Finite Group Databasebeta

← #32993275

Subgroups — #32993275

Order 18688.

910 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556526C211yesmaximal, characteristic, complemented
2C2×#17988589C211yesmaximal, characteristic
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988555C211yesmaximal, characteristic
5C2×#17988355C211yesmaximal, characteristic, complemented
6C2×#17988349C211yesmaximal, characteristic, complemented
7C2×#343×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9C22×#11924012C2211yescharacteristic
10C23×#10596831C2211yescharacteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C2×#14556526—22no—
13C2×#14556526—22no—
14C2×#14556526—22no—
15#17988589—22no—
16#17988589—22no—
17#17988586—22no—
18#17988586—22no—
19C2×#14556516—21no—
20C2×#14556514—21no—
21C2×#14556513—21no—
22C2×#14556511—21no—
23#17988555—22no—
24#17988555C414yescomplemented
25C2×#14556503—22no—
26#17988355—22no—
27#17988349—22no—
28C22×M16×C73C2211yescharacteristic
29C22×M16×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2211yescharacteristic
31#343×C73—22no—
32C2×#11924028—21no—
33C2×#11924028—21no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028C2×C412yes—
43C2×#11924028Q812yes—
44C22×Dic146D411yescharacteristic
45C22×D292D411yescharacteristic
46C22×C22⋊D73—21no—
47C23×Dic73—21no—
48C22×C4×D73—21no—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53C2×#11924012—21no—
54C2×#11924012D412yes—
55C22×#10596831—21no—
56C22×#10596831—21no—
57C22×#10596831—21no—
58C22×#10596831—21no—
59C22×#10596831—21no—
60C22×#10596831C2×C412yes—
61C22×#10596831D412yes—
62C22×#10596831D412yes—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73D412yes—
69#14556526—28no—
70#14556516—24no—
71#14556514—24no—
72#14556513—24no—
73#14556511—24no—
74#14556503—28no—
75C23×C4×C73—21no—
76C23×C4×C73C2311yescharacteristic
77C2×M16×C73—21no—
78C2×M16×C73—21no—
79C2×M16×C73—21no—
80C2×M16×C73—21no—
81C2×M16×C73—21no—
82C2×M16×C73—21no—
83C2×M16×C73—21no—
84C2×M16×C73—21no—
85C2×M16×C73—21no—
86C2×M16×C73—21no—
87C2×M16×C73C2×C412yes—
88C2×M16×C73D412yes—
89C22×C8×C73—21no—
90C22×C8×C73—21no—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73D412yes—
97C23×D73—21no—
98#11924028—22no—
99#11924028—22no—
100#11924028—22no—
101#11924028—22no—
102#11924028—22no—
103#11924028—22no—
104#11924028—22no—
105#11924028—22no—
106#11924028—41no—
107#11924028—41no—
108#11924028—41no—
109#11924028—41no—
110#11924028—41no—
111#11924028—41no—
112#11924028—41no—
113#11924028—41no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146C4⋊C412yes—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C4⋊C412yes—
126C2×C22⋊D73—22no—
127C2×C22⋊D73—22no—
128C2×C22⋊D73—41no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C2×C22⋊D73—41no—
132C22×Dic73—21no—
133C22×Dic73—22no—
134C22×Dic73—41no—
135C22×Dic73—41no—
136C22×Dic73—81no—
137C22×Dic73—81no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144#11924012—22no—
145#11924012—22no—
146#11924012—41no—
147#11924012—41no—
148C2×#10596831—22no—
149C2×#10596831—22no—
150C2×#10596831—22no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C2×#10596831—22no—
154C2×#10596831—22no—
155C2×#10596831—41no—
156C2×#10596831—41no—
157C2×#10596831C22⋊C414yes—
158C4⋊Dic73—22no—
159C4⋊Dic73—22no—
160C4⋊Dic73—41no—
161C4⋊Dic73—41no—
162C24×C73#4111yescharacteristic
163C22×C4×C73—22no—
164C22×C4×C73—41no—
165C22×C4×C73—41no—
166C22×C4×C73—81no—
167C22×C4×C73—81no—
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×Q811yescharacteristic
177C22×C4×C73#4111yescharacteristic
178C22×C4×C73#4111yescharacteristic
179C22×C4×C73#4111yescharacteristic
180C22×C4×C73#4111yescharacteristic
181C22×C4×C73#4111yescharacteristic
182M16×C73—22no—
183M16×C73—22no—
184M16×C73—22no—
185M16×C73—22no—
186M16×C73—41no—
187M16×C73—41no—
188M16×C73—41no—
189M16×C73—41no—
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C2×C8×C73—41no—
193C2×C8×C73—41no—
194C2×C8×C73—41no—
195C2×C8×C73—41no—
196C4⋊C4×C73—22no—
197C4⋊C4×C73—22no—
198C4⋊C4×C73—41no—
199C4⋊C4×C73—41no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73#11711yescharacteristic
205C23×C73#12111yescharacteristic
206C23×C73C22⋊Q811yescharacteristic
207C23×C73#12311yescharacteristic
208C23×C73#12311yescharacteristic
209C23×C73#12411yescharacteristic
210C23×C73#12411yescharacteristic
211C22×D73—22no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216C22×D73—41no—
217C22×D73—41no—
218C2×C4×C73—42no—
219C2×C4×C73—82no—
220C2×C4×C73—81no—
221C2×C4×C73—81no—
222C2×C4×C73—82no—
223C2×C4×C73C2×C22⋊C411yescharacteristic
224C2×C4×C73C2×C4⋊C411yescharacteristic
225C2×C4×C73C4×D411yescharacteristic
226C2×C4×C73C4×D411yescharacteristic
227C2×C4×C73C4×D411yescharacteristic
228C2×C4×C73C4×D411yescharacteristic
229C2×C4×C73#12011yescharacteristic
230C2×C4×C73#12011yescharacteristic
231C2×C4×C73#12011yescharacteristic
232C2×C4×C73#12111yescharacteristic
233C2×C4×C73#12111yescharacteristic
234C2×C4×C73#12111yescharacteristic
235C2×C4×C73#12111yescharacteristic
236C2×C4×C73#12111yescharacteristic
237C2×C4×C73#12111yescharacteristic
238C2×C4×C73#12111yescharacteristic
239C2×C4×C73C22⋊Q811yescharacteristic
240C2×C4×C73C22⋊Q811yescharacteristic
241C2×C4×C73C22⋊Q811yescharacteristic
242C2×C4×C73C22⋊Q811yescharacteristic
243C2×C4×C73C22⋊Q811yescharacteristic
244C2×C4×C73#12311yescharacteristic
245C2×C4×C73#12311yescharacteristic
246C2×C4×C73#12311yescharacteristic
247C2×C4×C73#12411yescharacteristic
248C2×C4×C73#12611yescharacteristic
249C2×C4×C73#12611yescharacteristic
250C2×C4×C73#12711yescharacteristic
251C22⋊D73—42no—
252C22⋊D73—42no—
253C22⋊D73—42no—
254C22⋊D73—42no—
255C2×Dic73—22no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—42no—
259C2×Dic73—82no—
260C2×Dic73—81no—
261C2×Dic73—81no—
262C2×Dic73—82no—
263D292—22no—
264D292—22no—
265D292—41no—
266D292—41no—
267C4×D73—42no—
268C4×D73—42no—
269C4×D73—42no—
270C4×D73—42no—
271Dic146—22no—
272Dic146—22no—
273Dic146—41no—
274Dic146—41no—
275C8×C73—42no—
276C8×C73—42no—
277#10596831—24no—
278#10596831—42no—
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—21no—
286C22×C73—21no—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73—21no—
290C22×C73—21no—
291C22×C73#38011yescharacteristic
292C22×C73#38811yescharacteristic
293C22×C73#38811yescharacteristic
294C22×C73#39311yescharacteristic
295C22×C73#39411yescharacteristic
296C22×C73#39411yescharacteristic
297C22×C73#39911yescharacteristic
298C22×C73#47111yescharacteristic
299C22×C73#47111yescharacteristic
300C22×C73#47211yescharacteristic
301C22×C73#47311yescharacteristic
302C2×D73—42no—
303C2×D73—42no—
304C2×D73—42no—
305C2×D73—81no—
306C2×D73—81no—
307C2×D73—81no—
308C2×D73—81no—
309C4×C73—82no—
310C4×C73#38511yescharacteristic
311C4×C73#38511yescharacteristic
312C4×C73#38611yescharacteristic
313C4×C73#39011yescharacteristic
314C4×C73#39211yescharacteristic
315C4×C73#39611yescharacteristic
316C4×C73#39611yescharacteristic
317C4×C73#39711yescharacteristic
318Dic73—42no—
319Dic73—82no—
320#11105—731nomaximal, Sylow
321C2×C73—21no—
322C2×C73—21no—
323C2×C73—21no—
324C2×C73—21no—
325C2×C73#143911yescharacteristic
326C2×C73#181511yescharacteristic
327C2×C73#181611yescharacteristic
328C2×C73#205111yescharacteristic
329C2×C73#205111yescharacteristic
330C2×C73#205211yescharacteristic
331C2×C73#205211yescharacteristic
332D73—82no—
333C22×#108—731no—
334C2×#417—731no—
335C2×#417—731no—
336C2×#408—731no—
337C2×#343—731no—
338C2×#343D7311yescharacteristic, complemented
339C2×#340—731no—
340C73#1110511yescharacteristic, complemented, Sylow
341C22×#41—731no—
342C22×M16—731no—
343C22×M16C2×D7311yescharacteristic
344C22×M16C2×D7311yescharacteristic
345C23×C8—731no—
346C22×C4⋊C4—731no—
347C22×C4⋊C4C2×D7311yescharacteristic
348C2×#108—1462no—
349C2×#108—1462no—
350C2×#108—1462no—
351#417—1462no—
352#417—1462no—
353#417—1462no—
354#417—1462no—
355C2×#103—1461no—
356C2×#103—1461no—
357C2×#102—1461no—
358C2×#102—1461no—
359#408—734no—
360#408—1462no—
361C2×C22⋊C8—1462no—
362#343—22no—
363#343—1462no—
364#340—1462no—
365C2×#41—732no—
366C2×#41—732no—
367C2×#41—1461no—
368C2×#41—1461no—
369C2×#41—1461no—
370C2×#41—1461no—
371C2×#41—1461no—
372C2×#41—1461no—
373C2×#41—1461no—
374C2×#41—1461no—
375C2×#41—1461no—
376C2×#41—1461no—
377C22×Q8—731no—
378C22×D4—731no—
379C22×D4—1461no—
380C23×C4—21no—
381C23×C4—1461no—
382C23×C4—1461no—
383C23×C4C22×D7311yescharacteristic
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—732no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16—1461no—
398C2×M16—1461no—
399C2×M16—1461no—
400C2×M16C4×D7312yes—
401C2×M16D29212yes—
402C22×C8—21no—
403C22×C8—21no—
404C22×C8—732no—
405C22×C8—732no—
406C22×C8—732no—
407C22×C8—1461no—
408C22×C8—1461no—
409C22×C8—1461no—
410C22×C8—1461no—
411C22×C8—1461no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—732no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4C22⋊D7312yes—
424#108—1468no—
425#103—1464no—
426#103—1464no—
427#102—1464no—
428#102—1464no—
429C22⋊C8—1468no—
430C24—1461no—
431C24#1192402811yescharacteristic
432#41—2921no—
433#41—2921no—
434#41—2921no—
435#41—2921no—
436#41—2921no—
437#41—2921no—
438#41—2921no—
439#41—2921no—
440#41—1462no—
441#41—1462no—
442#41—1462no—
443#41—1462no—
444#41—1462no—
445#41—1462no—
446#41—1462no—
447#41—1462no—
448C2×Q8—732no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×D4—2921no—
455C2×D4—2921no—
456C2×D4—2921no—
457C2×D4—2921no—
458C2×D4—732no—
459C2×D4—1462no—
460C2×D4—1462no—
461C2×D4—1461no—
462C2×D4—1461no—
463C2×D4—1461no—
464C2×D4—1461no—
465C2×D4—1461no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—2921no—
472C22×C4—2921no—
473C22×C4—2921no—
474C22×C4—2921no—
475C22×C4—2921no—
476C22×C4—2921no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—1461no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4C2×C4×D7311yescharacteristic
484C22×C4D4×D7311yescharacteristic
485C22×C4D4×D7311yescharacteristic
486C22×C4D4×D7311yescharacteristic
487C22×C4D4×D7311yescharacteristic
488C22×C4D4×D7311yescharacteristic
489C22×C4C2×C22⋊D7311yescharacteristic
490C22×C4C2×D29211yescharacteristic
491C22×C4Q8×D7311yescharacteristic
492C22×C4#1192402711yescharacteristic
493C22×C4#1192402811yescharacteristic
494C22×C4#1192402811yescharacteristic
495C22×C4#1192402811yescharacteristic
496C22×C4#1192402911yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—2921no—
506M16—2921no—
507M16—1462no—
508M16—1462no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—2921no—
516C2×C8—2921no—
517C2×C8—734no—
518C2×C8—1462no—
519C2×C8—1462no—
520C2×C8—1462no—
521C2×C8—1462no—
522C2×C8—1462no—
523C2×C8—1462no—
524C2×C8—1462no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—2921no—
530C4⋊C4—2921no—
531C4⋊C4—1462no—
532C4⋊C4—1462no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—2921no—
538C23—2921no—
539C23—2921no—
540C23—2921no—
541C23—2921no—
542C23—2921no—
543C23—1462no—
544C23#1455654811yescharacteristic
545C23#1455656411yescharacteristic
546C23#1455656911yescharacteristic
547C23#1455657411yescharacteristic
548C23#1455657611yescharacteristic
549C23#1455657711yescharacteristic
550C23#1455657911yescharacteristic
551Q8—2921no—
552Q8—2921no—
553Q8—1462no—
554Q8—1462no—
555D4—2922no—
556D4—2922no—
557D4—2922no—
558D4—2922no—
559D4—2921no—
560D4—2921no—
561D4—1462no—
562D4—1462no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—5842no—
576C2×C4—5841no—
577C2×C4—5841no—
578C2×C4—5842no—
579C2×C4—1462no—
580C2×C4C22⋊C4×D7311yescharacteristic
581C2×C4C4⋊C4×D7311yescharacteristic
582C2×C4#1455655111yescharacteristic
583C2×C4C4×C22⋊D7311yescharacteristic
584C2×C4#1455655311yescharacteristic
585C2×C4C4×D29211yescharacteristic
586C2×C4#1455655911yescharacteristic
587C2×C4#1455656111yescharacteristic
588C2×C4#1455656111yescharacteristic
589C2×C4#1455656311yescharacteristic
590C2×C4#1455656411yescharacteristic
591C2×C4#1455656411yescharacteristic
592C2×C4#1455656411yescharacteristic
593C2×C4#1455656411yescharacteristic
594C2×C4#1455656511yescharacteristic
595C2×C4#1455656611yescharacteristic
596C2×C4#1455656711yescharacteristic
597C2×C4#1455656911yescharacteristic
598C2×C4#1455656911yescharacteristic
599C2×C4#1455656911yescharacteristic
600C2×C4#1455657111yescharacteristic
601C2×C4#1455657311yescharacteristic
602C2×C4#1455657611yescharacteristic
603C2×C4#1455657611yescharacteristic
604C2×C4#1455657711yescharacteristic
605C2×C4#1455658511yescharacteristic
606C2×C4#1455658611yescharacteristic
607C2×C4#1455658711yescharacteristic
608C8—42no—
609C8—42no—
610C8—2922no—
611C8—1464no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—2922no—
625C22—2922no—
626C22—2922no—
627C22—5841no—
628C22—5841no—
629C22—5841no—
630C22—5841no—
631C22#1798844611yescharacteristic
632C22#1798847611yescharacteristic
633C22#1798847811yescharacteristic
634C22#1798850011yescharacteristic
635C22#1798850211yescharacteristic
636C22#1798850511yescharacteristic
637C22#1798852211yescharacteristic
638C22#1798884911yescharacteristic
639C22#1798885311yescharacteristic
640C22#1798885611yescharacteristic
641C22#1798886111yescharacteristic
642C4—82no—
643C4—2922no—
644C4—5842no—
645C4#1798845811yescharacteristic
646C4#1798846111yescharacteristic
647C4#1798846511yescharacteristic
648C4#1798848711yescharacteristic
649C4#1798849611yescharacteristic
650C4#1798850911yescharacteristic
651C4#1798851111yescharacteristic
652C4#1798851611yescharacteristic
653C2—21no—
654C2—21no—
655C2—21no—
656C2—21no—
657C2—5842no—
658C2#2260817511yescharacteristic
659C2#2260998911yescharacteristic
660C2#2260999411yescharacteristic
661C2#2261122311yescharacteristic
662C2#2261122611yescharacteristic
663C2#2261123011yescharacteristic
664C2#2261123311yescharacteristic
subgroupG/H
centerC23#14556564
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#343×C73C2
socleC23×C73#121
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenterC2×#343D73
hypocenterC73#11105
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
1. C22×#14556526C2semidirect
5. C2×#17988355C2semidirect
6. C2×#17988349C2semidirect
7. C2×#343×C73C2semidirect
24. #17988555C4semidirect
338. C2×#343D73semidirect
340. C73#11105semidirect

Maximal subgroups

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

subgroupindex
C22×#145565262
C2×#179885892
C2×#179885862
C2×#179885552
C2×#179883552
C2×#179883492
C2×#343×C732
#1110573