Finite Group Databasebeta

← #33008679

Subgroups — #33008679

Order 18688.

1862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556547C211yesmaximal, characteristic
2C2×#17988451C211yesmaximal, characteristic, complemented
3C2×#17988433C211yesmaximal, characteristic, complemented
4#22608182C212yesmaximal, complemented
5#22608174C212yesmaximal
6#22608168C211yesmaximal, characteristic
7#22608166C211yesmaximal, characteristic, complemented
8#22608164C211yesmaximal, characteristic, complemented
9#22608157C212yesmaximal, complemented
10#22608153C212yesmaximal
11C73×#1437C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C2×#14556547—24no—
14C2×#14556547—24no—
15C2×#14556547C418yescomplemented
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2212yes—
19C22×C22⋊Dic73C2212yes—
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23C22×C4×Dic73C2211yescharacteristic
24#17988451—24no—
25#17988433—24no—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C2×#14556498C2212yes—
35C22×C22⋊C4×C73C2212yes—
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C42×C73C2211yescharacteristic
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73—22no—
41C23×Dic73—22no—
42C23×Dic73—24no—
43C23×Dic73—24no—
44C23×Dic73C2×C414yes—
45C23×Dic73C2312yes—
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51#14556547—232no—
52C2×#11924007—22no—
53C2×#11924007—22no—
54C2×#11924007—24no—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007—42no—
58C2×#11924007—42no—
59C2×#11924007C2×C414yes—
60C2×#11924007C2×C414yes—
61C2×C22⋊Dic73—22no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—24no—
64C2×C22⋊Dic73—24no—
65C2×C22⋊Dic73—42no—
66C2×C22⋊Dic73C2×C414yes—
67C2×C4×Dic73—22no—
68C2×C4×Dic73—22no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73C2×C414yes—
72C2×C4×Dic73C2×C414yes—
73C2×C4×Dic73D414yes—
74C2×C4×Dic73D414yes—
75#14556498—24no—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—24no—
83#14556498—42no—
84C25×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—24no—
92C2×C22⋊C4×C73—24no—
93C2×C22⋊C4×C73—42no—
94C2×C22⋊C4×C73C2×C414yes—
95C2×C42×C73—22no—
96C2×C42×C73—24no—
97#95×C73—22no—
98#95×C73—42no—
99C22×Dic73—216no—
100C22×Dic73—216no—
101C22×Dic73—28no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—28no—
110C22×Dic73—28no—
111C22×Dic73—22no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73C22×C412yes—
129C22×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C412yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C22×Dic73#4114yes—
137C22×Dic73#4114yes—
138#11924007—28no—
139#11924007—28no—
140#11924007—48no—
141C22⋊Dic73—28no—
142C22⋊Dic73—48no—
143C4×Dic73—28no—
144C4×Dic73—28no—
145C4×Dic73C22⋊C4116yescomplemented
146C24×C73—21no—
147C24×C73—21no—
148C24×C73—22no—
149C24×C73—22no—
150C24×C73—22no—
151C24×C73—22no—
152C24×C73—22no—
153C24×C73C22×C412yes—
154C24×C73#4114yes—
155C24×C73C2411yescharacteristic
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73C22×C412yes—
176C22×C4×C73C22×C412yes—
177C22×C4×C73C2×D412yes—
178C22×C4×C73C2×D412yes—
179C22×C4×C73#4114yes—
180C22⋊C4×C73—28no—
181C22⋊C4×C73—48no—
182C42×C73—28no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—24no—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—24no—
190C23×C73—24no—
191C23×C73—24no—
192C23×C73—24no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—42no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73#11714yes—
204C23×C73C23×C411yescharacteristic
205C23×C73C22×D411yescharacteristic
206C23×C73C22×D411yescharacteristic
207C23×C73C2×#4112yes—
208C23×C73C2×#4112yes—
209C23×C73C2×#4112yes—
210C23×C73C2×#4112yes—
211C23×C73H(5,𝔽2)12yes—
212C23×C732−1+412yes—
213C2×C4×C73—24no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—42no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
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×C22⋊C414yes—
237C2×Dic73—232no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—22no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73C2×C22⋊C414yes—
270C2×Dic73C2×C22⋊C414yes—
271C2×Dic73#11714yes—
272C22×C73—28no—
273C22×C73—24no—
274C22×C73—42no—
275C22×C73—44no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73—44no—
280C22×C73—42no—
281C22×C73C22×C22⋊C411yescharacteristic
282C22×C73C2×#11711yescharacteristic
283C22×C73C2×#11711yescharacteristic
284C22×C73C4×#4112yes—
285C22×C73#52012yes—
286C22×C73#52511yescharacteristic
287C22×C73#52511yescharacteristic
288C22×C73#52512yes—
289C22×C73#52512yes—
290C22×C73#53212yes—
291C22×C73#53511yescharacteristic
292C22×C73#53611yescharacteristic
293C22×C73#53712yes—
294C22×C73#55112yes—
295C22×C73#55112yes—
296C22×C73#55212yes—
297C22×C73#55212yes—
298C22×C73#55212yes—
299C22×C73#55212yes—
300C22×C73#55512yes—
301C22×C73#55512yes—
302C4×C73—24no—
303C4×C73—42no—
304C4×C73—44no—
305C4×C73—82no—
306Dic73—24no—
307Dic73—24no—
308Dic73—24no—
309Dic73—44no—
310Dic73—42no—
311Dic73—82no—
312#13957—731nomaximal, Sylow
313C2×C73—44no—
314C2×C73#228011yescharacteristic
315C2×C73#229711yescharacteristic
316C2×C73#231111yescharacteristic
317C2×C73#234312yes—
318C2×C73#234612yes—
319C2×C73#260912yes—
320C2×C73#260912yes—
321C2×C73#261012yes—
322C2×C73#261112yes—
323C22×#117—731no—
324C2×#382—731no—
325C2×#376—731no—
326#1441—732no—
327#1439—732no—
328#1437—731no—
329#1437D7311yescharacteristic
330#1436—731no—
331#1436—731no—
332#1434—732no—
333C22⋊#95—732no—
334C73#1395711yescharacteristic, complemented, Sylow
335C24×C4—731no—
336C2×#117—738no—
337C2×#117—1464no—
338C2×#117—1464no—
339C22×C4⋊C4—732no—
340C22×C4⋊C4—731no—
341C22×C4⋊C4—731no—
342C22×C22⋊C4—732no—
343C22×C22⋊C4—731no—
344C22×C22⋊C4C2×D7312yes—
345C22×C22⋊C4C2×D7311yescharacteristic
346C22×C42—731no—
347C22×C42—731no—
348C22×C42—731no—
349C22×C42C2×D7311yescharacteristic
350#382—1464no—
351#376—1464no—
352C2×#95—732no—
353C2×#95—732no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—731no—
360C2×#95—732no—
361C2×#95C2×D7311yescharacteristic
362C2×#95C2×D7312yes—
363C25C22×D7311yescharacteristic
364C23×C4—731no—
365C23×C4—731no—
366C23×C4—731no—
367C23×C4—734no—
368C23×C4—731no—
369C23×C4—732no—
370C23×C4—732no—
371C23×C4—1462no—
372C23×C4—1462no—
373C23×C4—1464no—
374C23×C4—1464no—
375C23×C4C22×D7312yes—
376C23×C4C22×D7311yescharacteristic
377C23×C4C22×D7312yes—
378C23×C4C22×D7311yescharacteristic
379#117—14632no—
380C2×C4⋊C4—2922no—
381C2×C4⋊C4—2922no—
382C2×C4⋊C4—734no—
383C2×C4⋊C4—734no—
384C2×C4⋊C4—1464no—
385C2×C4⋊C4—1462no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4—1462no—
388C2×C4⋊C4—1462no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—2922no—
395C2×C22⋊C4—734no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1464no—
398C2×C22⋊C4—1462no—
399C2×C22⋊C4—1462no—
400C2×C22⋊C4C4×D7314yes—
401C2×C42—22no—
402C2×C42—24no—
403C2×C42—734no—
404C2×C42—734no—
405C2×C42—734no—
406C2×C42—734no—
407C2×C42—1462no—
408C2×C42—1462no—
409C2×C42—1464no—
410C2×C42—1462no—
411#95—22no—
412#95—42no—
413#95—2922no—
414#95—1464no—
415#95—1464no—
416#95—1464no—
417#95—1464no—
418#95—1464no—
419#95—1464no—
420#95—1462no—
421#95—1464no—
422C24—21no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D7312yes—
430C24#1192402814yes—
431C24C23×D7311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—732no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—732no—
456C22×C4—732no—
457C22×C4—732no—
458C22×C4—734no—
459C22×C4—732no—
460C22×C4—734no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1468no—
479C22×C4—1468no—
480C22×C4—14616no—
481C22×C4—14616no—
482C22×C4—1468no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4C2×C4×D7312yes—
491C22×C4C2×C4×D7312yes—
492C22×C4D4×D7312yes—
493C22×C4D4×D7312yes—
494C22×C4#1192402714yes—
495C4⋊C4—2928no—
496C4⋊C4—1468no—
497C4⋊C4—1468no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—2928no—
501C22⋊C4—1468no—
502C42—28no—
503C42—7316no—
504C42—1468no—
505C42—1468no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23#1455654814yes—
527C23C22×C4×D7311yescharacteristic
528C23C2×D4×D7311yescharacteristic
529C23C2×D4×D7311yescharacteristic
530C23#41×D7312yes—
531C23#41×D7312yes—
532C23C2×#1192402712yes—
533C23C2×#1192402812yes—
534C23#1455663612yes—
535C23#1455663812yes—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—734no—
574C2×C4—734no—
575C2×C4—734no—
576C2×C4—1464no—
577C2×C4—1464no—
578C2×C4—1462no—
579C2×C4—1464no—
580C2×C4—1464no—
581C2×C4—14632no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4—1464no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4—1462no—
589C2×C4—1464no—
590C2×C4—1464no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4C22⋊C4×D7314yes—
595C22—28no—
596C22—24no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22C2×C22⋊C4×D7311yescharacteristic
605C22#117×D7311yescharacteristic
606C22C2×#1455654811yescharacteristic
607C22C4×#1192402712yes—
608C22#1798906612yes—
609C22#1798909311yescharacteristic
610C22#1798909511yescharacteristic
611C22#1798909612yes—
612C22#1798909612yes—
613C22#1798913212yes—
614C22#1798915011yescharacteristic
615C22#1798915711yescharacteristic
616C22#1798916612yes—
617C22#1798926312yes—
618C22#1798926612yes—
619C22#1798927012yes—
620C22#1798927312yes—
621C22#1798927412yes—
622C22#1798927712yes—
623C22#1798929912yes—
624C22#1798930112yes—
625C4—24no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—2924no—
630C4—2922no—
631C4—5842no—
632C4—1464no—
633C4—1464no—
634C4—1464no—
635C2—44no—
636C2#2261231211yescharacteristic
637C2#2261241611yescharacteristic
638C2#2261252111yescharacteristic
639C2#2261278512yes—
640C2#2261281312yes—
641C2#2261560512yes—
642C2#2261560612yes—
643C2#2261561512yes—
644C2#2261562612yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1437C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1437D73
hypocenterC73#13957
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#17988451C2semidirect
3. C2×#17988433C2semidirect
4. #22608182C2semidirect
7. #22608166C2semidirect
8. #22608164C2semidirect
9. #22608157C2semidirect
15. C2×#14556547C4semidirect
145. C4×Dic73C22⋊C4semidirect
334. C73#13957semidirect

Maximal subgroups

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

subgroupindex
C22×#145565472
C2×#179884512
C2×#179884332
#226081822
#226081742
#226081682
#226081662
#226081642
#226081572
#226081532
C73×#14372
#1395773