Finite Group Databasebeta

← #14323936

Subgroups — #14323936

Order 1920.

1336 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91815C211yesmaximal, characteristic, complemented
2C2×C3×#4076C211yesmaximal, characteristic
3C2×#91434C211yesmaximal, characteristic, complemented
4C5×#5212C211yesmaximal, characteristic, complemented
5#11749518C212yesmaximal, complemented
6#11749489C212yesmaximal, complemented
7#11748900C212yesmaximal, complemented
8#11748865C211yesmaximal, characteristic, complemented
9#11748796C211yesmaximal, characteristic, complemented
10#11748772C212yesmaximal, complemented
11#11748699C211yesmaximal, characteristic, complemented
12#10605603—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C3×C22⋊D5C2212yes—
15C22×C3×D20C2211yescharacteristic
16C22×D5×Dic3C2212yes—
17C2×#6719C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19#91815—24no—
20C2×C4⋊Dic15C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×C4×Dic15C2211yescharacteristic
23C2×C4×Dic15C2211yescharacteristic
24C2×C5×#217C2212yes—
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C3×#4076C418yescomplemented
28C2×C3×#685C2212yes—
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×#6668C2212yes—
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91434—24no—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82633—51nomaximal, Hall
40C2×#4076D311yescharacteristic
41C2×#4072—31no—
42C2×#4039—31no—
43C5×#380—31no—
44#65277—31no—
45C4⋊C4×Dic5—31no—
46#65262—31no—
47#65246—32no—
48#65245—32no—
49#64959—32no—
50#64957—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5C2312yes—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C2×C3×C22⋊D5—42no—
65C2×C3×C22⋊D5—42no—
66C2×C3×C22⋊D5C2×C418yescomplemented
67C22×C3×Dic5C2312yes—
68C2×C3×D20—22no—
69C2×C3×D20—22no—
70C2×C3×D20C2×C414yes—
71C2×D5×Dic3—24no—
72C2×D5×Dic3—42no—
73C2×D5×Dic3—42no—
74#6719—22no—
75#6719—22no—
76C4⋊Dic15—22no—
77C4⋊Dic15—22no—
78C4×Dic15—22no—
79C4×Dic15D414yescomplemented
80C5×#217—42no—
81C5×C4⋊Dic3—22no—
82C5×C4⋊Dic3—22no—
83C3×#685C2×C418yescomplemented
84C3×C4⋊Dic5C2×C414yescomplemented
85#6668—42no—
86#6665—24no—
87#6665—42no—
88#6665—42no—
89Dic3×Dic5—42no—
90C2×C3×#121—51no—
91C2×D4×Dic3—51no—
92C2×#926—51no—
93#5221—52no—
94#5221—51no—
95#5220—52no—
96#5212D511yescharacteristic, complemented
97#4984—51no—
98Dic3×C4⋊C4—51no—
99#4951—52no—
100#4950—52no—
101C23×C4×C5C2×D311yescharacteristic
102C22×C22⋊D5C2×D312yes—
103C22×D20C2×D311yescharacteristic
104C22×C4×D5—32no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C4⋊C4—31no—
107C2×C5×C22⋊C4—32no—
108C2×C22⋊Dic5—31no—
109C2×C22⋊Dic5—31no—
110#4076Dic318yescomplemented
111C2×#685—32no—
112C2×#685—32no—
113C2×#685—32no—
114C2×#685C2×D312yes—
115#4072—64no—
116C2×C4⋊Dic5—31no—
117C2×C4⋊Dic5—31no—
118C2×C4⋊Dic5C2×D311yescharacteristic
119C2×#683—32no—
120C2×C4×Dic5—31no—
121C2×C4×Dic5—31no—
122C2×C4×Dic5—32no—
123#4039—64no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#2324—151noSylow
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5—21no—
133C23×C3×C5—21no—
134C23×C3×C5C22×C412yes—
135C23×C3×C5C2411yescharacteristic
136C22×C3×D5—22no—
137C22×C3×D5—22no—
138C22×C3×D5—22no—
139C22×C3×D5—22no—
140C22×C3×D5—22no—
141C22×C3×D5—22no—
142C22×C3×D5C22×C414yes—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—28no—
146C2×C3×C4×C5C22×C412yes—
147C2×C3×C4×C5C22×C412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5#4112yes—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15C2×D412yes—
161C2×Dic15C2×D412yes—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C3×C22⋊D5—48no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5C22×C414yes—
172C3×D20—28no—
173D5×Dic3—48no—
174C22×C3×D4—52no—
175C22×C3×D4—51no—
176C23×C3×C4C2×D511yescharacteristic
177C23×Dic3—52no—
178C3×#121—58no—
179C2×C3×C4⋊C4—51no—
180C2×C3×C22⋊C4—52no—
181C2×#217—51no—
182C2×#217—51no—
183C2×#217—52no—
184C2×#217—52no—
185C2×#217—52no—
186C2×#217C2×D512yes—
187D4×Dic3—104no—
188#926—104no—
189C2×C4⋊Dic3—51no—
190C2×C4⋊Dic3—51no—
191C2×C4⋊Dic3—52no—
192C2×C4⋊Dic3C2×D511yescharacteristic
193C2×C4⋊Dic3C2×D511yescharacteristic
194C2×C4×Dic3—51no—
195C2×C4×Dic3—51no—
196C2×C4×Dic3—52no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95C2×D512yes—
200C24×C5C22×D311yescharacteristic
201C23×D5C22×D312yes—
202C22×C4×C5—22no—
203C22×C4×C5—22no—
204C22×C4×C5—32no—
205C22×C4×C5—32no—
206C22×C4×C5C2×Dic314yescomplemented
207C22×C4×C5C22×D311yescharacteristic
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—42no—
210C2×C22⋊D5—42no—
211C2×C22⋊D5C2×Dic318yescomplemented
212C22×Dic5—31no—
213C22×Dic5—31no—
214C22×Dic5—31no—
215C22×Dic5—31no—
216C22×Dic5C22×D312yes—
217C2×D20—22no—
218C2×D20—22no—
219C2×D20C2×Dic314yes—
220C2×C4×D5—64no—
221C2×C4×D5—122no—
222C2×C4×D5—122no—
223C5×C4⋊C4—62no—
224C5×C4⋊C4—62no—
225C5×C22⋊C4—122no—
226C22⋊Dic5—62no—
227C22⋊Dic5—62no—
228#685—64no—
229#685—122no—
230#685—122no—
231#685C2×Dic318yescomplemented
232C4⋊Dic5—62no—
233C4⋊Dic5—62no—
234C4⋊Dic5C2×Dic314yescomplemented
235#683—122no—
236C4×Dic5—34no—
237C4×Dic5—62no—
238C4×Dic5—122no—
239C2×#121—51no—
240C2×C4×D4—151no—
241C2×#117—151no—
242#390—152no—
243#390—151no—
244#389—151no—
245#388—152no—
246#386—152no—
247#385—152no—
248#380—31no—
249C4×C4⋊C4—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—22no—
253C22×C3×C5—41no—
254C22×C3×C5—41no—
255C22×C3×C5—41no—
256C22×C3×C5—41no—
257C22×C3×C5C23×C411yescharacteristic
258C22×C3×C5C22×D411yescharacteristic
259C22×C3×C5C2×#4111yescharacteristic
260C22×C3×C5H(5,𝔽2)11yescharacteristic
261C22×C3×C5H(5,𝔽2)11yescharacteristic
262C22×C3×C5H(5,𝔽2)11yescharacteristic
263C22×C3×C52−1+411yescharacteristic
264C2×C3×D5—24no—
265C2×C3×D5—24no—
266C2×C3×D5—24no—
267C2×C3×D5—42no—
268C2×C3×D5—42no—
269C2×C3×D5—42no—
270C2×C3×D5—42no—
271C3×C4×C5—22no—
272C3×C4×C5C4×D414yes—
273Dic15—22no—
274Dic15—22no—
275Dic15—41no—
276Dic15—41no—
277C3×Dic5—24no—
278C5×Dic3—42no—
279C5×Dic3—42no—
280C24×C3—52no—
281C24×C3C22×D511yescharacteristic
282C2×C3×D4—58no—
283C2×C3×D4—54no—
284C2×C3×D4—102no—
285C2×C3×D4—102no—
286C2×C3×D4—202no—
287C2×C3×D4—202no—
288C22×C3×C4—22no—
289C22×C3×C4—22no—
290C22×C3×C4—52no—
291C22×C3×C4C4×D514yescomplemented
292C22×C3×C4C22×D511yescharacteristic
293C22×C3×C4C22×D511yescharacteristic
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—51no—
297C22×Dic3—51no—
298C22×Dic3—104no—
299C22×Dic3—202no—
300C22×Dic3—202no—
301C22×Dic3C22×D512yes—
302C22×Dic3C22×D512yes—
303C3×C4⋊C4—54no—
304C3×C22⋊C4—58no—
305#217—42no—
306#217—104no—
307#217—102no—
308#217—102no—
309#217—202no—
310#217—202no—
311C4⋊Dic3—22no—
312C4⋊Dic3—22no—
313C4⋊Dic3—102no—
314C4⋊Dic3—102no—
315C4⋊Dic3—202no—
316C4×Dic3—54no—
317C4×Dic3—102no—
318C4×Dic3—202no—
319C23×C5—21no—
320C23×C5—21no—
321C23×C5—21no—
322C23×C5—21no—
323C23×C5—21no—
324C23×C5—21no—
325C23×C5C22×Dic312yes—
326C23×C5C23×D311yescharacteristic
327C22×D5—22no—
328C22×D5—22no—
329C22×D5—22no—
330C22×D5—22no—
331C22×D5—22no—
332C22×D5—22no—
333C22×D5C22×Dic314yes—
334C2×C4×C5—21no—
335C2×C4×C5—21no—
336C2×C4×C5—28no—
337C2×C4×C5—62no—
338C2×C4×C5—62no—
339C2×C4×C5—62no—
340C2×C4×C5—62no—
341C2×C4×C5—62no—
342C2×C4×C5—62no—
343C2×C4×C5D3×D412yes—
344C2×C4×C5#23712yes—
345C2×C4×C5C22×Dic312yes—
346C2×C4×C5C22×Dic312yes—
347C22⋊D5—48no—
348C2×Dic5—22no—
349C2×Dic5—22no—
350C2×Dic5—32no—
351C2×Dic5—32no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—61no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358C2×Dic5—61no—
359C2×Dic5—61no—
360C2×Dic5—61no—
361C2×Dic5—61no—
362C2×Dic5C22×Dic314yes—
363D20—28no—
364C4×D5—128no—
365C22×D4—52no—
366C22×D4—51no—
367C23×C4—152no—
368C23×C4D3×D511yescharacteristic
369#121—58no—
370C4×D4—304no—
371#117—304no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—31no—
374C2×C4⋊C4—51no—
375C2×C4⋊C4—151no—
376C2×C4⋊C4—151no—
377C2×C4⋊C4—152no—
378C2×C22⋊C4—32no—
379C2×C22⋊C4—52no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—151no—
382C2×C22⋊C4—151no—
383C2×C22⋊C4—152no—
384C2×C22⋊C4—152no—
385C2×C42—151no—
386C2×C42—151no—
387C2×C42—152no—
388#95—32no—
389#95—152no—
390#95—152no—
391C2×C3×C5—42no—
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#52011yescharacteristic
395C2×C3×C5#53511yescharacteristic
396C2×C3×C5#53711yescharacteristic
397C2×C3×C5#54011yescharacteristic
398C2×C3×C5#54211yescharacteristic
399C3×D5—44no—
400C23×C3—21no—
401C23×C3—21no—
402C23×C3—21no—
403C23×C3—21no—
404C23×C3—21no—
405C23×C3—21no—
406C23×C3—54no—
407C23×C3—102no—
408C23×C3—102no—
409C23×C3—102no—
410C23×C3—102no—
411C23×C3—102no—
412C23×C3—102no—
413C23×C3C2×C4×D512yes—
414C23×C3C23×D511yescharacteristic
415C3×D4—108no—
416C3×D4—208no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—28no—
420C2×C3×C4—54no—
421C2×C3×C4—102no—
422C2×C3×C4—102no—
423C2×C3×C4C2×C4×D512yes—
424C2×C3×C4C2×C4×D512yes—
425C2×C3×C4D4×D512yes—
426C2×C3×C4#71112yes—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—52no—
434C2×Dic3—52no—
435C2×Dic3—101no—
436C2×Dic3—101no—
437C2×Dic3—101no—
438C2×Dic3—101no—
439C2×Dic3—101no—
440C2×Dic3—101no—
441C2×Dic3—101no—
442C2×Dic3—101no—
443C2×Dic3—101no—
444C2×Dic3—101no—
445C2×Dic3—208no—
446C22×C5—22no—
447C22×C5—22no—
448C22×C5—22no—
449C22×C5—41no—
450C22×C5—41no—
451C22×C5—41no—
452C22×C5—41no—
453C22×C5C2×D3×D411yescharacteristic
454C22×C5C2×#23711yescharacteristic
455C22×C5#100411yescharacteristic
456C22×C5#100411yescharacteristic
457C22×C5#100911yescharacteristic
458C22×C5#101011yescharacteristic
459C22×C5C23×Dic311yescharacteristic
460C2×D5—24no—
461C2×D5—24no—
462C2×D5—24no—
463C2×D5—42no—
464C2×D5—42no—
465C2×D5—42no—
466C2×D5—42no—
467C4×C5—22no—
468C4×C5—122no—
469C4×C5—122no—
470C4×C5D4×Dic314yes—
471Dic5—24no—
472Dic5—62no—
473Dic5—62no—
474Dic5—121no—
475Dic5—121no—
476C24—52no—
477C24C2×D3×D511yescharacteristic
478C2×D4—58no—
479C2×D4—54no—
480C2×D4—102no—
481C2×D4—102no—
482C2×D4—202no—
483C2×D4—202no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—52no—
489C22×C4—151no—
490C22×C4—151no—
491C22×C4—151no—
492C22×C4—151no—
493C22×C4—304no—
494C22×C4—602no—
495C22×C4—602no—
496C22×C4D5×Dic314yescomplemented
497C22×C4C2×D3×D511yescharacteristic
498C22×C4C2×D3×D511yescharacteristic
499C4⋊C4—54no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—302no—
503C4⋊C4—302no—
504C4⋊C4—602no—
505C22⋊C4—58no—
506C22⋊C4—122no—
507C22⋊C4—304no—
508C22⋊C4—302no—
509C22⋊C4—302no—
510C22⋊C4—602no—
511C22⋊C4—602no—
512C42—154no—
513C42—302no—
514C42—602no—
515C3×C5#232411yescharacteristic, complemented, Hall
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—104no—
524C22×C3—104no—
525C22×C3—104no—
526C22×C3—202no—
527C22×C3—202no—
528C22×C3—202no—
529C22×C3—202no—
530C22×C3C22×C4×D511yescharacteristic
531C22×C3C2×D4×D511yescharacteristic
532C22×C3C2×#71111yescharacteristic
533C22×C3#414411yescharacteristic
534C22×C3#414711yescharacteristic
535C22×C3#414911yescharacteristic
536C22×C3#414911yescharacteristic
537C3×C4—22no—
538C3×C4—104no—
539C3×C4#402714yes—
540Dic3—42no—
541Dic3—42no—
542Dic3—102no—
543Dic3—102no—
544Dic3—201no—
545Dic3—201no—
546C2×C5—42no—
547C2×C5#560311yescharacteristic
548C2×C5#560711yescharacteristic
549C2×C5#560811yescharacteristic
550C2×C5#561911yescharacteristic
551C2×C5C2×D4×Dic311yescharacteristic
552C2×C5#579811yescharacteristic
553C2×C5#582711yescharacteristic
554D5—44no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—54no—
562C23—102no—
563C23—102no—
564C23—102no—
565C23—102no—
566C23—102no—
567C23—102no—
568C23C2×D5×Dic312yes—
569C23C22×D3×D511yescharacteristic
570D4—108no—
571D4—208no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—54no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—102no—
583C2×C4—102no—
584C2×C4—152no—
585C2×C4—152no—
586C2×C4—301no—
587C2×C4—301no—
588C2×C4—301no—
589C2×C4—301no—
590C2×C4—301no—
591C2×C4—301no—
592C2×C4—301no—
593C2×C4—301no—
594C2×C4—301no—
595C2×C4—301no—
596C2×C4—608no—
597C2×C4C2×D5×Dic312yes—
598C2×C4C2×D5×Dic312yes—
599C2×C4#678012yes—
600C2×C4#679012yes—
601C2×C3—42no—
602C2×C3—204no—
603C2×C3C2×#402711yescharacteristic
604C2×C3#6583811yescharacteristic
605C2×C3#6585511yescharacteristic
606C2×C3#6594111yescharacteristic
607C2×C3#6595311yescharacteristic
608C2×C3#6596311yescharacteristic
609C2×C3#6597711yescharacteristic
610C5#8263311yescharacteristic, complemented, Sylow
611C22—22no—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—104no—
619C22—104no—
620C22—104no—
621C22—202no—
622C22—202no—
623C22—202no—
624C22—202no—
625C22#9200011yescharacteristic
626C22#9201711yescharacteristic
627C22#9201711yescharacteristic
628C22#9201811yescharacteristic
629C22C22×D5×Dic311yescharacteristic
630C22C2×#678011yescharacteristic
631C22C2×#679011yescharacteristic
632C4—22no—
633C4—104no—
634C4—122no—
635C4—122no—
636C4—302no—
637C4—302no—
638C4—601no—
639C4—601no—
640C4#9155914yes—
641C3#1060560311yescharacteristic, complemented, Sylow
642C2—42no—
643C2—204no—
644C2#1175437411yescharacteristic
645C2#1175448811yescharacteristic
646C2#1175449011yescharacteristic
647C2#1175449911yescharacteristic
648C2#1175460111yescharacteristic
649C2#1175467111yescharacteristic
650C2C2×#9155911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC2×C3×#4076C2
hypercenterC23×C4D3×D5
hypocenterC3×C5#2324
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91815C2semidirect
3. C2×#91434C2semidirect
4. C5×#5212C2semidirect
5. #11749518C2semidirect
6. #11749489C2semidirect
7. #11748900C2semidirect
8. #11748865C2semidirect
9. #11748796C2semidirect
10. #11748772C2semidirect
11. #11748699C2semidirect
27. C3×#4076C4semidirect
55. C22×C3×C4×C5C2×C4semidirect
66. C2×C3×C22⋊D5C2×C4semidirect
79. C4×Dic15D4semidirect
83. C3×#685C2×C4semidirect
84. C3×C4⋊Dic5C2×C4semidirect
96. #5212D5semidirect
110. #4076Dic3semidirect
206. C22×C4×C5C2×Dic3semidirect
211. C2×C22⋊D5C2×Dic3semidirect
231. #685C2×Dic3semidirect
234. C4⋊Dic5C2×Dic3semidirect
291. C22×C3×C4C4×D5semidirect
496. C22×C4D5×Dic3semidirect
515. C3×C5#2324semidirect
610. C5#82633semidirect
641. C3#10605603semidirect

Maximal subgroups

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

subgroupindex
C2×#918152
C2×C3×#40762
C2×#914342
C5×#52122
#117495182
#117494892
#117489002
#117488652
#117487962
#117487722
#117486992
#106056033
#826335