Finite Group Databasebeta

← #33012994

Subgroups — #33012994

Order 18688.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C211yesmaximal, characteristic
2C2×#17988484C212yesmaximal, complemented
3C2×#17988483C211yesmaximal, characteristic, complemented
4C2×#17988480C211yesmaximal, characteristic
5C2×#17988475C212yesmaximal
6#22608198C211yesmaximal, characteristic, complemented
7#22608197C212yesmaximal, complemented
8#22608194C211yesmaximal, characteristic, complemented
9#22608174C212yesmaximal
10#95×Dic73C211yesmaximal, characteristic, complemented
11C73×#1439C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924007C2212yes—
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2212yes—
19C22×C22⋊Dic73C2211yescharacteristic
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988523C418yescomplemented
23#17988484—28no—
24#17988483—24no—
25#17988480—24no—
26#17988475—28no—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2211yescharacteristic
33C2×#14556498C2212yes—
34C2×#14556498C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2312yes—
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2312yes—
45C23×Dic73C2311yescharacteristic
46C2×C4⋊Dic73—22no—
47C2×C4⋊Dic73—24no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—22no—
50C2×C4⋊Dic73C2×C414yes—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—42no—
58C2×#11924007—42no—
59C2×#11924007C2×C418yes—
60C2×C22⋊Dic73—24no—
61C2×C22⋊Dic73—24no—
62C2×C22⋊Dic73—24no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—42no—
66C2×C22⋊Dic73C2×C418yes—
67C2×C4×Dic73—42no—
68C2×C4×Dic73—42no—
69C2×C4×Dic73—42no—
70C2×C4×Dic73D418yes—
71C2×C4×Dic73D414yes—
72#14556498—24no—
73#14556498—22no—
74#14556498—24no—
75#14556498—24no—
76#14556498—24no—
77#14556498—22no—
78#14556498—24no—
79#14556498C2×C414yes—
80C25×C73C2311yescharacteristic
81C23×C4×C73C2312yes—
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73—22no—
86C2×C4⋊C4×C73—42no—
87C2×C22⋊C4×C73—22no—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—24no—
90C2×C22⋊C4×C73—42no—
91C2×C22⋊C4×C73—42no—
92C2×C22⋊C4×C73C2×C414yes—
93#95×C73—24no—
94#95×C73—22no—
95C22×Dic73—21no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—21no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—21no—
118C22×Dic73—21no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73C22×C414yes—
123C22×Dic73C22×C414yes—
124C22×Dic73C2×D412yes—
125C22×Dic73C2×D414yes—
126C22×Dic73C2×D414yes—
127C22×Dic73C2×D412yes—
128C22×Dic73#4114yes—
129C4⋊Dic73—28no—
130C4⋊Dic73—44no—
131#11924007—216no—
132#11924007—48no—
133#11924007—48no—
134C22⋊Dic73—216no—
135C22⋊Dic73—44no—
136C22⋊Dic73—44no—
137C4×Dic73—48no—
138C4×Dic73—44no—
139C24×C73—21no—
140C24×C73—22no—
141C24×C73—21no—
142C24×C73—22no—
143C24×C73—21no—
144C24×C73—22no—
145C24×C73—21no—
146C24×C73—21no—
147C24×C73—22no—
148C24×C73—21no—
149C24×C73C22×C412yes—
150C24×C73C2411yescharacteristic
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
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—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D414yes—
173C22×C4×C73C2×D412yes—
174C22×C4×C73#4112yes—
175C4⋊C4×C73—44no—
176C22⋊C4×C73—28no—
177C22⋊C4×C73—48no—
178C23×C73—22no—
179C23×C73—24no—
180C23×C73—22no—
181C23×C73—24no—
182C23×C73—22no—
183C23×C73—41no—
184C23×C73—42no—
185C23×C73—41no—
186C23×C73—42no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—41no—
199C23×C73—42no—
200C23×C73—41no—
201C23×C73C23×C411yescharacteristic
202C23×C73C22×D412yes—
203C23×C73C22×D411yescharacteristic
204C23×C73C2×#4112yes—
205C23×C73C2×#4111yescharacteristic
206C23×C73H(5,𝔽2)11yescharacteristic
207C23×C73H(5,𝔽2)12yes—
208C23×C732−1+411yescharacteristic
209C23×C732−1+411yescharacteristic
210C23×C732−1+412yes—
211C23×C732−1+411yescharacteristic
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—41no—
222C2×C4×C73—42no—
223C2×C4×C73—41no—
224C2×C4×C73—42no—
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×C73C4×D414yes—
236C2×Dic73—24no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—22no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—22no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—42no—
262C2×Dic73—41no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—42no—
268C2×Dic73C4×D418yes—
269C2×Dic73#12714yes—
270C22×C73—42no—
271C22×C73—44no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—44no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73—82no—
278C22×C73—81no—
279C22×C73—82no—
280C22×C73—81no—
281C22×C73C2×C4×D412yes—
282C22×C73C2×C4×D411yescharacteristic
283C22×C73#51911yescharacteristic
284C22×C73#52012yes—
285C22×C73#52011yescharacteristic
286C22×C73C2×#12711yescharacteristic
287C22×C73#53212yes—
288C22×C73#53211yescharacteristic
289C22×C73#53412yes—
290C22×C73#53411yescharacteristic
291C22×C73#53611yescharacteristic
292C22×C73#53612yes—
293C22×C73#53612yes—
294C22×C73#53611yescharacteristic
295C22×C73#53712yes—
296C22×C73#53711yescharacteristic
297C22×C73#53912yes—
298C22×C73#53912yes—
299C22×C73#53912yes—
300C22×C73#54112yes—
301C22×C73#54111yescharacteristic
302C22×C73#54212yes—
303C22×C73#54211yescharacteristic
304C4×C73—24no—
305C4×C73—42no—
306C4×C73—44no—
307C4×C73—82no—
308Dic73—28no—
309Dic73—24no—
310Dic73—44no—
311Dic73—42no—
312Dic73—81no—
313Dic73—81no—
314#14322—731nomaximal, Sylow
315C2×C73—82no—
316C2×C73C4×#12711yescharacteristic
317C2×C73#232212yes—
318C2×C73#232211yescharacteristic
319C2×C73#232311yescharacteristic
320C2×C73#232312yes—
321C2×C73#265611yescharacteristic
322C2×C73#265811yescharacteristic
323C2×C73#266312yes—
324C2×C73#266311yescharacteristic
325C2×C73#266412yes—
326C2×C73#266411yescharacteristic
327C2×#399—731no—
328C2×#389—732no—
329C2×#389—731no—
330C2×#388—732no—
331C2×#388—731no—
332#1445—732no—
333#1445—731no—
334#1444—731no—
335#1439—732no—
336#1439D7311yescharacteristic
337C4×#95—731no—
338C73#1432211yescharacteristic, complemented, Sylow
339C22×C4⋊C4—732no—
340C22×C4⋊C4—732no—
341C22×C4⋊C4—731no—
342C22×C4⋊C4—732no—
343C22×C4⋊C4—731no—
344C22×C4⋊C4C2×D7311yescharacteristic
345C22×C22⋊C4—731no—
346C22×C22⋊C4—732no—
347C22×C22⋊C4—731no—
348C22×C22⋊C4C2×D7311yescharacteristic
349C22×C22⋊C4C2×D7312yes—
350C22×C42—732no—
351C22×C42—731no—
352#399—738no—
353#389—1468no—
354#389—1464no—
355#388—1468no—
356#388—1464no—
357C2×#95—731no—
358C2×#95—732no—
359C2×#95—732no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95—731no—
363C2×#95—731no—
364C2×#95—732no—
365C2×#95C2×D7311yescharacteristic
366C2×#95C2×D7312yes—
367C25C22×D7311yescharacteristic
368C23×C4—731no—
369C23×C4—732no—
370C23×C4—732no—
371C23×C4—731no—
372C23×C4—731no—
373C23×C4—731no—
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—738no—
383C2×C4⋊C4—734no—
384C2×C4⋊C4—1464no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4—1464no—
387C2×C4⋊C4—1464no—
388C2×C4⋊C4—1462no—
389C2×C4⋊C4—1464no—
390C2×C4⋊C4—1464no—
391C2×C4⋊C4—1464no—
392C2×C4⋊C4—1462no—
393C2×C4⋊C4—1462no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—2922no—
400C2×C22⋊C4—738no—
401C2×C22⋊C4—1464no—
402C2×C22⋊C4—1462no—
403C2×C22⋊C4—1462no—
404C2×C22⋊C4—1464no—
405C2×C22⋊C4—1464no—
406C2×C22⋊C4C4×D7314yes—
407C2×C42—2922no—
408C2×C42—2922no—
409C2×C42—2922no—
410C2×C42—738no—
411C2×C42—734no—
412#95—24no—
413#95—22no—
414#95—734no—
415#95—1464no—
416#95—1462no—
417#95—1464no—
418#95—1464no—
419#95—1462no—
420#95—1464no—
421#95—1464no—
422C24—22no—
423C24—21no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24C2×C4×D7312yes—
433C24C23×D7311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
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—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—734no—
454C22×C4—734no—
455C22×C4—734no—
456C22×C4—732no—
457C22×C4—734no—
458C22×C4—734no—
459C22×C4—732no—
460C22×C4—1462no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1461no—
466C22×C4—1462no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1461no—
486C22×C4—1462no—
487C22×C4C2×C4×D7312yes—
488C22×C4C2×C4×D7312yes—
489C22×C4D4×D7314yes—
490C22×C4D4×D7312yes—
491C22×C4#1192402712yes—
492C4⋊C4—44no—
493C4⋊C4—2928no—
494C4⋊C4—2924no—
495C4⋊C4—2928no—
496C4⋊C4—14616no—
497C4⋊C4—1468no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—2924no—
501C22⋊C4—2924no—
502C22⋊C4—14616no—
503C42—2924no—
504C42—2928no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23C22×C4×D7311yescharacteristic
529C23C2×D4×D7312yes—
530C23C2×D4×D7311yescharacteristic
531C23#41×D7312yes—
532C23C2×#1192402711yescharacteristic
533C23#1455663611yescharacteristic
534C23#1455663612yes—
535C23#1455663812yes—
536C23#1455663811yescharacteristic
537C23#1455663811yescharacteristic
538C23#1455663911yescharacteristic
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
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—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2921no—
578C2×C4—734no—
579C2×C4—738no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1462no—
583C2×C4—1464no—
584C2×C4—1464no—
585C2×C4—1464no—
586C2×C4—1464no—
587C2×C4—1464no—
588C2×C4—1462no—
589C2×C4—1464no—
590C2×C4—1464no—
591C2×C4—1462no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4#1455655114yes—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—44no—
602C22—82no—
603C22—81no—
604C22—82no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C4×D4×D7312yes—
609C22C2×#1455655111yescharacteristic
610C22#1798906411yescharacteristic
611C22#1798906612yes—
612C22#1798906811yescharacteristic
613C22#127×D7311yescharacteristic
614C22#1798913212yes—
615C22#1798913611yescharacteristic
616C22#1798914712yes—
617C22#1798914711yescharacteristic
618C22#1798915912yes—
619C22#1798915911yescharacteristic
620C22#1798916011yescharacteristic
621C22#1798916012yes—
622C22#1798916612yes—
623C22#1798916611yescharacteristic
624C22#1798918012yes—
625C22#1798918212yes—
626C22#1798918312yes—
627C22#1798919412yes—
628C22#1798919711yescharacteristic
629C22#1798920112yes—
630C22#1798920411yescharacteristic
631C4—24no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—2924no—
636C4—2922no—
637C4—5841no—
638C4—5841no—
639C4—1468no—
640C4—1464no—
641C2—82no—
642C2#2261249511yescharacteristic
643C2#2261259812yes—
644C2#2261260511yescharacteristic
645C2#2261260911yescharacteristic
646C2#2261261012yes—
647C2#2261603211yescharacteristic
648C2#2261604011yescharacteristic
649C2#2261607812yes—
650C2#2261608711yescharacteristic
651C2#2261609112yes—
652C2#2261609911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14322
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#17988484C2semidirect
3. C2×#17988483C2semidirect
6. #22608198C2semidirect
7. #22608197C2semidirect
8. #22608194C2semidirect
10. #95×Dic73C2semidirect
22. #17988523C4semidirect
338. C73#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×#179884842
C2×#179884832
C2×#179884802
C2×#179884752
#226081982
#226081972
#226081942
#226081742
#95×Dic732
C73×#14392
#1432273