Finite Group Databasebeta

← #33246126

Subgroups — #33246126

Order 18688.

2542 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989055C211yesmaximal, characteristic
2#22612885C214yesmaximal, complemented
3#22612877C214yesmaximal, complemented
4#22612859C214yesmaximal, complemented
5#22612855C214yesmaximal, complemented
6#22612541C211yesmaximal, characteristic, complemented
7#22612540C211yesmaximal, characteristic, complemented
8#22612539C211yesmaximal, characteristic, complemented
9#22612520C212yesmaximal
10#22612337C211yesmaximal, characteristic, complemented
11#22612335C212yesmaximal, complemented
12#22612333C212yesmaximal, complemented
13#384×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×#14556548C2214yes—
24C2×#14556548C2211yescharacteristic
25C2×#14556548C2211yescharacteristic
26C2×#14556547C2214yes—
27C2×C4⋊C4×D73C2212yes—
28C2×C4⋊C4×D73C2214yes—
29C22×C4×Dic73C2211yescharacteristic
30C2×C42×D73C2211yescharacteristic
31C2×C42×D73C2211yescharacteristic
32#17988491C2214yes—
33#17988487C2214yes—
34#17988480C2214yes—
35#17988479C2214yes—
36#17988477C2214yes—
37#17988474C2214yes—
38#17988473C2214yes—
39#17988468C2214yes—
40#17988465C2214yes—
41#17988463C2214yes—
42#17988457C2211yescharacteristic
43#17988456C2212yes—
44#17988456C2214yes—
45#17988456C2212yes—
46#17988453C2211yescharacteristic
47#17988451C2214yes—
48#17988451C2214yes—
49#17988447C2212yes—
50#17988446C2214yes—
51#17988446C2214yes—
52#17988445C2214yes—
53#17988444C2212yes—
54#17988444C2212yes—
55C4⋊C4×Dic73C2212yes—
56C4⋊C4×Dic73C2212yes—
57C4×#11924007C2214yes—
58C4×#11924007C2214yes—
59C4×C4⋊Dic73C2212yes—
60#17988438C2214yes—
61#17988438C2214yes—
62#17988438C2214yes—
63C4×#11924005C2214yes—
64#17988434C2214yes—
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×D73C2312yes—
82C22×C4×D73C2312yes—
83C22×C4×D73C2314yes—
84C22×C4×D73C2311yescharacteristic
85#14556556C2×C4116yescomplemented
86#14556553C2×C4116yes—
87#14556551C2×C4116yes—
88#14556551C2×C4116yescomplemented
89#14556549—28no—
90#14556548—48no—
91#14556548C2×C418yes—
92#14556548C2×C418yescomplemented
93#14556547—48no—
94C2×C4⋊Dic73C2312yes—
95C2×C4⋊Dic73C2312yes—
96C2×C4⋊Dic73C2314yes—
97C2×#11924007C2314yes—
98C2×#11924007C2314yes—
99C2×#11924007C2314yes—
100C2×#11924007C2314yes—
101C2×#11924007C2314yes—
102C2×#11924007C2312yes—
103C2×#11924007C2312yes—
104C4⋊C4×D73—28no—
105C4⋊C4×D73—48no—
106C2×C22⋊Dic73C2314yes—
107C2×#11924005C2314yes—
108C2×#11924005C2314yes—
109C2×#11924005C2314yes—
110C2×#11924005C2312yes—
111C2×#11924005C2312yes—
112C2×C4×Dic73C2×C418yescomplemented
113C2×C4×Dic73C2314yes—
114C2×C4×Dic73C2314yes—
115C2×C4×Dic73C2314yes—
116C2×C4×Dic73C2314yes—
117C2×C4×Dic73C2312yes—
118C2×C4×Dic73C2312yes—
119C2×C4×Dic73C2312yes—
120C2×C4×Dic73C2311yescharacteristic
121C2×C4×Dic73C2311yescharacteristic
122C42×D73C2×C418yes—
123C42×D73C2×C418yescomplemented
124#14556498C2314yes—
125#14556498C2314yes—
126#14556498C2314yes—
127#14556498C2314yes—
128#14556498C2314yes—
129#14556498C2314yes—
130#14556498C2314yes—
131C23×C4×C73C2311yescharacteristic
132#117×C73C2×C418yescomplemented
133C2×C4⋊C4×C73C2312yes—
134C2×C4⋊C4×C73C2312yes—
135C2×C4⋊C4×C73C2314yes—
136C2×C4⋊C4×C73C2312yes—
137C2×C22⋊C4×C73C2314yes—
138C2×C22⋊C4×C73C2312yes—
139C2×C42×C73C2314yes—
140C2×C42×C73C2311yescharacteristic
141C2×C42×C73C2311yescharacteristic
142#95×C73C2314yes—
143#95×C73C2314yes—
144#95×C73C2314yes—
145C23×D73C2411yescharacteristic
146C23×D73C2411yescharacteristic
147#11924028—232no—
148C2×Dic146—22no—
149C2×Dic146—22no—
150C2×Dic146C22×C414yes—
151C2×D292—22no—
152C2×D292—22no—
153C2×D292C22×C414yes—
154C2×C22⋊D73—22no—
155C2×C22⋊D73—22no—
156C2×C22⋊D73—22no—
157C2×C22⋊D73—22no—
158C2×C22⋊D73C22×C414yes—
159C2×C22⋊D73C22×C414yes—
160C22×Dic73—24no—
161C22×Dic73—24no—
162C22×Dic73C22×C418yes—
163C22×Dic73C2414yes—
164C22×Dic73C2414yes—
165C22×Dic73C2411yescharacteristic
166C22×Dic73C2412yes—
167C22×Dic73C2411yescharacteristic
168C22×Dic73C2412yes—
169C2×C4×D73—22no—
170C2×C4×D73—22no—
171C2×C4×D73—24no—
172C2×C4×D73—24no—
173C2×C4×D73—24no—
174C2×C4×D73—24no—
175C2×C4×D73—28no—
176C2×C4×D73—44no—
177C2×C4×D73—44no—
178C2×C4×D73C22×C418yes—
179C2×C4×D73C22×C414yes—
180C2×C4×D73C22×C418yes—
181C2×C4×D73C22×C414yes—
182C2×C4×D73#4114yes—
183C2×C4×D73#4114yes—
184C4⋊Dic73—24no—
185C4⋊Dic73—24no—
186C4⋊Dic73—44no—
187#11924007—28no—
188#11924007—28no—
189#11924007—28no—
190#11924007—28no—
191#11924007—28no—
192#11924007C22×C418yes—
193#11924007C22×C418yes—
194C22⋊Dic73—44no—
195#11924005—28no—
196#11924005—44no—
197#11924005—44no—
198#11924005C22×C418yes—
199#11924005C22×C418yes—
200C4×Dic73—28no—
201C4×Dic73—44no—
202C4×Dic73—44no—
203C4×Dic73—44no—
204C4×Dic73C22×C414yes—
205C4×Dic73C22×C418yes—
206C4×Dic73C22×C418yes—
207C4×Dic73C22×C414yes—
208C4×Dic73C22×C418yes—
209C24×C73C2411yescharacteristic
210C22×C4×C73—22no—
211C22×C4×C73—22no—
212C22×C4×C73C22×C414yes—
213C22×C4×C73C2414yes—
214C22×C4×C73C2414yes—
215C22×C4×C73C2411yescharacteristic
216C22×C4×C73C2412yes—
217C22×C4×C73C2412yes—
218C22×C4×C73C2411yescharacteristic
219C4⋊C4×C73—24no—
220C4⋊C4×C73—24no—
221C4⋊C4×C73—44no—
222C4⋊C4×C73C22×C418yes—
223C22⋊C4×C73—44no—
224C22⋊C4×C73C22×C418yes—
225C42×C73—44no—
226C42×C73C22×C414yes—
227C42×C73C22×C414yes—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—22no—
231C23×C73—22no—
232C23×C73C23×C412yes—
233C23×C73C2511yescharacteristic
234C22×D73—22no—
235C22×D73—22no—
236C22×D73—21no—
237C22×D73—21no—
238C22×D73—22no—
239C22×D73—22no—
240C22×D73C23×C412yes—
241C22×D73C23×C412yes—
242C22×D73C2×#4112yes—
243C22×D73C2×#4112yes—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—28no—
251C2×C4×C73—24no—
252C2×C4×C73—24no—
253C2×C4×C73—24no—
254C2×C4×C73—24no—
255C2×C4×C73—24no—
256C2×C4×C73—24no—
257C2×C4×C73C23×C414yes—
258C2×C4×C73C23×C414yes—
259C2×C4×C73C23×C412yes—
260C2×C4×C73C23×C412yes—
261C2×C4×C73C2×#4112yes—
262C2×C4×C73C2×#4112yes—
263C22⋊D73—28no—
264C22⋊D73—28no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—22no—
268C2×Dic73—22no—
269C2×Dic73—216no—
270C2×Dic73—24no—
271C2×Dic73—24no—
272C2×Dic73—24no—
273C2×Dic73—24no—
274C2×Dic73—24no—
275C2×Dic73—24no—
276C2×Dic73C23×C414yes—
277C2×Dic73C23×C412yes—
278C2×Dic73C23×C412yes—
279C2×Dic73C23×C414yes—
280C2×Dic73C2×#4112yes—
281C2×Dic73C2×#4112yes—
282C2×Dic73H(5,𝔽2)14yes—
283C2×Dic732−1+414yes—
284D292—28no—
285C4×D73—28no—
286C4×D73—216no—
287C4×D73—216no—
288C4×D73—88no—
289C4×D73#117116yescomplemented
290Dic146—28no—
291C22×C73—22no—
292C22×C73—24no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—42no—
296C22×C73C24×C411yescharacteristic
297C22×C73C22×#4111yescharacteristic
298C22×C73C22×#4111yescharacteristic
299C22×C73C2×H(5,𝔽2)12yes—
300C22×C73C2×2−1+412yes—
301C2×D73—24no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—24no—
305C2×D73—22no—
306C2×D73—24no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—42no—
310C2×D73C2×#11714yes—
311C4×C73—22no—
312C4×C73—24no—
313C4×C73—24no—
314C4×C73—44no—
315C4×C73—44no—
316C4×C73C2×#11714yes—
317Dic73—22no—
318Dic73—24no—
319Dic73—44no—
320Dic73—44no—
321Dic73C2×#11714yes—
322Dic73#52018yes—
323#34188—731nomaximal, Sylow
324C2×C73—42no—
325C2×C73C22×#11711yescharacteristic
326C2×C73C2×#52012yes—
327C2×C73#350811yescharacteristic
328C2×C73#350911yescharacteristic
329C2×C73#351112yes—
330D73—24no—
331D73—42no—
332C2×C4×#41—731no—
333#2355—734no—
334#2354—734no—
335#2352—734no—
336#2352—734no—
337#2314—731no—
338#2314—731no—
339#2314—731no—
340#2314D7311yescharacteristic, complemented
341#2311—732no—
342#2285—731no—
343#2285—732no—
344#2285—732no—
345C2×#384—731no—
346#2274—732no—
347C73#3418811yescharacteristic, complemented, Sylow
348C22×#41—731no—
349C4×#41—7316no—
350C2×C4×Q8—732no—
351C2×C4×D4—732no—
352C2×C4×D4—732no—
353C2×C4×D4—732no—
354C2×#117—732no—
355C2×#117—734no—
356C2×#117—734no—
357C2×#117—731no—
358C2×#117—731no—
359C2×#117C2×D7311yescharacteristic
360C22×C4⋊C4—734no—
361C22×C4⋊C4—732no—
362C22×C42—731no—
363C22×C42—731no—
364C22×C42—731no—
365#391—734no—
366#390—734no—
367#388—734no—
368#388—734no—
369#388—734no—
370#388C2×D7314yes—
371#387—734no—
372#387—734no—
373#387—734no—
374#387C2×D7314yes—
375#386—734no—
376#386—734no—
377#384—734no—
378#384—732no—
379#384—732no—
380#384—731no—
381#384C2×D7311yescharacteristic
382#382—731no—
383#382—734no—
384#382—734no—
385#382C2×D7311yescharacteristic
386#380—732no—
387#380—732no—
388#380—732no—
389#380—734no—
390#380—734no—
391#380—734no—
392#380C2×D7312yes—
393C4×C4⋊C4—732no—
394C4×C4⋊C4—732no—
395C4×C4⋊C4—732no—
396C4×C4⋊C4—734no—
397C4×C4⋊C4—734no—
398C4×C4⋊C4—734no—
399C4×C4⋊C4—734no—
400C4×C4⋊C4—734no—
401C4×C4⋊C4C2×D7312yes—
402C4×C22⋊C4—734no—
403C4×C22⋊C4—734no—
404C2×#41—738no—
405C2×#41—1464no—
406C2×#41—1464no—
407C22×Q8—731no—
408C22×D4—731no—
409C22×D4—731no—
410C22×D4—731no—
411C23×C4—734no—
412C23×C4—731no—
413C23×C4—732no—
414C23×C4—732no—
415C23×C4—732no—
416C23×C4—731no—
417C23×C4C22×D7311yescharacteristic
418C4×Q8—7316no—
419C4×D4—7316no—
420C4×D4—7316no—
421C4×D4—7316no—
422#117—2928no—
423#117—2928no—
424#117—738no—
425#117—738no—
426#117—1468no—
427#117C4×D7318yescomplemented
428C2×C4⋊C4—2928no—
429C2×C4⋊C4—734no—
430C2×C4⋊C4—732no—
431C2×C4⋊C4—732no—
432C2×C4⋊C4—734no—
433C2×C4⋊C4—734no—
434C2×C4⋊C4—732no—
435C2×C4⋊C4—734no—
436C2×C4⋊C4—734no—
437C2×C4⋊C4—732no—
438C2×C4⋊C4—734no—
439C2×C4⋊C4—1468no—
440C2×C4⋊C4C22×D7312yes—
441C2×C4⋊C4C22×D7312yes—
442C2×C4⋊C4C22×D7314yes—
443C2×C4⋊C4C22×D7312yes—
444C2×C22⋊C4—734no—
445C2×C22⋊C4—734no—
446C2×C22⋊C4—734no—
447C2×C22⋊C4—732no—
448C2×C22⋊C4—732no—
449C2×C22⋊C4—734no—
450C2×C22⋊C4C22×D7314yes—
451C2×C22⋊C4C22×D7312yes—
452C2×C42—734no—
453C2×C42—734no—
454C2×C42—734no—
455C2×C42—738no—
456C2×C42—731no—
457C2×C42—732no—
458C2×C42—732no—
459C2×C42—732no—
460C2×C42—731no—
461C2×C42—738no—
462C2×C42—738no—
463C2×C42—734no—
464C2×C42C22×D7314yes—
465C2×C42C22×D7311yescharacteristic
466C2×C42C22×D7311yescharacteristic
467#95—734no—
468#95—734no—
469#95—734no—
470#95—734no—
471#95—734no—
472#95—734no—
473#95—734no—
474#95C22×D7314yes—
475#95C22×D7314yes—
476#95C22×D7314yes—
477C24—731no—
478C24—731no—
479C24C23×D7311yescharacteristic
480#41—14632no—
481C2×Q8—734no—
482C2×Q8—1462no—
483C2×Q8—1462no—
484C2×D4—734no—
485C2×D4—734no—
486C2×D4—734no—
487C2×D4—1462no—
488C2×D4—1462no—
489C2×D4—1462no—
490C2×D4—1462no—
491C2×D4—1462no—
492C2×D4—1462no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—2924no—
496C22×C4—2924no—
497C22×C4—731no—
498C22×C4—732no—
499C22×C4—734no—
500C22×C4—734no—
501C22×C4—734no—
502C22×C4—738no—
503C22×C4—731no—
504C22×C4—732no—
505C22×C4—734no—
506C22×C4—738no—
507C22×C4—738no—
508C22×C4—734no—
509C22×C4—734no—
510C22×C4—1468no—
511C22×C4—1464no—
512C22×C4—1464no—
513C22×C4—1464no—
514C22×C4—1464no—
515C22×C4—1464no—
516C22×C4—1464no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4C2×C4×D7314yes—
520C22×C4C23×D7314yes—
521C22×C4C23×D7314yes—
522C22×C4C23×D7311yescharacteristic
523C22×C4C23×D7312yes—
524C22×C4C23×D7312yes—
525C22×C4C23×D7311yescharacteristic
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—2924no—
530C4⋊C4—738no—
531C4⋊C4—738no—
532C4⋊C4—1468no—
533C4⋊C4—1468no—
534C4⋊C4—1468no—
535C4⋊C4—1468no—
536C4⋊C4—1468no—
537C4⋊C4—1464no—
538C4⋊C4—1464no—
539C4⋊C4C2×C4×D7318yes—
540C22⋊C4—44no—
541C22⋊C4—2924no—
542C22⋊C4—2924no—
543C22⋊C4—2924no—
544C22⋊C4—738no—
545C22⋊C4—738no—
546C22⋊C4—1468no—
547C22⋊C4C2×C4×D7318yes—
548C42—44no—
549C42—2924no—
550C42—2924no—
551C42—2924no—
552C42—734no—
553C42—734no—
554C42—738no—
555C42—738no—
556C42—738no—
557C42—1468no—
558C42C2×C4×D7314yes—
559C42C2×C4×D7314yes—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—732no—
565C23—732no—
566C23—732no—
567C23—732no—
568C23—1461no—
569C23—1461no—
570C23—1462no—
571C23—1462no—
572C23—1462no—
573C23—1462no—
574C23C22×C4×D7312yes—
575C23C24×D7311yescharacteristic
576Q8—1468no—
577D4—1468no—
578D4—1468no—
579D4—1468no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—5848no—
594C2×C4—734no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—734no—
599C2×C4—734no—
600C2×C4—7316no—
601C2×C4—734no—
602C2×C4—732no—
603C2×C4—14616no—
604C2×C4—14616no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4—1464no—
611C2×C4—14616no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1468no—
617C2×C4C22×C4×D7312yes—
618C2×C4C22×C4×D7314yes—
619C2×C4C22×C4×D7314yes—
620C2×C4C22×C4×D7312yes—
621C2×C4#41×D7312yes—
622C2×C4#41×D7312yes—
623C22—22no—
624C22—24no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—2921no—
629C22—2921no—
630C22—2922no—
631C22—734no—
632C22—1462no—
633C22—1464no—
634C22—1462no—
635C22—1462no—
636C22—1464no—
637C22—1464no—
638C22C23×C4×D7311yescharacteristic
639C22C2×#41×D7311yescharacteristic
640C22C2×#41×D7311yescharacteristic
641C22H(5,𝔽2)×D7312yes—
642C222−1+4×D7312yes—
643C4—22no—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—2924no—
649C4—2924no—
650C4—734no—
651C4—738no—
652C4—1462no—
653C4—1464no—
654C4#117×D7314yes—
655C2—42no—
656C2—2922no—
657C2—1464no—
658C2C2×#117×D7311yescharacteristic
659C2#520×D7312yes—
660C2#2262380311yescharacteristic
661C2#2262381311yescharacteristic
662C2#2262383612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2314C2
socle, cosocleC23×C73C25
prime radicalC22×#11924028C22
hypercenter#2314D73
hypocenterC73#34188
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #22612885C2semidirect
3. #22612877C2semidirect
4. #22612859C2semidirect
5. #22612855C2semidirect
6. #22612541C2semidirect
7. #22612540C2semidirect
8. #22612539C2semidirect
10. #22612337C2semidirect
11. #22612335C2semidirect
12. #22612333C2semidirect
13. #384×D73C2semidirect
14. #22612285C2semidirect
15. C73×#2314C2semidirect
17. #17989055C4semidirect
85. #14556556C2×C4semidirect
88. #14556551C2×C4semidirect
92. #14556548C2×C4semidirect
112. C2×C4×Dic73C2×C4semidirect
123. C42×D73C2×C4semidirect
132. #117×C73C2×C4semidirect
289. C4×D73#117semidirect
340. #2314D73semidirect
347. C73#34188semidirect
427. #117C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890552
#226128852
#226128772
#226128592
#226128552
#226125412
#226125402
#226125392
#226125202
#226123372
#226123352
#226123332
#384×D732
#226122852
C73×#23142
#3418873