Finite Group Databasebeta

← #33013256

Subgroups — #33013256

Order 18688.

1566 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988449C211yesmaximal, characteristic
2C2×#17988444C211yesmaximal, characteristic
3C2×#17988434C211yesmaximal, characteristic
4#22608195C211yesmaximal, characteristic, complemented
5#22608194C211yesmaximal, characteristic, complemented
6#22608188C211yesmaximal, characteristic, complemented
7#22608175C212yesmaximal
8#22608175C212yesmaximal
9#22608159C212yesmaximal, complemented
10#22608155C212yesmaximal, complemented
11C73×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924005C2212yes—
17C22×#11924005C2212yes—
18C22×#11924005C2211yescharacteristic
19C22×#11924005C2211yescharacteristic
20C22×C4×Dic73C2212yes—
21#17988449—24no—
22#17988444—24no—
23#17988434C418yescomplemented
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C2×#95×C73C2211yescharacteristic
35C2×#95×C73C2212yes—
36C2×#95×C73C2212yes—
37C2×#95×C73C2211yescharacteristic
38C24×D73C2311yescharacteristic
39C23×Dic73C2311yescharacteristic
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2312yes—
44C22×C4×D73—42no—
45C22×C4×D73—42no—
46C22×C4×D73C2×C414yes—
47C22×C4×D73D414yes—
48C22×C4×D73Q814yes—
49C2×C4⋊Dic73—22no—
50C2×C4⋊Dic73—42no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—42no—
54C2×#11924007—42no—
55C2×#11924007—42no—
56C2×#11924007—42no—
57C2×#11924005—24no—
58C2×#11924005—24no—
59C2×#11924005—24no—
60C2×#11924005—22no—
61C2×#11924005—22no—
62C2×#11924005—42no—
63C2×#11924005—42no—
64C2×#11924005—42no—
65C2×#11924005—42no—
66C2×#11924005C2×C418yes—
67C2×C4×Dic73—42no—
68C2×C4×Dic73—42no—
69C2×C4×Dic73C2×C418yes—
70#14556498—24no—
71#14556498—22no—
72#14556498—22no—
73#14556498—24no—
74#14556498—24no—
75#14556498—22no—
76#14556498—42no—
77#14556498—42no—
78#14556498C2×C414yes—
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2312yes—
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2312yes—
83C23×C4×C73C2311yescharacteristic
84C2×C4⋊C4×C73—22no—
85C2×C4⋊C4×C73—42no—
86#95×C73—22no—
87#95×C73—42no—
88#95×C73—42no—
89#95×C73C2×C414yes—
90C23×D73—22no—
91C23×D73—22no—
92C23×D73—21no—
93C23×D73—21no—
94C23×D73—22no—
95C23×D73—22no—
96C23×D73—21no—
97C23×D73—21no—
98C23×D73C22×C412yes—
99C23×D73C2×D412yes—
100C23×D73C2×Q812yes—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—22no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C414yes—
126C22×Dic73C2×D412yes—
127C22×Dic73C2×Q812yes—
128C22×Dic73#4114yes—
129C22×Dic73#4114yes—
130C2×C4×D73—48no—
131C2×C4×D73—44no—
132C2×C4×D73—44no—
133C2×C4×D73—48no—
134C2×C4×D73C4⋊C4116yescomplemented
135C4⋊Dic73—44no—
136#11924007—48no—
137#11924007—48no—
138#11924005—216no—
139#11924005—44no—
140#11924005—44no—
141#11924005—48no—
142C4×Dic73—48no—
143C24×C73C2411yescharacteristic
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73C22×C414yes—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×Q812yes—
173C4⋊C4×C73—44no—
174C23×C73C23×C411yescharacteristic
175C23×C73C22×D411yescharacteristic
176C23×C73C22×Q811yescharacteristic
177C23×C73C2×#4112yes—
178C23×C73C2×#4112yes—
179C23×C73H(5,𝔽2)12yes—
180C23×C73H(5,𝔽2)11yescharacteristic
181C23×C73H(5,𝔽2)12yes—
182C23×C73H(5,𝔽2)11yescharacteristic
183C23×C732−1+411yescharacteristic
184C23×C732−1+411yescharacteristic
185C22×D73—24no—
186C22×D73—24no—
187C22×D73—22no—
188C22×D73—22no—
189C22×D73—22no—
190C22×D73—22no—
191C22×D73—24no—
192C22×D73—22no—
193C22×D73—22no—
194C22×D73—22no—
195C22×D73—22no—
196C22×D73—24no—
197C22×D73—22no—
198C22×D73—22no—
199C22×D73—42no—
200C22×D73—42no—
201C22×D73—42no—
202C22×D73—42no—
203C22×D73—42no—
204C22×D73—42no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73C2×C4⋊C414yes—
208C2×C4×C73—24no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—41no—
220C2×C4×C73—42no—
221C2×C4×C73—41no—
222C2×C4×C73—42no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—41no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73C2×C4⋊C414yes—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—41no—
248C2×Dic73—42no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73C2×C4⋊C414yes—
265C2×Dic73#11718yes—
266C4×D73—416no—
267C22×C73C22×C4⋊C411yescharacteristic
268C22×C73C2×#11712yes—
269C22×C73#51812yes—
270C22×C73#52011yescharacteristic
271C22×C73#52011yescharacteristic
272C22×C73#53211yescharacteristic
273C22×C73#53211yescharacteristic
274C22×C73#53312yes—
275C22×C73#53411yescharacteristic
276C22×C73#53411yescharacteristic
277C22×C73#53511yescharacteristic
278C22×C73#53611yescharacteristic
279C22×C73#53812yes—
280C22×C73#53812yes—
281C22×C73#53912yes—
282C22×C73#53912yes—
283C22×C73#54012yes—
284C22×C73#54012yes—
285C22×C73#54212yes—
286C22×C73#54212yes—
287C22×C73#54312yes—
288C22×C73#54411yescharacteristic
289C22×C73#54411yescharacteristic
290C2×D73—24no—
291C2×D73—24no—
292C2×D73—24no—
293C2×D73—44no—
294C2×D73—44no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—42no—
299C2×D73—44no—
300C2×D73—44no—
301C4×C73—24no—
302C4×C73—44no—
303C4×C73—81no—
304C4×C73—82no—
305C4×C73—81no—
306Dic73—24no—
307Dic73—28no—
308Dic73—82no—
309Dic73—81no—
310Dic73—81no—
311#14355—731nomaximal, Sylow
312C2×C73#228511yescharacteristic
313C2×C73#231511yescharacteristic
314C2×C73#231711yescharacteristic
315C2×C73#231812yes—
316C2×C73#231912yes—
317C2×C73#266111yescharacteristic
318C2×C73#266111yescharacteristic
319C2×C73#266311yescharacteristic
320C2×C73#266311yescharacteristic
321C2×C73#266512yes—
322C2×C73#266712yes—
323D73—44no—
324C2×#381—731no—
325C2×#380—731no—
326C2×C4×C22⋊C4—731no—
327#1444—731no—
328#1444—731no—
329#1442—731no—
330#1442D7311yescharacteristic, complemented
331#1439—732no—
332#1439—732no—
333C4⋊#95—732no—
334#1434—732no—
335C73#1435511yescharacteristic, complemented, Sylow
336C24×C4—731no—
337C22×C4⋊C4—732no—
338C22×C4⋊C4—732no—
339C22×C4⋊C4—731no—
340C22×C4⋊C4C2×D7311yescharacteristic
341C22×C22⋊C4—732no—
342C22×C22⋊C4—731no—
343C22×C22⋊C4—731no—
344C22×C22⋊C4—732no—
345C22×C42—732no—
346#381—1464no—
347#380—1464no—
348C4×C22⋊C4—738no—
349C2×#95—732no—
350C2×#95—732no—
351C2×#95—731no—
352C2×#95—731no—
353C2×#95—732no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—731no—
357C2×#95—731no—
358C2×#95C2×D7311yescharacteristic
359C2×#95C2×D7312yes—
360C2×#95C2×D7312yes—
361C2×#95C2×D7311yescharacteristic
362C25—731no—
363C23×C4—2922no—
364C23×C4—2922no—
365C23×C4—734no—
366C23×C4—734no—
367C23×C4—734no—
368C23×C4—732no—
369C23×C4—731no—
370C23×C4—731no—
371C23×C4—732no—
372C23×C4—731no—
373C23×C4C22×D7311yescharacteristic
374C23×C4C22×D7312yes—
375C23×C4C22×D7311yescharacteristic
376C23×C4C22×D7312yes—
377C23×C4C22×D7311yescharacteristic
378C2×C4⋊C4—22no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—2922no—
381C2×C4⋊C4—2922no—
382C2×C4⋊C4—2922no—
383C2×C4⋊C4—2922no—
384C2×C4⋊C4—2922no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4—1464no—
387C2×C4⋊C4—1462no—
388C2×C22⋊C4—2922no—
389C2×C22⋊C4—2922no—
390C2×C22⋊C4—2922no—
391C2×C22⋊C4—2922no—
392C2×C22⋊C4—738no—
393C2×C22⋊C4—1464no—
394C2×C22⋊C4—1462no—
395C2×C22⋊C4—1462no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1464no—
398C2×C42—2922no—
399C2×C42—2922no—
400C2×C42—738no—
401#95—22no—
402#95—42no—
403#95—42no—
404#95—2922no—
405#95—2922no—
406#95—734no—
407#95—1464no—
408#95—1464no—
409#95—1464no—
410#95—1462no—
411#95—1462no—
412#95—1462no—
413#95C4×D7314yes—
414C24—732no—
415C24—732no—
416C24—732no—
417C24—1462no—
418C24—1462no—
419C24—1461no—
420C24—1462no—
421C24—1461no—
422C24—1462no—
423C24—1461no—
424C24—1461no—
425C24C23×D7311yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—2924no—
452C22×C4—2924no—
453C22×C4—2928no—
454C22×C4—2928no—
455C22×C4—7316no—
456C22×C4—734no—
457C22×C4—732no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—734no—
461C22×C4—734no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1461no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4C2×C4×D7314yes—
486C22×C4C2×C4×D7312yes—
487C22×C4D4×D7312yes—
488C22×C4Q8×D7312yes—
489C4⋊C4—44no—
490C4⋊C4—2924no—
491C4⋊C4—2928no—
492C4⋊C4—2928no—
493C22⋊C4—2924no—
494C22⋊C4—2924no—
495C22⋊C4—2928no—
496C22⋊C4—14616no—
497C42—2928no—
498C23—2922no—
499C23—2922no—
500C23—2922no—
501C23—2922no—
502C23—2922no—
503C23—2922no—
504C23—2922no—
505C23—2922no—
506C23—734no—
507C23—1462no—
508C23—1462no—
509C23—1464no—
510C23—1462no—
511C23—1462no—
512C23—1462no—
513C23—1462no—
514C23—1464no—
515C23—1462no—
516C23—1462no—
517C23—1464no—
518C23—1464no—
519C23—1462no—
520C23—1462no—
521C23C22×C4×D7311yescharacteristic
522C23C2×D4×D7311yescharacteristic
523C23C2×Q8×D7311yescharacteristic
524C23#41×D7312yes—
525C23#41×D7312yes—
526C23#1455663612yes—
527C23#1455663611yescharacteristic
528C23#1455663712yes—
529C23#1455663711yescharacteristic
530C23#1455663811yescharacteristic
531C23#1455663911yescharacteristic
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—29216no—
561C2×C4—2922no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—734no—
579C2×C4—738no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1464no—
583C2×C4—1464no—
584C2×C4—1464no—
585C2×C4—1464no—
586C2×C4—1464no—
587C2×C4—1464no—
588C2×C4—1464no—
589C2×C4—1464no—
590C2×C4C4⋊C4×D7314yes—
591C22—2922no—
592C22—2922no—
593C22—2922no—
594C22—2922no—
595C22—2924no—
596C22—2924no—
597C22—2924no—
598C22—2924no—
599C22—1464no—
600C22—1464no—
601C22—1464no—
602C22C2×C4⋊C4×D7311yescharacteristic
603C22#117×D7312yes—
604C22#1798905912yes—
605C22#1798906611yescharacteristic
606C22#1798906911yescharacteristic
607C22#1798913611yescharacteristic
608C22#1798913611yescharacteristic
609C22#1798914312yes—
610C22#1798914711yescharacteristic
611C22#1798914811yescharacteristic
612C22#1798915311yescharacteristic
613C22#1798916011yescharacteristic
614C22#1798917312yes—
615C22#1798917612yes—
616C22#1798918312yes—
617C22#1798918512yes—
618C22#1798918712yes—
619C22#1798918912yes—
620C22#1798920112yes—
621C22#1798920312yes—
622C22#1798921212yes—
623C22#1798921411yescharacteristic
624C22#1798921711yescharacteristic
625C4—24no—
626C4—44no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—5842no—
631C4—5841no—
632C4—5841no—
633C4—1464no—
634C4—1468no—
635C2—2924no—
636C2#2261233711yescharacteristic
637C2#2261254211yescharacteristic
638C2#2261255311yescharacteristic
639C2#2261256312yes—
640C2#2261257212yes—
641C2#2261606511yescharacteristic
642C2#2261606611yescharacteristic
643C2#2261608711yescharacteristic
644C2#2261608811yescharacteristic
645C2#2261610812yes—
646C2#2261613212yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1442C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1442D73
hypocenterC73#14355
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #22608195C2semidirect
5. #22608194C2semidirect
6. #22608188C2semidirect
9. #22608159C2semidirect
10. #22608155C2semidirect
11. C73×#1442C2semidirect
23. #17988434C4semidirect
134. C2×C4×D73C4⋊C4semidirect
330. #1442D73semidirect
335. C73#14355semidirect

Maximal subgroups

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

subgroupindex
C2×#179884492
C2×#179884442
C2×#179884342
#226081952
#226081942
#226081882
#226081752
#226081592
#226081552
C73×#14422
#1435573