Finite Group Databasebeta

← #33159085

Subgroups — #33159085

Order 18688.

680 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#22614364C211yesmaximal, characteristic
2#22611339C211yesmaximal, characteristic, complemented
3#22611326C211yesmaximal, characteristic, complemented
4#22611124C211yesmaximal, characteristic
5#22611124C211yesmaximal, characteristic
6#22610694C211yesmaximal, characteristic, complemented
7#22610671C211yesmaximal, characteristic, complemented
8#22610546C211yesmaximal, characteristic
9#22610516C211yesmaximal, characteristic, complemented
10#22610513C211yesmaximal, characteristic, complemented
11#22610199C211yesmaximal, characteristic, complemented
12#22610130C211yesmaximal, characteristic
13#22610026C211yesmaximal, characteristic
14#22609761C211yesmaximal, characteristic, complemented
15C73×#2362C211yesmaximal, characteristic
16C2×#14556524C2211yescharacteristic
17C2×#14556522C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28#17988529—21no—
29#17988510—21no—
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988480—21no—
33#17988474—21no—
34#17988457C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C22⋊Q8×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#388×C73—21no—
54#388×C73—21no—
55#387×C73—21no—
56C4×C4⋊C4×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×#11924007—21no—
70C2×#11924007—21no—
71C2×C22⋊Dic73—21no—
72C2×C22⋊Dic73—21no—
73C2×C4×Dic73—21no—
74#14556524—41no—
75#14556522—22no—
76#14556522—22no—
77#14556522—22no—
78C4⋊#10596831—22no—
79#14556515—22no—
80#14556515—22no—
81#14556515—22no—
82#14556515—41no—
83C22⋊#10596831—22no—
84C22⋊#10596831—41no—
85#14556502—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×Q8×C73C2311yescharacteristic
92C23×C4×C73C2311yescharacteristic
93C22⋊Q8×C73—22no—
94C22⋊Q8×C73—22no—
95C4×Q8×C73—22no—
96C4×Q8×C73—41no—
97#117×C73—22no—
98#117×C73—41no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73D411yescharacteristic
108C2×C22⋊C4×C73D411yescharacteristic
109C2×C42×C73—21no—
110C2×C42×C73—21no—
111C2×C42×C73—21no—
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C2×#10596831—21no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831#4112yes—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—81no—
145C4⋊Dic73—81no—
146#11924007—42no—
147#11924007—81no—
148C22⋊Dic73—81no—
149C22⋊Dic73—81no—
150C4×Dic73—81no—
151C24×C73C2×D411yescharacteristic
152C2×Q8×C73—21no—
153C2×Q8×C73—21no—
154C2×Q8×C73—21no—
155C2×Q8×C73—21no—
156C2×Q8×C73—21no—
157C2×Q8×C73#4112yes—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73—24no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73—42no—
181C4⋊C4×C73#4112yes—
182C4⋊C4×C73#4112yes—
183C22⋊C4×C73—24no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73—41no—
186C22⋊C4×C73—42no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—42no—
192C42×C73—42no—
193C42×C73—42no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73C22×D411yescharacteristic
202Q8×C73—22no—
203Q8×C73—22no—
204Q8×C73—41no—
205Q8×C73—41no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—22no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
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×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73#13712yes—
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73H(5,𝔽2)11yescharacteristic
242C2×C4×C732−1+411yescharacteristic
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—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255#10596831—22no—
256#10596831—41no—
257#10596831—41no—
258#10596831—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73—41no—
266C22×C73#52511yescharacteristic
267C22×C73#54611yescharacteristic
268C22×C73#54711yescharacteristic
269C22×C73C2×#13711yescharacteristic
270C22×C73#57511yescharacteristic
271C22×C73#57711yescharacteristic
272C22×C73#57711yescharacteristic
273C4×C73—22no—
274C4×C73—41no—
275C4×C73—42no—
276C4×C73—41no—
277C4×C73—42no—
278C4×C73—41no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—81no—
282C4×C73#52911yescharacteristic
283C4×C73#53911yescharacteristic
284C4×C73#55311yescharacteristic
285C4×C73#55611yescharacteristic
286Dic73—81no—
287Dic73—81no—
288Dic73—81no—
289Dic73—81no—
290#27060—731nomaximal, Sylow
291C2×C73—81no—
292C2×C73#246611yescharacteristic
293C2×C73#311611yescharacteristic
294C2×C73#312611yescharacteristic
295C2×C73#332111yescharacteristic
296C2×C73#333111yescharacteristic
297C2×C73#334411yescharacteristic
298C2×C73#335011yescharacteristic
299#2497—731no—
300#2362D7311yescharacteristic
301#2072—731no—
302#2068—731no—
303#2035—731no—
304#2035—731no—
305#1948—731no—
306#1944—731no—
307#1924—731no—
308#1920—731no—
309#1920—731no—
310#1865—731no—
311#1854—731no—
312#1825—731no—
313#1763—731no—
314C73#2706011yescharacteristic, complemented, Sylow
315C2×C22⋊Q8C2×D7311yescharacteristic
316C2×C4×Q8C2×D7311yescharacteristic
317C2×#117C2×D7311yescharacteristic
318C2×#107—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×#106—731no—
322C2×C4⋊C8—731no—
323C2×#103—731no—
324C2×#103—731no—
325C2×#103—731no—
326C2×#103—731no—
327C2×C22⋊C8—731no—
328C2×C22⋊C8—731no—
329C2×#97—731no—
330#400—1461no—
331#396—1461no—
332#391C2×D7311yescharacteristic
333#389—731no—
334#389—731no—
335#389C2×D7311yescharacteristic
336#388—21no—
337#388—21no—
338#388—1461no—
339#387—21no—
340#387—1461no—
341#384—731no—
342#380—731no—
343#380—731no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4—731no—
346C4×C4⋊C4C2×D7311yescharacteristic
347C4×C22⋊C4C2×D7311yescharacteristic
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#336—731no—
357#336—731no—
358C22×Q8C22×D7311yescharacteristic
359C23×C4C22×D7311yescharacteristic
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8—731no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C4×Q8—22no—
367C4×Q8—41no—
368#117—22no—
369#117—41no—
370C2×C4⋊C4—21no—
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—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4C22×D7311yescharacteristic
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4C22⋊D7311yescharacteristic
389C2×C22⋊C4C22⋊D7311yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—1461no—
394C2×C42C22×D7311yescharacteristic
395C2×C42C22×D7311yescharacteristic
396#107—2921no—
397#106—1462no—
398#106—1462no—
399#106—1462no—
400C4⋊C8—1462no—
401#103—2921no—
402#103—1462no—
403#103—1462no—
404#103—1462no—
405C22⋊C8—2921no—
406C22⋊C8—1462no—
407#97—2921no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95C22⋊D7311yescharacteristic
418#95C22⋊D7311yescharacteristic
419C24C2×C22⋊D7311yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#1192402812yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C2×C22⋊D7311yescharacteristic
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C23×D7311yescharacteristic
446C2×C8—732no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—2922no—
466C4⋊C4—5841no—
467C4⋊C4—5841no—
468C4⋊C4—5841no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4#1192402712yes—
478C4⋊C4#1192402812yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—5841no—
484C22⋊C4—5841no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—5841no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D7311yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4#1455662112yes—
548C2×C4#41×D7311yescharacteristic
549C2×C4C2×#1192402711yescharacteristic
550C2×C4C2×#1192402811yescharacteristic
551C2×C4C2×#1192402811yescharacteristic
552C2×C4#1455663611yescharacteristic
553C2×C4#1455663811yescharacteristic
554C8—2921no—
555C8—2921no—
556C8—2921no—
557C8—1462no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1798909811yescharacteristic
566C22#1798923111yescharacteristic
567C22#1798923611yescharacteristic
568C22C2×#1455662111yescharacteristic
569C22#1798938511yescharacteristic
570C22#1798939511yescharacteristic
571C22#1798939511yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—5841no—
582C4—5841no—
583C4—5841no—
584C4—5841no—
585C4#1798911711yescharacteristic
586C4#1798917911yescharacteristic
587C4#1798928711yescharacteristic
588C4#1798930711yescharacteristic
589C2—81no—
590C2#2261396111yescharacteristic
591C2#2262009311yescharacteristic
592C2#2262022211yescharacteristic
593C2#2262211711yescharacteristic
594C2#2262224711yescharacteristic
595C2#2262238611yescharacteristic
596C2#2262247611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2362C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2362D73
hypocenterC73#27060
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #22611339C2semidirect
3. #22611326C2semidirect
6. #22610694C2semidirect
7. #22610671C2semidirect
9. #22610516C2semidirect
10. #22610513C2semidirect
11. #22610199C2semidirect
14. #22609761C2semidirect
314. C73#27060semidirect

Maximal subgroups

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

subgroupindex
#226143642
#226113392
#226113262
#226111242
#226106942
#226106712
#226105462
#226105162
#226105132
#226101992
#226101302
#226100262
#226097612
C73×#23622
#2706073