Finite Group Databasebeta

← #34029539

Subgroups — #34029539

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626102C211yesmaximal, characteristic, complemented
2#22626101C211yesmaximal, characteristic, complemented
3#22626031C211yesmaximal, characteristic, complemented
4#22626029C211yesmaximal, characteristic, complemented
5#22626007C211yesmaximal, characteristic
6#22626003C211yesmaximal, characteristic
7#22626003C211yesmaximal, characteristic
8#22626003C211yesmaximal, characteristic
9#22625988C211yesmaximal, characteristic, complemented
10#22625986C211yesmaximal, characteristic, complemented
11#22625984C211yesmaximal, characteristic, complemented
12#22625984C211yesmaximal, characteristic, complemented
13#22625825C211yesmaximal, characteristic
14#22625790C211yesmaximal, characteristic
15#22613208C211yesmaximal, characteristic
16C2×#14556658C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×C4⋊#10596833C2211yescharacteristic
19C2×C4⋊#10596833C2211yescharacteristic
20C2×C4⋊#10596833C2211yescharacteristic
21C2×C4⋊#10596833C2211yescharacteristic
22C2×#14556648C2211yescharacteristic
23C2×#14556648C2211yescharacteristic
24C2×#14556648C2211yescharacteristic
25C2×#14556648C2211yescharacteristic
26C2×#14556646C2211yescharacteristic
27C2×C4×#10596833C2211yescharacteristic
28C2×C4×#10596833C2211yescharacteristic
29C2×C4×#10596833C2211yescharacteristic
30#17989453C2211yescharacteristic
31#17989453C2211yescharacteristic
32#17989453C2211yescharacteristic
33#17989453C2211yescharacteristic
34#17989453C2211yescharacteristic
35#17989453C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44C2×#14556556C2211yescharacteristic
45C2×#14556549C2211yescharacteristic
46#17988529C411yescharacteristic
47#17988522C411yescharacteristic
48#17988520C411yescharacteristic
49#17988509C411yescharacteristic
50#17988509C411yescharacteristic
51#17988508C411yescharacteristic
52#17988500C411yescharacteristic
53C4⋊C4×Dic73C2211yescharacteristic
54C4⋊C4×Dic73C2211yescharacteristic
55#17988438C2211yescharacteristic
56#17988434C2211yescharacteristic
57#17988434C2211yescharacteristic
58#398×C73C411yescharacteristic
59C22×#10596833C2311yescharacteristic
60C22×#10596833C2311yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67#14556658—41no—
68#14556658—41no—
69C4⋊#10596833—41no—
70C4⋊#10596833—41no—
71C4⋊#10596833—41no—
72C4⋊#10596833—41no—
73#14556648—41no—
74#14556648—41no—
75#14556648—41no—
76#14556648—41no—
77#14556646—41no—
78C4×#10596833—41no—
79C4×#10596833—41no—
80C4×#10596833—41no—
81C22×Dic146C2×C411yescharacteristic
82C22×C4×D73C2311yescharacteristic
83#14556556—41no—
84#14556556—41no—
85#14556549—41no—
86#14556549—41no—
87C2×C4⋊Dic73C2×C411yescharacteristic
88C2×C4⋊Dic73C2×C411yescharacteristic
89C2×C4⋊Dic73C2×C411yescharacteristic
90C2×#11924007C2×C411yescharacteristic
91C2×#11924007C2×C411yescharacteristic
92C2×#11924007C2×C411yescharacteristic
93C2×#11924007C2×C411yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C2×C4×Dic73C2311yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113C2×C4⋊C4×C73C2×C411yescharacteristic
114C2×C4⋊C4×C73C2×C411yescharacteristic
115C2×C4⋊C4×C73C2×C411yescharacteristic
116C2×C4⋊C4×C73C2×C411yescharacteristic
117#95×C73C2×C411yescharacteristic
118#95×C73C2×C411yescharacteristic
119#95×C73C2×C411yescharacteristic
120C2×#10596833—21no—
121C2×#10596833—21no—
122C2×#10596833—21no—
123C2×#10596833—21no—
124C2×#10596833—21no—
125C2×#10596833—21no—
126C2×#10596833—21no—
127C2×#10596833—21no—
128C2×#10596833—21no—
129C2×#10596833—21no—
130C2×#10596833—21no—
131C2×#10596833—21no—
132C2×#10596833—21no—
133C2×#10596833—21no—
134C2×#10596833—21no—
135C2×#10596833—21no—
136C2×#10596833—21no—
137C2×#10596833—21no—
138C2×#10596833—21no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
143C2×#10596833—21no—
144C23×D73C22×C411yescharacteristic
145C2×Dic146—22no—
146C2×Dic146—22no—
147C2×Dic146—41no—
148C22×Dic73C22×C411yescharacteristic
149C22×Dic73C22×C411yescharacteristic
150C22×Dic73C22×C411yescharacteristic
151C22×Dic73C22×C411yescharacteristic
152C22×Dic73C22×C411yescharacteristic
153C22×Dic73C22×C411yescharacteristic
154C22×Dic73C2411yescharacteristic
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C4⋊Dic73—22no—
162C4⋊Dic73—41no—
163C4⋊Dic73—41no—
164#11924007—22no—
165#11924007—41no—
166#11924007—41no—
167#11924007—41no—
168#11924005—22no—
169#11924005—22no—
170#11924005—22no—
171#11924005—41no—
172#11924005—41no—
173#11924005—41no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—41no—
196C4⋊C4×C73—41no—
197C23×C73C23×C411yescharacteristic
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73—21no—
203C22×D73—21no—
204C22×D73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
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—21no—
224C2×C4×C73#13112yes—
225C2×C4×C73#13112yes—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73#13112yes—
244C2×Dic73H(5,𝔽2)11yescharacteristic
245C2×Dic73H(5,𝔽2)11yescharacteristic
246C2×Dic73H(5,𝔽2)11yescharacteristic
247C2×Dic732−1+411yescharacteristic
248C2×Dic732−1+411yescharacteristic
249C2×Dic732−1+411yescharacteristic
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—41no—
254Dic146—44no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258#10596833—41no—
259#10596833—41no—
260#10596833—41no—
261#10596833—41no—
262#10596833—41no—
263C22×C73#51811yescharacteristic
264C22×C73#51911yescharacteristic
265C22×C73#52011yescharacteristic
266C22×C73C2×#13111yescharacteristic
267C22×C73C2×#13111yescharacteristic
268C22×C73C2×#13111yescharacteristic
269C22×C73#56811yescharacteristic
270C2×D73—41no—
271C2×D73—41no—
272C2×D73—41no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C2×D73—41no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—41no—
284Dic73—22no—
285Dic73—41no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289Dic73—41no—
290Dic73#56211yescharacteristic
291Dic73#56211yescharacteristic
292Dic73#56311yescharacteristic
293Dic73#56311yescharacteristic
294#23340—731nomaximal, Sylow
295C2×C73#237811yescharacteristic
296C2×C73#297711yescharacteristic
297C2×C73#298111yescharacteristic
298C2×C73#298511yescharacteristic
299C2×C73#298611yescharacteristic
300C2×C73#299311yescharacteristic
301C2×C73#299411yescharacteristic
302D73—81no—
303#2396—731no—
304#1984—731no—
305#1984—731no—
306#1940—731no—
307#1939—731no—
308#1924—731no—
309#1923—731no—
310#1923—731no—
311#1923—731no—
312#1917—731no—
313#1916—731no—
314#1916—731no—
315#1916—731no—
316#1818—731no—
317#1798—731no—
318C73#2334011yescharacteristic, complemented, Sylow
319C2×C4×Q8—731no—
320C2×#117—731no—
321C2×C4⋊C8—731no—
322C2×C4⋊C8—731no—
323C2×C4⋊C8—731no—
324C2×C4⋊C8—731no—
325C2×C4⋊C8—731no—
326C2×C4⋊C8—731no—
327C2×C22⋊C8—731no—
328C2×C22⋊C8—731no—
329C2×C22⋊C8—731no—
330C2×C22⋊C8—731no—
331C2×#97—731no—
332C2×C4×C8—731no—
333C2×C4×C8—731no—
334C2×C4×C8—731no—
335#400—731no—
336#399—731no—
337#398—731no—
338#398#6440811yescharacteristic
339#396—731no—
340#396—731no—
341#395—731no—
342#393—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—731no—
345C4×C4⋊C4—731no—
346C4×C22⋊C4—731no—
347C4×C22⋊C4—731no—
348#336—731no—
349#336—731no—
350#336—731no—
351#336—731no—
352#336—731no—
353#336—731no—
354#336—731no—
355#336—731no—
356#336—731no—
357#336—731no—
358#336—731no—
359#336—731no—
360#336—731no—
361#336—731no—
362C22×Q8—731no—
363C23×C4—731no—
364C22×C8—731no—
365C22×C8—731no—
366C22×C8—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370C22×C8—731no—
371C22×C8—731no—
372C4×Q8—2921no—
373C4×Q8—2921no—
374#117—2921no—
375#117—2921no—
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—731no—
383C2×C4⋊C4C2×#6440811yescharacteristic
384C2×C4⋊C4C2×#6440811yescharacteristic
385C2×C4⋊C4C2×#6440811yescharacteristic
386C2×C4⋊C4C2×#6440811yescharacteristic
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4—731no—
392C2×C22⋊C4—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—731no—
399C4⋊C8—2921no—
400C4⋊C8—2921no—
401C4⋊C8—2921no—
402C4⋊C8—2921no—
403C4⋊C8—2921no—
404C4⋊C8—2921no—
405C22⋊C8—2921no—
406C22⋊C8—2921no—
407C22⋊C8—2921no—
408C22⋊C8—2921no—
409#97—2921no—
410C4×C8—2921no—
411C4×C8—2921no—
412C4×C8—2921no—
413#95—731no—
414#95—731no—
415#95—731no—
416#95—731no—
417#95—731no—
418#95—731no—
419#95—731no—
420#95C2×#6440811yescharacteristic
421#95C2×#6440811yescharacteristic
422#95C2×#6440811yescharacteristic
423C24—731no—
424C2×Q8—2921no—
425C2×Q8—1462no—
426C2×Q8—1462no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4C22×#6440811yescharacteristic
441C22×C4C22×#6440811yescharacteristic
442C22×C4C22×#6440811yescharacteristic
443C22×C4C22×#6440811yescharacteristic
444C22×C4C22×#6440811yescharacteristic
445C22×C4C22×#6440811yescharacteristic
446C22×C4C22×#6440811yescharacteristic
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—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—2921no—
476C4⋊C4—2921no—
477C4⋊C4—2921no—
478C4⋊C4—2921no—
479C4⋊C4—2921no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C22⋊C4—2921no—
483C22⋊C4—2921no—
484C22⋊C4—2921no—
485C22⋊C4—1462no—
486C22⋊C4—1462no—
487C22⋊C4—1462no—
488C42—1461no—
489C42—1461no—
490C42—1461no—
491C42—1461no—
492C42—1461no—
493C42—1461no—
494C42—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C23—1461no—
501C23—1461no—
502C23—1461no—
503C23—1461no—
504C23—1461no—
505C23—1461no—
506C23—1461no—
507C23C23×#6440811yescharacteristic
508Q8—2924no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—731no—
533C2×C4—731no—
534C2×C4—731no—
535C2×C4—731no—
536C2×C4—731no—
537C2×C4—731no—
538C2×C4—732no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4#1455667812yes—
557C2×C4#1455667912yes—
558C8—2921no—
559C8—2921no—
560C8—2921no—
561C8—2921no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—2921no—
566C22—2921no—
567C22—2921no—
568C22—2921no—
569C22—2921no—
570C22—2921no—
571C22—2921no—
572C22—2921no—
573C22#1798965411yescharacteristic
574C22#1798965511yescharacteristic
575C22#1798965611yescharacteristic
576C22C2×#1455667811yescharacteristic
577C22D73⋊#13111yescharacteristic
578C22C2×#1455667911yescharacteristic
579C22#1798966611yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—2921no—
588C4—2921no—
589C4—2921no—
590C4—2921no—
591C4—2921no—
592C4—731no—
593C4—731no—
594C4—731no—
595C4—731no—
596C4—1462no—
597C2—5841no—
598C2#2262647311yescharacteristic
599C2#2262680611yescharacteristic
600C2#2262682111yescharacteristic
601C2#2262683811yescharacteristic
602C2#2262684211yescharacteristic
603C2#2262687711yescharacteristic
604C2#2262687911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#398×C73C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#398#64408
hypocenterC73#23340
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #22626102C2semidirect
2. #22626101C2semidirect
3. #22626031C2semidirect
4. #22626029C2semidirect
9. #22625988C2semidirect
10. #22625986C2semidirect
11. #22625984C2semidirect
12. #22625984C2semidirect
318. C73#23340semidirect

Maximal subgroups

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

subgroupindex
#226261022
#226261012
#226260312
#226260292
#226260072
#226260032
#226259882
#226259862
#226259842
#226258252
#226257902
#226132082
#2334073