Finite Group Databasebeta

← #14415971

Subgroups — #14415971

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×#91914C211yesmaximal, characteristic, complemented
2C2×#91911C211yesmaximal, characteristic, complemented
3#11752673C211yesmaximal, characteristic
4C4×#6758C211yesmaximal, characteristic, complemented
5C3×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
6#11751926C212yesmaximal, complemented
7#11751889C212yesmaximal, complemented
8#11751888C212yesmaximal, complemented
9#11751880C212yesmaximal, complemented
10#11751813C211yesmaximal, characteristic, complemented
11#11751801C211yesmaximal, characteristic, complemented
12#10617311—31nomaximal, Hall
13C2×#6773C2211yescharacteristic
14C2×#6772C2211yescharacteristic
15C2×#6758C2211yescharacteristic
16C2×#6758C2211yescharacteristic
17C2×C3×D5⋊C8C2211yescharacteristic
18C2×C3×D5⋊C8C2211yescharacteristic
19C2×#6740C2212yes—
20C2×#6739C2212yes—
21C2×#6738C2212yes—
22C2×D3×#170C2212yes—
23#91914—24no—
24#91911—24no—
25#91764C411yescharacteristic
26C5×#875C411yescharacteristic
27C3×C42×D5C2211yescharacteristic
28#91451C411yescharacteristic
29#91448C412yes—
30#91448C412yes—
31#91405C2212yes—
32#91405C2212yes—
33#91376C411yescharacteristic
34#91231C2212yes—
35C4×C3⋊#170C2212yes—
36C3×#4004C2211yescharacteristic
37C3×#4004C2211yescharacteristic
38C3×C4⋊#170C2211yescharacteristic
39C3×C4⋊#170C2211yescharacteristic
40#91171C2212yes—
41#91171C2212yes—
42(C2×D3)⋊#170C2212yes—
43(C2×D3)⋊#170C2212yes—
44#79004—51nomaximal, Hall
45C2×#4133—31no—
46C2×#4131—31no—
47#66002—31no—
48#65787—31no—
49#65776—31no—
50#65695—32no—
51#65694—32no—
52#65693—32no—
53#65692—32no—
54D5⋊(C4⋊C8)D311yescharacteristic, complemented
55C4×#700—31no—
56C2×D60C2×C411yescharacteristic
57C2×Dic30C2×C411yescharacteristic
58C2×C5×D12C2×C411yescharacteristic
59C2×C5×Dic6C2×C411yescharacteristic
60C2×C3×C4×D5C2×C411yescharacteristic
61C2×C3×C4×D5C2×C411yescharacteristic
62C2×C3×C4×D5C2311yescharacteristic
63C2×#1203C2×C412yes—
64C2×#1203C2×C412yes—
65C2×#1202C2×C412yes—
66C2×#1201C2×C412yes—
67C2×#1200C2312yes—
68C2×D3×Dic5C2312yes—
69#6773—22no—
70#6773—22no—
71#6772—22no—
72#6772—22no—
73C2×C3⋊#170C2312yes—
74C2×C3⋊#170C2312yes—
75#6758—22no—
76#6758D414yescomplemented
77C2×C3×#170C2311yescharacteristic
78C2×C3×#170C2311yescharacteristic
79C2×C3×#170C2311yescharacteristic
80C2×C3×#170C2311yescharacteristic
81C3×D5⋊C8—22no—
82C3×D5⋊C8—22no—
83#6740—24no—
84#6739—24no—
85#6738—24no—
86D3×#170—24no—
87C3×C42×C5C2×C411yescharacteristic
88#6714C2×C412yes—
89#6714C2×C412yes—
90C5×(C2×D3)⋊C4C2×C412yes—
91C5×(C2×D3)⋊C4C2×C412yes—
92C3×C4×Dic5C2×C411yescharacteristic
93C3×C4×Dic5C2311yescharacteristic
94C3×C4×Dic5C2311yescharacteristic
95#6668C2×C412yes—
96#6668C2×C412yes—
97C2×#901—51no—
98C2×#901—51no—
99#5477—51no—
100C2×C3×C4⋊C8—51no—
101#5110—51no—
102#5110—51no—
103C4×#208—51no—
104#4838—52no—
105#4836—52no—
106#4836—52no—
107#4834—52no—
108C2×#713—31no—
109C2×#711—31no—
110C2×#704—32no—
111C22×#170—32no—
112#4133—64no—
113#4131—64no—
114C2×#700—31no—
115C2×#700—31no—
116C2×#700—32no—
117C2×D5⋊C8—32no—
118C2×D5⋊C8C2×D311yescharacteristic
119C2×D5⋊C8C2×D311yescharacteristic
120C5×#124—31no—
121#4090—31no—
122#4086—31no—
123#4030—32no—
124#4030—32no—
125#4027—32no—
126#4027—32no—
127#4021—31no—
128C42×D5C2×D311yescharacteristic
129C22⋊#170—32no—
130C22⋊#170—32no—
131#4004C2×D311yescharacteristic
132#4004C2×D311yescharacteristic
133#4003—32no—
134#4003—32no—
135#4002—32no—
136C4⋊#170C2×D311yescharacteristic
137C4⋊#170C2×D311yescharacteristic
138C4×#170—32no—
139#2935—151noSylow
140C22×D15C22×C412yes—
141C22×C5×D3C22×C412yes—
142C22×C3×D5C22×C411yescharacteristic
143C2×C3×C4×C5C22×C411yescharacteristic
144C2×C3×C4×C5C22×C411yescharacteristic
145C2×C3×C4×C5C22×C411yescharacteristic
146C2×Dic15C22×C412yes—
147D60—22no—
148Dic30—22no—
149C2×C5×Dic3C22×C412yes—
150C5×D12—22no—
151C5×Dic6—22no—
152C2×C3×Dic5C22×C411yescharacteristic
153C2×C3×Dic5C22×C411yescharacteristic
154C2×C3×Dic5C2411yescharacteristic
155C3×C4×D5—22no—
156C3×C4×D5—41no—
157C3×C4×D5C2×D412yes—
158C3×C4×D5#4112yes—
159#1203—24no—
160#1203—42no—
161#1202—24no—
162#1201—24no—
163#1200—22no—
164#1200—22no—
165D3×Dic5—22no—
166D3×Dic5—22no—
167C3⋊#170—22no—
168C3⋊#170C2×D414yes—
169C3×#170—21no—
170C3×#170—21no—
171C3×#170—21no—
172C3×#170—21no—
173C2×#235—51no—
174C2×#235—51no—
175C22×C3×C8—51no—
176C22×C3×C8—51no—
177C2×C3×C42—51no—
178#930—52no—
179#930—52no—
180C2×#208—51no—
181C2×#208—51no—
182#901—104no—
183#901—104no—
184C2×ΓL(1,𝔽25)—52no—
185C2×ΓL(1,𝔽25)—52no—
186C2×D3×C8—52no—
187C2×D3×C8—52no—
188#875—51no—
189#875AGL(1,𝔽5)11yescharacteristic
190#874—51no—
191C4×D12—52no—
192C4×D12—52no—
193#869—51no—
194C3×C4⋊C8—51no—
195C3×C4⋊C8—51no—
196C3×C4⋊C8—51no—
197C3×C4⋊C8—51no—
198(C2×D3)⋊C8—52no—
199(C2×D3)⋊C8—52no—
200(C2×D3)⋊C8—52no—
201(C2×D3)⋊C8—52no—
202#803—52no—
203C4×C3⋊C8—52no—
204C2×C5×Q8—31no—
205C2×C5×D4—31no—
206C2×C22⋊D5—32no—
207C2×C22⋊D5—32no—
208C2×C22⋊D5—32no—
209C22×Dic5—32no—
210#713—62no—
211#713—62no—
212#711—62no—
213#711—62no—
214C2×D20—32no—
215C2×D20—31no—
216C2×C4×D5—32no—
217C2×C4×D5C4×D311yescharacteristic
218C2×C4×D5C4×D311yescharacteristic
219C2×C4×D5C22×D311yescharacteristic
220C2×Dic10—31no—
221#704—64no—
222C2×#170—32no—
223C2×#170—32no—
224C2×#170—64no—
225C2×#170C22×D311yescharacteristic
226C2×#170C22×D311yescharacteristic
227C2×#170C22×D311yescharacteristic
228C2×#170C22×D311yescharacteristic
229#700—34no—
230#700—64no—
231#700—62no—
232D5⋊C8—22no—
233D5⋊C8—22no—
234D5⋊C8—64no—
235C5×C22⋊C4—32no—
236C5×C22⋊C4—32no—
237C42×C5C4×D311yescharacteristic
238#685—32no—
239#685—32no—
240#683—32no—
241#683—32no—
242C4×Dic5C4×D311yescharacteristic
243C4×Dic5C22×D311yescharacteristic
244C4×Dic5C22×D311yescharacteristic
245C2×#131—151no—
246C2×#131—151no—
247#532—151no—
248#436—152no—
249#435—152no—
250#435—152no—
251C8×D4—152no—
252C2×C4⋊C8—51no—
253#408—151no—
254#408—151no—
255C4×M16—151no—
256C22×C3×C5C23×C411yescharacteristic
257C2×D15—22no—
258C2×D15—22no—
259C2×D15—22no—
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×C5—21no—
267C3×C4×C5—21no—
268C3×C4×C5C4×D412yes—
269Dic15—22no—
270C3×Dic5—21no—
271C3×Dic5#13112yescomplemented
272C3×Dic5C22×D411yescharacteristic
273C3×Dic5C2×#4111yescharacteristic
274C5×Dic3—22no—
275C22×C3×C4—51no—
276C22×C3×C4—51no—
277C22×C3×C4—51no—
278C2×#67—52no—
279C2×#67—52no—
280#235—102no—
281#235—102no—
282#235—102no—
283#235—102no—
284C2×D12—52no—
285C2×D12—51no—
286C2×D12—52no—
287C2×D12C2×AGL(1,𝔽5)11yescharacteristic
288C2×C4×D3—52no—
289C2×C4×D3—52no—
290C2×Dic6—51no—
291C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
292C2×C3×C8—51no—
293C2×C3×C8—51no—
294C2×C3×C8—51no—
295C2×C3×C8—51no—
296C2×C3×C8—102no—
297C2×C3×C8—102no—
298C3×C42—51no—
299C3×C42—51no—
300C3×C42—51no—
301C3×C42C2×AGL(1,𝔽5)11yescharacteristic
302(C2×D3)⋊C4—52no—
303(C2×D3)⋊C4—52no—
304(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
305(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
306C4⋊Dic3—52no—
307C4⋊Dic3—52no—
308#208—54no—
309#208—102no—
310C2×C3⋊C8—52no—
311C2×C3⋊C8—52no—
312ΓL(1,𝔽25)—104no—
313ΓL(1,𝔽25)—104no—
314D3×C8—104no—
315D3×C8—104no—
316C23×C5—32no—
317C22×D5—32no—
318C22×D5C2×C4×D311yescharacteristic
319C5×Q8—62no—
320C5×D4—62no—
321C2×C4×C5—32no—
322C2×C4×C5C2×C4×D311yescharacteristic
323C2×C4×C5C2×C4×D311yescharacteristic
324C2×C4×C5C2×C4×D311yescharacteristic
325C22⋊D5—64no—
326C22⋊D5—64no—
327C22⋊D5—122no—
328C2×Dic5—32no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5C2×C4×D311yescharacteristic
332C2×Dic5C2×C4×D311yescharacteristic
333C2×Dic5C23×D311yescharacteristic
334D20—64no—
335D20—62no—
336C4×D5—22no—
337C4×D5—41no—
338C4×D5—62no—
339C4×D5—62no—
340C4×D5D3×D412yes—
341C4×D5#23912yes—
342Dic10—62no—
343#170—21no—
344#170—21no—
345#170—21no—
346#170—21no—
347#170—34no—
348#170—62no—
349C2×#41—151no—
350C2×#41—151no—
351#131—304no—
352#131—304no—
353C2×M16—152no—
354C2×M16—152no—
355C2×M16—151no—
356C2×M16—151no—
357C22×C8—51no—
358C22×C8—51no—
359C22×C8—152no—
360C22×C8—152no—
361C4⋊Q8—151no—
362#127—151no—
363#124—31no—
364#124—151no—
365#121—152no—
366#121—152no—
367C4×D4—152no—
368C4×D4—152no—
369C2×C42—51no—
370C4⋊C8—51no—
371C4⋊C8—51no—
372C4⋊C8—51no—
373C4⋊C8—51no—
374C22⋊C8—152no—
375C22⋊C8—152no—
376C22⋊C8—152no—
377C22⋊C8—152no—
378#97—152no—
379C4×C8—152no—
380C2×C3×C5C2×C4×D411yescharacteristic
381C2×C3×C5C2×#13111yescharacteristic
382C2×C3×C5#56811yescharacteristic
383D15—42no—
384C3×D5—22no—
385C5×D3—42no—
386C23×C3—51no—
387C22×D3—52no—
388C22×D3C22×AGL(1,𝔽5)12yes—
389C2×C3×C4—51no—
390C2×C3×C4—51no—
391C2×C3×C4—52no—
392C2×C3×C4—52no—
393C2×C3×C4—51no—
394C2×C3×C4—102no—
395C2×C3×C4—201no—
396C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
397C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
398C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
399#67—104no—
400#67—104no—
401C2×Dic3—52no—
402C2×Dic3C22×AGL(1,𝔽5)12yes—
403D12—22no—
404D12—104no—
405D12—102no—
406D12—202no—
407C4×D3—102no—
408C4×D3—102no—
409C4×D3—102no—
410C4×D3—102no—
411Dic6—22no—
412Dic6—102no—
413C3×C8—101no—
414C3×C8—101no—
415C3×C8—101no—
416C3×C8—101no—
417C3⋊C8—54no—
418C3⋊C8—102no—
419C22×C5—62no—
420C22×C5—62no—
421C22×C5—62no—
422C22×C5C22×C4×D311yescharacteristic
423C2×D5—21no—
424C2×D5—21no—
425C2×D5—62no—
426C2×D5—62no—
427C2×D5—62no—
428C2×D5#89312yes—
429C4×C5—21no—
430C4×C5—21no—
431C4×C5—62no—
432C4×C5#89312yes—
433Dic5—21no—
434Dic5—62no—
435Dic5#90112yescomplemented
436Dic5C2×D3×D411yescharacteristic
437Dic5C2×#23911yescharacteristic
438#41—302no—
439#41—302no—
440#41—302no—
441#41—302no—
442C2×Q8—31no—
443C2×Q8—151no—
444C2×D4—31no—
445C2×D4—152no—
446C2×D4—152no—
447C2×D4—152no—
448C2×D4—152no—
449C2×D4—151no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—51no—
453C22×C4—152no—
454C22×C4—152no—
455M16—154no—
456M16—304no—
457M16—304no—
458M16—302no—
459C2×C8—51no—
460C2×C8—51no—
461C2×C8—51no—
462C2×C8—51no—
463C2×C8—102no—
464C2×C8—102no—
465C2×C8—152no—
466C2×C8—152no—
467C2×C8—304no—
468C2×C8—304no—
469C4⋊C4—152no—
470C4⋊C4—152no—
471C22⋊C4—32no—
472C22⋊C4—32no—
473C22⋊C4—152no—
474C22⋊C4—152no—
475C42—51no—
476C42—51no—
477C42—51no—
478C42D3×AGL(1,𝔽5)11yescharacteristic
479C3×C5#293511yescharacteristic, complemented, Hall
480C22×C3—52no—
481C22×C3—101no—
482C22×C3—101no—
483C22×C3C23×AGL(1,𝔽5)11yescharacteristic
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—102no—
488C2×D3—102no—
489C2×D3—102no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—51no—
493C3×C4—51no—
494C3×C4—52no—
495C3×C4—101no—
496C3×C4D4×AGL(1,𝔽5)12yes—
497Dic3—22no—
498Dic3—102no—
499C2×C5—122no—
500C2×C5C2×#89311yescharacteristic
501C2×C5C2×#90111yescharacteristic
502C2×C5#574511yescharacteristic
503D5—22no—
504D5—122no—
505C23—32no—
506C23—51no—
507C23—152no—
508Q8—62no—
509Q8—302no—
510D4—62no—
511D4—304no—
512D4—304no—
513D4—304no—
514D4—302no—
515D4—602no—
516C2×C4—32no—
517C2×C4—51no—
518C2×C4—51no—
519C2×C4—52no—
520C2×C4—52no—
521C2×C4—51no—
522C2×C4—102no—
523C2×C4—152no—
524C2×C4—201no—
525C2×C4—302no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
530C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
531C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
532C8—101no—
533C8—101no—
534C8—101no—
535C8—101no—
536C8—154no—
537C8—302no—
538C2×C3—102no—
539C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
540C2×C3D5⋊#13111yescharacteristic
541C2×C3#6626411yescharacteristic
542D3—42no—
543D3—202no—
544C5#7900411yescharacteristic, complemented, Sylow
545C22—52no—
546C22—62no—
547C22—62no—
548C22—62no—
549C22—101no—
550C22—101no—
551C22—302no—
552C22—302no—
553C22—302no—
554C22C22×D3×AGL(1,𝔽5)11yescharacteristic
555C4—21no—
556C4—21no—
557C4—52no—
558C4—51no—
559C4—51no—
560C4—62no—
561C4—101no—
562C4—302no—
563C4#9192112yes—
564C3#1061731111yescharacteristic, complemented, Sylow
565C2—102no—
566C2—122no—
567C2—602no—
568C2D5⋊#90111yescharacteristic
569C2#1175682311yescharacteristic
570C2C2×#9192111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radical#11752673C2
cosocleC2×C3×Dic5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2935
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#91914C2semidirect
2. C2×#91911C2semidirect
4. C4×#6758C2semidirect
5. C3×D5⋊(C4⋊C8)C2semidirect
6. #11751926C2semidirect
7. #11751889C2semidirect
8. #11751888C2semidirect
9. #11751880C2semidirect
10. #11751813C2semidirect
11. #11751801C2semidirect
54. D5⋊(C4⋊C8)D3semidirect
76. #6758D4semidirect
271. C3×Dic5#131semidirect
435. Dic5#901semidirect
479. C3×C5#2935semidirect
544. C5#79004semidirect
564. C3#10617311semidirect

Maximal subgroups

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

subgroupindex
C2×#919142
C2×#919112
#117526732
C4×#67582
C3×D5⋊(C4⋊C8)2
#117519262
#117518892
#117518882
#117518802
#117518132
#117518012
#106173113
#790045