Finite Group Databasebeta

← #33089001

Subgroups — #33089001

Order 18688.

966 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×#17988688C211yesmaximal, characteristic
2#22612406C211yesmaximal, characteristic, complemented
3C4×D4×Dic73C211yesmaximal, characteristic
4#22610575C211yesmaximal, characteristic, complemented
5#22610562C211yesmaximal, characteristic, complemented
6#22610522C211yesmaximal, characteristic, complemented
7#22610503C211yesmaximal, characteristic, complemented
8#22610267C211yesmaximal, characteristic
9#22610190C211yesmaximal, characteristic
10#22610178C211yesmaximal, characteristic
11#22609827C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic
14#22609678C211yesmaximal, characteristic, complemented
15C73×#1877C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C22×C4×Dic73—21no—
20#17988688C418yescomplemented
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988472—21no—
34#17988470—21no—
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39#17988438—21no—
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73—21no—
43#17988432—21no—
44C42×Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×SD16×C73C2211yescharacteristic
54C2×C4×D4×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#391×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64Q8×Dic73—22no—
65Q8×Dic73C2×C414yescomplemented
66D4×Dic73—22no—
67D4×Dic73—22no—
68D4×Dic73—22no—
69D4×Dic73C2×C414yes—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—44no—
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556525C2×C414yescomplemented
92#14556524—22no—
93#14556522—22no—
94#14556515—22no—
95#14556515C2×C414yescomplemented
96#14556512—22no—
97#14556512C2×C414yes—
98#14556501—22no—
99#14556501C2×C414yescomplemented
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×Q8×C73C2311yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C23×C4×C73—21no—
111C2×SD16×C73—22no—
112C2×SD16×C73—22no—
113C2×SD16×C73C2×C414yes—
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C4×D4×C73—22no—
117C4×D4×C73—22no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#103×C73—22no—
123#102×C73—22no—
124#95×C73—21no—
125#95×C73—21no—
126C22×Dic73—24no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—24no—
131C22×Dic73—21no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831C22×C412yes—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—24no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73C22×C412yes—
153C4⋊Dic73C22×C412yes—
154C4⋊Dic73#4112yes—
155#11924007—42no—
156#11924007—42no—
157C22⋊Dic73—24no—
158C22⋊Dic73—24no—
159C4×Dic73—22no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—24no—
163C4×Dic73—22no—
164C4×Dic73—41no—
165C4×Dic73—41no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C4×Dic73C22×C412yes—
171C4×Dic73#4112yes—
172C24×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73C22×C412yes—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73C22×C412yes—
183C2×D4×C73#4112yes—
184C2×D4×C73#4112yes—
185C22×C4×C73—24no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—42no—
190C22×C4×C73—42no—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2411yescharacteristic
194SD16×C73—28no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73C22×C412yes—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
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
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
220D4×C73—22no—
221D4×C73—22no—
222D4×C73—22no—
223D4×C73#11714yes—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—48no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73AGL(1,ℤ/8)12yes—
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—21no—
255C2×Dic73—22no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—48no—
259C2×Dic73—48no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73C4×D412yes—
270C2×Dic73C4×D412yes—
271C2×Dic73#13712yes—
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×C73C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C73C2×#13711yescharacteristic
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—22no—
302Dic73—42no—
303Dic73—41no—
304Dic73—42no—
305Dic73—42no—
306Dic73—81no—
307#21121—731nomaximal, Sylow
308C2×C73—44no—
309C2×C73#229011yescharacteristic
310C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
311C2×C73C4×#13711yescharacteristic
312C2×C73#311011yescharacteristic
313C2×C73#311111yescharacteristic
314C2×C73#311411yescharacteristic
315C2×C73#311611yescharacteristic
316C2×#440—731no—
317#2295—731no—
318C42×D4—731no—
319#1928—731no—
320#1927—731no—
321#1921—731no—
322#1918—731no—
323#1877—731no—
324#1877D7311yescharacteristic
325#1864—731no—
326#1862—731no—
327#1776—731no—
328#1763—731no—
329#1762—731no—
330#1745—731no—
331C73#2112111yescharacteristic, complemented, Sylow
332C22×SD16C2×D7311yescharacteristic
333C2×C4×Q8—731no—
334C2×C4×D4—731no—
335C2×C4×D4—731no—
336C2×C4×D4C2×D7311yescharacteristic
337C22×C42—1461no—
338#440—738no—
339C2×#107—731no—
340C2×#107—731no—
341C2×#106—731no—
342C2×#103—731no—
343C2×#103—731no—
344C2×#103C2×D7311yescharacteristic
345C2×#102—731no—
346C2×#102—731no—
347C2×#102C2×D7311yescharacteristic
348C2×#97—731no—
349C2×#97—731no—
350#391—731no—
351#391C2×D7311yescharacteristic
352#389—731no—
353#389—731no—
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×Q8C22×D7311yescharacteristic
377C22×D4C22×D7311yescharacteristic
378C23×C4—21no—
379C23×C4—1461no—
380C23×C4—1461no—
381C2×SD16—22no—
382C2×SD16—22no—
383C2×SD16C4×D7314yes—
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—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
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#107—734no—
422#107—1462no—
423#106—1462no—
424#103—22no—
425#103—734no—
426#103—1462no—
427#102—22no—
428#102—734no—
429#102—1462no—
430#97—734no—
431#97—1462no—
432#95—21no—
433#95—21no—
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—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D7312yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D7312yes—
453C2×D4#1192402712yes—
454C2×D4#1192402712yes—
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—28no—
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—2922no—
495C4⋊C4—2922no—
496C4⋊C4—732no—
497C4⋊C4—732no—
498C4⋊C4—732no—
499C4⋊C4—1464no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
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—732no—
518C42—732no—
519C42—1462no—
520C42—1464no—
521C42—1462no—
522C42—1461no—
523C42—1461no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D7311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1455654714yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
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—1461no—
570C2×C4—1461no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1461no—
574C2×C4—1462no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4#1455661812yes—
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4C2×#1192402711yescharacteristic
586C2×C4C2×#1192402711yescharacteristic
587C2×C4C2×#1192402711yescharacteristic
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#1798909711yescharacteristic
602C22C2×#1455661811yescharacteristic
603C22#137×D7311yescharacteristic
604C22#1798938311yescharacteristic
605C22#1798938411yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—2922no—
612C4—2922no—
613C4—2922no—
614C4—2921no—
615C4—5841no—
616C4—1462no—
617C4—1462no—
618C4—1462no—
619C4C2×#1455654711yescharacteristic
620C4C4×#1192402711yescharacteristic
621C4#1798911411yescharacteristic
622C4#1798911811yescharacteristic
623C2—44no—
624C2#2261237111yescharacteristic
625C2C4×#1455661811yescharacteristic
626C2#2261875311yescharacteristic
627C2#2262002911yescharacteristic
628C2#2262003711yescharacteristic
629C2#2262007011yescharacteristic
630C2#2262008611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1877C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1877D73
hypocenterC73#21121
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #22612406C2semidirect
4. #22610575C2semidirect
5. #22610562C2semidirect
6. #22610522C2semidirect
7. #22610503C2semidirect
11. #22609827C2semidirect
12. #22609763C2semidirect
14. #22609678C2semidirect
20. #17988688C4semidirect
65. Q8×Dic73C2×C4semidirect
91. #14556525C2×C4semidirect
95. #14556515C2×C4semidirect
99. #14556501C2×C4semidirect
331. C73#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#179886882
#226124062
C4×D4×Dic732
#226105752
#226105622
#226105222
#226105032
#226102672
#226101902
#226101782
#226098272
#226097632
#226097582
#226096782
C73×#18772
#2112173