Finite Group Databasebeta

← #33094049

Subgroups — #33094049

Order 18688.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988945C211yesmaximal, characteristic
2#22612502C211yesmaximal, characteristic, complemented
3#22612478C211yesmaximal, characteristic
4#22610902C211yesmaximal, characteristic, complemented
5#22610746C211yesmaximal, characteristic
6#22610631C211yesmaximal, characteristic
7#22610621C211yesmaximal, characteristic, complemented
8#22610620C211yesmaximal, characteristic, complemented
9#22610063C211yesmaximal, characteristic, complemented
10#22609750C211yesmaximal, characteristic, complemented
11#22609750C211yesmaximal, characteristic, complemented
12#102×Dic73C211yesmaximal, characteristic
13#102×Dic73C211yesmaximal, characteristic
14#97×Dic73C211yesmaximal, characteristic, complemented
15C73×#1936C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556578C2211yescharacteristic
18C2×#14556556—21no—
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21#17988945C418yescomplemented
22C2×#14556516C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988491—21no—
35#17988485C2211yescharacteristic
36#17988482—21no—
37#17988475—21no—
38#17988475—21no—
39#17988456C2211yescharacteristic
40#17988452C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43#17988438—21no—
44C22⋊C4×Dic73—21no—
45C22⋊C4×Dic73—21no—
46C42×Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146—21no—
61C22×Dic146C2311yescharacteristic
62C23×Dic73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556590—22no—
66#14556590C2×C414yescomplemented
67#14556578—22no—
68#14556578C2×C414yes—
69#14556556—22no—
70#14556556—22no—
71#14556556—44no—
72D4×Dic73—22no—
73D4×Dic73—22no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73D411yescharacteristic
90C2×C4×Dic73D411yescharacteristic
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556516—22no—
95#14556516C2×C414yescomplemented
96#14556514—22no—
97#14556514C2×C414yescomplemented
98#14556512—22no—
99#14556512C2×C414yes—
100#14556502—22no—
101#14556501—22no—
102#14556501C2×C414yescomplemented
103C8×Dic73—22no—
104C8×Dic73—22no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C22×D4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#102×C73—22no—
123#102×C73C2×C414yes—
124#97×C73—22no—
125#95×C73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—24no—
132C2×Dic146—42no—
133C2×Dic146—42no—
134C2×Dic146C22×C412yes—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C22×Dic73C2×D411yescharacteristic
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831C22×C412yes—
151C2×#10596831#4112yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73C22×C412yes—
158#11924007—24no—
159#11924007—42no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163C22⋊Dic73—24no—
164C22⋊Dic73—24no—
165C22⋊Dic73—42no—
166C4×Dic73—22no—
167C4×Dic73—22no—
168C4×Dic73—41no—
169C4×Dic73—41no—
170C4×Dic73—42no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73C22×C412yes—
174C4×Dic73#4112yes—
175C4×Dic73#4112yes—
176C4×Dic73#4112yes—
177C24×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73C22×C412yes—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73C22×C412yes—
195C2×C8×C73#4112yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73C22×C412yes—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73—22no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73C22×D411yescharacteristic
215D4×C73—22no—
216D4×C73—22no—
217D4×C73—41no—
218D4×C73—41no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
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×C73C23×C411yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—84no—
259C2×Dic73C4×D412yes—
260C2×Dic73C4×D412yes—
261C2×Dic73#12412yes—
262C2×Dic73#12412yes—
263C2×Dic73AGL(1,ℤ/8)12yes—
264C2×Dic73#13712yes—
265Dic146—22no—
266Dic146—22no—
267Dic146—48no—
268Dic146—41no—
269Dic146—41no—
270C8×C73—22no—
271C8×C73—22no—
272#10596831—22no—
273#10596831—22no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73C2×C4×D411yescharacteristic
282C22×C73C2×#12411yescharacteristic
283C22×C73#53211yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73C2×#13711yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57511yescharacteristic
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—42no—
291C4×C73—81no—
292C4×C73—81no—
293C4×C73C4×#4111yescharacteristic
294C4×C73C4×#4111yescharacteristic
295C4×C73#52911yescharacteristic
296C4×C73#52911yescharacteristic
297Dic73—22no—
298Dic73—22no—
299Dic73—22no—
300Dic73—42no—
301Dic73—42no—
302Dic73—81no—
303Dic73—81no—
304Dic73#48914yescomplemented
305C4×#489—731nomaximal, Sylow
306C2×C73—82no—
307C2×C73C4×#12411yescharacteristic
308C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
309C2×C73C4×#13711yescharacteristic
310C2×C73C2×#48911yescharacteristic
311C2×C73#313511yescharacteristic
312C2×C73#317111yescharacteristic
313C2×C73#317211yescharacteristic
314C2×#489—731no—
315C4×C4⋊Q8—731no—
316C4×#124—731no—
317#1993—731no—
318#1959—731no—
319#1936—731no—
320#1936D7311yescharacteristic
321#1935—731no—
322#1935—731no—
323#1836—731no—
324C4×#103—731no—
325C4×#103—731no—
326C4×#102—731no—
327C4×#102—731no—
328C4×#97—731no—
329C73C4×#48911yescharacteristic, complemented, Sylow
330C2×C4⋊Q8—731no—
331C2×#124—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8—1461no—
334C2×C4×D4—731no—
335#489—738no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×#103—731no—
340C2×#102—731no—
341C2×#102—731no—
342C2×#102C2×D7311yescharacteristic
343C2×#102C2×D7311yescharacteristic
344C2×#97—731no—
345C2×#97—731no—
346C2×#97—731no—
347C2×#97C2×D7311yescharacteristic
348C2×C4×C8—731no—
349C2×C4×C8—731no—
350#391—731no—
351#391—1461no—
352#390C2×D7311yescharacteristic
353#389—731no—
354#389—1461no—
355#389C2×D7311yescharacteristic
356#388—1461no—
357#388—1461no—
358#384—731no—
359#382—731no—
360C4×C4⋊C4—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4—1461no—
363C4×C22⋊C4—1461no—
364C4×C22⋊C4—1461no—
365C43—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340C2×D7311yescharacteristic
373#340C2×D7311yescharacteristic
374C22×Q8—731no—
375C22×Q8—1461no—
376C22×D4C22×D7311yescharacteristic
377C23×C4—1461no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C4⋊Q8—734no—
383C4⋊Q8—1462no—
384#124—734no—
385#124—1462no—
386C4×Q8—2924no—
387C4×Q8—1462no—
388C4×Q8—1462no—
389C4×D4—1462no—
390C4×D4—1462no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—731no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418#103—734no—
419#103—734no—
420#103—1462no—
421#103—1462no—
422#102—22no—
423#102—734no—
424#102—1462no—
425#102C4×D7314yes—
426#97—22no—
427#97—734no—
428#97—1462no—
429#97—1462no—
430C4×C8—1462no—
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—
440C24—21no—
441C2×Q8—2922no—
442C2×Q8—2922no—
443C2×Q8—732no—
444C2×Q8—1464no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D7312yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4D4×D7311yescharacteristic
471C22×C4D4×D7311yescharacteristic
472C22×C4C23×D7311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—732no—
478C2×C8—732no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8C2×C4×D7312yes—
484C2×C8#1192402712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—732no—
496C4⋊C4—1464no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4C2×C4×D7312yes—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—2922no—
506C22⋊C4—1464no—
507C22⋊C4—1464no—
508C42—21no—
509C42—21no—
510C42—2921no—
511C42—2921no—
512C42—2922no—
513C42—2922no—
514C42—2922no—
515C42—732no—
516C42—732no—
517C42—732no—
518C42—732no—
519C42—1462no—
520C42—1462no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D7311yescharacteristic
529Q8—2928no—
530Q8—2921no—
531Q8—2921no—
532Q8—1462no—
533Q8—1462no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—5844no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—732no—
568C2×C4—732no—
569C2×C4—732no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4#41×D7311yescharacteristic
581C2×C4#41×D7311yescharacteristic
582C2×C4#41×D7311yescharacteristic
583C2×C4C2×#1192402711yescharacteristic
584C8—22no—
585C8—22no—
586C8—1462no—
587C8—1462no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D7311yescharacteristic
596C22#124×D7311yescharacteristic
597C22#1798913211yescharacteristic
598C22AGL(1,ℤ/8)×D7311yescharacteristic
599C22#137×D7311yescharacteristic
600C22#1798938311yescharacteristic
601C22#1798938311yescharacteristic
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—2922no—
608C4—2922no—
609C4—5841no—
610C4—5841no—
611C4—734no—
612C4—1462no—
613C4—1462no—
614C4—1462no—
615C4C4×#1192402711yescharacteristic
616C4#1798905511yescharacteristic
617C4#1798911411yescharacteristic
618C4#1798912011yescharacteristic
619C2—82no—
620C2#2261247211yescharacteristic
621C2#2261874211yescharacteristic
622C2#2261875311yescharacteristic
623C2#489×D7311yescharacteristic
624C2#2262027511yescharacteristic
625C2#2262052111yescharacteristic
626C2#2262053011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1936C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1936D73
hypocenterC73C4×#489
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #22612502C2semidirect
4. #22610902C2semidirect
7. #22610621C2semidirect
8. #22610620C2semidirect
9. #22610063C2semidirect
10. #22609750C2semidirect
11. #22609750C2semidirect
14. #97×Dic73C2semidirect
21. #17988945C4semidirect
66. #14556590C2×C4semidirect
95. #14556516C2×C4semidirect
97. #14556514C2×C4semidirect
102. #14556501C2×C4semidirect
304. Dic73#489semidirect
329. C73C4×#489semidirect

Maximal subgroups

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

subgroupindex
C2×#179889452
#226125022
#226124782
#226109022
#226107462
#226106312
#226106212
#226106202
#226100632
#226097502
#102×Dic732
#97×Dic732
C73×#19362
C4×#48973