Finite Group Databasebeta

← #14323959

Subgroups — #14323959

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×C3×#4041C211yesmaximal, characteristic
2C2×Dic3×D20C211yesmaximal, characteristic, complemented
3C2×#91357C211yesmaximal, characteristic, complemented
4#11751085C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6#11748900C212yesmaximal, complemented
7#11748896C212yesmaximal, complemented
8#11748796C211yesmaximal, characteristic, complemented
9#11748785C212yesmaximal, complemented
10#11748770C212yesmaximal, complemented
11#11748759C211yesmaximal, characteristic, complemented
12#10605494—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C22×C3×D20C2212yes—
15C22×C3×C4×D5C2211yescharacteristic
16C22×D5×Dic3C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4⋊Dic15C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×#685C2212yes—
25C3×#4041C418yescomplemented
26C2×#6668C2211yescharacteristic
27C2×#6668C2211yescharacteristic
28C2×#6665C2211yescharacteristic
29C2×#6665C2212yes—
30C2×#6665C2211yescharacteristic
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34Dic3×D20—24no—
35#91357—24no—
36#91115C2212yes—
37C5×C3⋊#95C2212yes—
38#90994C2212yes—
39#82633—51nomaximal, Hall
40C2×#4041D311yescharacteristic
41C2×C4×D20—31no—
42C2×#4018—31no—
43C4×C5×C4⋊C4—31no—
44#65277—31no—
45#65276—31no—
46#65274—31no—
47#65229—32no—
48#65228—32no—
49#64957—32no—
50#64954—32no—
51C23×C3×D5C2312yes—
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C5×Dic3C2311yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C2×C3×D20—22no—
63C2×C3×D20—22no—
64C2×C3×D20—42no—
65C2×C3×D20—42no—
66C2×C3×D20C2×C414yes—
67C2×C3×D20C2×C418yescomplemented
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5C2×C414yescomplemented
71C2×D5×Dic3—24no—
72C2×D5×Dic3—42no—
73C2×D5×Dic3—42no—
74C3×C5×C4⋊C4C2×C414yescomplemented
75C4⋊Dic15—22no—
76C4⋊Dic15—42no—
77C4×Dic15—22no—
78C5×C4⋊Dic3—22no—
79C4×C5×Dic3—42no—
80C4×C5×Dic3D414yescomplemented
81C3×#685C2×C418yescomplemented
82#6668—22no—
83#6668—22no—
84#6665—24no—
85#6665—22no—
86#6665—22no—
87#6665—42no—
88#6665—42no—
89#6665—42no—
90C2×C3×#121—51no—
91C2×D4×Dic3—51no—
92C2×#926—51no—
93#5221—51no—
94#5221—52no—
95#5220—52no—
96#5212—51no—
97#4984—51no—
98Dic3×C4⋊C4D511yescharacteristic, complemented
99#4951—52no—
100#4950—52no—
101C22×D20C2×D311yescharacteristic
102C22×D20C2×D312yes—
103C22×C4×D5—32no—
104C22×C4×D5C2×D311yescharacteristic
105C2×C5×C4⋊C4—31no—
106C2×C5×C4⋊C4C2×D311yescharacteristic
107C2×C42×C5—31no—
108C2×C42×C5—32no—
109C2×#685—31no—
110C2×#685—31no—
111C2×#685—32no—
112C2×#685—32no—
113C2×#685—32no—
114C2×#685—32no—
115C2×#685C2×D312yes—
116C2×C4⋊Dic5—31no—
117C2×C4⋊Dic5—32no—
118C2×#683—31no—
119C2×#683—31no—
120C2×C4×Dic5—31no—
121#4041Dic318yescomplemented
122C4×D20—64no—
123#4018—64no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#2324—151noSylow
128C23×C3×C5C2411yescharacteristic
129C22×C3×D5—21no—
130C22×C3×D5—22no—
131C22×C3×D5—21no—
132C22×C3×D5—22no—
133C22×C3×D5—21no—
134C22×C3×D5—22no—
135C22×C3×D5—22no—
136C22×C3×D5—22no—
137C22×C3×D5—21no—
138C22×C3×D5—21no—
139C22×C3×D5—22no—
140C22×C3×D5—21no—
141C22×C3×D5C22×C414yes—
142C22×C3×D5C22×C412yes—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5C22×C414yes—
146C2×C3×C4×C5C22×C412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—22no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—22no—
155C2×Dic15—21no—
156C2×Dic15—22no—
157C2×Dic15—21no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3C2×D412yes—
166C2×C5×Dic3C2×D412yes—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5C22×C412yes—
170C3×D20—28no—
171C3×D20—48no—
172C3×C4×D5—28no—
173D5×Dic3—48no—
174C22×C3×D4—51no—
175C22×C3×D4—52no—
176C23×C3×C4—51no—
177C23×Dic3—52no—
178C3×#121—58no—
179C2×C3×C4⋊C4C2×D511yescharacteristic
180C2×C3×C22⋊C4—52no—
181C2×#217—52no—
182C2×#217—51no—
183C2×#217—51no—
184C2×#217—52no—
185C2×#217—52no—
186C2×#217—52no—
187D4×Dic3—104no—
188#926—104no—
189C2×C4⋊Dic3—51no—
190C2×C4⋊Dic3—51no—
191C2×C4⋊Dic3—51no—
192C2×C4⋊Dic3—52no—
193C2×C4⋊Dic3C2×D511yescharacteristic
194C2×C4×Dic3—51no—
195C2×C4×Dic3C2×D512yes—
196C2×C4×Dic3C2×D511yescharacteristic
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95C2×D512yes—
200C23×D5C22×D312yes—
201C23×D5C22×D311yescharacteristic
202C22×C4×C5—31no—
203C22×C4×C5—31no—
204C22×C4×C5—32no—
205C22×C4×C5C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C22×Dic5—32no—
208C22×Dic5—31no—
209C22×Dic5—31no—
210C22×Dic5C22×D311yescharacteristic
211C2×D20—22no—
212C2×D20—22no—
213C2×D20—42no—
214C2×D20—42no—
215C2×D20C2×Dic314yes—
216C2×D20C2×Dic318yescomplemented
217C2×C4×D5—22no—
218C2×C4×D5—22no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C2×C4×D5—122no—
222C2×C4×D5C2×Dic314yescomplemented
223C5×C4⋊C4—62no—
224C5×C4⋊C4C2×Dic314yescomplemented
225C42×C5—34no—
226C42×C5—122no—
227#685—62no—
228#685—62no—
229#685—64no—
230#685—122no—
231#685—122no—
232#685—122no—
233#685C2×Dic318yescomplemented
234C4⋊Dic5—62no—
235C4⋊Dic5—122no—
236#683—62no—
237#683—62no—
238C4×Dic5—62no—
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—151no—
249C4×C4⋊C4—31no—
250C22×C3×C5C23×C411yescharacteristic
251C22×C3×C5C22×D411yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5H(5,𝔽2)11yescharacteristic
254C22×C3×C5H(5,𝔽2)11yescharacteristic
255C22×C3×C5H(5,𝔽2)11yescharacteristic
256C22×C3×C52−1+411yescharacteristic
257C2×C3×D5—22no—
258C2×C3×D5—24no—
259C2×C3×D5—22no—
260C2×C3×D5—24no—
261C2×C3×D5—22no—
262C2×C3×D5—24no—
263C2×C3×D5—42no—
264C2×C3×D5—41no—
265C2×C3×D5—41no—
266C2×C3×D5—42no—
267C2×C3×D5—41no—
268C2×C3×D5—42no—
269C2×C3×D5—41no—
270C2×C3×D5—42no—
271C3×C4×C5—24no—
272C3×C4×C5C4×D414yes—
273Dic15—42no—
274Dic15—41no—
275Dic15—41no—
276C3×Dic5—22no—
277C5×Dic3—22no—
278C5×Dic3—22no—
279C5×Dic3—42no—
280C24×C3—51no—
281C24×C3—52no—
282C2×C3×D4—54no—
283C2×C3×D4—58no—
284C2×C3×D4—102no—
285C2×C3×D4—102no—
286C2×C3×D4—202no—
287C2×C3×D4—202no—
288C22×C3×C4—51no—
289C22×C3×C4—54no—
290C22×C3×C4—102no—
291C22×C3×C4—102no—
292C22×C3×C4C22×D512yes—
293C22×C3×C4C22×D511yescharacteristic
294C22×Dic3—51no—
295C22×Dic3—52no—
296C22×Dic3—51no—
297C22×Dic3—104no—
298C22×Dic3—202no—
299C22×Dic3—202no—
300C22×Dic3C22×D512yes—
301C22×Dic3C22×D511yescharacteristic
302C22×Dic3C22×D511yescharacteristic
303C3×C4⋊C4C4×D514yescomplemented
304C3×C22⋊C4—58no—
305#217—102no—
306#217—102no—
307#217—104no—
308#217—202no—
309#217—202no—
310#217—202no—
311C4⋊Dic3—22no—
312C4⋊Dic3—102no—
313C4⋊Dic3—102no—
314C4⋊Dic3—102no—
315C4⋊Dic3—202no—
316C4×Dic3—42no—
317C4×Dic3—102no—
318C4×Dic3D2014yescomplemented
319C23×C5C23×D311yescharacteristic
320C22×D5—21no—
321C22×D5—22no—
322C22×D5—21no—
323C22×D5—22no—
324C22×D5—22no—
325C22×D5—22no—
326C22×D5—21no—
327C22×D5—21no—
328C22×D5—21no—
329C22×D5—22no—
330C22×D5—22no—
331C22×D5—21no—
332C22×D5C22×Dic314yes—
333C22×D5C22×Dic312yes—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—32no—
337C2×C4×C5—32no—
338C2×C4×C5—61no—
339C2×C4×C5—61no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5—62no—
343C2×C4×C5—62no—
344C2×C4×C5—62no—
345C2×C4×C5D3×D412yes—
346C2×C4×C5#23712yes—
347C2×C4×C5C22×Dic314yes—
348C2×C4×C5C22×Dic312yes—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—61no—
352C2×Dic5—61no—
353C2×Dic5—62no—
354C2×Dic5—61no—
355C2×Dic5—62no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358C2×Dic5—62no—
359C2×Dic5—61no—
360C2×Dic5C22×Dic312yes—
361D20—28no—
362D20—48no—
363C4×D5—28no—
364C4×D5—128no—
365C22×D4—51no—
366C22×D4—52no—
367C23×C4—51no—
368C23×C4—152no—
369#121—58no—
370C4×D4—304no—
371#117—304no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—151no—
374C2×C4⋊C4—151no—
375C2×C4⋊C4—151no—
376C2×C4⋊C4—152no—
377C2×C4⋊C4D3×D511yescharacteristic
378C2×C22⋊C4—52no—
379C2×C22⋊C4—151no—
380C2×C22⋊C4—151no—
381C2×C22⋊C4—152no—
382C2×C22⋊C4—152no—
383C2×C22⋊C4—152no—
384C2×C22⋊C4—152no—
385C2×C42—31no—
386C2×C42—32no—
387C2×C42—151no—
388#95—32no—
389#95—152no—
390#95—152no—
391C2×C3×C5C2×C4×D411yescharacteristic
392C2×C3×C5#51811yescharacteristic
393C2×C3×C5#52011yescharacteristic
394C2×C3×C5#53511yescharacteristic
395C2×C3×C5#53711yescharacteristic
396C2×C3×C5#54011yescharacteristic
397C2×C3×C5#54211yescharacteristic
398C3×D5—42no—
399C3×D5—44no—
400C23×C3—54no—
401C23×C3—52no—
402C23×C3—101no—
403C23×C3—102no—
404C23×C3—101no—
405C23×C3—102no—
406C23×C3—102no—
407C23×C3—101no—
408C23×C3—101no—
409C23×C3—102no—
410C23×C3—101no—
411C23×C3—102no—
412C23×C3—102no—
413C23×C3—101no—
414C23×C3C23×D511yescharacteristic
415C3×D4—108no—
416C3×D4—208no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—52no—
420C2×C3×C4—101no—
421C2×C3×C4—101no—
422C2×C3×C4—108no—
423C2×C3×C4C2×C4×D514yes—
424C2×C3×C4C2×C4×D512yes—
425C2×C3×C4C2×D2012yes—
426C2×C3×C4#70912yes—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—101no—
435C2×Dic3—101no—
436C2×Dic3—102no—
437C2×Dic3—101no—
438C2×Dic3—101no—
439C2×Dic3—101no—
440C2×Dic3—102no—
441C2×Dic3—101no—
442C2×Dic3—102no—
443C2×Dic3—208no—
444C2×Dic3C2×D2012yes—
445C2×Dic3C2×D2012yes—
446C22×C5C2×D3×D411yescharacteristic
447C22×C5C2×#23711yescharacteristic
448C22×C5#100411yescharacteristic
449C22×C5#100411yescharacteristic
450C22×C5#100911yescharacteristic
451C22×C5#101011yescharacteristic
452C22×C5C23×Dic311yescharacteristic
453C2×D5—22no—
454C2×D5—24no—
455C2×D5—22no—
456C2×D5—24no—
457C2×D5—22no—
458C2×D5—24no—
459C2×D5—41no—
460C2×D5—42no—
461C2×D5—42no—
462C2×D5—41no—
463C2×D5—41no—
464C2×D5—42no—
465C2×D5—41no—
466C2×D5—42no—
467C4×C5—24no—
468C4×C5—62no—
469C4×C5—62no—
470C4×C5—122no—
471C4×C5D4×Dic314yes—
472Dic5—22no—
473Dic5—122no—
474Dic5—121no—
475Dic5—121no—
476C24—52no—
477C24—51no—
478C2×D4—54no—
479C2×D4—58no—
480C2×D4—102no—
481C2×D4—102no—
482C2×D4—202no—
483C2×D4—202no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—32no—
487C22×C4—51no—
488C22×C4—54no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—151no—
492C22×C4—152no—
493C22×C4—151no—
494C22×C4—304no—
495C22×C4—602no—
496C22×C4—602no—
497C22×C4C2×D3×D512yes—
498C22×C4C2×D3×D511yescharacteristic
499C4⋊C4—62no—
500C4⋊C4—302no—
501C4⋊C4—302no—
502C4⋊C4—302no—
503C4⋊C4—602no—
504C4⋊C4D5×Dic314yescomplemented
505C22⋊C4—58no—
506C22⋊C4—302no—
507C22⋊C4—302no—
508C22⋊C4—304no—
509C22⋊C4—602no—
510C22⋊C4—602no—
511C22⋊C4—602no—
512C42—34no—
513C42—122no—
514C42—302no—
515C3×C5#232411yescharacteristic, complemented, Hall
516C22×C3—102no—
517C22×C3—104no—
518C22×C3—102no—
519C22×C3—104no—
520C22×C3—104no—
521C22×C3—102no—
522C22×C3—201no—
523C22×C3—202no—
524C22×C3—202no—
525C22×C3—201no—
526C22×C3—201no—
527C22×C3—202no—
528C22×C3—201no—
529C22×C3—202no—
530C22×C3C22×C4×D511yescharacteristic
531C22×C3C22×D2011yescharacteristic
532C22×C3C2×#70911yescharacteristic
533C22×C3#414411yescharacteristic
534C22×C3#414711yescharacteristic
535C22×C3#414911yescharacteristic
536C22×C3#414911yescharacteristic
537C3×C4—24no—
538C3×C4—102no—
539C3×C4C4×D2014yes—
540Dic3—22no—
541Dic3—22no—
542Dic3—42no—
543Dic3—201no—
544Dic3—201no—
545Dic3—202no—
546C2×C5#560311yescharacteristic
547C2×C5#560711yescharacteristic
548C2×C5#560811yescharacteristic
549C2×C5#561911yescharacteristic
550C2×C5C2×D4×Dic311yescharacteristic
551C2×C5#579811yescharacteristic
552C2×C5#582711yescharacteristic
553D5—42no—
554D5—44no—
555C23—54no—
556C23—52no—
557C23—101no—
558C23—102no—
559C23—101no—
560C23—102no—
561C23—102no—
562C23—101no—
563C23—102no—
564C23—101no—
565C23—102no—
566C23—101no—
567C23—101no—
568C23—102no—
569C23C22×D3×D511yescharacteristic
570D4—108no—
571D4—208no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—52no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—101no—
585C2×C4—101no—
586C2×C4—108no—
587C2×C4—301no—
588C2×C4—301no—
589C2×C4—302no—
590C2×C4—301no—
591C2×C4—301no—
592C2×C4—302no—
593C2×C4—301no—
594C2×C4—301no—
595C2×C4—302no—
596C2×C4—608no—
597C2×C4#676512yes—
598C2×C4D3×D2012yes—
599C2×C4C2×D5×Dic314yes—
600C2×C4C2×D5×Dic312yes—
601C2×C3—202no—
602C2×C3—204no—
603C2×C3C2×C4×D2011yescharacteristic
604C2×C3#6584411yescharacteristic
605C2×C3#6585911yescharacteristic
606C2×C3#6587911yescharacteristic
607C2×C3#6589611yescharacteristic
608C2×C3#6590811yescharacteristic
609C2×C3#6591611yescharacteristic
610C5#8263311yescharacteristic, complemented, Sylow
611C22—102no—
612C22—104no—
613C22—102no—
614C22—104no—
615C22—104no—
616C22—102no—
617C22—202no—
618C22—201no—
619C22—201no—
620C22—202no—
621C22—201no—
622C22—202no—
623C22—201no—
624C22—202no—
625C22C2×#676511yescharacteristic
626C22C2×D3×D2011yescharacteristic
627C22#9202511yescharacteristic
628C22#9202611yescharacteristic
629C22#9202811yescharacteristic
630C22#9203511yescharacteristic
631C22C22×D5×Dic311yescharacteristic
632C4—24no—
633C4—62no—
634C4—62no—
635C4—102no—
636C4—122no—
637C4—602no—
638C4—601no—
639C4—601no—
640C4Dic3×D2014yes—
641C3#1060549411yescharacteristic, complemented, Sylow
642C2—202no—
643C2—204no—
644C2#1175344111yescharacteristic
645C2#1175364411yescharacteristic
646C2#1175364711yescharacteristic
647C2#1175388211yescharacteristic
648C2C2×Dic3×D2011yescharacteristic
649C2#1175491011yescharacteristic
650C2#1175521711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×C3×#4041C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2324
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×D20C2semidirect
3. C2×#91357C2semidirect
4. #11751085C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
6. #11748900C2semidirect
7. #11748896C2semidirect
8. #11748796C2semidirect
9. #11748785C2semidirect
10. #11748770C2semidirect
11. #11748759C2semidirect
25. C3×#4041C4semidirect
67. C2×C3×D20C2×C4semidirect
70. C2×C3×C4×D5C2×C4semidirect
74. C3×C5×C4⋊C4C2×C4semidirect
80. C4×C5×Dic3D4semidirect
81. C3×#685C2×C4semidirect
98. Dic3×C4⋊C4D5semidirect
121. #4041Dic3semidirect
216. C2×D20C2×Dic3semidirect
222. C2×C4×D5C2×Dic3semidirect
224. C5×C4⋊C4C2×Dic3semidirect
233. #685C2×Dic3semidirect
303. C3×C4⋊C4C4×D5semidirect
318. C4×Dic3D20semidirect
504. C4⋊C4D5×Dic3semidirect
515. C3×C5#2324semidirect
610. C5#82633semidirect
641. C3#10605494semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40412
C2×Dic3×D202
C2×#913572
#117510852
C5×Dic3×C4⋊C42
#117489002
#117488962
#117487962
#117487852
#117487702
#117487592
#106054943
#826335