Finite Group Databasebeta

← #14417608

Subgroups — #14417608

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66262C211yesmaximal, characteristic
2#11756812C211yesmaximal, characteristic
3C2×D5×#216C211yesmaximal, characteristic
4#11752441C211yesmaximal, characteristic
5#11751968C214yesmaximal, complemented
6AGL(1,𝔽5)×#216C214yesmaximal, complemented
7#11751726C211yesmaximal, characteristic
8#11751711C211yesmaximal, characteristic
9#11751695C211yesmaximal, characteristic
10#10617434—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13#91989C2211yescharacteristic
14D5⋊#208C2211yescharacteristic
15C3×Q8×AGL(1,𝔽5)—22no—
16C3×Q8×AGL(1,𝔽5)C2214yes—
17C3×#4130—22no—
18C3×#4130C2211yescharacteristic
19C3×#4130C2211yescharacteristic
20#91923—22no—
21ΓL(1,𝔽16)⋊Q8—22no—
22AGL(1,𝔽5)×Dic6C2214yes—
23C2×#6718C411yescharacteristic, complemented
24C2×C5×#216C411yescharacteristic, complemented
25D5×#216C412yescomplemented
26D5×#216C412yescomplemented
27D5×#216C2212yes—
28D5×#216C2212yes—
29C2×#6662C411yescharacteristic, complemented
30C2×#6661C411yescharacteristic, complemented
31C2×D5×C3⋊C8C2211yescharacteristic
32#91238C2214yes—
33#91160C2212yes—
34#91159C2212yes—
35#91153C2214yes—
36#91149C2212yes—
37AGL(1,𝔽5)×C3⋊C8C2212yes—
38#82549—51nomaximal, Hall
39#66262—31no—
40#66262D311yescharacteristic
41C2×D5×Q16—31no—
42#65782—31no—
43#65782—31no—
44#65739—34no—
45Q16×AGL(1,𝔽5)—34no—
46#65721—31no—
47#65718—31no—
48C2×C3×C5×Q8C2×C411yescharacteristic
49C2×Dic30C2×C411yescharacteristic
50C2×C5×Dic6C2×C411yescharacteristic
51C3×Q8×D5—21no—
52C3×Q8×D5—21no—
53C3×Q8×D5C2×C412yes—
54C3×Q8×D5C2312yes—
55C2×C3×C4×D5—21no—
56C2×C3×C4×D5C2311yescharacteristic
57C2×C3×Dic10—21no—
58C2×C3×Dic10C2×C411yescharacteristic
59C2×#1204—21no—
60C2×D5×Dic3—21no—
61D5×Dic6—21no—
62D5×Dic6—21no—
63D5×Dic6C2×C412yes—
64D5×Dic6C2312yes—
65#6763—21no—
66#6762—21no—
67C4⋊ΓL(1,𝔽16)C2312yes—
68C4×ΓL(1,𝔽16)—21no—
69#6758—21no—
70D5⋊(C3⋊C8)C2312yes—
71C3×C22⋊AGL(1,𝔽5)—21no—
72C3×C22⋊AGL(1,𝔽5)D412yes—
73C3×C4⋊AGL(1,𝔽5)—24no—
74C3×C4⋊AGL(1,𝔽5)D412yes—
75C3×C4⋊AGL(1,𝔽5)C2312yes—
76C3×C4×AGL(1,𝔽5)—24no—
77C3×C4×AGL(1,𝔽5)—21no—
78C3×C4×AGL(1,𝔽5)C2312yes—
79Dic3⋊AGL(1,𝔽5)—24no—
80Dic3×AGL(1,𝔽5)—24no—
81#6718C2×C412yes—
82#6718C2×C412yes—
83C2×C3⋊#168C2×C411yescharacteristic
84C5×#216C2×C412yes—
85C5×#216C2×C412yes—
86C2×C5×C3⋊C8C2×C411yescharacteristic
87#6662C2×C412yes—
88#6662C2×C412yes—
89#6661C2×C412yes—
90#6661C2×C412yes—
91D5×C3⋊C8C2×C411yescharacteristic
92D5×C3⋊C8C2×C411yescharacteristic
93D5×C3⋊C8C2311yescharacteristic
94D5×C3⋊C8C2311yescharacteristic
95C3×#519—51no—
96C22×#216—51no—
97#5516—51no—
98#5225—51no—
99#5030—54no—
100C4×#216—54no—
101#5004—51no—
102#4998—51no—
103#4997—51no—
104C2×Q8×D5—31no—
105C2×Q8×D5C2×D311yescharacteristic
106Q8×AGL(1,𝔽5)—22no—
107Q8×AGL(1,𝔽5)—34no—
108Q8×AGL(1,𝔽5)—62no—
109Q8×AGL(1,𝔽5)C2×D314yes—
110#4130—22no—
111#4130—31no—
112#4130—62no—
113#4130C2×D311yescharacteristic
114#4130C2×D311yescharacteristic
115D5⋊M16—31no—
116C2×C5×Q16—31no—
117C2×#689—31no—
118C2×#689—31no—
119D5×Q16—32no—
120D5×Q16—32no—
121D5×Q16—32no—
122D5×Q16—32no—
123C2×Dic20—31no—
124C2×C8×D5—31no—
125#4009—34no—
126#4009—34no—
127#3994—32no—
128#3993—32no—
129#3992—32no—
130C8×AGL(1,𝔽5)—32no—
131#2950—151noSylow
132C22×C3×D5C2×D411yescharacteristic
133C2×ΓL(1,𝔽16)—22no—
134C2×C3×AGL(1,𝔽5)—22no—
135C2×C3×AGL(1,𝔽5)C2×D414yes—
136C3×C5×Q8—21no—
137C3×C5×Q8C22×C412yes—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5C22×C411yescharacteristic
140C2×Dic15—21no—
141Dic30—21no—
142Dic30C22×C412yes—
143C2×C5×Dic3—21no—
144C5×Dic6—21no—
145C5×Dic6C22×C412yes—
146C2×C3×Dic5—21no—
147C2×C3×Dic5#4111yescharacteristic
148C3×C4×D5—22no—
149C3×C4×D5—22no—
150C3×C4×D5C22×C411yescharacteristic
151C3×C4×D5C2×D411yescharacteristic
152C3×C4×D5#4111yescharacteristic
153C3×C4×D5C2411yescharacteristic
154C3×Dic10—21no—
155C3×Dic10—22no—
156C3×Dic10—22no—
157C3×Dic10C22×C412yes—
158#1204—22no—
159#1204—22no—
160D5×Dic3—22no—
161D5×Dic3—22no—
162C3⋊#170—22no—
163C3⋊#168C22×C411yescharacteristic
164C3⋊#168C22×C411yescharacteristic
165C5×C3⋊C8C22×C411yescharacteristic
166C5×C3⋊C8C22×C411yescharacteristic
167C22×C3×Q8—51no—
168C22×Dic6—51no—
169C3×C4×Q8—54no—
170C3×C4×Q8—102no—
171C3×#117—51no—
172C3×#117—51no—
173C3×#117—102no—
174C2×#216—51no—
175C2×#216—51no—
176C2×#216—52no—
177C2×#216—52no—
178C2×#216—52no—
179C2×#216—52no—
180C2×#216—51no—
181C2×#216AGL(1,𝔽5)11yescharacteristic, complemented
182#926—51no—
183C2×#208—51no—
184C22×C3⋊C8—51no—
185#887—102no—
186#877—102no—
187C4×Dic6—54no—
188#835—54no—
189#810—54no—
190#808—52no—
191#807—52no—
192#803—52no—
193C4×C3⋊C8—52no—
194C2×C5×Q8—31no—
195C2×C5×Q8C4×D311yescharacteristic
196Q8×D5—21no—
197Q8×D5—21no—
198Q8×D5—32no—
199Q8×D5—32no—
200Q8×D5—61no—
201Q8×D5—61no—
202Q8×D5C4×D312yes—
203Q8×D5C22×D312yes—
204C2×C4×D5—21no—
205C2×C4×D5—61no—
206C2×C4×D5C22×D311yescharacteristic
207C2×Dic10—21no—
208C2×Dic10—31no—
209C2×Dic10—61no—
210C2×Dic10C4×D311yescharacteristic
211C22⋊AGL(1,𝔽5)—21no—
212C22⋊AGL(1,𝔽5)—61no—
213C22⋊AGL(1,𝔽5)#6712yes—
214#704—61no—
215C4⋊AGL(1,𝔽5)—24no—
216C4⋊AGL(1,𝔽5)—32no—
217C4⋊AGL(1,𝔽5)—64no—
218C4⋊AGL(1,𝔽5)#6712yes—
219C4⋊AGL(1,𝔽5)C22×D312yes—
220C4×AGL(1,𝔽5)—24no—
221C4×AGL(1,𝔽5)—21no—
222C4×AGL(1,𝔽5)—64no—
223C4×AGL(1,𝔽5)—61no—
224C4×AGL(1,𝔽5)C22×D312yes—
225#700—61no—
226D5⋊C8—32no—
227C5×Q16—32no—
228C5×Q16—32no—
229C2×C5×C8—31no—
230#689—32no—
231#689—32no—
232#689—32no—
233#689—32no—
234C2×#168—31no—
235Dic20—32no—
236Dic20—32no—
237C8×D5—31no—
238C8×D5—31no—
239C8×D5—31no—
240C8×D5—31no—
241C22×Q16—151no—
242#519—51no—
243#519—151no—
244#441—154no—
245C4×Q16—154no—
246#427—151no—
247#419—151no—
248#419—151no—
249#405—151no—
250C22×C3×C5C4×D411yescharacteristic
251C2×C3×D5—22no—
252C2×C3×D5C4×D411yescharacteristic
253C2×C3×D5C22×D411yescharacteristic
254ΓL(1,𝔽16)—42no—
255C3×AGL(1,𝔽5)—24no—
256C3×AGL(1,𝔽5)—42no—
257C3×C4×C5—22no—
258C3×C4×C5C4×D411yescharacteristic
259C3×C4×C5C23×C411yescharacteristic
260Dic15—22no—
261C3×Dic5—22no—
262C3×Dic5C4×D411yescharacteristic
263C3×Dic5C2×#4111yescharacteristic
264C5×Dic3—22no—
265C2×C3×Q8—52no—
266C2×C3×Q8—52no—
267C2×C3×Q8—51no—
268C2×C3×Q8—101no—
269C2×C3×Q8—101no—
270C2×C3×Q8—101no—
271C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
272C22×C3×C4—51no—
273C22×C3×C4—101no—
274C22×Dic3—101no—
275C2×Dic6—51no—
276C2×Dic6—52no—
277C2×Dic6—52no—
278C2×Dic6—101no—
279C2×Dic6—101no—
280C2×Dic6—101no—
281C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
282C3×C4⋊C4—52no—
283C3×C4⋊C4—52no—
284C3×C4⋊C4—104no—
285C3×C22⋊C4—52no—
286C3×C22⋊C4—101no—
287C3×C42—52no—
288C3×C42—101no—
289C3×C42—104no—
290#217—101no—
291#216—52no—
292#216—52no—
293#216—52no—
294#216—52no—
295#216—52no—
296#216—52no—
297#216C2×AGL(1,𝔽5)12yes—
298#216C2×AGL(1,𝔽5)12yes—
299C4⋊Dic3—52no—
300Dic3⋊C4—104no—
301C4×Dic3—104no—
302C4×Dic3—101no—
303#208—101no—
304#208—101no—
305C2×C3⋊C8—51no—
306C2×C3⋊C8—51no—
307C2×C3⋊C8—51no—
308C2×C3⋊C8—51no—
309C2×C3⋊C8—51no—
310C2×C3⋊C8—52no—
311C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
312C22×D5C2×#6711yescharacteristic
313C2×AGL(1,𝔽5)—22no—
314C2×AGL(1,𝔽5)—62no—
315C2×AGL(1,𝔽5)C2×#6714yes—
316C5×Q8—21no—
317C5×Q8—32no—
318C5×Q8—61no—
319C5×Q8C2×C4×D312yes—
320C2×C4×C5—21no—
321C2×C4×C5—61no—
322C2×C4×C5C2×C4×D311yescharacteristic
323C2×Dic5—21no—
324C2×Dic5—61no—
325C2×Dic5#23511yescharacteristic
326C4×D5—22no—
327C4×D5—22no—
328C4×D5—62no—
329C4×D5—62no—
330C4×D5C2×C4×D311yescharacteristic
331C4×D5#23511yescharacteristic
332C4×D5C2×#6711yescharacteristic
333C4×D5C23×D311yescharacteristic
334Dic10—21no—
335Dic10—22no—
336Dic10—22no—
337Dic10—32no—
338Dic10—61no—
339Dic10—62no—
340Dic10—62no—
341Dic10C2×C4×D312yes—
342#170—62no—
343C5×C8—31no—
344C5×C8—31no—
345#168—31no—
346#168—31no—
347C22×Q8—51no—
348C22×Q8—151no—
349C2×Q16—31no—
350C2×Q16—151no—
351C2×Q16—152no—
352C2×Q16—152no—
353C2×Q16—152no—
354C2×Q16—152no—
355C2×Q16—151no—
356C2×Q16—151no—
357C2×M16—151no—
358C22×C8—151no—
359C4×Q8—54no—
360C4×Q8—102no—
361C4×Q8—154no—
362C4×Q8—302no—
363#117—51no—
364#117—51no—
365#117—102no—
366#117—151no—
367#117—302no—
368#107—152no—
369#106—152no—
370#103—154no—
371#103—154no—
372#97—152no—
373C4×C8—152no—
374C2×C3×C5—21no—
375C2×C3×C5C2×C4×D411yescharacteristic
376C3×D5—21no—
377C3×D5#57812yes—
378C23×C3—51no—
379C3×Q8—21no—
380C3×Q8—52no—
381C3×Q8—102no—
382C3×Q8—102no—
383C3×Q8—101no—
384C3×Q8C22×AGL(1,𝔽5)12yes—
385C2×C3×C4—21no—
386C2×C3×C4—51no—
387C2×C3×C4—51no—
388C2×C3×C4—51no—
389C2×C3×C4—51no—
390C2×C3×C4—51no—
391C2×C3×C4—54no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—101no—
395C2×C3×C4—102no—
396C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3—101no—
401C2×Dic3—102no—
402Dic6—21no—
403Dic6—52no—
404Dic6—102no—
405Dic6—102no—
406Dic6—101no—
407Dic6C22×AGL(1,𝔽5)12yes—
408C3⋊C8—51no—
409C3⋊C8—51no—
410C3⋊C8—102no—
411C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
412C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
413C22×C5C4×#6711yescharacteristic
414C2×D5—22no—
415C2×D5C4×#6711yescharacteristic
416C2×D5C22×#6711yescharacteristic
417AGL(1,𝔽5)—24no—
418AGL(1,𝔽5)—42no—
419AGL(1,𝔽5)—122no—
420C4×C5—22no—
421C4×C5—62no—
422C4×C5C4×#6711yescharacteristic
423C4×C5C22×C4×D311yescharacteristic
424Dic5—22no—
425Dic5—62no—
426Dic5C4×#6711yescharacteristic
427Dic5C2×#23511yescharacteristic
428C2×Q8—31no—
429C2×Q8—52no—
430C2×Q8—52no—
431C2×Q8—51no—
432C2×Q8—101no—
433C2×Q8—101no—
434C2×Q8—101no—
435C2×Q8—152no—
436C2×Q8—152no—
437C2×Q8—151no—
438C2×Q8—301no—
439C2×Q8—301no—
440C2×Q8—301no—
441C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
442C22×C4—51no—
443C22×C4—101no—
444C22×C4—301no—
445Q16—32no—
446Q16—32no—
447Q16—152no—
448Q16—152no—
449Q16—152no—
450Q16—152no—
451Q16—152no—
452Q16—152no—
453M16—301no—
454M16—301no—
455C2×C8—31no—
456C2×C8—151no—
457C2×C8—151no—
458C2×C8—151no—
459C2×C8—151no—
460C2×C8—151no—
461C2×C8—152no—
462C4⋊C4—52no—
463C4⋊C4—52no—
464C4⋊C4—104no—
465C4⋊C4—152no—
466C4⋊C4—304no—
467C22⋊C4—52no—
468C22⋊C4—101no—
469C22⋊C4—301no—
470C42—52no—
471C42—101no—
472C42—104no—
473C42—301no—
474C42—304no—
475C3×C5#295011yescharacteristic, complemented, Hall
476C22×C3—51no—
477C22×C3—51no—
478C22×C3—102no—
479C22×C3D4×AGL(1,𝔽5)11yescharacteristic
480C3×C4—22no—
481C3×C4—51no—
482C3×C4—51no—
483C3×C4—104no—
484C3×C4—102no—
485C3×C4—202no—
486C3×C4D4×AGL(1,𝔽5)11yescharacteristic
487C3×C4C23×AGL(1,𝔽5)11yescharacteristic
488Dic3—22no—
489Dic3—102no—
490Dic3—202no—
491C2×C5—21no—
492C2×C5C2×C4×#6711yescharacteristic
493D5—21no—
494D5#583812yes—
495C23—51no—
496Q8—21no—
497Q8—32no—
498Q8—52no—
499Q8—61no—
500Q8—102no—
501Q8—102no—
502Q8—101no—
503Q8—152no—
504Q8—302no—
505Q8—302no—
506Q8—301no—
507Q8C2×D3×AGL(1,𝔽5)12yes—
508C2×C4—21no—
509C2×C4—51no—
510C2×C4—51no—
511C2×C4—51no—
512C2×C4—51no—
513C2×C4—51no—
514C2×C4—54no—
515C2×C4—61no—
516C2×C4—102no—
517C2×C4—102no—
518C2×C4—101no—
519C2×C4—102no—
520C2×C4—302no—
521C2×C4—302no—
522C2×C4—301no—
523C2×C4—302no—
524C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
525C8—31no—
526C8—31no—
527C8—151no—
528C8—151no—
529C8—302no—
530C2×C3—21no—
531C2×C3—52no—
532C2×C3—101no—
533C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
534C5#8254911yescharacteristic, complemented, Sylow
535C22—51no—
536C22—51no—
537C22—102no—
538C22AGL(1,𝔽5)×#6711yescharacteristic
539C4—22no—
540C4—51no—
541C4—51no—
542C4—62no—
543C4—104no—
544C4—102no—
545C4—202no—
546C4—302no—
547C4—602no—
548C4AGL(1,𝔽5)×#6711yescharacteristic
549C4C22×D3×AGL(1,𝔽5)11yescharacteristic
550C3#1061743411yescharacteristic, complemented, Sylow
551C2—21no—
552C2—52no—
553C2—101no—
554C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
centerC2C2×AGL(1,𝔽5)×#67
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5C2×C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC3×C4×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2950
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
5. #11751968C2semidirect
6. AGL(1,𝔽5)×#216C2semidirect
23. C2×#6718C4semidirect
24. C2×C5×#216C4semidirect
25. D5×#216C4semidirect
26. D5×#216C4semidirect
29. C2×#6662C4semidirect
30. C2×#6661C4semidirect
181. C2×#216AGL(1,𝔽5)semidirect
475. C3×C5#2950semidirect
534. C5#82549semidirect
550. C3#10617434semidirect

Maximal subgroups

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

subgroupindex
C3×#662622
#117568122
C2×D5×#2162
#117524412
#117519682
AGL(1,𝔽5)×#2162
#117517262
#117517112
#117516952
#106174343
#825495