Finite Group Databasebeta

← #33088949

Subgroups — #33088949

Order 18688.

978 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988678C211yesmaximal, characteristic
2#22612402C211yesmaximal, characteristic, complemented
3C4×D4×Dic73C211yesmaximal, characteristic
4#22610566C211yesmaximal, characteristic, complemented
5#22610564C211yesmaximal, characteristic, complemented
6#22610519C211yesmaximal, characteristic, complemented
7#22610513C211yesmaximal, characteristic, complemented
8#22610265C211yesmaximal, characteristic
9#22610263C211yesmaximal, characteristic
10#22610194C211yesmaximal, characteristic
11#106×Dic73C211yesmaximal, characteristic, complemented
12#22609750C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic
14#22609675C211yesmaximal, characteristic, complemented
15C73×#1864C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20C22×C4×Dic73—21no—
21#17988678C418yescomplemented
22C2×#14556523C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988493C2211yescharacteristic
31#17988492C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988474—21no—
34#17988468—21no—
35C4⋊C4×Dic73—21no—
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73—21no—
43#17988432—21no—
44C42×Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#106×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C23×Dic73—21no—
62C23×Dic73—21no—
63C2×#11924016—22no—
64C2×#11924016—22no—
65C2×#11924016C2×C414yes—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556556—22no—
69#14556556C2×C414yescomplemented
70D4×Dic73—22no—
71D4×Dic73—22no—
72D4×Dic73—22no—
73D4×Dic73C2×C414yes—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—44no—
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73D411yescharacteristic
90C2×C4×Dic73D411yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93#14556523—22no—
94#14556522C2×C414yescomplemented
95#14556516—22no—
96#14556516C2×C414yescomplemented
97#14556514—22no—
98#14556512—22no—
99C8×Dic73—22no—
100C8×Dic73C2×C414yescomplemented
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110C22×D4×C73C2311yescharacteristic
111C23×C4×C73—21no—
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C4×D4×C73—22no—
115C4×D4×C73—22no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#106×C73—22no—
123#102×C73—22no—
124#102×C73C2×C414yes—
125#95×C73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146C22×C412yes—
132C22×Dic73—24no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—24no—
137C22×Dic73—21no—
138C22×Dic73—42no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C22×Dic73C2×D411yescharacteristic
146#11924016—28no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831C22×C412yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73C22×C412yes—
159C4⋊Dic73#4112yes—
160#11924007—24no—
161C22⋊Dic73—24no—
162C22⋊Dic73—24no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—24no—
166C4×Dic73—22no—
167C4×Dic73—41no—
168C4×Dic73—41no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73SD1614yescomplemented
174C4×Dic73C22×C412yes—
175C4×Dic73#4112yes—
176C24×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73C22×C412yes—
181C2×D4×C73#4112yes—
182C2×D4×C73#4112yes—
183C22×C4×C73—24no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C412yes—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73C22×C412yes—
204C4⋊C4×C73#4112yes—
205C22⋊C4×C73—24no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73—22no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73C22×D411yescharacteristic
216D4×C73—22no—
217D4×C73—22no—
218D4×C73—22no—
219D4×C73#11714yes—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—48no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73C2×SD1612yes—
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—21no—
249C2×Dic73—22no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—48no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—48no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73C4×D412yes—
264C2×Dic73C4×D412yes—
265C2×Dic73C2×SD1612yes—
266C2×Dic73C2×SD1612yes—
267C2×Dic73#13512yes—
268Dic146—22no—
269Dic146—22no—
270Dic146—41no—
271Dic146—41no—
272C8×C73—22no—
273C8×C73—41no—
274#10596831—22no—
275#10596831—41no—
276C22×C73—24no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73C2×C4×D411yescharacteristic
284C22×C73#52511yescharacteristic
285C22×C73#52511yescharacteristic
286C22×C73C22×SD1611yescharacteristic
287C22×C73C2×#13511yescharacteristic
288C22×C73#57511yescharacteristic
289C22×C73#57511yescharacteristic
290C4×C73—22no—
291C4×C73—42no—
292C4×C73—41no—
293C4×C73—42no—
294C4×C73—81no—
295C4×C73C2×#11711yescharacteristic
296C4×C73C4×#4111yescharacteristic
297C4×C73#52911yescharacteristic
298C4×C73#52911yescharacteristic
299Dic73—22no—
300Dic73—22no—
301Dic73—42no—
302Dic73—41no—
303Dic73—42no—
304Dic73—42no—
305Dic73—81no—
306Dic73C4×SD1614yes—
307#21117—731nomaximal, Sylow
308C2×C73—44no—
309C2×C73#229011yescharacteristic
310C2×C73C2×C4×SD1611yescharacteristic
311C2×C73#294311yescharacteristic
312C2×C73#310111yescharacteristic
313C2×C73#310111yescharacteristic
314C2×C73#310211yescharacteristic
315C2×C73#310211yescharacteristic
316C2×C4×SD16—731no—
317#2295—731no—
318C42×D4—731no—
319#1927—731no—
320#1927—731no—
321#1920—731no—
322#1920—731no—
323#1877—731no—
324#1877—731no—
325#1864—731no—
326#1864D7311yescharacteristic
327C4×#106—731no—
328C4×#103—731no—
329C4×#102—731no—
330#1744—731no—
331C73#2111711yescharacteristic, complemented, Sylow
332C22×SD16—731no—
333C2×C4×Q8—731no—
334C2×C4×D4—731no—
335C2×C4×D4—731no—
336C2×C4×D4C2×D7311yescharacteristic
337C22×C42—1461no—
338C4×SD16—738no—
339C2×#106—731no—
340C2×#106—731no—
341C2×#106C2×D7311yescharacteristic
342C2×#103—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#102—731no—
346C2×#102C2×D7311yescharacteristic
347C2×#102C2×D7311yescharacteristic
348C2×C4×C8—731no—
349C2×C4×C8—731no—
350#391—731no—
351#391—731no—
352#389—731no—
353#389C2×D7311yescharacteristic
354#387—1461no—
355#387—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362C4×C22⋊C4—1461no—
363C4×C22⋊C4—1461no—
364#376—1461no—
365C43—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340C2×D7311yescharacteristic
372#336—731no—
373#336—731no—
374#336—731no—
375#336C2×D7311yescharacteristic
376C22×Q8—731no—
377C22×D4C22×D7311yescharacteristic
378C23×C4—21no—
379C23×C4—1461no—
380C23×C4—1461no—
381C2×SD16—734no—
382C2×SD16—1462no—
383C2×SD16—1462no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8C22×D7311yescharacteristic
387C22×C8C22×D7311yescharacteristic
388C4×Q8—734no—
389C4×Q8—1462no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—734no—
393C4×D4—1462no—
394C4×D4—1462no—
395C4×D4—1462no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—2924no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42—1461no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421#106—22no—
422#106—734no—
423#106—1462no—
424#103—734no—
425#103—1462no—
426#103—1462no—
427#102—22no—
428#102—1462no—
429#102C4×D7314yes—
430C4×C8—734no—
431C4×C8—1462no—
432#95—21no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442C24—21no—
443C2×Q8—732no—
444C2×Q8—1461no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D7312yes—
453C2×D4#1192402712yes—
454C2×D4#1192402912yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—1461no—
470C22×C4—1464no—
471C22×C4—1464no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4D4×D7311yescharacteristic
476C22×C4D4×D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478SD16—1468no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—732no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8C2×C4×D7312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—2922no—
497C4⋊C4—732no—
498C4⋊C4—732no—
499C4⋊C4—1464no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4C2×C4×D7312yes—
505C4⋊C4#1192402712yes—
506C22⋊C4—24no—
507C22⋊C4—1464no—
508C22⋊C4—1464no—
509C42—21no—
510C42—21no—
511C42—2921no—
512C42—2921no—
513C42—2922no—
514C42—2922no—
515C42—2922no—
516C42—2922no—
517C42—734no—
518C42—732no—
519C42—732no—
520C42—1462no—
521C42—1464no—
522C42—1461no—
523C42—1461no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D7311yescharacteristic
532Q8—2921no—
533Q8—2921no—
534Q8—1462no—
535Q8—1462no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1455654914yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2928no—
562C2×C4—2928no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—732no—
567C2×C4—732no—
568C2×C4—732no—
569C2×C4—732no—
570C2×C4—732no—
571C2×C4—1461no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1461no—
575C2×C4—1462no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4SD16×D7312yes—
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4C2×#1192402711yescharacteristic
586C2×C4C2×#1192402711yescharacteristic
587C2×C4C2×#1192402911yescharacteristic
588C8—22no—
589C8—41no—
590C8—2921no—
591C8—1462no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D7311yescharacteristic
600C22#1798909311yescharacteristic
601C22#1798910011yescharacteristic
602C22C2×SD16×D7311yescharacteristic
603C22#135×D7311yescharacteristic
604C22#1798938311yescharacteristic
605C22#1798938311yescharacteristic
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—2922no—
612C4—2921no—
613C4—2922no—
614C4—2922no—
615C4—5841no—
616C4—734no—
617C4—1462no—
618C4—1462no—
619C4C2×#1455654911yescharacteristic
620C4C4×#1192402711yescharacteristic
621C4#1798911411yescharacteristic
622C4#1798912311yescharacteristic
623C2—44no—
624C2#2261236711yescharacteristic
625C2C4×SD16×D7311yescharacteristic
626C2#2261872811yescharacteristic
627C2#2261993411yescharacteristic
628C2#2261994111yescharacteristic
629C2#2261994211yescharacteristic
630C2#2261995411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1864C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1864D73
hypocenterC73#21117
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #22612402C2semidirect
4. #22610566C2semidirect
5. #22610564C2semidirect
6. #22610519C2semidirect
7. #22610513C2semidirect
11. #106×Dic73C2semidirect
12. #22609750C2semidirect
14. #22609675C2semidirect
21. #17988678C4semidirect
69. #14556556C2×C4semidirect
94. #14556522C2×C4semidirect
96. #14556516C2×C4semidirect
100. C8×Dic73C2×C4semidirect
173. C4×Dic73SD16semidirect
331. C73#21117semidirect

Maximal subgroups

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

subgroupindex
C2×#179886782
#226124022
C4×D4×Dic732
#226105662
#226105642
#226105192
#226105132
#226102652
#226102632
#226101942
#106×Dic732
#226097502
#102×Dic732
#226096752
C73×#18642
#2111773