Finite Group Databasebeta

← #33144278

Subgroups — #33144278

Order 18688.

744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614959C211yesmaximal, characteristic, complemented
2#22612903C211yesmaximal, characteristic
3#22611315C211yesmaximal, characteristic
4#22611067C211yesmaximal, characteristic
5#22611065C211yesmaximal, characteristic
6#22610881C211yesmaximal, characteristic, complemented
7#22610878C211yesmaximal, characteristic, complemented
8#22610700C211yesmaximal, characteristic, complemented
9#22610651C211yesmaximal, characteristic, complemented
10#22610520C211yesmaximal, characteristic, complemented
11#22610511C211yesmaximal, characteristic, complemented
12#22610063C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#22609758C211yesmaximal, characteristic
15C73×#2065C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556590—21no—
18C2×#14556590C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26#17988508—21no—
27#17988507—21no—
28#17988490C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988484—21no—
31#17988481—21no—
32#17988475—21no—
33#17988456C2211yescharacteristic
34#17988452C2211yescharacteristic
35#17988448—21no—
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×#11924007C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#106×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56#390×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×Dic146—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556591—22no—
65#14556591—41no—
66#14556590—22no—
67#14556590—41no—
68#14556590—44no—
69D4×Dic73—22no—
70D4×Dic73Q814yescomplemented
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
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×Dic73C2311yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90#14556524—22no—
91#14556520—22no—
92#14556520—41no—
93#14556512—22no—
94#14556512—41no—
95#14556501—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104C22×D4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C42×C73C2311yescharacteristic
114#106×C73—22no—
115C4⋊C8×C73—41no—
116#102×C73—22no—
117#102×C73—41no—
118#95×C73—21no—
119C2×Dic146—42no—
120C2×Dic146—42no—
121C2×Dic146—42no—
122C2×Dic146—42no—
123C2×Dic146—81no—
124C2×Dic146—81no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—24no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—81no—
131C22×Dic73—81no—
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—42no—
149C4⋊Dic73—42no—
150C4⋊Dic73C2×Q812yes—
151C4⋊Dic73#4112yes—
152#11924007—22no—
153#11924007—22no—
154#11924007—42no—
155#11924007—42no—
156#11924007—81no—
157C22⋊Dic73—24no—
158C22⋊Dic73—42no—
159C22⋊Dic73—81no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—42no—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73C2×Q812yes—
166C4×Dic73#4112yes—
167C24×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73C2×Q812yes—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2411yescharacteristic
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—42no—
192C4⋊C4×C73#4112yes—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—81no—
195C42×C73—21no—
196C42×C73—21no—
197C23×C73—22no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73C22×D411yescharacteristic
205D4×C73—22no—
206D4×C73—22no—
207D4×C73—41no—
208D4×C73—41no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—22no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73#13712yes—
224C2×C4×C73C22×Q811yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73H(5,𝔽2)11yescharacteristic
229C2×C4×C732−1+411yescharacteristic
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—22no—
237C2×Dic73—21no—
238C2×Dic73—22no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—84no—
252C2×Dic73C22⋊Q812yes—
253C2×Dic73C22⋊Q812yes—
254C2×Dic73AGL(1,ℤ/8)12yes—
255Dic146—84no—
256Dic146—84no—
257C8×C73—41no—
258C8×C73—41no—
259#10596831—41no—
260#10596831—41no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73C2×C22⋊Q811yescharacteristic
269C22×C73#54611yescharacteristic
270C22×C73#54811yescharacteristic
271C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C73C2×#13711yescharacteristic
273C22×C73#57611yescharacteristic
274C22×C73#57711yescharacteristic
275C4×C73—22no—
276C4×C73—42no—
277C4×C73—41no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73#53311yescharacteristic
281C4×C73#54411yescharacteristic
282C4×C73#55511yescharacteristic
283C4×C73#55611yescharacteristic
284Dic73—22no—
285Dic73—22no—
286Dic73—22no—
287Dic73—42no—
288Dic73—42no—
289Dic73—41no—
290Dic73—81no—
291Dic73—81no—
292#26036—731nomaximal, Sylow
293C2×C73—82no—
294C2×C73#243911yescharacteristic
295C2×C73#307911yescharacteristic
296C2×C73#308111yescharacteristic
297C2×C73#331811yescharacteristic
298C2×C73#332011yescharacteristic
299C2×C73#337311yescharacteristic
300C2×C73#337511yescharacteristic
301#2551—731no—
302#2357—731no—
303#2067—731no—
304#2065D7311yescharacteristic
305#2023—731no—
306#2023—731no—
307#1988—731no—
308#1987—731no—
309#1949—731no—
310#1940—731no—
311#1921—731no—
312#1919—731no—
313#1836—731no—
314#1802—731no—
315#1762—731no—
316C73#2603611yescharacteristic, complemented, Sylow
317C2×C4⋊Q8—731no—
318C2×C4⋊Q8—731no—
319C2×C4⋊Q8—1461no—
320C2×C4×D4—731no—
321C2×#107—731no—
322C2×#106C2×D7311yescharacteristic
323C2×C4⋊C8—731no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8C2×D7311yescharacteristic
326C2×#102—731no—
327C2×#102—731no—
328C2×#102C2×D7311yescharacteristic
329C2×#102C2×D7311yescharacteristic
330C2×#97—731no—
331#395—1461no—
332#395—1461no—
333#390—731no—
334#390C2×D7311yescharacteristic
335#389—731no—
336#389—1461no—
337#389C2×D7311yescharacteristic
338#388—1461no—
339#388—1461no—
340#384—731no—
341#382—731no—
342#381—1461no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—731no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4—731no—
347C4×C22⋊C4—1461no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340C2×D7311yescharacteristic
358#336—731no—
359#336—731no—
360C22×Q8—1461no—
361C22×Q8—1461no—
362C22×D4C22×D7311yescharacteristic
363C23×C4—1461no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8C22×D7311yescharacteristic
367C22×C8C22×D7311yescharacteristic
368C4⋊Q8—2921no—
369C4⋊Q8—2924no—
370C4⋊Q8—2921no—
371C4⋊Q8—1462no—
372C4⋊Q8—1462no—
373C4×D4—734no—
374C4×D4—1462no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—1461no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42C22×D7311yescharacteristic
401#107—1462no—
402#106—22no—
403C4⋊C8—41no—
404C4⋊C8—2921no—
405C4⋊C8—1462no—
406#102—22no—
407#102—41no—
408#102—2921no—
409#102—1462no—
410#97—1462no—
411#95—21no—
412#95—731no—
413#95—731no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420C24—21no—
421C2×Q8—2922no—
422C2×Q8—2922no—
423C2×Q8—2922no—
424C2×Q8—2922no—
425C2×Q8—5841no—
426C2×Q8—5841no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4Q8×D7312yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—5841no—
437C22×C4—5841no—
438C22×C4—731no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1464no—
447C22×C4D4×D7311yescharacteristic
448C22×C4D4×D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—2922no—
469C4⋊C4—2922no—
470C4⋊C4—2922no—
471C4⋊C4—2922no—
472C4⋊C4—5841no—
473C4⋊C4—732no—
474C4⋊C4—732no—
475C4⋊C4—1462no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1462no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4#1192402712yes—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—2922no—
487C22⋊C4—5841no—
488C22⋊C4—1464no—
489C42—21no—
490C42—21no—
491C42—2922no—
492C42—5841no—
493C42—5841no—
494C42—732no—
495C42—732no—
496C42—1461no—
497C42—1461no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×D4×D7311yescharacteristic
506Q8—5844no—
507Q8—5844no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—5844no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—732no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4#1455662012yes—
552C2×C4C2×Q8×D7311yescharacteristic
553C2×C4#41×D7311yescharacteristic
554C2×C4#41×D7311yescharacteristic
555C2×C4C2×#1192402711yescharacteristic
556C2×C4#1455663611yescharacteristic
557C2×C4#1455663811yescharacteristic
558C8—41no—
559C8—41no—
560C8—2921no—
561C8—2921no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C22⋊Q8×D7311yescharacteristic
570C22#1798922611yescharacteristic
571C22#1798924311yescharacteristic
572C22AGL(1,ℤ/8)×D7311yescharacteristic
573C22C2×#1455662011yescharacteristic
574C22#1798939111yescharacteristic
575C22#1798939411yescharacteristic
576C4—22no—
577C4—41no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—2922no—
582C4—2922no—
583C4—2921no—
584C4—5841no—
585C4—5841no—
586C4—1462no—
587C4—1462no—
588C4—1462no—
589C4#1798913911yescharacteristic
590C4#1798921611yescharacteristic
591C4#1798930111yescharacteristic
592C4#1798930411yescharacteristic
593C2—82no—
594C2#2261361711yescharacteristic
595C2#2261974311yescharacteristic
596C2#2261976111yescharacteristic
597C2#2262206711yescharacteristic
598C2#2262210811yescharacteristic
599C2#2262271811yescharacteristic
600C2#2262274911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2065C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2065D73
hypocenterC73#26036
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. #22614959C2semidirect
6. #22610881C2semidirect
7. #22610878C2semidirect
8. #22610700C2semidirect
9. #22610651C2semidirect
10. #22610520C2semidirect
11. #22610511C2semidirect
12. #22610063C2semidirect
70. D4×Dic73Q8semidirect
316. C73#26036semidirect

Maximal subgroups

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

subgroupindex
#226149592
#226129032
#226113152
#226110672
#226110652
#226108812
#226108782
#226107002
#226106512
#226105202
#226105112
#226100632
#226099292
#226097582
C73×#20652
#2603673