8—21no—
333C3×Q8C4×AGL(1,𝔽5)12yes—
334C3×D4—52no—
335C3×D4—102no—
336C3×D4—102no—
337C3×D4—101no—
338C2×C3×C4—21no—
339C2×C3×C4—51no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—51no—
343C2×C3×C4—51no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4—201no—
348C2×C3×C4—201no—
349C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—51no—
352C2×Dic3—51no—
353C2×Dic3—102no—
354C2×Dic3—101no—
355C2×Dic3—201no—
356C2×Dic3—201no—
357C2×Dic3—202no—
358C2×Dic3—202no—
359C2×Dic3—201no—
360C2×Dic3—201no—
361C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
362C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
363C3×C8—102no—
364C3⋊C8—102no—
365C22×C5D3×C22⋊C411yescharacteristic
366C2×D5—21no—
367C2×D5—21no—
368C2×D5—22no—
369C2×D5—41no—
370C2×D5—41no—
371C2×D5#67⋊C411yescharacteristic
372C2×D5D4×Dic311yescharacteristic
373AGL(1,𝔽5)—42no—
374AGL(1,𝔽5)—122no—
375C4×C5—22no—
376C4×C5—61no—
377C4×C5—61no—
378C4×C5—121no—
379C4×C5#87711yescharacteristic
380C4×C5C2×C4×Dic311yescharacteristic
381Dic5—61no—
382Dic5—61no—
383Dic5—121no—
384Dic5#67⋊C411yescharacteristic
385Dic5D4×Dic311yescharacteristic
386#41—52no—
387#41—52no—
388#41—101no—
389#41—101no—
390C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
391C2×D4—51no—
392C2×D4—101no—
393C22×C4—51no—
394C22×C4—101no—
395C22×C4—101no—
396C22×C4—151no—
397C22×C4—151no—
398C22×C4—301no—
399C22×C4—301no—
400C22×C4—301no—
401M16—52no—
402M16—101no—
403M16—152no—
404M16—301no—
405C2×C8—101no—
406C2×C8—301no—
407C4⋊C4—31no—
408C4⋊C4—61no—
409C4⋊C4—151no—
410C4⋊C4—301no—
411C22⋊C4—301no—
412C22⋊C4—301no—
413C22⋊C4—301no—
414C42—31no—
415C42—52no—
416C42—61no—
417C42—101no—
418C42—151no—
419C42—151no—
420C42—152no—
421C42—151no—
422C42—301no—
423C42—301no—
424C42—301no—
425C3×C5#175911yescharacteristic, complemented, Hall
426C22×C3—51no—
427C22×C3—51no—
428C22×C3—101no—
429C22×C3—101no—
430C22×C3—102no—
431C22×C3—201no—
432C22×C3—201no—
433C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
434C3×C4—22no—
435C3×C4—51no—
436C3×C4—51no—
437C3×C4—202no—
438C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
439C3×C4#413011yescharacteristic
440Dic3—21no—
441Dic3—21no—
442Dic3—41no—
443Dic3—101no—
444Dic3—101no—
445Dic3—202no—
446Dic3—201no—
447C2×C5#482111yescharacteristic
448C2×C5#482111yescharacteristic
449C2×C5Dic3×C22⋊C411yescharacteristic
450D5—21no—
451D5—21no—
452D5—42no—
453C23—51no—
454C23—101no—
455Q8—21no—
456Q8Dic3×AGL(1,𝔽5)12yes—
457D4—52no—
458D4—102no—
459D4—102no—
460D4—101no—
461C2×C4—21no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—51no—
465C2×C4—51no—
466C2×C4—51no—
467C2×C4—51no—
468C2×C4—51no—
469C2×C4—61no—
470C2×C4—102no—
471C2×C4—102no—
472C2×C4—102no—
473C2×C4—151no—
474C2×C4—151no—
475C2×C4—201no—
476C2×C4—201no—
477C2×C4—302no—
478C2×C4—301no—
479C2×C4—601no—
480C2×C4—601no—
481C2×C4—602no—
482C2×C4—602no—
483C2×C4—601no—
484C2×C4—601no—
485C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
486C8—102no—
487C8—302no—
488C2×C3—101no—
489C2×C3—101no—
490C2×C3—202no—
491C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
492C2×C3#6578411yescharacteristic
493C2×C3#6579511yescharacteristic
494C5#7035411yescharacteristic, complemented, Sylow
495C22—51no—
496C22—51no—
497C22—101no—
498C22—101no—
499C22—102no—
500C22—201no—
501C22—201no—
502C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
503C4—22no—
504C4—51no—
505C4—51no—
506C4—61no—
507C4—61no—
508C4—121no—
509C4—202no—
510C4—301no—
511C4—301no—
512C4—602no—
513C4—601no—
514C4C2×Dic3×AGL(1,𝔽5)11yescharacteristic
515C4#9192311yescharacteristic
516C3#1060445011yescharacteristic, complemented, Sylow
517C2—101no—
518C2—101no—
519C2—202no—
520C2#1175197311yescharacteristic
521C2#1175197411yescharacteristic
522C2Dic3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1759
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
4. Dic3×#700C2semidirect
5. #11751748C2semidirect
6. #11746526C2semidirect
7. #11746524C2semidirect
14. Q8×Dic15C4semidirect
15. C5×Q8×Dic3C4semidirect
17. #91434C4semidirect
19. Dic3×D20C4semidirect
24. #91239C4semidirect
25. C3×#4010C4semidirect
55. C4×ΓL(1,𝔽16)C2×C4semidirect
57. #6758C2×C4semidirect
60. C3×C4×AGL(1,𝔽5)C2×C4semidirect
62. C3×#700C2×C4semidirect
100. #4010Dic3semidirect
145. Q8×Dic3AGL(1,𝔽5)semidirect
182. C4×AGL(1,𝔽5)C2×Dic3semidirect
186. #700C2×Dic3semidirect
425. C3×C5#1759semidirect
494. C5#70354semidirect
516. C3#10604450semidirect

Maximal subgroups

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

subgroupindex
Dic3×#7132
C2×#912392
C2×C3×#40102
Dic3×#7002
#117517482
#117465262
#117465242
#106044503
#703545