Finite Group Databasebeta

← #33258736

Subgroups — #33258736

Order 18688.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618246C214yesmaximal, complemented
2#22615884C214yesmaximal, complemented
3#22614485C214yesmaximal, complemented
4#22614310C214yesmaximal, complemented
5#22614192C214yesmaximal, complemented
6#22614177C214yesmaximal, complemented
7#22612543C211yesmaximal, characteristic, complemented
8#22612543C211yesmaximal, characteristic, complemented
9#22612506C211yesmaximal, characteristic
10#22612506C211yesmaximal, characteristic
11C2×#17988466C211yesmaximal, characteristic
12C2×#17988460C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊C4×C73C211yesmaximal, characteristic, complemented
14C23×C22⋊D73C2211yescharacteristic
15C2×#14556585C2214yescomplemented
16C22×#11924007C2212yescomplemented
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2211yescharacteristic
19C22×C22⋊Dic73C2211yescharacteristic
20C22×#11924005C2212yes—
21C73⋊#401C2214yescomplemented
22#17988528C2214yescomplemented
23#17988528C2214yescomplemented
24#17988528C2214yescomplemented
25#17988516C2214yescomplemented
26#17988516C2214yescomplemented
27#17988516C2214yescomplemented
28#17988511C2214yescomplemented
29#17988478C2214yescomplemented
30#17988478C2214yescomplemented
31#17988478C2214yescomplemented
32#17988471C2214yescomplemented
33#17988471C2214yescomplemented
34#17988471C2214yescomplemented
35#17988466C2214yes—
36#17988466C2214yes—
37#17988466C2212yes—
38#17988466C2212yes—
39#17988466C2214yes—
40#17988464C2212yes—
41#17988464C2214yes—
42#17988464C2212yes—
43#17988460C2214yes—
44#17988460C2214yescomplemented
45#17988460C2214yescomplemented
46#17988460C2212yescomplemented
47#17988460C2212yescomplemented
48#17988459C2214yes—
49#17988459C2214yescomplemented
50#17988455C2214yescomplemented
51#17988448C2212yes—
52#17988448C2212yes—
53#17988445C2214yes—
54#17988445C2212yescomplemented
55#17988445C2212yescomplemented
56#17988442C2212yes—
57#17988442C2212yes—
58C2×#14556498C2212yes—
59C2×#14556498C2212yescomplemented
60C24×C4×C73C2211yescharacteristic
61C22×C22⋊C4×C73C2211yescharacteristic
62C22×C22⋊C4×C73C2211yescharacteristic
63C22×C42×C73C2212yescomplemented
64C4×C22⋊C4×C73C2214yes—
65C4×C22⋊C4×C73C2214yescomplemented
66C2×#95×C73C2212yescomplemented
67C24×D73C2311yescharacteristic
68C22×C22⋊D73—24no—
69C22×C22⋊D73—24no—
70C22×C22⋊D73—24no—
71C22×C22⋊D73C2312yes—
72C22×C22⋊D73C2312yes—
73C23×Dic73C2311yescharacteristic
74C23×Dic73C2312yes—
75C23×Dic73C2311yescharacteristic
76C23×Dic73C2312yes—
77C23×Dic73C2311yescharacteristic
78#14556585—48no—
79C2×#11924007—44no—
80C2×#11924007C2314yes—
81C2×#11924007C2314yes—
82C2×#11924007C2314yes—
83C2×#11924007C2314yes—
84C2×#11924007C2314yes—
85C2×#11924007C2314yes—
86C2×C22⋊Dic73—24no—
87C2×C22⋊Dic73—24no—
88C2×C22⋊Dic73—24no—
89C2×C22⋊Dic73C2312yes—
90C2×C22⋊Dic73C2312yes—
91C2×C22⋊Dic73C2312yes—
92C2×C22⋊Dic73C2312yes—
93C2×C22⋊Dic73C2312yes—
94C2×C22⋊Dic73C2312yes—
95C2×#11924005—44no—
96C2×#11924005C2314yes—
97C2×#11924005C2314yes—
98C2×#11924005C2314yes—
99C2×#11924005C2314yes—
100C2×#11924005C2314yes—
101C2×#11924005C2314yes—
102#14556498C2314yes—
103#14556498C2314yes—
104#14556498C2314yes—
105#14556498C2314yes—
106#14556498C2314yes—
107#14556498C2314yes—
108#14556498C2314yes—
109#14556498C2314yes—
110#14556498C2314yes—
111#14556498C2314yes—
112#14556498C2314yes—
113#14556498C2314yes—
114C25×C73C2311yescharacteristic
115C23×C4×C73—24no—
116C23×C4×C73—44no—
117C23×C4×C73C2311yescharacteristic
118C23×C4×C73C2312yes—
119C23×C4×C73C2312yes—
120C23×C4×C73C2312yes—
121C23×C4×C73C2312yes—
122C23×C4×C73C2311yescharacteristic
123C2×C22⋊C4×C73—24no—
124C2×C22⋊C4×C73—24no—
125C2×C22⋊C4×C73C2312yes—
126C2×C22⋊C4×C73C2312yes—
127C2×C22⋊C4×C73C2312yes—
128C2×C22⋊C4×C73C2312yes—
129C2×C42×C73—44no—
130C2×C42×C73C2314yes—
131C2×C42×C73C2314yes—
132#95×C73C2314yes—
133#95×C73C2314yes—
134C23×D73—22no—
135C23×D73—22no—
136C23×D73—22no—
137C23×D73—42no—
138C23×D73—42no—
139C23×D73C2412yes—
140C2×C22⋊D73—24no—
141C2×C22⋊D73—24no—
142C2×C22⋊D73—24no—
143C2×C22⋊D73—24no—
144C2×C22⋊D73—24no—
145C2×C22⋊D73—24no—
146C2×C22⋊D73—416no—
147C22×Dic73—24no—
148C22×Dic73—22no—
149C22×Dic73—22no—
150C22×Dic73—22no—
151C22×Dic73—24no—
152C22×Dic73—22no—
153C22×Dic73—22no—
154C22×Dic73—42no—
155C22×Dic73—42no—
156C22×Dic73—42no—
157C22×Dic73—42no—
158C22×Dic73—42no—
159C22×Dic73—42no—
160C22×Dic73C2412yes—
161C22×Dic73C2412yes—
162C22×Dic73C2414yes—
163C22×Dic73C2414yes—
164C22×Dic73C2412yes—
165#11924007—28no—
166#11924007—44no—
167#11924007—44no—
168#11924007—44no—
169#11924007—44no—
170#11924007—44no—
171C22⋊Dic73—24no—
172C22⋊Dic73—24no—
173C22⋊Dic73—24no—
174C22⋊Dic73—24no—
175C22⋊Dic73—24no—
176C22⋊Dic73—24no—
177#11924005—28no—
178#11924005—44no—
179#11924005—44no—
180#11924005—44no—
181#11924005—44no—
182#11924005—44no—
183C24×C73—42no—
184C24×C73—42no—
185C24×C73#4114yes—
186C24×C73#4114yes—
187C24×C73#4114yes—
188C24×C73C2411yescharacteristic
189C24×C73C2411yescharacteristic
190C24×C73C2411yescharacteristic
191C22×C4×C73—22no—
192C22×C4×C73—24no—
193C22×C4×C73—24no—
194C22×C4×C73—24no—
195C22×C4×C73—24no—
196C22×C4×C73—24no—
197C22×C4×C73—22no—
198C22×C4×C73—24no—
199C22×C4×C73—24no—
200C22×C4×C73—42no—
201C22×C4×C73—42no—
202C22×C4×C73—42no—
203C22×C4×C73—42no—
204C22×C4×C73—42no—
205C22×C4×C73—42no—
206C22×C4×C73—42no—
207C22×C4×C73—416no—
208C22×C4×C73C2412yes—
209C22×C4×C73C2414yes—
210C22×C4×C73C2414yes—
211C22×C4×C73C2412yes—
212C22⋊C4×C73—24no—
213C22⋊C4×C73—24no—
214C22⋊C4×C73—24no—
215C22⋊C4×C73—24no—
216C42×C73—28no—
217C42×C73—44no—
218C23×C73—22no—
219C23×C73—22no—
220C23×C73—22no—
221C23×C73—22no—
222C23×C73—22no—
223C23×C73—22no—
224C23×C73—44no—
225C23×C73—48no—
226C23×C73—44no—
227C23×C73—48no—
228C23×C73#126116yescomplemented
229C23×C73C2×#4112yes—
230C23×C73C2×#4112yes—
231C23×C73C2×#4112yes—
232C23×C73C2×#4112yes—
233C23×C73C2×#4112yes—
234C23×C73C2×#4112yes—
235C23×C73C2×#4112yes—
236C23×C73C2×#4112yes—
237C23×C73C2×#4112yes—
238C23×C73C2511yescharacteristic
239C22×D73—22no—
240C22×D73—22no—
241C22×D73—22no—
242C22×D73—24no—
243C22×D73—24no—
244C22×D73—48no—
245C22×D73—44no—
246C22×D73—82no—
247C22×D73—82no—
248C22×D73—84no—
249C2×C4×C73—28no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—24no—
253C2×C4×C73—24no—
254C2×C4×C73—22no—
255C2×C4×C73—24no—
256C2×C4×C73—24no—
257C2×C4×C73—22no—
258C2×C4×C73—22no—
259C2×C4×C73—24no—
260C2×C4×C73—24no—
261C2×C4×C73—48no—
262C2×C4×C73—48no—
263C2×C4×C73—48no—
264C2×C4×C73—48no—
265C2×C4×C73—44no—
266C2×C4×C73H(5,𝔽2)14yes—
267C22⋊D73—48no—
268C22⋊D73—48no—
269C2×Dic73—24no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—24no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—22no—
276C2×Dic73—24no—
277C2×Dic73—24no—
278C2×Dic73—24no—
279C2×Dic73—24no—
280C2×Dic73—24no—
281C2×Dic73—24no—
282C2×Dic73—48no—
283C2×Dic73—44no—
284C2×Dic73—82no—
285C2×Dic73—82no—
286C2×Dic73—84no—
287C22×C73—24no—
288C22×C73—24no—
289C22×C73—24no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—24no—
297C22×C73—24no—
298C22×C73—24no—
299C22×C73—416no—
300C22×C73C2×#12614yes—
301C22×C73C2×#12614yes—
302C22×C73C2×#12614yes—
303C22×C73C22×#4111yescharacteristic
304C22×C73C22×#4111yescharacteristic
305C22×C73C22×#4111yescharacteristic
306C22×C73C2×H(5,𝔽2)12yes—
307C22×C73#58512yes—
308C2×D73—42no—
309C2×D73—44no—
310C2×D73—42no—
311C2×D73—42no—
312C2×D73—44no—
313C2×D73—88no—
314C4×C73—24no—
315C4×C73—44no—
316C4×C73—42no—
317C4×C73—44no—
318Dic73—42no—
319Dic73—44no—
320Dic73—42no—
321Dic73—42no—
322Dic73—44no—
323#34995—731nomaximal, Sylow
324C2×C73—24no—
325C2×C73—24no—
326C2×C73—24no—
327C2×C73C22×#12611yescharacteristic
328C2×C73#345912yes—
329C2×C73#349011yescharacteristic
330C2×C73#349111yescharacteristic
331C2×C73#350512yes—
332D73—82no—
333#2852—734no—
334#2643—734no—
335#2510—734no—
336#2493—734no—
337#2481—734no—
338#2480—734no—
339#2315—731no—
340#2315—731no—
341#2308—731no—
342#2308—731no—
343C2×#386—731no—
344C2×#385—731no—
345C2×C4×C22⋊C4D7311yescharacteristic, complemented
346C73#3499511yescharacteristic, complemented, Sylow
347C23×D4—731no—
348C24×C4C2×D7311yescharacteristic
349C2×#126—734no—
350C22×C4⋊C4—732no—
351C22×C22⋊C4—732no—
352C22×C22⋊C4—731no—
353C22×C22⋊C4—731no—
354C22×C22⋊C4—731no—
355C22×C22⋊C4C2×D7311yescharacteristic
356C22×C22⋊C4C2×D7311yescharacteristic
357C22×C42C2×D7312yescomplemented
358#401—734no—
359#400—734no—
360#400—734no—
361#400—734no—
362#397—734no—
363#397—734no—
364#397—734no—
365#396—734no—
366#388—734no—
367#388—734no—
368#388—734no—
369#387—734no—
370#387—734no—
371#387—734no—
372#386—734no—
373#386—734no—
374#386—732no—
375#386—732no—
376#386—734no—
377#386—732no—
378#386—734no—
379#386—732no—
380#385—734no—
381#385—734no—
382#385—732no—
383#385—732no—
384#385—734no—
385#385—734no—
386#385—734no—
387#383—734no—
388#381—732no—
389#381—732no—
390#380—734no—
391#380—732no—
392#380—732no—
393#379—732no—
394#379—732no—
395C4×C22⋊C4C2×D7314yes—
396C4×C22⋊C4C2×D7314yescomplemented
397C2×#95—732no—
398C2×#95—732no—
399C2×#95C2×D7312yescomplemented
400C25—731no—
401C25C22×D7311yescharacteristic
402C22×D4—732no—
403C22×D4—732no—
404C22×D4—1464no—
405C22×D4—1464no—
406C22×D4—1464no—
407C23×C4—24no—
408C23×C4—44no—
409C23×C4—732no—
410C23×C4—731no—
411C23×C4—732no—
412C23×C4—731no—
413C23×C4—731no—
414C23×C4C22×D7311yescharacteristic
415C23×C4C22×D7312yes—
416C23×C4C22×D7312yes—
417C23×C4C22×D7312yes—
418C23×C4C22×D7312yes—
419C23×C4C22×D7311yescharacteristic
420#126—2928no—
421C2×C4⋊C4—2924no—
422C2×C4⋊C4—734no—
423C2×C4⋊C4—734no—
424C2×C4⋊C4—734no—
425C2×C4⋊C4—734no—
426C2×C4⋊C4—734no—
427C2×C4⋊C4—734no—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—2924no—
431C2×C22⋊C4—734no—
432C2×C22⋊C4—734no—
433C2×C22⋊C4—734no—
434C2×C22⋊C4—734no—
435C2×C22⋊C4—734no—
436C2×C22⋊C4—732no—
437C2×C22⋊C4—732no—
438C2×C22⋊C4—732no—
439C2×C22⋊C4—732no—
440C2×C22⋊C4—732no—
441C2×C22⋊C4—732no—
442C2×C22⋊C4—734no—
443C2×C22⋊C4—1464no—
444C2×C22⋊C4—1464no—
445C2×C22⋊C4—1464no—
446C2×C22⋊C4C22×D7312yes—
447C2×C22⋊C4C22×D7312yes—
448C2×C22⋊C4C22×D7312yes—
449C2×C22⋊C4C22×D7312yes—
450C2×C42—44no—
451C2×C42C22×D7314yes—
452C2×C42C22×D7314yes—
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#95—734no—
465#95C22×D7314yes—
466#95C22×D7314yes—
467C24—42no—
468C24—42no—
469C24—2922no—
470C24—2922no—
471C24—732no—
472C24—1462no—
473C24—1462no—
474C24—1462no—
475C24#1192402814yes—
476C24#1192402814yes—
477C24#1192402814yes—
478C24C23×D7311yescharacteristic
479C24C23×D7311yescharacteristic
480C24C23×D7311yescharacteristic
481C2×D4—29216no—
482C2×D4—1464no—
483C2×D4—1464no—
484C2×D4—1464no—
485C2×D4—1464no—
486C2×D4—1464no—
487C2×D4—1464no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—416no—
505C22×C4—2922no—
506C22×C4—2922no—
507C22×C4—2922no—
508C22×C4—2922no—
509C22×C4—2922no—
510C22×C4—2922no—
511C22×C4—732no—
512C22×C4—732no—
513C22×C4—734no—
514C22×C4—734no—
515C22×C4—732no—
516C22×C4—1464no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1464no—
521C22×C4—1462no—
522C22×C4—1462no—
523C22×C4C23×D7312yes—
524C22×C4C23×D7314yes—
525C22×C4C23×D7314yes—
526C22×C4C23×D7312yes—
527C4⋊C4—2924no—
528C4⋊C4—2924no—
529C4⋊C4—2924no—
530C4⋊C4—2924no—
531C4⋊C4—2924no—
532C4⋊C4—1468no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—2924no—
538C22⋊C4—2924no—
539C22⋊C4—2924no—
540C22⋊C4—2924no—
541C22⋊C4—2924no—
542C22⋊C4—1464no—
543C22⋊C4—1464no—
544C22⋊C4—1464no—
545C22⋊C4—1464no—
546C22⋊C4—1464no—
547C22⋊C4—1464no—
548C22⋊C4—1468no—
549C42—28no—
550C42—44no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—2928no—
562C23—2924no—
563C23—5844no—
564C23—5842no—
565C23—5842no—
566C23—1462no—
567C23—1462no—
568C23—1462no—
569C23—1464no—
570C23—1464no—
571C23#14556585116yescomplemented
572C23C2×#1192402812yes—
573C23C2×#1192402812yes—
574C23C2×#1192402812yes—
575C23C2×#1192402812yes—
576C23C2×#1192402812yes—
577C23C2×#1192402812yes—
578C23C2×#1192402812yes—
579C23C2×#1192402812yes—
580C23C2×#1192402812yes—
581C23C24×D7311yescharacteristic
582D4—2928no—
583D4—2928no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—2928no—
602C2×C4—2924no—
603C2×C4—5844no—
604C2×C4—5842no—
605C2×C4—5842no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1464no—
614C2×C4—1464no—
615C2×C4—1464no—
616C2×C4—1464no—
617C2×C4—1464no—
618C2×C4—1464no—
619C2×C4#1455663614yes—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—416no—
633C22—2924no—
634C22—2924no—
635C22—2922no—
636C22—2922no—
637C22—2922no—
638C22—5848no—
639C22C2×#1455658514yes—
640C22C2×#1455658514yes—
641C22C2×#1455658514yes—
642C22C22×#1192402811yescharacteristic
643C22C22×#1192402811yescharacteristic
644C22C22×#1192402811yescharacteristic
645C22C2×#1455663612yes—
646C22#1798942312yes—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—2924no—
652C4—2924no—
653C4—2922no—
654C4—2922no—
655C4—2922no—
656C2—24no—
657C2—24no—
658C2—24no—
659C2—5842no—
660C2C22×#1455658511yescharacteristic
661C2#2262334012yes—
662C2#2262362011yescharacteristic
663C2#2262362911yescharacteristic
664C2#2262377012yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC2×C4×C22⋊C4×C73C2
prime radicalC23×C22⋊D73C22
hypercenterC2×C4×C22⋊C4D73
hypocenterC73#34995
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22618246C2semidirect
2. #22615884C2semidirect
3. #22614485C2semidirect
4. #22614310C2semidirect
5. #22614192C2semidirect
6. #22614177C2semidirect
7. #22612543C2semidirect
8. #22612543C2semidirect
12. C2×#17988460C2semidirect
13. C2×C4×C22⋊C4×C73C2semidirect
15. C2×#14556585C22semidirect
16. C22×#11924007C22semidirect
21. C73⋊#401C22semidirect
22. #17988528C22semidirect
23. #17988528C22semidirect
24. #17988528C22semidirect
25. #17988516C22semidirect
26. #17988516C22semidirect
27. #17988516C22semidirect
28. #17988511C22semidirect
29. #17988478C22semidirect
30. #17988478C22semidirect
31. #17988478C22semidirect
32. #17988471C22semidirect
33. #17988471C22semidirect
34. #17988471C22semidirect
44. #17988460C22semidirect
45. #17988460C22semidirect
46. #17988460C22semidirect
47. #17988460C22semidirect
49. #17988459C22semidirect
50. #17988455C22semidirect
54. #17988445C22semidirect
55. #17988445C22semidirect
59. C2×#14556498C22semidirect
63. C22×C42×C73C22semidirect
65. C4×C22⋊C4×C73C22semidirect
66. C2×#95×C73C22semidirect
228. C23×C73#126semidirect
345. C2×C4×C22⋊C4D73semidirect
346. C73#34995semidirect
357. C22×C42C2×D73semidirect
396. C4×C22⋊C4C2×D73semidirect
399. C2×#95C2×D73semidirect
571. C23#14556585semidirect

Maximal subgroups

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

subgroupindex
#226182462
#226158842
#226144852
#226143102
#226141922
#226141772
#226125432
#226125062
C2×#179884662
C2×#179884602
C2×C4×C22⋊C4×C732
#3499573