Finite Group Databasebeta

← #14415963

Subgroups — #14415963

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91990C211yesmaximal, characteristic, complemented
2C2×#91911C211yesmaximal, characteristic, complemented
3#11754793C211yesmaximal, characteristic
4C3×#65776C211yesmaximal, characteristic, complemented
5#11752053C212yesmaximal, complemented
6#11752039C212yesmaximal, complemented
7#67⋊#170C212yesmaximal, complemented
8#11751992C212yesmaximal, complemented
9D5⋊(Dic3⋊C8)C211yesmaximal, characteristic, complemented
10#11751811C211yesmaximal, characteristic, complemented
11Dic3×#700C211yesmaximal, characteristic, complemented
12#10617363—31nomaximal, Hall
13C2×C3×#711C2211yescharacteristic
14C2×#6772C2211yescharacteristic
15C2×#6762C2212yes—
16C22×C3⋊#170C2212yes—
17#91990—24no—
18C2×#6758C2211yescharacteristic
19C2×D5⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×#700C2211yescharacteristic
21C2×C3×D5⋊C8C2211yescharacteristic
22C2×#6739C2212yes—
23C2×D3×#170C2212yes—
24#91911—24no—
25#91825C411yescharacteristic
26C5×#940C411yescharacteristic
27Dic5×#67C2212yes—
28Dic5×#67C2212yes—
29#91549C412yes—
30#91549C412yes—
31C4×D5×Dic3C2211yescharacteristic
32#91389C411yescharacteristic
33#91370C411yescharacteristic
34C3×C22⋊#170C2212yes—
35C3×C22⋊#170C2212yes—
36#91177C2211yescharacteristic
37#91177C2211yescharacteristic
38Dic3⋊#170C2211yescharacteristic
39Dic3⋊#170C2211yescharacteristic
40(C2×D3)⋊#170C2212yes—
41(C2×D3)⋊#170C2212yes—
42#91169C2212yes—
43Dic3×#170C2212yes—
44#78660—51nomaximal, Hall
45C2×#4131—31no—
46C2×#4131—31no—
47#66050—31no—
48#65776—31no—
49#65776D311yescharacteristic, complemented
50#65774—32no—
51#65773—32no—
52#65773—32no—
53D4×#170—32no—
54D5⋊(C4⋊C8)—31no—
55C4×#700—31no—
56C2×C3×C5×D4C2×C411yescharacteristic
57C2×Dic30C2×C411yescharacteristic
58C2×C5×#67C2×C412yes—
59C2×C5×#67C2×C412yes—
60C2×C5×D12C2×C411yescharacteristic
61C2×C3×C22⋊D5C2×C412yes—
62C22×C3×Dic5C2312yes—
63C3×#711—22no—
64C3×#711—22no—
65C2×C3×C4×D5C2311yescharacteristic
66C2×C3×Dic10C2×C411yescharacteristic
67C2×#1201C2×C412yes—
68C2×D3×Dic5C2312yes—
69C2×D5×Dic3C2×C411yescharacteristic
70C2×D5×Dic3C2×C411yescharacteristic
71#6772—22no—
72#6772—22no—
73#6762—24no—
74C2×C3⋊#170—24no—
75C2×C3⋊#170C2311yescharacteristic
76C2×C3⋊#170C2311yescharacteristic
77C2×C3⋊#170C2312yes—
78#6758D414yescomplemented
79D5⋊(C3⋊C8)—22no—
80C2×C3×#170C2311yescharacteristic
81C2×C3×#170C2311yescharacteristic
82C2×C3×#170C2312yes—
83C3×#700—22no—
84C3×D5⋊C8—22no—
85#6739—24no—
86D3×#170—24no—
87#6719C2×C412yes—
88#6719C2×C412yes—
89C4×Dic15C2×C411yescharacteristic
90C4×C5×Dic3C2×C411yescharacteristic
91Dic3⋊Dic5C2×C412yes—
92Dic3⋊Dic5C2×C412yes—
93(C2×D3)⋊Dic5C2×C412yes—
94(C2×D3)⋊Dic5C2×C412yes—
95Dic3×Dic5C2311yescharacteristic
96Dic3×Dic5C2311yescharacteristic
97#5832—51no—
98C2×#948—51no—
99C2×#901—51no—
100C3×#408—51no—
101#5129—51no—
102Dic3×M16—51no—
103C2×Dic3⋊C8—51no—
104#4730—52no—
105#4729—52no—
106#4728—52no—
107#67⋊C8—52no—
108C2×#711—31no—
109C2×#711C2×D311yescharacteristic
110C2×#704—32no—
111C2×#704—32no—
112C22×#170—32no—
113C22×#170—32no—
114#4131—64no—
115#4131—64no—
116C2×#700—31no—
117C2×#700C2×D311yescharacteristic
118C2×D5⋊C8—31no—
119C2×D5⋊C8C2×D311yescharacteristic
120C5×#127—31no—
121#4084—32no—
122#4084—32no—
123#4082—31no—
124#4082—31no—
125D4×Dic5—32no—
126D4×Dic5—32no—
127C42×D5—31no—
128#4015—31no—
129C22⋊#170—32no—
130C22⋊#170—32no—
131C22⋊#170C2×D312yes—
132C22⋊#170C2×D312yes—
133#4004—31no—
134#4004—31no—
135#4002—32no—
136C4⋊#170—31no—
137C4⋊#170—31no—
138C4×#170—32no—
139#2935—151noSylow
140C23×C3×C5C22×C412yes—
141C22×C5×D3C22×C412yes—
142C22×C3×D5C22×C411yescharacteristic
143C3×C5×D4—22no—
144C2×C3×C4×C5C22×C411yescharacteristic
145C2×Dic15C22×C411yescharacteristic
146C2×Dic15C22×C411yescharacteristic
147C2×Dic15C22×C412yes—
148Dic30—22no—
149C5×#67—24no—
150C5×#67—42no—
151C2×C5×Dic3C22×C411yescharacteristic
152C2×C5×Dic3C22×C411yescharacteristic
153C5×D12—22no—
154C3×C22⋊D5—24no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5C22×C412yes—
158C2×C3×Dic5C2411yescharacteristic
159C3×C4×D5C2×D412yes—
160C3×C4×D5#4112yes—
161C3×Dic10—22no—
162#1201—24no—
163D3×Dic5—22no—
164D3×Dic5—22no—
165D5×Dic3—22no—
166D5×Dic3—41no—
167C3⋊#170—21no—
168C3⋊#170—21no—
169C3⋊#170C2×D414yes—
170C3×#170—21no—
171C3×#170—21no—
172C3×#170—22no—
173C2×C3×#41—51no—
174C2×#235—51no—
175C2×C3×M16—51no—
176C22×C3×C8—51no—
177#948—104no—
178#947—51no—
179#944—51no—
180#940AGL(1,𝔽5)11yescharacteristic
181#939—52no—
182#939—52no—
183#936—51no—
184C4×#67—52no—
185C4×#67—52no—
186C2×C4×Dic3—51no—
187C2×#208—51no—
188C2×#208—52no—
189C22×C3⋊C8—51no—
190C22×C3⋊C8—52no—
191#901—104no—
192C2×ΓL(1,𝔽25)—52no—
193C2×D3×C8—52no—
194C3×C22⋊C8—52no—
195C3×C22⋊C8—52no—
196(C2×D3)⋊C8—52no—
197(C2×D3)⋊C8—52no—
198#815—52no—
199Dic3⋊C8—51no—
200Dic3⋊C8—51no—
201Dic3⋊C8—51no—
202Dic3⋊C8—51no—
203C8×Dic3—52no—
204C2×C5×D4—32no—
205C2×C5×D4—32no—
206C2×C5×D4—31no—
207C2×C5×D4C4×D311yescharacteristic
208C2×C22⋊D5—32no—
209C2×C22⋊D5C4×D312yes—
210C22×Dic5—32no—
211C22×Dic5C22×D312yes—
212#711—22no—
213#711—22no—
214#711—62no—
215#711—62no—
216C2×C4×D5—31no—
217C2×C4×D5—31no—
218C2×C4×D5C22×D311yescharacteristic
219C2×Dic10—31no—
220C2×Dic10C4×D311yescharacteristic
221#704—64no—
222#704—64no—
223C2×#170—32no—
224C2×#170—31no—
225C2×#170—31no—
226C2×#170—64no—
227C2×#170—64no—
228C2×#170C22×D311yescharacteristic
229C2×#170C22×D311yescharacteristic
230C2×#170C22×D312yes—
231#700—22no—
232#700—34no—
233D5⋊C8—22no—
234D5⋊C8—62no—
235C42×C5—31no—
236C22⋊Dic5—32no—
237C22⋊Dic5—32no—
238C22⋊Dic5—32no—
239C22⋊Dic5—32no—
240C4⋊Dic5—32no—
241C4⋊Dic5—32no—
242C4×Dic5—31no—
243C4×Dic5—31no—
244C4×Dic5—31no—
245C2×#131—151no—
246C2×#131—151no—
247#532—151no—
248#436—152no—
249#435—152no—
250#435—152no—
251C8×D4—152no—
252C2×C4⋊C8—151no—
253#408—51no—
254#408—151no—
255C4×M16—151no—
256C22×C3×C5—22no—
257C22×C3×C5—22no—
258C22×C3×C5—22no—
259C22×C3×C5C23×C411yescharacteristic
260C2×C5×D3—22no—
261C2×C5×D3—22no—
262C2×C5×D3—22no—
263C2×C3×D5—21no—
264C2×C3×D5—21no—
265C2×C3×D5C4×D412yes—
266C3×C4×C5C4×D412yes—
267Dic15—22no—
268Dic15—21no—
269Dic15—21no—
270C3×Dic5—22no—
271C3×Dic5C22×D411yescharacteristic
272C3×Dic5C2×#4111yescharacteristic
273C5×Dic3—21no—
274C5×Dic3#13112yescomplemented
275C3×#41—102no—
276C3×#41—102no—
277C2×C3×Q8—51no—
278C2×C3×D4—52no—
279C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
280C22×C3×C4—51no—
281C22×C3×C4—52no—
282C2×#67—52no—
283C2×#67C2×AGL(1,𝔽5)12yes—
284C2×#67C2×AGL(1,𝔽5)12yes—
285C22×Dic3—51no—
286C22×Dic3—51no—
287#235—102no—
288#235—102no—
289C2×D12C2×AGL(1,𝔽5)11yescharacteristic
290C2×C4×D3—52no—
291C2×Dic6—51no—
292C3×M16—102no—
293C2×C3×C8—52no—
294C2×C3×C8—51no—
295C2×C3×C8—51no—
296C2×C3×C8—102no—
297#217—52no—
298#217—52no—
299(C2×D3)⋊C4—52no—
300(C2×D3)⋊C4—52no—
301Dic3⋊C4—52no—
302Dic3⋊C4—52no—
303C4×Dic3—51no—
304C4×Dic3—51no—
305C4×Dic3—51no—
306C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
307#208—54no—
308#208—104no—
309C2×C3⋊C8—52no—
310C2×C3⋊C8—51no—
311C2×C3⋊C8—51no—
312C2×C3⋊C8—104no—
313C2×C3⋊C8—102no—
314ΓL(1,𝔽25)—104no—
315D3×C8—104no—
316C23×C5—32no—
317C23×C5C2×C4×D312yes—
318C22×D5C2×C4×D311yescharacteristic
319C5×D4—22no—
320C5×D4—64no—
321C5×D4—62no—
322C5×D4—122no—
323C2×C4×C5—31no—
324C2×C4×C5—31no—
325C2×C4×C5C2×C4×D311yescharacteristic
326C22⋊D5—24no—
327C22⋊D5—64no—
328C2×Dic5—22no—
329C2×Dic5—22no—
330C2×Dic5—31no—
331C2×Dic5—31no—
332C2×Dic5—32no—
333C2×Dic5—62no—
334C2×Dic5—62no—
335C2×Dic5C2×C4×D312yes—
336C2×Dic5C23×D311yescharacteristic
337C4×D5—62no—
338C4×D5—121no—
339C4×D5D3×D412yes—
340C4×D5#23712yes—
341Dic10—22no—
342Dic10—62no—
343#170—21no—
344#170—21no—
345#170—22no—
346#170—34no—
347#170—61no—
348#170—61no—
349C2×#41—51no—
350C2×#41—151no—
351#131—304no—
352#131—304no—
353C2×M16—51no—
354C2×M16—152no—
355C2×M16—151no—
356C2×M16—152no—
357C22×C8—51no—
358C22×C8—152no—
359C22×C8—151no—
360C22×C8—152no—
361C4⋊Q8—151no—
362#127—31no—
363#124—151no—
364#124—151no—
365#121—152no—
366#121—152no—
367C4×D4—152no—
368C4×D4—152no—
369C2×C42—151no—
370C4⋊C8—151no—
371C4⋊C8—151no—
372C4⋊C8—151no—
373C4⋊C8—151no—
374C22⋊C8—52no—
375C22⋊C8—52no—
376C22⋊C8—152no—
377C22⋊C8—152no—
378#97—152no—
379C4×C8—152no—
380C2×C3×C5—42no—
381C2×C3×C5C2×C4×D411yescharacteristic
382C2×C3×C5C2×#13111yescharacteristic
383C2×C3×C5#56811yescharacteristic
384C3×D5—22no—
385C5×D3—42no—
386C23×C3—51no—
387C23×C3C22×AGL(1,𝔽5)12yes—
388C22×D3C22×AGL(1,𝔽5)12yes—
389C3×Q8—102no—
390C3×D4—22no—
391C3×D4—104no—
392C2×C3×C4—51no—
393C2×C3×C4—52no—
394C2×C3×C4—52no—
395C2×C3×C4—52no—
396C2×C3×C4—102no—
397C2×C3×C4—102no—
398C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
399#67—24no—
400#67—42no—
401#67—104no—
402C2×Dic3—51no—
403C2×Dic3—51no—
404C2×Dic3—52no—
405C2×Dic3—102no—
406C2×Dic3—201no—
407C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
408C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
409D12—22no—
410C4×D3—102no—
411C4×D3—102no—
412Dic6—102no—
413C3×C8—101no—
414C3×C8—101no—
415C3×C8—102no—
416C3⋊C8—54no—
417C3⋊C8—101no—
418C3⋊C8—101no—
419C22×C5—22no—
420C22×C5—22no—
421C22×C5—22no—
422C22×C5—62no—
423C22×C5—62no—
424C22×C5—62no—
425C22×C5C22×C4×D311yescharacteristic
426C2×D5—21no—
427C2×D5—21no—
428C2×D5#67⋊C412yes—
429C4×C5—32no—
430C4×C5—61no—
431C4×C5#67⋊C412yes—
432Dic5—22no—
433Dic5—62no—
434Dic5—61no—
435Dic5—61no—
436Dic5C2×D3×D411yescharacteristic
437Dic5C2×#23711yescharacteristic
438#41—102no—
439#41—102no—
440#41—302no—
441#41—302no—
442C2×Q8—51no—
443C2×Q8—151no—
444C2×D4—32no—
445C2×D4—32no—
446C2×D4—31no—
447C2×D4—52no—
448C2×D4—152no—
449C2×D4D3×AGL(1,𝔽5)11yescharacteristic
450C22×C4—51no—
451C22×C4—52no—
452C22×C4—151no—
453C22×C4—151no—
454C22×C4—152no—
455M16—102no—
456M16—154no—
457M16—304no—
458M16—304no—
459C2×C8—52no—
460C2×C8—51no—
461C2×C8—51no—
462C2×C8—102no—
463C2×C8—152no—
464C2×C8—151no—
465C2×C8—151no—
466C2×C8—304no—
467C2×C8—304no—
468C2×C8—302no—
469C4⋊C4—152no—
470C4⋊C4—152no—
471C22⋊C4—152no—
472C22⋊C4—152no—
473C22⋊C4—152no—
474C22⋊C4—152no—
475C42—31no—
476C42—151no—
477C42—151no—
478C42—151no—
479C3×C5#293511yescharacteristic, complemented, Hall
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—52no—
484C22×C3—101no—
485C22×C3—101no—
486C22×C3C23×AGL(1,𝔽5)11yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C3×C4—51no—
491C3×C4—51no—
492C3×C4—102no—
493C3×C4D4×AGL(1,𝔽5)12yes—
494Dic3—21no—
495Dic3—101no—
496Dic3—101no—
497Dic3—102no—
498Dic3#413112yescomplemented
499C2×C5—42no—
500C2×C5—122no—
501C2×C5C2×#67⋊C411yescharacteristic
502C2×C5D3×#13111yescharacteristic
503C2×C5#575011yescharacteristic
504D5—22no—
505C23—32no—
506C23—51no—
507C23C2×D3×AGL(1,𝔽5)12yes—
508Q8—102no—
509Q8—302no—
510D4—22no—
511D4—64no—
512D4—62no—
513D4—104no—
514D4—122no—
515D4—304no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—51no—
519C2×C4—52no—
520C2×C4—52no—
521C2×C4—52no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—151no—
525C2×C4—151no—
526C2×C4—152no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4—601no—
531C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
532C8—101no—
533C8—101no—
534C8—102no—
535C8—154no—
536C8—301no—
537C8—301no—
538C2×C3—42no—
539C2×C3—102no—
540C2×C3C2×#413111yescharacteristic
541C2×C3#6625611yescharacteristic
542C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
543D3—42no—
544C5#7866011yescharacteristic, complemented, Sylow
545C22—22no—
546C22—22no—
547C22—22no—
548C22—52no—
549C22—62no—
550C22—62no—
551C22—62no—
552C22—101no—
553C22—101no—
554C22C22×D3×AGL(1,𝔽5)11yescharacteristic
555C4—32no—
556C4—51no—
557C4—51no—
558C4—61no—
559C4—102no—
560C4—301no—
561C4—301no—
562C4—302no—
563C4#67⋊AGL(1,𝔽5)12yes—
564C3#1061736311yescharacteristic, complemented, Sylow
565C2—42no—
566C2—102no—
567C2—122no—
568C2D3×#413111yescharacteristic
569C2#1175684111yescharacteristic
570C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×D4C2×C4
prime radical#11754793C2
cosocleC2×C3×Dic5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2935
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#91990C2semidirect
2. C2×#91911C2semidirect
4. C3×#65776C2semidirect
5. #11752053C2semidirect
6. #11752039C2semidirect
7. #67⋊#170C2semidirect
8. #11751992C2semidirect
9. D5⋊(Dic3⋊C8)C2semidirect
10. #11751811C2semidirect
11. Dic3×#700C2semidirect
49. #65776D3semidirect
78. #6758D4semidirect
274. C5×Dic3#131semidirect
479. C3×C5#2935semidirect
498. Dic3#4131semidirect
544. C5#78660semidirect
564. C3#10617363semidirect

Maximal subgroups

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

subgroupindex
C2×#919902
C2×#919112
#117547932
C3×#657762
#117520532
#117520392
#67⋊#1702
#117519922
D5⋊(Dic3⋊C8)2
#117518112
Dic3×#7002
#106173633
#786605