Finite Group Databasebeta

← #33246136

Subgroups — #33246136

Order 18688.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989055C211yesmaximal, characteristic
2#22612901C214yesmaximal, complemented
3#22612893C214yesmaximal, complemented
4#22612870C214yesmaximal, complemented
5#22612866C214yesmaximal, complemented
6#22612541C211yesmaximal, characteristic, complemented
7#22612540C211yesmaximal, characteristic, complemented
8#22612539C211yesmaximal, characteristic, complemented
9#22612520C212yesmaximal
10#22612336C211yesmaximal, characteristic, complemented
11#22612327C212yesmaximal, complemented
12#22612326C212yesmaximal, complemented
13#382×D73C211yesmaximal, characteristic, complemented
14#22612285C212yesmaximal, complemented
15C73×#2314C211yesmaximal, characteristic, complemented
16C22×#11924028C2211yescharacteristic
17#17989055C4116yescomplemented
18C2×#14556556C2212yes—
19C2×#14556553C2212yes—
20C2×#14556551C2212yes—
21C2×#14556551C2212yes—
22C2×#14556549C2212yes—
23C2×#14556548C2211yescharacteristic
24C2×#14556548C2211yescharacteristic
25C2×C4⋊C4×D73C2212yes—
26C22×C4×Dic73C2211yescharacteristic
27C2×C42×D73C2211yescharacteristic
28C2×C42×D73C2211yescharacteristic
29#17988491C2214yes—
30#17988487C2214yes—
31#17988480C2214yes—
32#17988479C2214yes—
33#17988477C2214yes—
34#17988474C2214yes—
35#17988473C2214yes—
36#17988468C2214yes—
37#17988465C2214yes—
38#17988463C2214yes—
39#17988457C2211yescharacteristic
40#17988454C2214yes—
41#17988454C2214yes—
42#17988453C2211yescharacteristic
43#17988452C2212yes—
44#17988452C2212yes—
45#17988451C2214yes—
46#17988449C2214yes—
47#17988448C2214yes—
48#17988447C2212yes—
49#17988444C2212yes—
50#17988444C2212yes—
51#17988444C2214yes—
52C4⋊C4×Dic73C2212yes—
53C4⋊C4×Dic73C2212yes—
54C4⋊C4×Dic73C2214yes—
55C4×#11924007C2214yes—
56C4×C4⋊Dic73C2212yes—
57#17988438C2214yes—
58C22⋊C4×Dic73C2214yes—
59C4×#11924005C2214yes—
60C4×#11924005C2214yes—
61#17988434C2214yes—
62#17988433C2214yes—
63#17988432C2214yes—
64#95×D73C2214yes—
65C2×#117×C73C2211yescharacteristic
66#388×C73C2214yes—
67#387×C73C2214yes—
68#384×C73C2211yescharacteristic
69#382×C73C2211yescharacteristic
70#380×C73C2212yes—
71C4×C4⋊C4×C73C2212yes—
72C2×#11924028—24no—
73C2×#11924028—24no—
74C2×#11924028C2×C418yes—
75C22×Dic146C2311yescharacteristic
76C22×D292C2311yescharacteristic
77C22×C22⋊D73C2311yescharacteristic
78C22×C22⋊D73C2311yescharacteristic
79C23×Dic73C2312yes—
80C22×C4×D73C2311yescharacteristic
81C22×C4×D73C2314yes—
82C22×C4×D73C2312yes—
83C22×C4×D73C2312yes—
84C22×C4×D73C2311yescharacteristic
85#14556556C2×C4116yescomplemented
86#14556553C2×C4116yes—
87#14556551C2×C4116yes—
88#14556551C2×C4116yescomplemented
89#14556549—28no—
90#14556548C2×C418yes—
91#14556548C2×C418yescomplemented
92C2×C4⋊Dic73C2312yes—
93C2×C4⋊Dic73C2312yes—
94C2×C4⋊Dic73C2314yes—
95C2×#11924007C2314yes—
96C2×#11924007C2312yes—
97C2×#11924007C2312yes—
98C4⋊C4×D73—28no—
99C2×C22⋊Dic73C2314yes—
100C2×#11924005C2314yes—
101C2×#11924005C2314yes—
102C2×#11924005C2314yes—
103C2×#11924005C2312yes—
104C2×#11924005C2312yes—
105C2×C4×Dic73C2×C418yescomplemented
106C2×C4×Dic73C2314yes—
107C2×C4×Dic73C2314yes—
108C2×C4×Dic73C2314yes—
109C2×C4×Dic73C2314yes—
110C2×C4×Dic73C2312yes—
111C2×C4×Dic73C2312yes—
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2312yes—
114C2×C4×Dic73C2311yescharacteristic
115C42×D73C2×C418yes—
116C42×D73C2×C418yescomplemented
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2314yes—
124#14556498C2314yes—
125#14556498C2314yes—
126#14556498C2314yes—
127#14556498C2314yes—
128C23×C4×C73C2311yescharacteristic
129#117×C73C2×C418yescomplemented
130C2×C4⋊C4×C73C2312yes—
131C2×C4⋊C4×C73C2312yes—
132C2×C4⋊C4×C73C2314yes—
133C2×C4⋊C4×C73C2312yes—
134C2×C22⋊C4×C73C2314yes—
135C2×C22⋊C4×C73C2312yes—
136C2×C42×C73C2314yes—
137C2×C42×C73C2311yescharacteristic
138C2×C42×C73C2311yescharacteristic
139#95×C73C2314yes—
140#95×C73C2314yes—
141#95×C73C2314yes—
142C23×D73C2411yescharacteristic
143C23×D73C2411yescharacteristic
144#11924028—232no—
145C2×Dic146—22no—
146C2×Dic146—22no—
147C2×Dic146C22×C414yes—
148C2×D292—22no—
149C2×D292—22no—
150C2×D292C22×C414yes—
151C2×C22⋊D73—22no—
152C2×C22⋊D73—22no—
153C2×C22⋊D73—22no—
154C2×C22⋊D73—22no—
155C2×C22⋊D73C22×C414yes—
156C2×C22⋊D73C22×C414yes—
157C22×Dic73—42no—
158C22×Dic73—42no—
159C22×Dic73C22×C418yes—
160C22×Dic73C2414yes—
161C22×Dic73C2414yes—
162C22×Dic73C2412yes—
163C22×Dic73C2411yescharacteristic
164C22×Dic73C2412yes—
165C22×Dic73C2411yescharacteristic
166C2×C4×D73—22no—
167C2×C4×D73—22no—
168C2×C4×D73—28no—
169C2×C4×D73—42no—
170C2×C4×D73—42no—
171C2×C4×D73—42no—
172C2×C4×D73—42no—
173C2×C4×D73—44no—
174C2×C4×D73—44no—
175C2×C4×D73C22×C414yes—
176C2×C4×D73C22×C418yes—
177C2×C4×D73C22×C418yes—
178C2×C4×D73C22×C414yes—
179C2×C4×D73#4114yes—
180C2×C4×D73#4114yes—
181C4⋊Dic73—24no—
182C4⋊Dic73—24no—
183C4⋊Dic73—44no—
184#11924007—28no—
185#11924007C22×C418yes—
186#11924007C22×C418yes—
187C22⋊Dic73—44no—
188#11924005—28no—
189#11924005—44no—
190#11924005—44no—
191#11924005C22×C418yes—
192#11924005C22×C418yes—
193C4×Dic73—28no—
194C4×Dic73—44no—
195C4×Dic73—44no—
196C4×Dic73—44no—
197C4×Dic73C22×C414yes—
198C4×Dic73C22×C418yes—
199C4×Dic73C22×C418yes—
200C4×Dic73C22×C414yes—
201C4×Dic73C22×C418yes—
202C24×C73C2411yescharacteristic
203C22×C4×C73—22no—
204C22×C4×C73—22no—
205C22×C4×C73C22×C414yes—
206C22×C4×C73C2414yes—
207C22×C4×C73C2414yes—
208C22×C4×C73C2411yescharacteristic
209C22×C4×C73C2412yes—
210C22×C4×C73C2412yes—
211C22×C4×C73C2411yescharacteristic
212C4⋊C4×C73—24no—
213C4⋊C4×C73—24no—
214C4⋊C4×C73—44no—
215C4⋊C4×C73C22×C418yes—
216C22⋊C4×C73—44no—
217C22⋊C4×C73C22×C418yes—
218C42×C73—44no—
219C42×C73C22×C414yes—
220C42×C73C22×C414yes—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—22no—
224C23×C73—22no—
225C23×C73C23×C412yes—
226C23×C73C2511yescharacteristic
227C22×D73—22no—
228C22×D73—22no—
229C22×D73—21no—
230C22×D73—21no—
231C22×D73—22no—
232C22×D73—22no—
233C22×D73C23×C412yes—
234C22×D73C23×C412yes—
235C22×D73C2×#4112yes—
236C22×D73C2×#4112yes—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—28no—
244C2×C4×C73—24no—
245C2×C4×C73—24no—
246C2×C4×C73—24no—
247C2×C4×C73—24no—
248C2×C4×C73—24no—
249C2×C4×C73—24no—
250C2×C4×C73C23×C414yes—
251C2×C4×C73C23×C414yes—
252C2×C4×C73C23×C412yes—
253C2×C4×C73C23×C412yes—
254C2×C4×C73C2×#4112yes—
255C2×C4×C73C2×#4112yes—
256C22⋊D73—28no—
257C22⋊D73—28no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—24no—
267C2×Dic73—24no—
268C2×Dic73—24no—
269C2×Dic73—24no—
270C2×Dic73—48no—
271C2×Dic73C23×C414yes—
272C2×Dic73C23×C414yes—
273C2×Dic73C23×C412yes—
274C2×Dic73C23×C412yes—
275C2×Dic73C2×#4112yes—
276C2×Dic73C2×#4112yes—
277D292—28no—
278C4×D73—28no—
279C4×D73—48no—
280C4×D73—48no—
281C4×D73—88no—
282C4×D73#117116yes—
283Dic146—28no—
284C22×C73—22no—
285C22×C73—24no—
286C22×C73—42no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73C24×C411yescharacteristic
290C22×C73C22×#4111yescharacteristic
291C22×C73C22×#4111yescharacteristic
292C22×C73#58512yes—
293C22×C73#58512yes—
294C2×D73—24no—
295C2×D73—24no—
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—22no—
299C2×D73—24no—
300C2×D73—42no—
301C2×D73—41no—
302C2×D73—41no—
303C2×D73C2×#11714yes—
304C4×C73—22no—
305C4×C73—24no—
306C4×C73—24no—
307C4×C73—44no—
308C4×C73—44no—
309C4×C73C2×#11714yes—
310Dic73—22no—
311Dic73—24no—
312Dic73—24no—
313Dic73—44no—
314Dic73—44no—
315Dic73C2×#11714yes—
316#34189—731nomaximal, Sylow
317C2×C73—42no—
318C2×C73C22×#11711yescharacteristic
319C2×C73#342812yes—
320C2×C73#351012yes—
321C2×C73#351211yescharacteristic
322C2×C73#351211yescharacteristic
323D73—24no—
324D73—42no—
325C2×C4×#41—731no—
326#2357—734no—
327#2356—734no—
328#2353—734no—
329#2353—734no—
330#2314—731no—
331#2314—731no—
332#2314—731no—
333#2314D7311yescharacteristic, complemented
334#2311—732no—
335#2285—731no—
336#2283—732no—
337#2283—732no—
338C2×#382—731no—
339#2274—732no—
340C73#3418911yescharacteristic, complemented, Sylow
341C22×#41—731no—
342C4×#41—7316no—
343C2×C4×Q8—732no—
344C2×C4×D4—732no—
345C2×C4×D4—732no—
346C2×C4×D4—732no—
347C2×#117—732no—
348C2×#117—731no—
349C2×#117—731no—
350C2×#117C2×D7311yescharacteristic
351C22×C4⋊C4—732no—
352C22×C42—731no—
353C22×C42—731no—
354C22×C42—731no—
355#391—734no—
356#390—734no—
357#388—734no—
358#388—734no—
359#388—734no—
360#388C2×D7314yes—
361#387—734no—
362#387—734no—
363#387—734no—
364#387C2×D7314yes—
365#386—734no—
366#386—734no—
367#384—731no—
368#384C2×D7311yescharacteristic
369#383—734no—
370#383—734no—
371#382—732no—
372#382—732no—
373#382—731no—
374#382—734no—
375#382C2×D7311yescharacteristic
376#381—734no—
377#381—734no—
378#380—732no—
379#380—732no—
380#380—732no—
381#380—734no—
382#380C2×D7312yes—
383C4×C4⋊C4—732no—
384C4×C4⋊C4—732no—
385C4×C4⋊C4—734no—
386C4×C4⋊C4—732no—
387C4×C4⋊C4—734no—
388C4×C4⋊C4—734no—
389C4×C4⋊C4C2×D7312yes—
390C4×C22⋊C4—734no—
391C4×C22⋊C4—734no—
392C4×C22⋊C4—734no—
393C4×C22⋊C4—734no—
394#376—734no—
395#376—734no—
396C2×#95—734no—
397C2×#41—738no—
398C2×#41—1464no—
399C2×#41—1464no—
400C22×Q8—731no—
401C22×D4—731no—
402C22×D4—731no—
403C22×D4—731no—
404C23×C4—732no—
405C23×C4—732no—
406C23×C4—732no—
407C23×C4—734no—
408C23×C4—731no—
409C23×C4—731no—
410C23×C4C22×D7311yescharacteristic
411C4×Q8—7316no—
412C4×D4—7316no—
413C4×D4—7316no—
414C4×D4—7316no—
415#117—738no—
416#117—738no—
417#117—1468no—
418#117C4×D7318yescomplemented
419C2×C4⋊C4—734no—
420C2×C4⋊C4—732no—
421C2×C4⋊C4—732no—
422C2×C4⋊C4—732no—
423C2×C4⋊C4—734no—
424C2×C4⋊C4—732no—
425C2×C4⋊C4—1468no—
426C2×C4⋊C4C22×D7312yes—
427C2×C4⋊C4C22×D7312yes—
428C2×C4⋊C4C22×D7314yes—
429C2×C4⋊C4C22×D7312yes—
430C2×C22⋊C4—734no—
431C2×C22⋊C4—734no—
432C2×C22⋊C4—734no—
433C2×C22⋊C4—732no—
434C2×C22⋊C4—732no—
435C2×C22⋊C4—734no—
436C2×C22⋊C4C22×D7314yes—
437C2×C22⋊C4C22×D7312yes—
438C2×C42—734no—
439C2×C42—734no—
440C2×C42—734no—
441C2×C42—738no—
442C2×C42—731no—
443C2×C42—732no—
444C2×C42—732no—
445C2×C42—732no—
446C2×C42—731no—
447C2×C42—738no—
448C2×C42—738no—
449C2×C42—734no—
450C2×C42C22×D7314yes—
451C2×C42C22×D7311yescharacteristic
452C2×C42C22×D7311yescharacteristic
453#95—734no—
454#95—734no—
455#95—734no—
456#95—734no—
457#95—734no—
458#95—734no—
459#95—734no—
460#95—734no—
461#95—734no—
462#95—734no—
463#95—734no—
464#95C22×D7314yes—
465#95C22×D7314yes—
466#95C22×D7314yes—
467C24—731no—
468C24—731no—
469C24C23×D7311yescharacteristic
470#41—14632no—
471C2×Q8—734no—
472C2×Q8—1462no—
473C2×Q8—1462no—
474C2×D4—734no—
475C2×D4—734no—
476C2×D4—734no—
477C2×D4—1462no—
478C2×D4—1462no—
479C2×D4—1462no—
480C2×D4—1462no—
481C2×D4—1462no—
482C2×D4—1462no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—2924no—
490C22×C4—2924no—
491C22×C4—2922no—
492C22×C4—2922no—
493C22×C4—738no—
494C22×C4—738no—
495C22×C4—738no—
496C22×C4—731no—
497C22×C4—734no—
498C22×C4—734no—
499C22×C4—734no—
500C22×C4—732no—
501C22×C4—731no—
502C22×C4—732no—
503C22×C4—734no—
504C22×C4—734no—
505C22×C4—734no—
506C22×C4—1468no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4C2×C4×D7314yes—
510C22×C4C23×D7314yes—
511C22×C4C23×D7314yes—
512C22×C4C23×D7311yescharacteristic
513C22×C4C23×D7312yes—
514C22×C4C23×D7312yes—
515C22×C4C23×D7311yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—2924no—
520C4⋊C4—738no—
521C4⋊C4—738no—
522C4⋊C4—1468no—
523C4⋊C4—1464no—
524C4⋊C4—1464no—
525C4⋊C4C2×C4×D7318yes—
526C22⋊C4—44no—
527C22⋊C4—2924no—
528C22⋊C4—2924no—
529C22⋊C4—2924no—
530C22⋊C4—738no—
531C22⋊C4—738no—
532C22⋊C4—1468no—
533C22⋊C4C2×C4×D7318yes—
534C42—44no—
535C42—2924no—
536C42—2924no—
537C42—2924no—
538C42—734no—
539C42—734no—
540C42—738no—
541C42—738no—
542C42—738no—
543C42—1468no—
544C42C2×C4×D7314yes—
545C42C2×C4×D7314yes—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—732no—
551C23—732no—
552C23—732no—
553C23—732no—
554C23—1461no—
555C23—1461no—
556C23—1462no—
557C23—1462no—
558C23—1462no—
559C23—1462no—
560C23C22×C4×D7312yes—
561C23C24×D7311yescharacteristic
562Q8—1468no—
563D4—1468no—
564D4—1468no—
565D4—1468no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—2928no—
580C2×C4—2928no—
581C2×C4—2928no—
582C2×C4—5848no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—732no—
586C2×C4—7316no—
587C2×C4—734no—
588C2×C4—734no—
589C2×C4—732no—
590C2×C4—1464no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1468no—
603C2×C4C22×C4×D7312yes—
604C2×C4C22×C4×D7314yes—
605C2×C4C22×C4×D7314yes—
606C2×C4C22×C4×D7312yes—
607C2×C4#41×D7312yes—
608C2×C4#41×D7312yes—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—2922no—
615C22—2921no—
616C22—2921no—
617C22—734no—
618C22—1462no—
619C22—1464no—
620C22—1464no—
621C22—1462no—
622C22—1462no—
623C22—1464no—
624C22C23×C4×D7311yescharacteristic
625C22C2×#41×D7311yescharacteristic
626C22C2×#41×D7311yescharacteristic
627C22#1798942212yes—
628C22#1798942412yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—2924no—
635C4—2924no—
636C4—734no—
637C4—1462no—
638C4—1464no—
639C4—1464no—
640C4#117×D7314yes—
641C2—42no—
642C2—2922no—
643C2—1464no—
644C2C2×#117×D7311yescharacteristic
645C2#2262313112yes—
646C2#2262382312yes—
647C2#2262385011yescharacteristic
648C2#2262385411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2314C2
socle, cosocleC23×C73C25
prime radicalC22×#11924028C22
hypercenter#2314D73
hypocenterC73#34189
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #22612901C2semidirect
3. #22612893C2semidirect
4. #22612870C2semidirect
5. #22612866C2semidirect
6. #22612541C2semidirect
7. #22612540C2semidirect
8. #22612539C2semidirect
10. #22612336C2semidirect
11. #22612327C2semidirect
12. #22612326C2semidirect
13. #382×D73C2semidirect
14. #22612285C2semidirect
15. C73×#2314C2semidirect
17. #17989055C4semidirect
85. #14556556C2×C4semidirect
88. #14556551C2×C4semidirect
91. #14556548C2×C4semidirect
105. C2×C4×Dic73C2×C4semidirect
116. C42×D73C2×C4semidirect
129. #117×C73C2×C4semidirect
333. #2314D73semidirect
340. C73#34189semidirect
418. #117C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890552
#226129012
#226128932
#226128702
#226128662
#226125412
#226125402
#226125392
#226125202
#226123362
#226123272
#226123262
#382×D732
#226122852
C73×#23142
#3418973