Finite Group Databasebeta

← #14326084

Subgroups — #14326084

Order 1920.

1856 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6773C211yesmaximal, characteristic
2C2×#91782C211yesmaximal, characteristic, complemented
3C2×#91428C211yesmaximal, characteristic, complemented
4C5×#4973C211yesmaximal, characteristic, complemented
5C3×#65276C211yesmaximal, characteristic, complemented
6#11748917C212yesmaximal, complemented
7#11748889C212yesmaximal, complemented
8#11748888C212yesmaximal, complemented
9C4×#6665C211yesmaximal, characteristic, complemented
10#11748700C212yesmaximal, complemented
11#11748607C211yesmaximal, characteristic, complemented
12#10608344—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×C5×Dic6C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C22×#1202C2212yes—
18C22×#1200C2212yes—
19C22×#1200C2212yes—
20C2×#6773—24no—
21C2×#6773—24no—
22C2×#6773C418yescomplemented
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×#6714C2212yes—
25C2×C4×Dic15C2211yescharacteristic
26#91782—24no—
27C2×C4×C5×Dic3C2211yescharacteristic
28C2×C3×#685C2212yes—
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×#6667C2212yes—
31C2×#6665C2211yescharacteristic
32C2×#6665C2211yescharacteristic
33#91428—24no—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C3×#3963C2212yes—
37#90994C2211yescharacteristic
38#90994C2212yes—
39#90994C2211yescharacteristic
40#90994C2212yes—
41#74008—51nomaximal, Hall
42C22×#713—31no—
43C2×#4039—31no—
44C2×C4×D20—31no—
45C5×#391—31no—
46#65519—31no—
47#65276D311yescharacteristic, complemented
48C4×#685—31no—
49#64954—32no—
50#64954—32no—
51#64953—32no—
52D5×#95—32no—
53C23×D15C2312yes—
54C23×C3×D5C2311yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C2×D60C2×C414yes—
60C2×D60D414yes—
61C2×D60D414yes—
62C2×C4×D15—24no—
63C2×C4×D15—24no—
64C2×C4×D15—42no—
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2312yes—
67C22×C5×Dic3C2311yescharacteristic
68C2×C5×Dic6—22no—
69C2×C5×Dic6—22no—
70C2×C5×Dic6C2×C414yescomplemented
71C22×C3×Dic5C2312yes—
72C22×C3×Dic5C2311yescharacteristic
73C2×C3×C4×D5—22no—
74C2×C3×C4×D5—22no—
75C2×C3×C4×D5C2×C414yescomplemented
76C2×#1202—24no—
77C2×#1202—24no—
78C2×#1202C2×C418yescomplemented
79C2×#1200—24no—
80C2×#1200—24no—
81C2×#1200—24no—
82C2×#1200—24no—
83C2×#1200—42no—
84C2×#1200C2×C418yescomplemented
85#6773—232no—
86C3×C5×C4⋊C4—22no—
87#6714—24no—
88C4×Dic15—22no—
89C4×C5×Dic3—22no—
90C3×#685—42no—
91C3×C4⋊Dic5—22no—
92#6667—24no—
93#6665—22no—
94#6665—22no—
95C22×#235—51no—
96C2×#893—51no—
97C2×#893—51no—
98C3×#380—51no—
99C4×#217—51no—
100#4973—51no—
101#4973D511yescharacteristic, complemented
102#4956—52no—
103#4956—52no—
104#4664—52no—
105D3×#95—52no—
106C22×C5×Q8—31no—
107C2×#713—38no—
108C2×#713—64no—
109C2×#713—64no—
110C22×D20—31no—
111C22×D20—32no—
112C22×C4×D5—32no—
113C22×C4×D5—32no—
114C22×C4×D5—32no—
115C22×C4×D5C2×D311yescharacteristic
116C2×C5×C4⋊C4C2×D311yescharacteristic
117C2×C42×C5—31no—
118C2×#685—31no—
119C2×#685—31no—
120C2×#685—32no—
121C2×#685—32no—
122C2×#685C2×D312yes—
123C2×C4⋊Dic5C2×D311yescharacteristic
124C2×C4×Dic5—31no—
125#4039—64no—
126C4×D20—64no—
127C5×#95—32no—
128C5×#95—32no—
129#3963—31no—
130#3963—32no—
131#3963—31no—
132#3963—32no—
133#3963C2×D312yes—
134#2339—151noSylow
135C23×C3×C5C2411yescharacteristic
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15C22×C414yes—
141C22×D15C2×D414yes—
142C22×D15C2×D414yes—
143C22×C3×D5—21no—
144C22×C3×D5—21no—
145C22×C3×D5—21no—
146C22×C3×D5—21no—
147C22×C3×D5—21no—
148C22×C3×D5—21no—
149C22×C3×D5C22×C412yes—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5C22×C412yes—
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5C2×D412yes—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15—21no—
161C2×Dic15#4112yescomplemented
162D60C22⋊C4116yescomplemented
163C4×D15—48no—
164C2×C5×Dic3—21no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—21no—
168C2×C5×Dic3—21no—
169C2×C5×Dic3—21no—
170C2×C5×Dic3—21no—
171C2×C5×Dic3C22×C414yes—
172C2×C5×Dic3#4112yescomplemented
173C5×Dic6—28no—
174C2×C3×Dic5—21no—
175C2×C3×Dic5—21no—
176C2×C3×Dic5—22no—
177C2×C3×Dic5—22no—
178C2×C3×Dic5—22no—
179C2×C3×Dic5C22×C412yes—
180C3×C4×D5—28no—
181#1202—216no—
182#1200—216no—
183#1200—48no—
184C23×C3×C4—51no—
185C22×#67—52no—
186C2×#235—58no—
187C2×#235—104no—
188C2×#235—104no—
189C22×D12—51no—
190C22×C4×D3—52no—
191C22×C4×D3—52no—
192C22×C4×D3—52no—
193C22×Dic6C2×D511yescharacteristic
194C2×C3×C4⋊C4—51no—
195C2×C3×C4⋊C4C2×D511yescharacteristic
196C2×C3×C22⋊C4—52no—
197C2×#217—51no—
198C2×#217—51no—
199C2×(C2×D3)⋊C4—52no—
200C2×(C2×D3)⋊C4—52no—
201C2×C4×Dic3—51no—
202C2×C4×Dic3C2×D511yescharacteristic
203#893—104no—
204#893—104no—
205C3×#95—52no—
206C3⋊#95—51no—
207C3⋊#95—52no—
208C3⋊#95—51no—
209C3⋊#95—52no—
210C3⋊#95C2×D512yes—
211C3⋊#95C2×D512yes—
212C23×D5—32no—
213C23×D5C22×D311yescharacteristic
214C2×C5×Q8—34no—
215C2×C5×Q8—62no—
216C2×C5×Q8—62no—
217C22×C4×C5—31no—
218C22×C4×C5—32no—
219C22×C4×C5—31no—
220C22×C4×C5C22×D312yes—
221C22×C4×C5C22×D311yescharacteristic
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5C22×D312yes—
225C22×Dic5C22×D311yescharacteristic
226#713—632no—
227C2×D20—34no—
228C2×D20—34no—
229C2×D20—34no—
230C2×D20—38no—
231C2×D20—64no—
232C2×D20—64no—
233C2×C4×D5—22no—
234C2×C4×D5—22no—
235C2×C4×D5—38no—
236C2×C4×D5—64no—
237C2×C4×D5—64no—
238C2×C4×D5—64no—
239C2×C4×D5—64no—
240C2×C4×D5—64no—
241C2×C4×D5—64no—
242C2×C4×D5—122no—
243C2×C4×D5—122no—
244C2×C4×D5C4×D314yescomplemented
245C5×C4⋊C4—22no—
246C42×C5—62no—
247#685—42no—
248#685—64no—
249#685—62no—
250#685—64no—
251#685—62no—
252C4⋊Dic5—22no—
253C4×Dic5—62no—
254C22×#41—151no—
255C2×C4×D4—151no—
256C2×C4×D4—151no—
257#391—31no—
258#391—151no—
259#386—152no—
260#386—152no—
261#381—152no—
262#380—51no—
263C4×C22⋊C4—151no—
264C2×#95—152no—
265C22×C3×C5C23×C411yescharacteristic
266C22×C3×C5C22×D411yescharacteristic
267C22×C3×C5C22×D411yescharacteristic
268C22×C3×C5C2×#4111yescharacteristic
269C22×C3×C5C2×#4111yescharacteristic
270C22×C3×C5H(5,𝔽2)11yescharacteristic
271C22×C3×C52−1+411yescharacteristic
272C2×D15—24no—
273C2×D15—24no—
274C2×D15—24no—
275C2×D15—24no—
276C2×D15—24no—
277C2×D15—24no—
278C2×D15C2×C22⋊C418yes—
279C2×C3×D5—22no—
280C2×C3×D5—22no—
281C2×C3×D5—22no—
282C2×C3×D5—41no—
283C2×C3×D5—41no—
284C2×C3×D5—41no—
285C2×C3×D5—41no—
286C3×C4×C5—42no—
287C3×C4×C5C2×C22⋊C414yes—
288Dic15—22no—
289Dic15—41no—
290C3×Dic5—22no—
291C3×Dic5—42no—
292C5×Dic3—24no—
293C5×Dic3—22no—
294C5×Dic3—41no—
295C24×C3—51no—
296C23×D3—52no—
297C22×C3×C4—51no—
298C22×C3×C4—54no—
299C22×C3×C4—52no—
300C22×C3×C4—102no—
301C22×C3×C4—102no—
302C22×C3×C4C22×D512yes—
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—58no—
305C2×#67—104no—
306C2×#67—104no—
307C22×Dic3—51no—
308C22×Dic3—51no—
309C22×Dic3C22×D511yescharacteristic
310C22×Dic3C22×D512yes—
311C22×Dic3C22×D511yescharacteristic
312#235—1032no—
313C2×D12—54no—
314C2×D12—54no—
315C2×D12—54no—
316C2×C4×D3—58no—
317C2×C4×D3—104no—
318C2×C4×D3—104no—
319C2×C4×D3—104no—
320C2×C4×D3—104no—
321C2×C4×D3—104no—
322C2×C4×D3—104no—
323C2×C4×D3—202no—
324C2×C4×D3—202no—
325C2×Dic6—22no—
326C2×Dic6—22no—
327C2×Dic6C4×D514yescomplemented
328C3×C4⋊C4—22no—
329C3×C4⋊C4—102no—
330C3×C22⋊C4—202no—
331#217—102no—
332#217—102no—
333(C2×D3)⋊C4—104no—
334(C2×D3)⋊C4—104no—
335C4×Dic3—22no—
336C4×Dic3—102no—
337C23×C5C23×D311yescharacteristic
338C22×D5—21no—
339C22×D5—21no—
340C22×D5—21no—
341C22×D5—21no—
342C22×D5—21no—
343C22×D5—21no—
344C22×D5—34no—
345C22×D5—34no—
346C22×D5—34no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5—62no—
351C22×D5C2×C4×D312yes—
352C5×Q8—68no—
353C2×C4×C5—22no—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—34no—
357C2×C4×C5—32no—
358C2×C4×C5—61no—
359C2×C4×C5—62no—
360C2×C4×C5—62no—
361C2×C4×C5—61no—
362C2×C4×C5—61no—
363C2×C4×C5—61no—
364C2×C4×C5—61no—
365C2×C4×C5C2×C4×D312yes—
366C2×C4×C5D3×D412yes—
367C2×C4×C5D3×D412yes—
368C2×Dic5—21no—
369C2×Dic5—21no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—22no—
373C2×Dic5—32no—
374C2×Dic5—61no—
375C2×Dic5—61no—
376C2×Dic5—61no—
377C2×Dic5—61no—
378C2×Dic5—61no—
379C2×Dic5C2×C4×D312yes—
380D20—316no—
381D20—616no—
382C4×D5—28no—
383C4×D5—616no—
384C4×D5—128no—
385C4×D5—128no—
386C2×#41—158no—
387C2×#41—304no—
388C2×#41—304no—
389C22×Q8—31no—
390C22×D4—152no—
391C22×D4—151no—
392C23×C4—51no—
393C23×C4—152no—
394C23×C4—152no—
395C23×C4—152no—
396C4×D4—304no—
397C4×D4—304no—
398C2×C4⋊C4—51no—
399C2×C4⋊C4D3×D511yescharacteristic
400C2×C22⋊C4—52no—
401C2×C22⋊C4—152no—
402C2×C22⋊C4—151no—
403C2×C22⋊C4—151no—
404C2×C22⋊C4—152no—
405C2×C42—31no—
406C2×C42—151no—
407#95—32no—
408#95—32no—
409#95—52no—
410#95—151no—
411#95—152no—
412#95—151no—
413#95—152no—
414C2×C3×C5C22×C22⋊C411yescharacteristic
415C2×C3×C5C4×#4111yescharacteristic
416C2×C3×C5#52011yescharacteristic
417C2×C3×C5#54611yescharacteristic
418C2×C3×C5#54611yescharacteristic
419C2×C3×C5#54711yescharacteristic
420C2×C3×C5#54711yescharacteristic
421D15—28no—
422C3×D5—42no—
423C23×C3—52no—
424C23×C3—101no—
425C23×C3—101no—
426C23×C3—101no—
427C23×C3—101no—
428C23×C3—101no—
429C23×C3—101no—
430C23×C3C23×D511yescharacteristic
431C22×D3—54no—
432C22×D3—54no—
433C22×D3—54no—
434C22×D3—102no—
435C22×D3—102no—
436C22×D3—102no—
437C22×D3—102no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—52no—
442C2×C3×C4—101no—
443C2×C3×C4—101no—
444C2×C3×C4—108no—
445C2×C3×C4—102no—
446C2×C3×C4—102no—
447C2×C3×C4—102no—
448C2×C3×C4C2×C4×D512yes—
449C2×C3×C4D4×D512yes—
450C2×C3×C4D4×D512yes—
451#67—1016no—
452C2×Dic3—21no—
453C2×Dic3—22no—
454C2×Dic3—22no—
455C2×Dic3—21no—
456C2×Dic3—21no—
457C2×Dic3—21no—
458C2×Dic3—21no—
459C2×Dic3—52no—
460C2×Dic3—101no—
461C2×Dic3—101no—
462C2×Dic3—101no—
463C2×Dic3—101no—
464C2×Dic3—101no—
465C2×Dic3C2×C4×D514yes—
466C2×Dic3#71312yescomplemented
467D12—516no—
468C4×D3—1016no—
469C4×D3—208no—
470C4×D3—208no—
471Dic6—28no—
472C22×C5C22×C4×D311yescharacteristic
473C22×C5C2×D3×D411yescharacteristic
474C22×C5C2×D3×D411yescharacteristic
475C22×C5#100411yescharacteristic
476C22×C5#100711yescharacteristic
477C22×C5D3×#4111yescharacteristic
478C22×C5D3×#4111yescharacteristic
479C2×D5—22no—
480C2×D5—22no—
481C2×D5—22no—
482C2×D5—38no—
483C2×D5—41no—
484C2×D5—41no—
485C2×D5—41no—
486C2×D5—41no—
487C2×D5—64no—
488C2×D5—64no—
489C2×D5—64no—
490C2×D5—64no—
491C2×D5—64no—
492C2×D5—64no—
493C4×C5—42no—
494C4×C5—64no—
495C4×C5—62no—
496C4×C5—121no—
497C4×C5D3×C22⋊C414yes—
498Dic5—22no—
499Dic5—42no—
500Dic5—62no—
501Dic5—121no—
502C24—51no—
503C24—152no—
504#41—3032no—
505C2×Q8—34no—
506C2×Q8—62no—
507C2×Q8—62no—
508C2×D4—158no—
509C2×D4—154no—
510C2×D4—154no—
511C2×D4—154no—
512C2×D4—304no—
513C2×D4—304no—
514C22×C4—32no—
515C22×C4—31no—
516C22×C4—31no—
517C22×C4—51no—
518C22×C4—54no—
519C22×C4—52no—
520C22×C4—102no—
521C22×C4—102no—
522C22×C4—158no—
523C22×C4—151no—
524C22×C4—151no—
525C22×C4—304no—
526C22×C4—304no—
527C22×C4—304no—
528C22×C4—304no—
529C22×C4—304no—
530C22×C4—304no—
531C22×C4—602no—
532C22×C4—602no—
533C22×C4C2×D3×D512yes—
534C22×C4C2×D3×D511yescharacteristic
535C4⋊C4—22no—
536C4⋊C4—102no—
537C22⋊C4—202no—
538C22⋊C4—304no—
539C22⋊C4—302no—
540C22⋊C4—304no—
541C22⋊C4—302no—
542C42—62no—
543C42—302no—
544C3×C5#233911yescharacteristic, complemented, Hall
545C22×C3—102no—
546C22×C3—102no—
547C22×C3—102no—
548C22×C3—201no—
549C22×C3—201no—
550C22×C3—201no—
551C22×C3—201no—
552C22×C3C22×C4×D511yescharacteristic
553C22×C3C2×D4×D511yescharacteristic
554C22×C3C2×D4×D511yescharacteristic
555C22×C3C2×#71311yescharacteristic
556C22×C3#414711yescharacteristic
557C22×C3D5×#4111yescharacteristic
558C22×C3#414911yescharacteristic
559C2×D3—58no—
560C2×D3—104no—
561C2×D3—104no—
562C2×D3—104no—
563C2×D3—104no—
564C2×D3—104no—
565C2×D3—104no—
566C3×C4—42no—
567C3×C4—102no—
568C3×C4—202no—
569C3×C4D5×C22⋊C414yes—
570Dic3—24no—
571Dic3—22no—
572Dic3—41no—
573Dic3—102no—
574Dic3—201no—
575C2×C5C2×D3×C22⋊C411yescharacteristic
576C2×C5#550411yescharacteristic
577C2×C5#552211yescharacteristic
578C2×C5#561211yescharacteristic
579C2×C5#561211yescharacteristic
580C2×C5#563111yescharacteristic
581C2×C5#563111yescharacteristic
582D5—42no—
583D5—68no—
584C23—52no—
585C23—101no—
586C23—101no—
587C23—101no—
588C23—101no—
589C23—101no—
590C23—101no—
591C23—154no—
592C23—154no—
593C23—154no—
594C23—302no—
595C23—302no—
596C23—302no—
597C23—302no—
598C23C22×D3×D511yescharacteristic
599Q8—68no—
600D4—1516no—
601D4—3016no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—34no—
606C2×C4—32no—
607C2×C4—52no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—101no—
616C2×C4—101no—
617C2×C4—108no—
618C2×C4—102no—
619C2×C4—102no—
620C2×C4—102no—
621C2×C4—152no—
622C2×C4—3016no—
623C2×C4—301no—
624C2×C4—301no—
625C2×C4—301no—
626C2×C4—301no—
627C2×C4—301no—
628C2×C4—608no—
629C2×C4—608no—
630C2×C4C4×D3×D512yes—
631C2×C4#677712yes—
632C2×C4#679012yes—
633C2×C3—202no—
634C2×C3C2×D5×C22⋊C411yescharacteristic
635C2×C3C4×#71311yescharacteristic
636C2×C3#6590811yescharacteristic
637C2×C3#6596411yescharacteristic
638C2×C3#6596511yescharacteristic
639C2×C3#6601011yescharacteristic
640C2×C3#6606011yescharacteristic
641D3—108no—
642C5#7400811yescharacteristic, complemented, Sylow
643C22—102no—
644C22—102no—
645C22—102no—
646C22—158no—
647C22—201no—
648C22—201no—
649C22—201no—
650C22—201no—
651C22—304no—
652C22—304no—
653C22—304no—
654C22—304no—
655C22—304no—
656C22—304no—
657C22C2×C4×D3×D511yescharacteristic
658C22C2×#677711yescharacteristic
659C22#9202411yescharacteristic
660C22#9202611yescharacteristic
661C22#9202911yescharacteristic
662C22D3×#71311yescharacteristic
663C22C2×#679011yescharacteristic
664C4—42no—
665C4—64no—
666C4—62no—
667C4—102no—
668C4—121no—
669C4—202no—
670C4—302no—
671C4—601no—
672C4#9147314yes—
673C3#1060834411yescharacteristic, complemented, Sylow
674C2—202no—
675C2—308no—
676C2#1175333611yescharacteristic
677C2#1175340311yescharacteristic
678C2#1175370411yescharacteristic
679C2#1175388811yescharacteristic
680C2C2×#9147311yescharacteristic
681C2#1175495811yescharacteristic
682C2#1175523911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6773C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2339
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×#91782C2semidirect
3. C2×#91428C2semidirect
4. C5×#4973C2semidirect
5. C3×#65276C2semidirect
6. #11748917C2semidirect
7. #11748889C2semidirect
8. #11748888C2semidirect
9. C4×#6665C2semidirect
10. #11748700C2semidirect
11. #11748607C2semidirect
22. C2×#6773C4semidirect
47. #65276D3semidirect
70. C2×C5×Dic6C2×C4semidirect
75. C2×C3×C4×D5C2×C4semidirect
78. C2×#1202C2×C4semidirect
84. C2×#1200C2×C4semidirect
101. #4973D5semidirect
161. C2×Dic15#41semidirect
162. D60C22⋊C4semidirect
172. C2×C5×Dic3#41semidirect
244. C2×C4×D5C4×D3semidirect
327. C2×Dic6C4×D5semidirect
466. C2×Dic3#713semidirect
544. C3×C5#2339semidirect
642. C5#74008semidirect
673. C3#10608344semidirect

Maximal subgroups

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

subgroupindex
C22×#67732
C2×#917822
C2×#914282
C5×#49732
C3×#652762
#117489172
#117488892
#117488882
C4×#66652
#117487002
#117486072
#106083443
#740085