Finite Group Databasebeta

← #33131124

Subgroups — #33131124

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614851C211yesmaximal, characteristic
2#22612843C211yesmaximal, characteristic, complemented
3#22611492C211yesmaximal, characteristic, complemented
4#22611488C211yesmaximal, characteristic, complemented
5#22611337C211yesmaximal, characteristic
6#22610865C211yesmaximal, characteristic, complemented
7#22610688C211yesmaximal, characteristic, complemented
8#22610553C211yesmaximal, characteristic
9#22610548C211yesmaximal, characteristic
10#22610505C211yesmaximal, characteristic, complemented
11#22610503C211yesmaximal, characteristic, complemented
12#22610057C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2071C211yesmaximal, characteristic
16C2×#14556568—21no—
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556556—21no—
19C2×D4×Dic73C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C8×Dic73C2211yescharacteristic
26#17988510—21no—
27#17988510—21no—
28#17988490C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483—21no—
32#17988482—21no—
33#17988482—21no—
34#17988456C2211yescharacteristic
35#17988456C2211yescharacteristic
36#17988456C2211yescharacteristic
37#17988445—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#102×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#384×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146D411yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556568—44no—
64C4×Dic146—22no—
65C4×Dic146—22no—
66#14556556—44no—
67D4×Dic73—22no—
68D4×Dic73—22no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73D411yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007D411yescharacteristic
83C2×#11924007D411yescharacteristic
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556525—22no—
93#14556524—22no—
94#14556520—22no—
95#14556512—22no—
96#14556512—22no—
97C8×Dic73—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C2×C4⋊C4×C73—21no—
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#102×C73—22no—
115#102×C73—22no—
116C2×Dic146—22no—
117C2×Dic146—22no—
118C2×Dic146D814yes—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—22no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145C4⋊Dic73#4112yes—
146#11924007—22no—
147#11924007—22no—
148#11924007—42no—
149#11924007—42no—
150#11924007—42no—
151#11924007—42no—
152#11924007—81no—
153#11924007—81no—
154#11924007—81no—
155C22⋊Dic73—42no—
156C22⋊Dic73—42no—
157C22⋊Dic73—81no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—42no—
163C4×Dic73#4112yes—
164C4×Dic73#4112yes—
165C24×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73#4112yes—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—42no—
191C4⋊C4×C73—81no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—81no—
194C42×C73—21no—
195C42×C73—21no—
196C23×C73—22no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73C22×D411yescharacteristic
204D4×C73—22no—
205D4×C73—22no—
206D4×C73—41no—
207D4×C73—41no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
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×C73C2×D812yes—
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—22no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—84no—
249C2×Dic73#12312yes—
250C2×Dic73#12312yes—
251C2×Dic73C2×D812yes—
252C2×Dic73C2×D812yes—
253C2×Dic73#13712yes—
254Dic146—28no—
255C8×C73—41no—
256C8×C73—41no—
257#10596831—41no—
258#10596831—41no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—42no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73C2×#12311yescharacteristic
267C22×C73#54811yescharacteristic
268C22×C73#54811yescharacteristic
269C22×C73C22×D811yescharacteristic
270C22×C73C2×#13711yescharacteristic
271C22×C73#57711yescharacteristic
272C22×C73#57711yescharacteristic
273C4×C73—22no—
274C4×C73—41no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73—81no—
278C4×C73#52911yescharacteristic
279C4×C73#54111yescharacteristic
280C4×C73#55211yescharacteristic
281C4×C73#55611yescharacteristic
282Dic73—22no—
283Dic73—22no—
284Dic73—41no—
285Dic73—42no—
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289Dic73#48014yes—
290#24990—731nomaximal, Sylow
291C2×C73—82no—
292C2×C73#242111yescharacteristic
293C2×C73C2×#48011yescharacteristic
294C2×C73#309911yescharacteristic
295C2×C73#335811yescharacteristic
296C2×C73#335811yescharacteristic
297C2×C73#336911yescharacteristic
298C2×C73#337211yescharacteristic
299#2538—731no—
300Q8×C4⋊C4—731no—
301#2099—731no—
302#2099—731no—
303#2071—731no—
304#2071D7311yescharacteristic
305#1984—731no—
306#1947—731no—
307#1925—731no—
308#1925—731no—
309#1918—731no—
310#1918—731no—
311#1834—731no—
312#1802—731no—
313C4×#102—731no—
314C73#2499011yescharacteristic, complemented, Sylow
315C2×C22⋊Q8—1461no—
316C2×C4×Q8—731no—
317C2×C4×Q8—1461no—
318C2×C4×D4—731no—
319C2×#107—731no—
320C2×#107—731no—
321C2×C4⋊C8—731no—
322C2×#102—731no—
323C2×#102—731no—
324C2×#102C2×D7311yescharacteristic
325C2×#102C2×D7311yescharacteristic
326C2×C4×C8—731no—
327#396—1461no—
328#396—1461no—
329#390—731no—
330#390C2×D7311yescharacteristic
331#389—731no—
332#389—731no—
333#389—1461no—
334#389—1461no—
335#389—1461no—
336#384—731no—
337#384—731no—
338#384—731no—
339#384C2×D7311yescharacteristic
340#380—1461no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343C4×C4⋊C4—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340C2×D7311yescharacteristic
353#340C2×D7311yescharacteristic
354#336—731no—
355#336—731no—
356#336—731no—
357#336C2×D7311yescharacteristic
358C22×Q8—731no—
359C22×D4C22×D7311yescharacteristic
360C23×C4—1461no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8C22×D7311yescharacteristic
364C22×C8C22×D7311yescharacteristic
365C22⋊Q8—2924no—
366C4×Q8—2924no—
367C4×Q8—1462no—
368C4×Q8—1462no—
369C4×D4—1462no—
370C4×D4—1462no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C22⋊C4—1461no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42—1461no—
401C2×C42C22×D7311yescharacteristic
402#107—1462no—
403#107—1462no—
404C4⋊C8—1462no—
405#102—22no—
406#102—22no—
407#102—1462no—
408#102—1462no—
409C4×C8—1462no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415C24—21no—
416C2×Q8—734no—
417C2×Q8—1462no—
418C2×Q8—1462no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4#1192402712yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—2922no—
429C22×C4—2922no—
430C22×C4—2922no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4D4×D7311yescharacteristic
440C22×C4D4×D7311yescharacteristic
441C22×C4C23×D7311yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—2922no—
463C4⋊C4—2922no—
464C4⋊C4—2922no—
465C4⋊C4—2922no—
466C4⋊C4—2922no—
467C4⋊C4—5841no—
468C4⋊C4—5841no—
469C4⋊C4—5841no—
470C4⋊C4—732no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—2922no—
484C22⋊C4—2922no—
485C22⋊C4—5841no—
486C42—21no—
487C42—21no—
488C42—2922no—
489C42—2922no—
490C42—2922no—
491C42—732no—
492C42—732no—
493C42—1461no—
494C42—1461no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×D4×D7311yescharacteristic
503Q8—1468no—
504D4—22no—
505D4—22no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—5844no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—732no—
537C2×C4—732no—
538C2×C4—732no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1462no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4D8×D7312yes—
548C2×C4#41×D7311yescharacteristic
549C2×C4#41×D7311yescharacteristic
550C2×C4#41×D7311yescharacteristic
551C2×C4C2×#1192402711yescharacteristic
552C2×C4#1455663811yescharacteristic
553C2×C4#1455663811yescharacteristic
554C8—41no—
555C8—41no—
556C8—2921no—
557C8—2921no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22#123×D7311yescharacteristic
566C22#1798924311yescharacteristic
567C22#1798924511yescharacteristic
568C22C2×D8×D7311yescharacteristic
569C22#137×D7311yescharacteristic
570C22#1798939311yescharacteristic
571C22#1798939711yescharacteristic
572C4—22no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—2921no—
578C4—2922no—
579C4—5841no—
580C4—5841no—
581C4—5841no—
582C4—734no—
583C4—1462no—
584C4—1462no—
585C4#1798911411yescharacteristic
586C4#1798919411yescharacteristic
587C4#1798927411yescharacteristic
588C4#1798930411yescharacteristic
589C2—82no—
590C2#2261342711yescharacteristic
591C2#480×D7311yescharacteristic
592C2#2261992111yescharacteristic
593C2#2262256111yescharacteristic
594C2#2262256311yescharacteristic
595C2#2262266811yescharacteristic
596C2#2262270711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2071C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2071D73
hypocenterC73#24990
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #22612843C2semidirect
3. #22611492C2semidirect
4. #22611488C2semidirect
6. #22610865C2semidirect
7. #22610688C2semidirect
10. #22610505C2semidirect
11. #22610503C2semidirect
12. #22610057C2semidirect
314. C73#24990semidirect

Maximal subgroups

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

subgroupindex
#226148512
#226128432
#226114922
#226114882
#226113372
#226108652
#226106882
#226105532
#226105482
#226105052
#226105032
#226100572
#226099292
#102×Dic732
C73×#20712
#2499073