Finite Group Databasebeta

← #33012425

Subgroups — #33012425

Order 18688.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988462C211yesmaximal, characteristic
2C2×#17988459C211yesmaximal, characteristic
3C22×#14556498C211yesmaximal, characteristic
4#22608200C212yesmaximal, complemented
5#22608192C212yesmaximal, complemented
6#22608189C212yesmaximal, complemented
7#22608183C212yesmaximal, complemented
8#22608154C211yesmaximal, characteristic
9#22608153C211yesmaximal, characteristic
10#22608153C211yesmaximal, characteristic
11C73×C22⋊#95C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C24×Dic73C2211yescharacteristic
14C22×C4⋊Dic73C2212yes—
15C22×#11924007C2211yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2211yescharacteristic
19#17988462C418yescomplemented
20#17988459—24no—
21C2×#14556498—24no—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C24×C4×C73C2211yescharacteristic
34C22×C22⋊C4×C73C2211yescharacteristic
35C22×C22⋊C4×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2212yes—
38C23×Dic73—22no—
39C23×Dic73—22no—
40C23×Dic73—22no—
41C23×Dic73—22no—
42C23×Dic73—42no—
43C23×Dic73—42no—
44C23×Dic73—42no—
45C23×Dic73—42no—
46C23×Dic73C2×C414yes—
47C23×Dic73D414yes—
48C23×Dic73C2312yes—
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53C23×Dic73C2311yescharacteristic
54C2×C4⋊Dic73—42no—
55C2×C4⋊Dic73—42no—
56C2×C4⋊Dic73C2×C418yescomplemented
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—42no—
60C2×#11924007—42no—
61C2×C22⋊Dic73—22no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73C2×C414yes—
67#14556498—24no—
68#14556498—24no—
69#14556498—22no—
70#14556498—22no—
71#14556498—42no—
72#14556498—42no—
73#14556498—42no—
74#14556498—42no—
75#14556498—42no—
76#14556498—42no—
77#14556498C2×C418yescomplemented
78C25×C73C2311yescharacteristic
79C23×C4×C73—22no—
80C23×C4×C73—22no—
81C23×C4×C73—42no—
82C23×C4×C73—42no—
83C23×C4×C73Q814yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73C2×C414yes—
94#95×C73—42no—
95#95×C73—42no—
96C22×Dic73—28no—
97C22×Dic73—28no—
98C22×Dic73—22no—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
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—48no—
128C22×Dic73—44no—
129C22×Dic73—44no—
130C22×Dic73—44no—
131C22×Dic73—44no—
132C22×Dic73—48no—
133C22×Dic73—48no—
134C22×Dic73—48no—
135C22×Dic73C22×C412yes—
136C22×Dic73C22×C414yes—
137C22×Dic73C22×C412yes—
138C22×Dic73C2×D412yes—
139C22×Dic73C2×D412yes—
140C4⋊Dic73—48no—
141#11924007—44no—
142#11924007—44no—
143C22⋊Dic73—28no—
144C22⋊Dic73—44no—
145C24×C73—22no—
146C24×C73—22no—
147C24×C73—22no—
148C24×C73—22no—
149C24×C73C22×C412yes—
150C24×C73C2×D412yes—
151C24×C73C2×Q812yes—
152C24×C73#4112yes—
153C24×C73#4112yes—
154C24×C73#4112yes—
155C24×C73#4112yes—
156C24×C73C2411yescharacteristic
157C22×C4×C73—28no—
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×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—48no—
181C22×C4×C73—44no—
182C22×C4×C73—44no—
183C22×C4×C73—48no—
184C22×C4×C73C22×C414yes—
185C22×C4×C73C2×Q814yes—
186C22⋊C4×C73—28no—
187C22⋊C4×C73—44no—
188C23×C73—24no—
189C23×C73—24no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—22no—
193C23×C73—22no—
194C23×C73—24no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—24no—
198C23×C73—24no—
199C23×C73—24no—
200C23×C73—24no—
201C23×C73—24no—
202C23×C73—24no—
203C23×C73—24no—
204C23×C73#11714yes—
205C23×C73C4×D414yes—
206C23×C73C4×Q814yes—
207C23×C73C22⋊Q814yes—
208C23×C73#12314yes—
209C23×C73#12514yes—
210C23×C73#12614yes—
211C23×C73C23×C411yescharacteristic
212C23×C73C22×D411yescharacteristic
213C23×C73C22×Q811yescharacteristic
214C23×C73C2×#4111yescharacteristic
215C23×C73C2×#4111yescharacteristic
216C23×C73C2×#4111yescharacteristic
217C23×C73C2×#4111yescharacteristic
218C23×C73H(5,𝔽2)12yes—
219C23×C73H(5,𝔽2)12yes—
220C23×C73H(5,𝔽2)12yes—
221C23×C732−1+412yes—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—416no—
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×C73—42no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—416no—
261C2×Dic73—416no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—42no—
265C2×Dic73—41no—
266C2×Dic73—42no—
267C2×Dic73—41no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73—42no—
279C2×Dic73—42no—
280C2×Dic73—42no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C22×C73—28no—
284C22×C73—24no—
285C22×C73—24no—
286C22×C73—28no—
287C22×C73—24no—
288C22×C73—24no—
289C22×C73—24no—
290C22×C73—24no—
291C22×C73—28no—
292C22×C73—28no—
293C22×C73#38718yes—
294C22×C73C2×#11711yescharacteristic
295C22×C73C2×C4×D411yescharacteristic
296C22×C73C2×C4×Q811yescharacteristic
297C22×C73#51812yes—
298C22×C73#52012yes—
299C22×C73C2×C22⋊Q811yescharacteristic
300C22×C73C2×#12311yescharacteristic
301C22×C73C2×#12511yescharacteristic
302C22×C73C2×#12611yescharacteristic
303C22×C73#53411yescharacteristic
304C22×C73#53411yescharacteristic
305C22×C73#53511yescharacteristic
306C22×C73#53611yescharacteristic
307C22×C73#53812yes—
308C22×C73#53812yes—
309C22×C73#53812yes—
310C22×C73#53812yes—
311C22×C73#53912yes—
312C22×C73#53912yes—
313C22×C73#53912yes—
314C22×C73#53912yes—
315C22×C73#54312yes—
316C22×C73#54412yes—
317C4×C73—44no—
318C4×C73—44no—
319C4×C73—81no—
320C4×C73—81no—
321Dic73—42no—
322Dic73—44no—
323Dic73—42no—
324Dic73—42no—
325Dic73—42no—
326Dic73—82no—
327#14265—731nomaximal, Sylow
328C2×C73—28no—
329C2×C73C2×#38711yescharacteristic
330C2×C73#231812yes—
331C2×C73#232111yescharacteristic
332C2×C73#232211yescharacteristic
333C2×C73#232912yes—
334C2×C73#262211yescharacteristic
335C2×C73#262311yescharacteristic
336C2×C73#262611yescharacteristic
337C2×C73#262711yescharacteristic
338C2×C73#264312yes—
339C2×C73#264612yes—
340C2×#385—731no—
341C2×#385—731no—
342C22×#95—731no—
343#1445—732no—
344#1443—732no—
345#1442—732no—
346#1441—732no—
347C22⋊#95—731no—
348C22⋊#95—731no—
349C22⋊#95—731no—
350C22⋊#95D7311yescharacteristic
351C73#1426511yescharacteristic, complemented, Sylow
352C24×C4—731no—
353C24×C4—731no—
354C24×C4C2×D7311yescharacteristic
355C22×C4⋊C4—732no—
356C22×C4⋊C4—731no—
357C22×C4⋊C4—731no—
358C22×C22⋊C4—731no—
359C22×C22⋊C4—731no—
360C22×C22⋊C4C2×D7311yescharacteristic
361C22×C22⋊C4C2×D7311yescharacteristic
362#385—738no—
363#385—1464no—
364C2×#95—732no—
365C2×#95—732no—
366C2×#95—732no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—732no—
373C2×#95—731no—
374C2×#95—731no—
375C2×#95—1464no—
376C2×#95C2×D7312yes—
377C2×#95C2×D7312yes—
378C25C22×D7311yescharacteristic
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—42no—
382C23×C4—42no—
383C23×C4—2922no—
384C23×C4—2922no—
385C23×C4—2922no—
386C23×C4—2922no—
387C23×C4—734no—
388C23×C4—734no—
389C23×C4—731no—
390C23×C4—731no—
391C23×C4—732no—
392C23×C4—731no—
393C23×C4—731no—
394C23×C4—732no—
395C23×C4—1462no—
396C23×C4—1462no—
397C23×C4—1462no—
398C23×C4—1462no—
399C23×C4Dic14614yes—
400C23×C4C22×D7311yescharacteristic
401C23×C4C22×D7311yescharacteristic
402C23×C4C22×D7312yes—
403C23×C4C22×D7312yes—
404C2×C4⋊C4—2922no—
405C2×C4⋊C4—2922no—
406C2×C4⋊C4—2922no—
407C2×C4⋊C4—2922no—
408C2×C4⋊C4—738no—
409C2×C4⋊C4—1462no—
410C2×C4⋊C4—1462no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—734no—
417C2×C22⋊C4—1462no—
418C2×C22⋊C4—1462no—
419C2×C22⋊C4—1462no—
420C2×C22⋊C4—1462no—
421C2×C22⋊C4—1462no—
422C2×C22⋊C4C4×D7314yes—
423#95—42no—
424#95—42no—
425#95—2922no—
426#95—2922no—
427#95—2922no—
428#95—2922no—
429#95—2922no—
430#95—2922no—
431#95—738no—
432#95—1464no—
433#95—1464no—
434#95—1462no—
435#95—1462no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24C2×C4×D7312yes—
441C24D4×D7312yes—
442C24C2×Dic14612yes—
443C24#1192402712yes—
444C24#1192402712yes—
445C24#1192402812yes—
446C24#1192402912yes—
447C24C23×D7311yescharacteristic
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—2924no—
476C22×C4—2924no—
477C22×C4—2928no—
478C22×C4—2928no—
479C22×C4—2928no—
480C22×C4—2924no—
481C22×C4—2924no—
482C22×C4—2928no—
483C22×C4—732no—
484C22×C4—734no—
485C22×C4—732no—
486C22×C4—732no—
487C22×C4—732no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1468no—
496C22×C4—1468no—
497C22×C4—1462no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4C2×C4×D7314yes—
520C22×C4C2×Dic14614yes—
521C4⋊C4—2928no—
522C4⋊C4—2924no—
523C4⋊C4—2924no—
524C22⋊C4—28no—
525C22⋊C4—44no—
526C22⋊C4—2924no—
527C22⋊C4—1468no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23#1455654714yes—
545C23#1455655314yes—
546C23C4×Dic14614yes—
547C23#1455656814yes—
548C23#1455657314yes—
549C23#1455658214yes—
550C23#1455658614yes—
551C23C22×C4×D7311yescharacteristic
552C23C2×D4×D7311yescharacteristic
553C23C22×Dic14611yescharacteristic
554C23C2×#1192402711yescharacteristic
555C23C2×#1192402711yescharacteristic
556C23C2×#1192402811yescharacteristic
557C23C2×#1192402911yescharacteristic
558C23#1455663612yes—
559C23#1455663612yes—
560C23#1455663712yes—
561C23#1455663812yes—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—416no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—29216no—
588C2×C4—29216no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—2922no—
606C2×C4—2922no—
607C2×C4—2922no—
608C2×C4—2922no—
609C2×C4—2922no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1464no—
617C2×C4—1464no—
618C2×C4—1464no—
619C2×C4—1464no—
620C2×C4—1464no—
621C2×C4—1462no—
622C2×C4—1462no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—24no—
632C22—24no—
633C22#1798847018yes—
634C22C2×#1455654711yescharacteristic
635C22C2×#1455655311yescharacteristic
636C22C2×C4×Dic14611yescharacteristic
637C22#1798905912yes—
638C22#1798906612yes—
639C22C2×#1455656811yescharacteristic
640C22C2×#1455657311yescharacteristic
641C22C2×#1455658211yescharacteristic
642C22C2×#1455658611yescharacteristic
643C22#1798914711yescharacteristic
644C22#1798914811yescharacteristic
645C22#1798915211yescharacteristic
646C22#1798915911yescharacteristic
647C22#1798917012yes—
648C22#1798917212yes—
649C22#1798917412yes—
650C22#1798917712yes—
651C22#1798917912yes—
652C22#1798918112yes—
653C22#1798918212yes—
654C22#1798918612yes—
655C22#1798921312yes—
656C22#1798921512yes—
657C4—44no—
658C4—44no—
659C4—81no—
660C4—81no—
661C4—2924no—
662C4—2922no—
663C4—2922no—
664C4—2922no—
665C4—2922no—
666C4—5842no—
667C2—28no—
668C2C2×#1798847011yescharacteristic
669C2#2261255712yes—
670C2#2261258811yescharacteristic
671C2#2261259911yescharacteristic
672C2#2261265912yes—
673C2#2261570811yescharacteristic
674C2#2261571511yescharacteristic
675C2#2261574011yescharacteristic
676C2#2261574811yescharacteristic
677C2#2261588712yes—
678C2#2261590412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#14265
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #22608200C2semidirect
5. #22608192C2semidirect
6. #22608189C2semidirect
7. #22608183C2semidirect
19. #17988462C4semidirect
56. C2×C4⋊Dic73C2×C4semidirect
77. #14556498C2×C4semidirect
351. C73#14265semidirect

Maximal subgroups

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

subgroupindex
C2×#179884622
C2×#179884592
C22×#145564982
#226082002
#226081922
#226081892
#226081832
#226081542
#226081532
C73×C22⋊#952
#1426573