Finite Group Databasebeta

← #14331643

Subgroups — #14331643

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6769C211yesmaximal, characteristic, complemented
2C2×C5×#67⋊C4C211yesmaximal, characteristic, complemented
3C2×#91556C212yesmaximal, complemented
4#11751061C211yesmaximal, characteristic, complemented
5C3×#65236C211yesmaximal, characteristic, complemented
6#11749523C212yesmaximal, complemented
7#11748881C212yesmaximal, complemented
8#11748790C211yesmaximal, characteristic, complemented
9#11748756C212yesmaximal, complemented
10#11748719C211yesmaximal, characteristic, complemented
11#11748572C211yesmaximal, characteristic, complemented
12#10614131—31nomaximal, Hall
13C22×#1220C2212yes—
14C22×D60C2211yescharacteristic
15C22×C5×#67C2211yescharacteristic
16C23×C5×Dic3C2211yescharacteristic
17C22×C4×C5×D3C2211yescharacteristic
18C22×C3×Dic10C2211yescharacteristic
19C22×#1203C2212yes—
20C22×#1200C2212yes—
21C2×#6769—24no—
22C2×#6769—28no—
23C2×C3×C5×C22⋊C4C2211yescharacteristic
24C2×#6714C2212yes—
25C2×C5×(C2×D3)⋊C4C2211yescharacteristic
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C4×C5×Dic3C2211yescharacteristic
28C5×#67⋊C4—24no—
29C2×C3×C22⋊Dic5C2212yes—
30#91556—28no—
31C2×#6669C2212yes—
32C2×#6667C2212yes—
33C2×#6667C2212yes—
34C2×(C2×D3)⋊Dic5C2212yes—
35C2×Dic3×Dic5C2212yes—
36#91115C2211yescharacteristic
37C3×#3963C2212yes—
38C3×#3963C2211yescharacteristic
39#90994C2212yes—
40#76777—51nomaximal, Hall
41C22×#709—31no—
42C2×C4×C5×D4—31no—
43C2×C4×C22⋊D5—32no—
44#65512—32no—
45C4×C22⋊Dic5—31no—
46#65280—31no—
47#65268—31no—
48#65249—31no—
49#65248—32no—
50#65236D311yescharacteristic, complemented
51C4×#685—32no—
52C24×C3×C5C2311yescharacteristic
53C23×D15C2312yes—
54C23×C5×D3C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—24no—
58C2×#1220—24no—
59C2×#1220—24no—
60C22×Dic15C2312yes—
61C2×D60—24no—
62C2×D60—22no—
63C2×C5×#67—22no—
64C2×C5×#67—22no—
65C2×C5×#67—22no—
66C22×C5×Dic3—22no—
67C22×C5×Dic3—24no—
68C22×C5×Dic3C2311yescharacteristic
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2311yescharacteristic
71C2×C4×C5×D3—24no—
72C2×C4×C5×D3D414yescomplemented
73C22×C3×Dic5C2312yes—
74C22×C3×Dic5C2312yes—
75C2×C3×Dic10—22no—
76C2×C3×Dic10—24no—
77C2×#1203—24no—
78C2×#1203—24no—
79C2×#1203—24no—
80C2×#1200—24no—
81C2×#1200—24no—
82C2×#1200D418yescomplemented
83#6769—232no—
84C3×C5×C22⋊C4—22no—
85#6714—24no—
86C5×(C2×D3)⋊C4—22no—
87C5×Dic3⋊C4—22no—
88C4×C5×Dic3—22no—
89C3×C22⋊Dic5—24no—
90#6669—24no—
91#6667—24no—
92#6667—24no—
93(C2×D3)⋊Dic5—24no—
94Dic3×Dic5—24no—
95C22×#239—51no—
96C2×#67⋊C4—52no—
97C2×#67⋊C4D511yescharacteristic, complemented
98C3×#393—51no—
99#5230—51no—
100#4960—52no—
101#4960—51no—
102#4672—52no—
103#4672—51no—
104#4666—51no—
105#4666—52no—
106C22×C5×D4—31no—
107C23×C4×C5—31no—
108C23×C4×C5—31no—
109C22×C22⋊D5—32no—
110C22×C22⋊D5—32no—
111C2×#709—68no—
112C2×#709—64no—
113C22×D20—31no—
114C22×C4×D5—32no—
115C22×Dic10C2×D311yescharacteristic
116C4×C5×D4—64no—
117C2×C5×C4⋊C4—31no—
118C2×C5×C22⋊C4—31no—
119C2×C5×C22⋊C4C2×D311yescharacteristic
120C2×C42×C5—31no—
121C2×C22⋊Dic5—32no—
122C2×C22⋊Dic5C2×D312yes—
123C4×C22⋊D5—68no—
124C2×#685—32no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#683—32no—
128C2×C4×Dic5—32no—
129#3963—31no—
130#3963—32no—
131#3963C2×D311yescharacteristic
132#3963C2×D312yes—
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—21no—
136C23×C3×C5—21no—
137C23×C3×C5—22no—
138C23×C3×C5#4112yescomplemented
139C23×C3×C5C2411yescharacteristic
140C22×D15—22no—
141C22×D15—22no—
142C22×D15—22no—
143C22×D15—22no—
144C22×D15—22no—
145C22×D15—22no—
146C22×D15C2×D414yes—
147C22×C5×D3—22no—
148C22×C5×D3—22no—
149C22×C5×D3—21no—
150C22×C5×D3—21no—
151C22×C5×D3C2×D412yes—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5C2×D412yes—
156#1220—216no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15#4114yescomplemented
160D60—28no—
161C5×#67—28no—
162C2×C5×Dic3—21no—
163C2×C5×Dic3—21no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—21no—
166C2×C5×Dic3—44no—
167C2×C5×Dic3C2×D414yes—
168C2×C5×Dic3C2×D412yes—
169C2×C5×Dic3#4112yescomplemented
170C4×C5×D3—28no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5—22no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—22no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5C2×D414yes—
177C3×Dic10—28no—
178#1203—216no—
179#1200—216no—
180C22×C3×Q8—51no—
181C22×#67—52no—
182C22×#67C2×D511yescharacteristic
183C23×Dic3C2×D511yescharacteristic
184C2×#239—108no—
185C2×#239—104no—
186C22×D12—52no—
187C22×D12—51no—
188C22×C4×D3—52no—
189C22×C4×D3C2×D511yescharacteristic
190C2×C3×C22⋊C4—52no—
191C2×C3×C22⋊C4C2×D511yescharacteristic
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4C2×D511yescharacteristic
197C2×Dic3⋊C4—52no—
198C2×Dic3⋊C4C2×D511yescharacteristic
199C2×C4×Dic3—52no—
200C2×C4×Dic3C2×D511yescharacteristic
201#67⋊C4—24no—
202#67⋊C4—108no—
203C3×#95—52no—
204C3×#95—51no—
205C3⋊#95—51no—
206C3⋊#95—52no—
207C24×C5—31no—
208C24×C5C22×D311yescharacteristic
209C23×D5—32no—
210C2×C5×D4—62no—
211C2×C5×D4—62no—
212C2×C5×D4—62no—
213C22×C4×C5—31no—
214C22×C4×C5—31no—
215C22×C4×C5—34no—
216C22×C4×C5—31no—
217C22×C4×C5—64no—
218C22×C4×C5—62no—
219C22×C4×C5—64no—
220C22×C4×C5C22×D311yescharacteristic
221C22×C4×C5C22×D311yescharacteristic
222C2×C22⋊D5—64no—
223C2×C22⋊D5—64no—
224C2×C22⋊D5—64no—
225C2×C22⋊D5—64no—
226C2×C22⋊D5—64no—
227C2×C22⋊D5—64no—
228C22×Dic5—32no—
229C22×Dic5C22×D312yes—
230C22×Dic5C22×D312yes—
231#709—632no—
232C2×D20—64no—
233C2×D20—62no—
234C2×C4×D5—38no—
235C2×C4×D5—64no—
236C2×C4×D5—64no—
237C2×Dic10—22no—
238C2×Dic10—24no—
239C5×C4⋊C4—62no—
240C5×C22⋊C4—22no—
241C5×C22⋊C4—62no—
242C42×C5—62no—
243C22⋊Dic5—24no—
244C22⋊Dic5—64no—
245#685—64no—
246#685—64no—
247#685—64no—
248#683—64no—
249C4×Dic5—64no—
250C22×#41—151no—
251C2×C4×D4—31no—
252C2×C4×D4—152no—
253#395—151no—
254#394—152no—
255#394—151no—
256#393—51no—
257#392—152no—
258#392—151no—
259C4×C22⋊C4—152no—
260C4×C22⋊C4—151no—
261C22×C3×C5—22no—
262C22×C3×C5—24no—
263C22×C3×C5—41no—
264C22×C3×C5—42no—
265C22×C3×C5—41no—
266C22×C3×C5C22×D412yes—
267C22×C3×C5C22×D411yescharacteristic
268C22×C3×C5C2×#4111yescharacteristic
269C22×C3×C5C2×#4111yescharacteristic
270C22×C3×C5C2×#4112yes—
271C2×D15—24no—
272C2×D15—24no—
273C2×D15—24no—
274C2×D15—42no—
275C2×D15—42no—
276C2×D15—42no—
277C2×D15—42no—
278C2×C5×D3—22no—
279C2×C5×D3—22no—
280C2×C5×D3—22no—
281C2×C5×D3—42no—
282C2×C5×D3—42no—
283C3×C4×C5—22no—
284C3×C4×C5—41no—
285Dic15—24no—
286C3×Dic5—24no—
287C3×Dic5—42no—
288C5×Dic3—22no—
289C5×Dic3—41no—
290C5×Dic3#12014yes—
291C24×C3C22×D511yescharacteristic
292C23×D3—52no—
293C23×D3C22×D511yescharacteristic
294C2×C3×Q8—104no—
295C2×C3×Q8—102no—
296C22×C3×C4—52no—
297C22×C3×C4—52no—
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D511yescharacteristic
300C2×#67—22no—
301C2×#67—22no—
302C2×#67—22no—
303C2×#67—104no—
304C2×#67—104no—
305C2×#67—104no—
306C22×Dic3—24no—
307C22×Dic3—22no—
308C22×Dic3—52no—
309C22×Dic3C22×D511yescharacteristic
310C22×Dic3C22×D511yescharacteristic
311C22×Dic3C22×D511yescharacteristic
312#239—1032no—
313C2×D12—104no—
314C2×D12—104no—
315C2×D12—104no—
316C2×D12—104no—
317C2×D12—102no—
318C2×C4×D3—24no—
319C2×C4×D3—58no—
320C2×C4×D3—104no—
321C2×C4×D3—104no—
322C2×C4×D3C22⋊D514yescomplemented
323C3×C22⋊C4—22no—
324C3×C22⋊C4—104no—
325(C2×D3)⋊C4—22no—
326(C2×D3)⋊C4—104no—
327(C2×D3)⋊C4—104no—
328(C2×D3)⋊C4—104no—
329(C2×D3)⋊C4—104no—
330Dic3⋊C4—22no—
331Dic3⋊C4—104no—
332C4×Dic3—22no—
333C4×Dic3—104no—
334C23×C5—22no—
335C23×C5—21no—
336C23×C5—22no—
337C23×C5—21no—
338C23×C5—32no—
339C23×C5—62no—
340C23×C5—61no—
341C23×C5—62no—
342C23×C5—61no—
343C23×C5#23912yescomplemented
344C23×C5C23×D311yescharacteristic
345C22×D5—34no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5—62no—
351C22×D5—62no—
352C5×D4—68no—
353C2×C4×C5—22no—
354C2×C4×C5—21no—
355C2×C4×C5—22no—
356C2×C4×C5—34no—
357C2×C4×C5—32no—
358C2×C4×C5—32no—
359C2×C4×C5—68no—
360C2×C4×C5—61no—
361C2×C4×C5—61no—
362C2×C4×C5—62no—
363C2×C4×C5—61no—
364C2×C4×C5—124no—
365C2×C4×C5D3×D412yes—
366C22⋊D5—616no—
367C22⋊D5—616no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—22no—
373C2×Dic5—34no—
374C2×Dic5—62no—
375C2×Dic5—62no—
376C2×Dic5D3×D414yes—
377D20—68no—
378C4×D5—616no—
379Dic10—28no—
380C2×#41—304no—
381C2×#41—308no—
382C22×Q8—51no—
383C22×D4—31no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C23×C4—31no—
388C23×C4—31no—
389C23×C4—152no—
390C4×D4—64no—
391C4×D4—308no—
392C2×C4⋊C4—31no—
393C2×C4⋊C4—152no—
394C2×C22⋊C4—31no—
395C2×C22⋊C4—52no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4D3×D511yescharacteristic
401C2×C42—31no—
402C2×C42—152no—
403#95—52no—
404#95—51no—
405#95—152no—
406#95—151no—
407C2×C3×C5—42no—
408C2×C3×C5C2×#12011yescharacteristic
409C2×C3×C5#52511yescharacteristic
410C2×C3×C5#52512yes—
411C2×C3×C5#53212yes—
412C2×C3×C5#53211yescharacteristic
413D15—44no—
414C5×D3—42no—
415C23×C3—21no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—21no—
419C23×C3#70912yescomplemented
420C23×C3C23×D511yescharacteristic
421C22×D3—21no—
422C22×D3—21no—
423C22×D3—22no—
424C22×D3—22no—
425C22×D3—54no—
426C22×D3—102no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3—102no—
432C22×D3C2×C22⋊D512yes—
433C3×Q8—108no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—54no—
438C2×C3×C4—102no—
439C2×C3×C4—102no—
440C2×C3×C4—102no—
441C2×C3×C4—102no—
442C2×C3×C4—102no—
443C2×C3×C4C2×C22⋊D512yes—
444#67—28no—
445#67—1016no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—22no—
449C2×Dic3—21no—
450C2×Dic3—44no—
451C2×Dic3—54no—
452C2×Dic3—102no—
453C2×Dic3—102no—
454C2×Dic3#70912yescomplemented
455C2×Dic3D4×D514yes—
456C2×Dic3C2×C22⋊D512yes—
457D12—108no—
458D12—1016no—
459C4×D3—28no—
460C4×D3—1016no—
461C22×C5—24no—
462C22×C5—22no—
463C22×C5—41no—
464C22×C5—42no—
465C22×C5—41no—
466C22×C5—62no—
467C22×C5—62no—
468C22×C5—62no—
469C22×C5—122no—
470C22×C5—122no—
471C22×C5C2×D3×D412yes—
472C22×C5C2×D3×D411yescharacteristic
473C22×C5C2×#23911yescharacteristic
474C22×C5D3×#4111yescharacteristic
475C22×C5D3×#4112yes—
476C2×D5—64no—
477C2×D5—64no—
478C2×D5—64no—
479C2×D5—122no—
480C2×D5—122no—
481C2×D5—122no—
482C2×D5—122no—
483C4×C5—22no—
484C4×C5—34no—
485C4×C5—41no—
486C4×C5—62no—
487C4×C5—121no—
488Dic5—24no—
489Dic5—42no—
490Dic5—64no—
491C24—31no—
492C24—152no—
493C24C2×D3×D511yescharacteristic
494#41—3032no—
495C2×Q8—104no—
496C2×Q8—102no—
497C2×D4—62no—
498C2×D4—62no—
499C2×D4—62no—
500C2×D4—304no—
501C2×D4—304no—
502C2×D4—304no—
503C2×D4—304no—
504C2×D4—304no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—302no—
508C22×C4—31no—
509C22×C4—31no—
510C22×C4—34no—
511C22×C4—31no—
512C22×C4—52no—
513C22×C4—52no—
514C22×C4—64no—
515C22×C4—62no—
516C22×C4—64no—
517C22×C4—158no—
518C22×C4—152no—
519C22×C4—304no—
520C22×C4—304no—
521C22×C4C2×D3×D511yescharacteristic
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—62no—
524C4⋊C4—304no—
525C22⋊C4—22no—
526C22⋊C4—62no—
527C22⋊C4—104no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—62no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—22no—
536C22×C3—24no—
537C22×C3—41no—
538C22×C3—42no—
539C22×C3—41no—
540C22×C3C2×#70911yescharacteristic
541C22×C3C2×#70911yescharacteristic
542C22×C3C2×D4×D512yes—
543C22×C3D5×#4112yes—
544C22×C3C22×C22⋊D511yescharacteristic
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—104no—
551C2×D3—104no—
552C2×D3—104no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—202no—
557C3×C4—22no—
558C3×C4—41no—
559C3×C4—104no—
560C3×C4—202no—
561Dic3—22no—
562Dic3—41no—
563Dic3—104no—
564Dic3#408314yes—
565C2×C5—42no—
566C2×C5—122no—
567C2×C5D3×#12011yescharacteristic
568C2×C5#562711yescharacteristic
569C2×C5#562712yes—
570C2×C5#566811yescharacteristic
571C2×C5#566812yes—
572D5—124no—
573C23—21no—
574C23—22no—
575C23—22no—
576C23—21no—
577C23—32no—
578C23—61no—
579C23—61no—
580C23—62no—
581C23—62no—
582C23—154no—
583C23—302no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23#676912yescomplemented
590C23C22×D3×D511yescharacteristic
591Q8—108no—
592D4—68no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—34no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—54no—
603C2×C4—68no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—61no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—102no—
611C2×C4—102no—
612C2×C4—102no—
613C2×C4—124no—
614C2×C4—154no—
615C2×C4—3016no—
616C2×C4—302no—
617C2×C4—302no—
618C2×C4D3×C22⋊D512yes—
619C2×C3—42no—
620C2×C3#6588112yes—
621C2×C3#6588212yes—
622C2×C3#6612511yescharacteristic
623C2×C3C2×#408311yescharacteristic
624C2×C3#6616611yescharacteristic
625D3—42no—
626D3—204no—
627C5#7677711yescharacteristic, complemented, Sylow
628C22—22no—
629C22—24no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—62no—
634C22—62no—
635C22—62no—
636C22—122no—
637C22—122no—
638C22—304no—
639C22—304no—
640C22—304no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—602no—
645C22C2×#676911yescharacteristic
646C22D3×#70911yescharacteristic
647C22D3×D4×D512yes—
648C22#9202412yes—
649C22C2×D3×C22⋊D511yescharacteristic
650C4—22no—
651C4—34no—
652C4—41no—
653C4—62no—
654C4—104no—
655C4—121no—
656C4—202no—
657C4—304no—
658C3#1061413111yescharacteristic, complemented, Sylow
659C2—42no—
660C2—122no—
661C2—604no—
662C2#1175284912yes—
663C2#1175285512yes—
664C2#1175435211yescharacteristic
665C2#1175481411yescharacteristic
666C2D3×#408311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FittingC2×C3×C5×C22⋊C4C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6769C2semidirect
2. C2×C5×#67⋊C4C2semidirect
3. C2×#91556C2semidirect
4. #11751061C2semidirect
5. C3×#65236C2semidirect
6. #11749523C2semidirect
7. #11748881C2semidirect
8. #11748790C2semidirect
9. #11748756C2semidirect
10. #11748719C2semidirect
11. #11748572C2semidirect
50. #65236D3semidirect
72. C2×C4×C5×D3D4semidirect
82. C2×#1200D4semidirect
97. C2×#67⋊C4D5semidirect
138. C23×C3×C5#41semidirect
159. C2×Dic15#41semidirect
169. C2×C5×Dic3#41semidirect
322. C2×C4×D3C22⋊D5semidirect
343. C23×C5#239semidirect
419. C23×C3#709semidirect
454. C2×Dic3#709semidirect
534. C3×C5#2388semidirect
589. C23#6769semidirect
627. C5#76777semidirect
658. C3#10614131semidirect

Maximal subgroups

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

subgroupindex
C22×#67692
C2×C5×#67⋊C42
C2×#915562
#117510612
C3×#652362
#117495232
#117488812
#117487902
#117487562
#117487192
#117485722
#106141313
#767775