Finite Group Databasebeta

← #14415314

Subgroups — #14415314

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91913C212yesmaximal, complemented
2#11753680C211yesmaximal, characteristic
3#11752361C211yesmaximal, characteristic, complemented
4C3×#65705C211yesmaximal, characteristic, complemented
5#11751921C211yesmaximal, characteristic, complemented
6#11751913C211yesmaximal, characteristic, complemented
7#11751910C212yesmaximal, complemented
8#11751896C212yesmaximal, complemented
9#11751879C212yesmaximal, complemented
10#11751867C211yesmaximal, characteristic, complemented
11C4×D3×#170C211yesmaximal, characteristic, complemented
12#10617227—31nomaximal, Hall
13C2×#6772C2212yes—
14C2×D5⋊(C3⋊C8)C2212yes—
15C2×C3×#700C2212yes—
16C2×#6739C2212yes—
17C2×#6739C2212yes—
18C2×D3×#170C2212yes—
19C2×D3×#170C2211yescharacteristic
20C2×D3×#170C2211yescharacteristic
21#91913—28no—
22#91777C411yescharacteristic
23C5×#895C411yescharacteristic
24C3×#4039C2211yescharacteristic
25#91463C411yescharacteristic
26#91462C412yes—
27#91456C411yescharacteristic
28#91453C412yes—
29#91440C2212yes—
30Dic5×D12C2211yescharacteristic
31C4×D3×Dic5C2211yescharacteristic
32#91232C2212yes—
33C4⋊(C3⋊#170)C2211yescharacteristic
34C4×C3⋊#170C2211yescharacteristic
35C3×#4003C2212yes—
36C3×C4⋊#170C2211yescharacteristic
37C3×C4×#170C2211yescharacteristic
38#91177C2212yes—
39(C2×D3)⋊#170C2212yes—
40(C2×D3)⋊#170C2212yes—
41Dic3×#170C2211yescharacteristic
42Dic3×#170C2211yescharacteristic
43#79538—51nomaximal, Hall
44C2×#4131—32no—
45#65930—31no—
46#65775—32no—
47#65774—31no—
48D4×#170—31no—
49#65758—31no—
50C2×C4×#170—31no—
51#65705D311yescharacteristic, complemented
52#65702—31no—
53#65695—32no—
54#65692—32no—
55C2×Dic30C2×C412yes—
56C2×C5×D12C2×C411yescharacteristic
57C2×C5×D12C2×C412yes—
58C2×C4×C5×D3C2×C411yescharacteristic
59C2×C3×D20C2×C411yescharacteristic
60C2×C3×C4×D5C2312yes—
61C2×#1201C2×C412yes—
62C2×#1201C2×C412yes—
63C2×#1201C2×C412yes—
64C2×D3×Dic5C2×C411yescharacteristic
65C2×D3×Dic5C2312yes—
66C2×D3×Dic5C2311yescharacteristic
67#6772—24no—
68#6772—24no—
69C2×C3⋊#170C2311yescharacteristic
70C2×C3⋊#170C2312yes—
71C2×C3⋊#170C2311yescharacteristic
72D5⋊(C3⋊C8)—24no—
73C2×C3×#170C2312yes—
74C2×C3×#170C2311yescharacteristic
75C2×C3×#170C2311yescharacteristic
76C3×#700—24no—
77#6739—24no—
78#6739—24no—
79D3×#170—22no—
80D3×#170—24no—
81D3×#170D414yescomplemented
82C3×C5×C4⋊C4C2×C411yescharacteristic
83C4⋊Dic15C2×C411yescharacteristic
84#6712C2×C412yes—
85C4×Dic15C2×C411yescharacteristic
86C5×(C2×D3)⋊C4C2×C412yes—
87C3×#685C2×C412yes—
88C3×C4×Dic5C2311yescharacteristic
89(C2×D3)⋊Dic5C2×C412yes—
90#6665C2×C412yes—
91Dic3×Dic5C2×C411yescharacteristic
92Dic3×Dic5C2311yescharacteristic
93C2×#907—52no—
94#5548—51no—
95C3×#436—51no—
96#5128—52no—
97D4×C3⋊C8—51no—
98#4837—51no—
99#4834—51no—
100#4727—52no—
101#67⋊C8—52no—
102#4688—51no—
103C4×D3×C8—51no—
104C2×#711—32no—
105C2×#704—32no—
106C2×#704—32no—
107C22×#170—31no—
108C22×#170—32no—
109C22×#170—31no—
110#4131—68no—
111C2×#700C2×D312yes—
112C2×D5⋊C8—32no—
113C5×#121—31no—
114#4086—31no—
115#4085—32no—
116D4×Dic5—31no—
117#4076—31no—
118C2×C4×Dic5—31no—
119#4039C2×D311yescharacteristic
120#4034—31no—
121#4031—32no—
122#4027—32no—
123C22⋊#170—32no—
124C22⋊#170—32no—
125#4004—32no—
126#4003—32no—
127#4003C2×D312yes—
128C4⋊#170—31no—
129C4⋊#170C2×D311yescharacteristic
130C4×#170—31no—
131C4×#170—31no—
132C4×#170—31no—
133C4×#170C2×D311yescharacteristic
134#2931—151noSylow
135C22×C5×D3C22×C412yes—
136C22×C5×D3C22×C411yescharacteristic
137C22×C3×D5C22×C412yes—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C22×C411yescharacteristic
140C2×Dic15C22×C412yes—
141C2×Dic15C22×C411yescharacteristic
142C2×Dic15C22×C411yescharacteristic
143Dic30—24no—
144C2×C5×Dic3C22×C411yescharacteristic
145C5×D12—24no—
146C5×D12—22no—
147C4×C5×D3—22no—
148C2×C3×Dic5C22×C411yescharacteristic
149C2×C3×Dic5C2411yescharacteristic
150C3×D20—22no—
151C3×C4×D5—22no—
152C3×C4×D5—22no—
153#1201—24no—
154#1201—24no—
155#1201—24no—
156D3×Dic5—22no—
157D3×Dic5—22no—
158D3×Dic5—22no—
159D3×Dic5C2×D412yes—
160D3×Dic5#4112yes—
161C3⋊#170—22no—
162C3⋊#170—21no—
163C3⋊#170C2×D412yes—
164C3×#170—22no—
165C3×#170—21no—
166C3×#170C2×D412yes—
167C2×#235—52no—
168C2×C3×M16—52no—
169C3×C4×D4—51no—
170#940—51no—
171#938—51no—
172C4×#67—52no—
173C22×C3⋊C8—52no—
174#907—108no—
175C2×ΓL(1,𝔽25)—52no—
176C2×ΓL(1,𝔽25)—52no—
177C2×D3×C8—51no—
178C2×D3×C8—51no—
179C2×D3×C8—52no—
180#895AGL(1,𝔽5)11yescharacteristic
181#888—51no—
182#885—52no—
183#884—52no—
184C4×D12—51no—
185C42×D3—51no—
186C3×C4⋊C8—51no—
187C3×C22⋊C8—52no—
188C3×C4×C8—51no—
189#830—52no—
190(C2×D3)⋊C8—52no—
191(C2×D3)⋊C8—52no—
192Dic3⋊C8—52no—
193C8×Dic3—51no—
194C8×Dic3—51no—
195C4⋊(C3⋊C8)—51no—
196C4×C3⋊C8—51no—
197C2×C5×D4—32no—
198C2×C5×D4—31no—
199C22×C4×C5—31no—
200C2×C22⋊D5—32no—
201C2×C22⋊D5—32no—
202C2×C22⋊D5—32no—
203C22×Dic5—31no—
204C22×Dic5—31no—
205C22×Dic5—32no—
206#711—64no—
207#711—64no—
208C2×D20C4×D311yescharacteristic
209C2×C4×D5C22×D312yes—
210C2×Dic10—32no—
211#704—64no—
212#704—64no—
213C2×#170—31no—
214C2×#170—34no—
215C2×#170—32no—
216C2×#170—31no—
217C2×#170—64no—
218C2×#170—62no—
219C2×#170C22×D312yes—
220C2×#170C22×D311yescharacteristic
221C2×#170C22×D311yescharacteristic
222#700—24no—
223D5⋊C8—64no—
224C5×C4⋊C4C4×D311yescharacteristic
225C5×C22⋊C4—32no—
226C22⋊Dic5—32no—
227#685—32no—
228#685C4×D312yes—
229C4⋊Dic5—31no—
230#683—32no—
231C4×Dic5—31no—
232C4×Dic5—31no—
233C4×Dic5—31no—
234C4×Dic5C22×D311yescharacteristic
235C2×#131—152no—
236#532—151no—
237#436—51no—
238#436—151no—
239#436—152no—
240C8×D4—152no—
241C8×D4—151no—
242C8×D4—151no—
243#423—151no—
244#408—152no—
245C2×C4×C8—151no—
246C22×C3×C5C23×C411yescharacteristic
247C2×C5×D3—21no—
248C2×C5×D3—21no—
249C2×C5×D3—22no—
250C2×C5×D3—22no—
251C2×C5×D3—22no—
252C2×C5×D3C4×D412yes—
253C2×C3×D5—22no—
254C2×C3×D5—22no—
255C2×C3×D5—22no—
256C3×C4×C5—22no—
257C3×C4×C5#13112yescomplemented
258Dic15—22no—
259Dic15C4×D412yes—
260Dic15#13112yescomplemented
261C3×Dic5—21no—
262C3×Dic5C22×D411yescharacteristic
263C3×Dic5C2×#4111yescharacteristic
264C5×Dic3—21no—
265C2×C3×D4—51no—
266C22×C3×C4—52no—
267C2×#67—52no—
268C2×#67—52no—
269C2×#67—52no—
270#235—104no—
271#235—104no—
272C2×D12C2×AGL(1,𝔽5)11yescharacteristic
273C2×D12C2×AGL(1,𝔽5)12yes—
274C2×C4×D3—51no—
275C2×C4×D3—52no—
276C2×C4×D3—51no—
277C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
278C2×Dic6—52no—
279C3×M16—104no—
280C2×C3×C8—51no—
281C2×C3×C8—51no—
282C2×C3×C8—52no—
283C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
284C3×C22⋊C4—52no—
285C3×C42—51no—
286#217—52no—
287(C2×D3)⋊C4—52no—
288(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
289C4⋊Dic3—51no—
290Dic3⋊C4—52no—
291C4×Dic3—51no—
292C4×Dic3—51no—
293C4×Dic3—51no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—52no—
297C2×C3⋊C8—104no—
298ΓL(1,𝔽25)—104no—
299ΓL(1,𝔽25)—104no—
300D3×C8—54no—
301D3×C8—104no—
302D3×C8—102no—
303C23×C5—31no—
304C23×C5—32no—
305C22×D5C2×C4×D312yes—
306C5×D4—64no—
307C5×D4—62no—
308C2×C4×C5—31no—
309C2×C4×C5—62no—
310C2×C4×C5C2×C4×D312yes—
311C2×C4×C5C2×C4×D311yescharacteristic
312C22⋊D5—64no—
313C22⋊D5—64no—
314C22⋊D5—64no—
315C2×Dic5—31no—
316C2×Dic5—31no—
317C2×Dic5—32no—
318C2×Dic5—32no—
319C2×Dic5—32no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5—62no—
323C2×Dic5C2×C4×D311yescharacteristic
324C2×Dic5C23×D311yescharacteristic
325D20—22no—
326C4×D5—22no—
327C4×D5—22no—
328Dic10—64no—
329#170—22no—
330#170—21no—
331#170—32no—
332#170—62no—
333#170—61no—
334#170D3×D412yes—
335C2×#41—152no—
336#131—308no—
337C2×M16—52no—
338C2×M16—152no—
339C2×M16—152no—
340C22×C8—151no—
341C22×C8—152no—
342C22×C8—151no—
343C22×C8—152no—
344C4⋊Q8—151no—
345#127—151no—
346#124—152no—
347#121—31no—
348#121—152no—
349#121—151no—
350C4×D4—51no—
351C4×D4—151no—
352C4×D4—152no—
353C2×C42—151no—
354C4⋊C8—51no—
355C4⋊C8—151no—
356C4⋊C8—152no—
357C22⋊C8—52no—
358C22⋊C8—152no—
359C22⋊C8—152no—
360C22⋊C8—152no—
361C4×C8—51no—
362C4×C8—151no—
363C4×C8—151no—
364C4×C8—151no—
365C2×C3×C5C2×C4×D411yescharacteristic
366C2×C3×C5C2×#13111yescharacteristic
367C2×C3×C5C2×#13111yescharacteristic
368C3×D5—42no—
369C5×D3—22no—
370C5×D3—42no—
371C23×C3—52no—
372C22×D3C22×AGL(1,𝔽5)12yes—
373C22×D3C22×AGL(1,𝔽5)11yescharacteristic
374C3×D4—102no—
375C2×C3×C4—51no—
376C2×C3×C4—51no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4C22×AGL(1,𝔽5)12yes—
380C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
381#67—104no—
382#67—104no—
383#67—104no—
384C2×Dic3—51no—
385C2×Dic3—51no—
386C2×Dic3—52no—
387C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
388D12—24no—
389D12—22no—
390C4×D3—22no—
391C4×D3—52no—
392C4×D3—52no—
393C4×D3—102no—
394C4×D3—102no—
395C4×D3—102no—
396Dic6—104no—
397C3×C8—52no—
398C3×C8—102no—
399C3×C8—101no—
400C3⋊C8—52no—
401C3⋊C8—101no—
402C3⋊C8—102no—
403C22×C5—32no—
404C22×C5—61no—
405C22×C5—61no—
406C22×C5—62no—
407C22×C5—62no—
408C22×C5—62no—
409C22×C5C22×C4×D311yescharacteristic
410C2×D5—22no—
411C2×D5—22no—
412C2×D5—22no—
413C4×C5—22no—
414C4×C5—61no—
415C4×C5#90712yescomplemented
416Dic5—21no—
417Dic5—32no—
418Dic5—32no—
419Dic5—62no—
420Dic5C2×D3×D411yescharacteristic
421Dic5D3×#4111yescharacteristic
422#41—304no—
423#41—304no—
424C2×Q8—152no—
425C2×D4—31no—
426C2×D4—32no—
427C2×D4—51no—
428C2×D4—152no—
429C2×D4—152no—
430C2×D4—152no—
431C22×C4—31no—
432C22×C4—52no—
433C22×C4—151no—
434C22×C4—152no—
435C22×C4—151no—
436M16—104no—
437M16—304no—
438M16—304no—
439C2×C8—51no—
440C2×C8—51no—
441C2×C8—52no—
442C2×C8—154no—
443C2×C8—151no—
444C2×C8—151no—
445C2×C8—152no—
446C2×C8—304no—
447C2×C8—302no—
448C2×C8—304no—
449C4⋊C4—152no—
450C4⋊C4—151no—
451C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
452C22⋊C4—32no—
453C22⋊C4—52no—
454C22⋊C4—152no—
455C22⋊C4—152no—
456C42—51no—
457C42—151no—
458C42—151no—
459C42—151no—
460C3×C5#293111yescharacteristic, complemented, Hall
461C22×C3—102no—
462C22×C3—102no—
463C22×C3—102no—
464C22×C3C23×AGL(1,𝔽5)11yescharacteristic
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3D4×AGL(1,𝔽5)12yes—
471C3×C4—22no—
472C3×C4—51no—
473C3×C4—51no—
474C3×C4—101no—
475C3×C4#413112yescomplemented
476Dic3—21no—
477Dic3—52no—
478Dic3—52no—
479Dic3—102no—
480C2×C5—62no—
481C2×C5—122no—
482C2×C5C4×D3×D411yescharacteristic
483C2×C5C2×#90711yescharacteristic
484C2×C5D3×#13111yescharacteristic
485D5—42no—
486C23—31no—
487C23—32no—
488C23—52no—
489Q8—304no—
490D4—64no—
491D4—62no—
492D4—102no—
493D4—304no—
494D4—304no—
495D4—304no—
496C2×C4—31no—
497C2×C4—51no—
498C2×C4—51no—
499C2×C4—62no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—152no—
503C2×C4—152no—
504C2×C4—151no—
505C2×C4—151no—
506C2×C4—152no—
507C2×C4—302no—
508C2×C4—302no—
509C2×C4—302no—
510C2×C4C2×D3×AGL(1,𝔽5)12yes—
511C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
512C8—52no—
513C8—102no—
514C8—101no—
515C8—152no—
516C8—302no—
517C8—301no—
518C2×C3—202no—
519C2×C3C2×#413111yescharacteristic
520C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
521C2×C3D5⋊#13111yescharacteristic
522D3—22no—
523D3—42no—
524C5#7953811yescharacteristic, complemented, Sylow
525C22—32no—
526C22—61no—
527C22—61no—
528C22—62no—
529C22—62no—
530C22—62no—
531C22—102no—
532C22—102no—
533C22—102no—
534C22C22×D3×AGL(1,𝔽5)11yescharacteristic
535C4—22no—
536C4—51no—
537C4—51no—
538C4—61no—
539C4—101no—
540C4—152no—
541C4—152no—
542C4—302no—
543C4#9191312yescomplemented
544C3#1061722711yescharacteristic, complemented, Sylow
545C2—62no—
546C2—122no—
547C2—202no—
548C2C2×#9191311yescharacteristic
549C2D3×D4×AGL(1,𝔽5)11yescharacteristic
550C2#1175684911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C4⋊C4C2×C4
prime radical#11753680C2
cosocleC2×C3×Dic5C24
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2931
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91913C2semidirect
3. #11752361C2semidirect
4. C3×#65705C2semidirect
5. #11751921C2semidirect
6. #11751913C2semidirect
7. #11751910C2semidirect
8. #11751896C2semidirect
9. #11751879C2semidirect
10. #11751867C2semidirect
11. C4×D3×#170C2semidirect
51. #65705D3semidirect
81. D3×#170D4semidirect
257. C3×C4×C5#131semidirect
260. Dic15#131semidirect
415. C4×C5#907semidirect
460. C3×C5#2931semidirect
475. C3×C4#4131semidirect
524. C5#79538semidirect
543. C4#91913semidirect
544. C3#10617227semidirect

Maximal subgroups

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

subgroupindex
C2×#919132
#117536802
#117523612
C3×#657052
#117519212
#117519132
#117519102
#117518962
#117518792
#117518672
C4×D3×#1702
#106172273
#795385