Finite Group Databasebeta

← #33052148

Subgroups — #33052148

Order 18688.

1012 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989188C211yesmaximal, characteristic
2#22618960C211yesmaximal, characteristic, complemented
3#22608292C212yesmaximal, complemented
4#22608290C212yesmaximal, complemented
5#22608280C212yesmaximal, complemented
6#22608277C212yesmaximal, complemented
7#22608271C212yesmaximal, complemented
8#22608263C212yesmaximal, complemented
9C73×#2973C211yesmaximal, characteristic, complemented
10#17989188—24no—
11#17989188—24no—
12C2×#14556587C411yescharacteristic
13C2×#14556582C411yescharacteristic
14C2×#14556574C2211yescharacteristic
15C2×#14556574C2211yescharacteristic
16C2×#14556573C411yescharacteristic
17C2×#14556573C411yescharacteristic
18C2×#14556566C411yescharacteristic
19C2×#14556565C2211yescharacteristic
20C2×#14556565C2211yescharacteristic
21C2×#14556564C411yescharacteristic
22C2×#14556564C411yescharacteristic
23C2×#14556562C411yescharacteristic
24C2×C4×D292C2211yescharacteristic
25C2×#14556548C2211yescharacteristic
26#17988659—24no—
27#17988644C2212yes—
28#17988551C2212yes—
29C2×C22⋊#10596831C2212yes—
30#17988312C2214yes—
31#17988310C2214yes—
32#17988310C2214yes—
33#17988310C2214yes—
34C2×C4×D4×C73C2211yescharacteristic
35#435×C73—24no—
36#432×C73C2212yes—
37#407×C73C2212yes—
38C2×C22⋊C8×C73C2212yes—
39C22×D292C2×C411yescharacteristic
40C22×D292C2×C411yescharacteristic
41C22×D292C2×C411yescharacteristic
42C22×C22⋊D73C2×C411yescharacteristic
43C22×C22⋊D73C2×C411yescharacteristic
44C22×C22⋊D73C2×C411yescharacteristic
45C22×C22⋊D73C2×C411yescharacteristic
46C22×C22⋊D73C2×C411yescharacteristic
47C22×C22⋊D73C2×C411yescharacteristic
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C2×#11924012—22no—
52C22×#10596831—22no—
53#14556587—24no—
54#14556582—24no—
55#14556574—22no—
56#14556574—22no—
57#14556574—22no—
58#14556574—22no—
59#14556573—24no—
60#14556573—24no—
61#14556566—24no—
62#14556565—22no—
63#14556565—22no—
64#14556565—22no—
65#14556565—22no—
66#14556564—24no—
67#14556564—24no—
68#14556562—24no—
69C4×D292—22no—
70C4×D292—22no—
71#14556548—22no—
72#14556548—22no—
73C2×C4⋊Dic73C2×C411yescharacteristic
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×C4⋊Dic73C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2311yescharacteristic
80C2×C22⋊Dic73C2×C411yescharacteristic
81C2×C22⋊Dic73C2×C411yescharacteristic
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2311yescharacteristic
86C2×#11924005C2311yescharacteristic
87C2×#11924005C2311yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C4⋊#10596831—22no—
90C22⋊#10596831—22no—
91C22⋊#10596831—22no—
92C22⋊#10596831C2314yes—
93#14556502—22no—
94C22×D4×C73C2×C411yescharacteristic
95C23×C4×C73C2311yescharacteristic
96C23×C4×C73C2311yescharacteristic
97C2×M16×C73—22no—
98C22×C8×C73—22no—
99C4×D4×C73D414yes—
100C4×D4×C73D414yes—
101C2×C4⋊C4×C73C2×C411yescharacteristic
102C2×C22⋊C4×C73C2×C411yescharacteristic
103C2×C22⋊C4×C73C2×C411yescharacteristic
104C2×C42×C73C2311yescharacteristic
105C4⋊C8×C73—22no—
106C22⋊C8×C73—22no—
107C22⋊C8×C73—22no—
108C22⋊C8×C73C2314yes—
109#97×C73—22no—
110C23×D73C22×C411yescharacteristic
111C23×D73C22×C411yescharacteristic
112C23×D73C22×C411yescharacteristic
113C2×D292—22no—
114C2×D292—22no—
115C2×D292—22no—
116C2×D292—42no—
117C2×D292—81no—
118C2×D292—81no—
119C2×C22⋊D73—22no—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—22no—
122C2×C22⋊D73—22no—
123C2×C22⋊D73—22no—
124C2×C22⋊D73—22no—
125C2×C22⋊D73—81no—
126C2×C22⋊D73—81no—
127C2×C22⋊D73—81no—
128C2×C22⋊D73—81no—
129C2×C22⋊D73—81no—
130C2×C22⋊D73—81no—
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C2×C4×D73—22no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—42no—
140C2×C4×D73—81no—
141C2×C4×D73—81no—
142#11924012—82no—
143C2×#10596831—24no—
144C2×#10596831—82no—
145C4⋊Dic73—22no—
146C4⋊Dic73—22no—
147C4⋊Dic73—22no—
148#11924007—22no—
149#11924007—22no—
150#11924007—22no—
151#11924007—21no—
152#11924007—21no—
153C22⋊Dic73—22no—
154C22⋊Dic73—22no—
155#11924005—22no—
156#11924005—22no—
157#11924005—22no—
158#11924005—21no—
159#11924005—21no—
160#11924005—21no—
161#11924005—21no—
162#11924005—21no—
163#11924005—21no—
164C4×Dic73—22no—
165C24×C73C22×C411yescharacteristic
166C24×C73C22×C411yescharacteristic
167C2×D4×C73—42no—
168C2×D4×C73C22⋊C414yes—
169C22×C4×C73—81no—
170C22×C4×C73—81no—
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D412yes—
177C22×C4×C73C2×D412yes—
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2411yescharacteristic
180M16×C73—82no—
181C2×C8×C73—24no—
182C2×C8×C73—82no—
183C4⋊C4×C73C22⋊C414yes—
184C22⋊C4×C73C22⋊C414yes—
185C22⋊C4×C73C22⋊C414yes—
186C42×C73C2×D412yes—
187C42×C73C2×D412yes—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73C2×C22⋊C412yes—
195C23×C73C2×C22⋊C412yes—
196C23×C73C23×C411yescharacteristic
197C22×D73—21no—
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
207C22×D73—41no—
208C22×D73—41no—
209D4×C73—44no—
210C2×C4×C73—22no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—41no—
219C2×C4×C73C2×C22⋊C412yes—
220C2×C4×C73C2×C22⋊C412yes—
221C2×C4×C73C2×C22⋊C412yes—
222C2×C4×C73C2×C22⋊C412yes—
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C22×D411yescharacteristic
225C22⋊D73—82no—
226C22⋊D73—82no—
227C22⋊D73—82no—
228C22⋊D73—82no—
229C22⋊D73—82no—
230C22⋊D73—82no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—22no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241D292—44no—
242D292—82no—
243D292—82no—
244C4×D73—44no—
245C4×D73—82no—
246C4×D73—82no—
247C8×C73—44no—
248#10596831—44no—
249C22×C73—21no—
250C22×C73—21no—
251C22×C73—22no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—82no—
258C22×C73—82no—
259C22×C73C22×C22⋊C411yescharacteristic
260C22×C73#56812yes—
261C2×D73—41no—
262C2×D73—41no—
263C2×D73—41no—
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C2×D73—41no—
270C2×D73—82no—
271C2×D73—82no—
272C2×D73—82no—
273C4×C73—22no—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—22no—
279Dic73—41no—
280Dic73—41no—
281Dic73—41no—
282Dic73—41no—
283Dic73—41no—
284#17408—731nomaximal, Sylow
285C2×C73—22no—
286C2×C73—42no—
287C2×C73—42no—
288C2×C73#287911yescharacteristic
289C2×C73#288311yescharacteristic
290C2×C73#288811yescharacteristic
291D73—81no—
292D73—81no—
293D73—81no—
294C2×#540—731no—
295#2973—731no—
296#2973D7311yescharacteristic, complemented
297#1471—732no—
298#1471—732no—
299#1468—732no—
300#1468—732no—
301#1466—732no—
302#1464—732no—
303C73#1740811yescharacteristic, complemented, Sylow
304#540—1464no—
305#540—1464no—
306C2×#127—731no—
307C2×#125—731no—
308C2×#123—731no—
309C2×#123—731no—
310C2×#123—731no—
311C2×#123—731no—
312C2×#121—731no—
313C2×#121—731no—
314C2×#121—731no—
315C2×#121—731no—
316C2×#121—731no—
317C2×#121—731no—
318C2×C4×D4—731no—
319C2×C4×D4C2×D7311yescharacteristic
320C2×#117—731no—
321#435—24no—
322#435—1464no—
323#432—732no—
324#432C2×D7312yes—
325#407—732no—
326#407C2×D7312yes—
327C2×C22⋊C8—732no—
328C2×C22⋊C8C2×D7312yes—
329#324—734no—
330#323—734no—
331#323—734no—
332#323—734no—
333C22×D4—731no—
334C22×D4—731no—
335C22×D4—731no—
336C22×D4—731no—
337C22×D4—731no—
338C22×D4—731no—
339C22×D4—731no—
340C22×D4—731no—
341C22×D4—731no—
342C22×D4C4×D7311yescharacteristic
343C23×C4—731no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4C22×D7311yescharacteristic
347C23×C4C22×D7311yescharacteristic
348C2×M16—22no—
349C2×M16—1462no—
350C22×C8—22no—
351C22×C8—1462no—
352#127—1464no—
353#125—1464no—
354#123—1462no—
355#123—1462no—
356#123—1462no—
357#123—1462no—
358#123—1464no—
359#123—1464no—
360#121—1464no—
361#121—1462no—
362#121—1462no—
363#121—1462no—
364#121—1462no—
365#121—1464no—
366#121—1464no—
367#121—1464no—
368C4×D4—1462no—
369C4×D4—1462no—
370C4×D4D29214yes—
371C4×D4C22⋊D7314yes—
372#117—1462no—
373#117—1462no—
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—731no—
381C2×C4⋊C4C4×D7311yescharacteristic
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4C4×D7311yescharacteristic
391C2×C22⋊C4C4×D7311yescharacteristic
392C2×C42—731no—
393C2×C42C22×D7311yescharacteristic
394C4⋊C8—22no—
395C4⋊C8—1462no—
396C22⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—734no—
399C22⋊C8—1462no—
400C22⋊C8—1462no—
401C22⋊C8C22×D7314yes—
402#97—22no—
403#97—1462no—
404C24—731no—
405C24—731no—
406C24—731no—
407C24C2×C4×D7311yescharacteristic
408C24C2×C4×D7311yescharacteristic
409C2×D4—42no—
410C2×D4—2922no—
411C2×D4—5841no—
412C2×D4—5841no—
413C2×D4—5841no—
414C2×D4—5841no—
415C2×D4—5841no—
416C2×D4—5841no—
417C2×D4—5841no—
418C2×D4—5841no—
419C2×D4—1462no—
420C2×D4—1462no—
421C2×D4—1462no—
422C2×D4—1462no—
423C2×D4—1462no—
424C2×D4—1462no—
425C2×D4—1462no—
426C2×D4—1462no—
427C2×D4—1462no—
428C2×D4#1192400514yes—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—2922no—
432C22×C4—5841no—
433C22×C4—5841no—
434C22×C4—731no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4C2×C4×D7311yescharacteristic
443C22×C4C2×C4×D7311yescharacteristic
444C22×C4C2×C4×D7311yescharacteristic
445C22×C4C2×C4×D7311yescharacteristic
446C22×C4C2×C22⋊D7312yes—
447C22×C4C2×C22⋊D7312yes—
448C22×C4C2×D29212yes—
449C22×C4C2×D29212yes—
450C22×C4C23×D7311yescharacteristic
451M16—82no—
452M16—5842no—
453C2×C8—24no—
454C2×C8—82no—
455C2×C8—5842no—
456C2×C8—1464no—
457C4⋊C4—1462no—
458C4⋊C4—1462no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C4⋊C4—1462no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4#1192400514yes—
466C22⋊C4—1462no—
467C22⋊C4—1462no—
468C22⋊C4—1462no—
469C22⋊C4—1462no—
470C22⋊C4—1462no—
471C22⋊C4—1461no—
472C22⋊C4—1461no—
473C22⋊C4—1461no—
474C22⋊C4—1461no—
475C22⋊C4—1461no—
476C22⋊C4—1461no—
477C22⋊C4#1192400514yes—
478C22⋊C4#1192400514yes—
479C42—1462no—
480C42C2×C22⋊D7312yes—
481C42C2×D29212yes—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—2921no—
489C23—2921no—
490C23—2921no—
491C23—2921no—
492C23—2921no—
493C23—2921no—
494C23—1461no—
495C23—1461no—
496C23—1461no—
497C23—1462no—
498C23—1462no—
499C23—1462no—
500C23C2×#1192400512yes—
501C23C2×#1192400512yes—
502C23C22×C4×D7311yescharacteristic
503D4—44no—
504D4—2924no—
505D4—5842no—
506D4—5842no—
507D4—5842no—
508D4—5842no—
509D4—5842no—
510D4—5842no—
511D4—5842no—
512D4—5842no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2924no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—5842no—
528C2×C4—5842no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1462no—
535C2×C4C2×#1192400512yes—
536C2×C4C2×#1192400512yes—
537C2×C4C2×#1192400512yes—
538C2×C4C2×#1192400512yes—
539C2×C4C22×C22⋊D7311yescharacteristic
540C2×C4C22×D29211yescharacteristic
541C8—44no—
542C8—2924no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—82no—
552C22—82no—
553C22—2921no—
554C22—2921no—
555C22—2921no—
556C22—2921no—
557C22—2921no—
558C22—2921no—
559C22—2921no—
560C22—2921no—
561C22—2921no—
562C22—5842no—
563C22—5842no—
564C22—5842no—
565C22C22×#1192400511yescharacteristic
566C22#1798935312yes—
567C4—22no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—2921no—
574C4—2921no—
575C4—2921no—
576C4—2921no—
577C4—2921no—
578C2—22no—
579C2—42no—
580C2—42no—
581C2—5841no—
582C2—5841no—
583C2—5841no—
584C2#2261837511yescharacteristic
585C2#2261839111yescharacteristic
586C2#2261840911yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2973C2
socleC22×C73C22×C22⋊C4
prime radicalC2×#17989188C2
cosocleC22×C4×C73C24
hypercenter#2973D73
hypocenterC73#17408
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #22618960C2semidirect
3. #22608292C2semidirect
4. #22608290C2semidirect
5. #22608280C2semidirect
6. #22608277C2semidirect
7. #22608271C2semidirect
8. #22608263C2semidirect
9. C73×#2973C2semidirect
296. #2973D73semidirect
303. C73#17408semidirect

Maximal subgroups

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

subgroupindex
C2×#179891882
#226189602
#226082922
#226082902
#226082802
#226082772
#226082712
#226082632
C73×#29732
#1740873