Finite Group Databasebeta

← #33012524

Subgroups — #33012524

Order 18688.

1926 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988461C212yesmaximal
2C2×#17988446C211yesmaximal, characteristic
3C2×#17988444C211yesmaximal, characteristic
4C2×#95×D73C211yesmaximal, characteristic
5#22608196C211yesmaximal, characteristic, complemented
6#22608195C211yesmaximal, characteristic, complemented
7#22608190C211yesmaximal, characteristic, complemented
8#22608185C212yesmaximal, complemented
9#22608182C212yesmaximal, complemented
10#22608152C212yesmaximal
11C73×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C23×C4×D73C2211yescharacteristic
14C23×C4×D73C2211yescharacteristic
15C22×C4⋊Dic73C2212yes—
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×#11924005C2212yes—
20C22×#11924005C2212yes—
21#17988461—28no—
22#17988446C418yescomplemented
23#17988444—24no—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2211yescharacteristic
33#95×D73—24no—
34C22×C4⋊C4×C73C2211yescharacteristic
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C24×D73C2311yescharacteristic
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C22×C4×D73—22no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—22no—
49C22×C4×D73—42no—
50C22×C4×D73—42no—
51C22×C4×D73—42no—
52C22×C4×D73—42no—
53C22×C4×D73—42no—
54C22×C4×D73—42no—
55C22×C4×D73C2×C414yes—
56C22×C4×D73D414yes—
57C22×C4×D73D414yes—
58C22×C4×D73Q814yes—
59C22×C4×D73Q814yes—
60C2×C4⋊Dic73—24no—
61C2×C4⋊Dic73—22no—
62C2×C4⋊Dic73—42no—
63C2×C4⋊Dic73—42no—
64C2×C4⋊Dic73—42no—
65C2×#11924007—24no—
66C2×#11924007—42no—
67C2×#11924007—42no—
68C2×#11924007—42no—
69C2×#11924007—42no—
70C2×#11924007C2×C418yescomplemented
71C2×#11924005—24no—
72C2×#11924005—24no—
73C2×#11924005—24no—
74C2×#11924005—24no—
75C2×#11924005—24no—
76C2×#11924005C2×C418yes—
77#14556498—22no—
78#14556498—22no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—42no—
83#14556498—42no—
84#14556498—42no—
85#14556498C2×C414yescomplemented
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2311yescharacteristic
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73—42no—
93#95×C73—22no—
94#95×C73—24no—
95#95×C73—42no—
96#95×C73C2×C414yescomplemented
97C23×D73—22no—
98C23×D73—22no—
99C23×D73—22no—
100C23×D73—22no—
101C23×D73C22×C412yes—
102C23×D73C2×D412yes—
103C23×D73C2×D412yes—
104C23×D73C2×Q812yes—
105C23×D73C2×Q812yes—
106C23×D73#4112yes—
107C23×D73#4112yes—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—21no—
115C22×Dic73—21no—
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×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73C22×C414yes—
132C22×Dic73C22×C412yes—
133C22×Dic73C2×D412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×Q812yes—
136C22×Dic73C2×Q812yes—
137C2×C4×D73—28no—
138C2×C4×D73—28no—
139C2×C4×D73—48no—
140C2×C4×D73—44no—
141C2×C4×D73—44no—
142C2×C4×D73—48no—
143C2×C4×D73—48no—
144C2×C4×D73—48no—
145C2×C4×D73—44no—
146C2×C4×D73—44no—
147C2×C4×D73—44no—
148C2×C4×D73—44no—
149C2×C4×D73—48no—
150C2×C4×D73—48no—
151C2×C4×D73C4⋊C4116yescomplemented
152C4⋊Dic73—48no—
153C4⋊Dic73—44no—
154#11924007—48no—
155#11924007—48no—
156#11924005—216no—
157#11924005—48no—
158C24×C73C2411yescharacteristic
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
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×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73C22×C414yes—
183C22×C4×C73C22×C412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D412yes—
186C22×C4×C73C2×Q812yes—
187C22×C4×C73C2×Q812yes—
188C4⋊C4×C73—44no—
189C23×C73C23×C411yescharacteristic
190C23×C73C22×D411yescharacteristic
191C23×C73C22×D411yescharacteristic
192C23×C73C22×Q811yescharacteristic
193C23×C73C22×Q811yescharacteristic
194C23×C73C2×#4111yescharacteristic
195C23×C73C2×#4111yescharacteristic
196C23×C73H(5,𝔽2)12yes—
197C23×C73H(5,𝔽2)12yes—
198C23×C73H(5,𝔽2)12yes—
199C23×C732−1+412yes—
200C22×D73—24no—
201C22×D73—24no—
202C22×D73—22no—
203C22×D73—22no—
204C22×D73—24no—
205C22×D73—22no—
206C22×D73—22no—
207C22×D73—24no—
208C22×D73—24no—
209C22×D73—24no—
210C22×D73—24no—
211C22×D73—24no—
212C22×D73—24no—
213C22×D73—24no—
214C22×D73—24no—
215C22×D73—24no—
216C22×D73C2×C4⋊C414yes—
217C22×D73C4×D414yes—
218C22×D73C4×Q814yes—
219C22×D73#12114yes—
220C22×D73C22⋊Q814yes—
221C22×D73#12514yes—
222C22×D73C4⋊Q814yes—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—42no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73C2×C4⋊C414yes—
252C2×Dic73—24no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—24no—
262C2×Dic73—24no—
263C2×Dic73—42no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
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×Dic73C2×C4⋊C414yes—
281C4×D73—416no—
282C4×D73—416no—
283C4×D73—416no—
284C22×C73C22×C4⋊C411yescharacteristic
285C22×C73C2×C4×D411yescharacteristic
286C22×C73C2×C4×Q811yescharacteristic
287C22×C73#51812yes—
288C22×C73#52012yes—
289C22×C73C2×#12111yescharacteristic
290C22×C73C2×C22⋊Q811yescharacteristic
291C22×C73C2×#12511yescharacteristic
292C22×C73C2×C4⋊Q811yescharacteristic
293C22×C73#53411yescharacteristic
294C22×C73#53411yescharacteristic
295C22×C73#53412yes—
296C22×C73#53511yescharacteristic
297C22×C73#53611yescharacteristic
298C22×C73#53712yes—
299C22×C73#53812yes—
300C22×C73#53812yes—
301C22×C73#53912yes—
302C22×C73#53912yes—
303C22×C73#54312yes—
304C22×C73#54312yes—
305C22×C73#54412yes—
306C22×C73#54412yes—
307C2×D73—24no—
308C2×D73—24no—
309C2×D73—28no—
310C2×D73—28no—
311C2×D73—24no—
312C2×D73—24no—
313C2×D73—24no—
314C2×D73—24no—
315C2×D73—28no—
316C2×D73—28no—
317C2×D73#38918yes—
318C4×C73—24no—
319C4×C73—44no—
320C4×C73—42no—
321C4×C73—42no—
322C4×C73—82no—
323Dic73—24no—
324Dic73—44no—
325Dic73—42no—
326Dic73—42no—
327Dic73—82no—
328#14275—731nomaximal, Sylow
329C2×C73C2×#38911yescharacteristic
330C2×C73#231511yescharacteristic
331C2×C73#231711yescharacteristic
332C2×C73#232512yes—
333C2×C73#232912yes—
334C2×C73#261711yescharacteristic
335C2×C73#261811yescharacteristic
336C2×C73#262611yescharacteristic
337C2×C73#262711yescharacteristic
338C2×C73#264612yes—
339C2×C73#265912yes—
340D73—28no—
341C2×#385—732no—
342C2×#380—731no—
343C2×#380—731no—
344C22×#95—731no—
345#1444—731no—
346#1444—731no—
347#1442—731no—
348#1442D7311yescharacteristic, complemented
349#1441—732no—
350#1441—732no—
351C22⋊#95—732no—
352C73#1427511yescharacteristic, complemented, Sylow
353C24×C4—731no—
354C24×C4—731no—
355C24×C4—731no—
356C22×C4⋊C4—732no—
357C22×C4⋊C4—732no—
358C22×C4⋊C4—732no—
359C22×C4⋊C4—731no—
360C22×C4⋊C4C2×D7311yescharacteristic
361C22×C22⋊C4—732no—
362C22×C22⋊C4—732no—
363#385—1468no—
364#380—738no—
365#380—1464no—
366C2×#95—732no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—731no—
370C2×#95—731no—
371C2×#95—731no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95—731no—
375C2×#95—1464no—
376C2×#95C2×D7312yes—
377C2×#95C2×D7311yescharacteristic
378C2×#95C2×D7312yes—
379C2×#95C2×D7311yescharacteristic
380C25—731no—
381C23×C4—2922no—
382C23×C4—2922no—
383C23×C4—2922no—
384C23×C4—2922no—
385C23×C4—2922no—
386C23×C4—2922no—
387C23×C4—734no—
388C23×C4—734no—
389C23×C4—734no—
390C23×C4—734no—
391C23×C4—734no—
392C23×C4—731no—
393C23×C4—732no—
394C23×C4—731no—
395C23×C4—731no—
396C23×C4—732no—
397C23×C4—1462no—
398C23×C4—1462no—
399C23×C4—1462no—
400C23×C4—1462no—
401C23×C4C22×D7312yes—
402C23×C4C22×D7312yes—
403C23×C4C22×D7311yescharacteristic
404C23×C4C22×D7311yescharacteristic
405C23×C4C22×D7311yescharacteristic
406C2×C4⋊C4—22no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—2922no—
409C2×C4⋊C4—2922no—
410C2×C4⋊C4—2922no—
411C2×C4⋊C4—2922no—
412C2×C4⋊C4—2922no—
413C2×C4⋊C4—2922no—
414C2×C4⋊C4—2922no—
415C2×C4⋊C4—738no—
416C2×C4⋊C4—1464no—
417C2×C4⋊C4—1462no—
418C2×C4⋊C4—1464no—
419C2×C22⋊C4—738no—
420C2×C22⋊C4—1464no—
421C2×C22⋊C4—1464no—
422C2×C22⋊C4—1464no—
423C2×C22⋊C4—1464no—
424C2×C22⋊C4—1464no—
425#95—22no—
426#95—24no—
427#95—42no—
428#95—2922no—
429#95—2922no—
430#95—2922no—
431#95—734no—
432#95—1462no—
433#95—1462no—
434#95—1464no—
435#95—1464no—
436#95—1462no—
437#95C4×D7314yescomplemented
438C24—732no—
439C24—732no—
440C24—732no—
441C24—732no—
442C24—732no—
443C24—732no—
444C24—732no—
445C24—1462no—
446C24—1462no—
447C24—1462no—
448C24—1462no—
449C24C23×D7311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—2924no—
474C22×C4—2924no—
475C22×C4—2928no—
476C22×C4—2928no—
477C22×C4—2928no—
478C22×C4—2928no—
479C22×C4—2924no—
480C22×C4—2924no—
481C22×C4—2924no—
482C22×C4—2924no—
483C22×C4—2928no—
484C22×C4—2928no—
485C22×C4—7316no—
486C22×C4—732no—
487C22×C4—732no—
488C22×C4—732no—
489C22×C4—734no—
490C22×C4—732no—
491C22×C4—732no—
492C22×C4—1468no—
493C22×C4—1468no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1461no—
506C22×C4—1461no—
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×C4C2×C4×D7314yes—
518C22×C4C2×C4×D7312yes—
519C22×C4D4×D7312yes—
520C22×C4D4×D7312yes—
521C22×C4Q8×D7312yes—
522C22×C4Q8×D7312yes—
523C4⋊C4—44no—
524C4⋊C4—2928no—
525C4⋊C4—2924no—
526C4⋊C4—2928no—
527C4⋊C4—2928no—
528C22⋊C4—2928no—
529C22⋊C4—14616no—
530C23—734no—
531C23—734no—
532C23—734no—
533C23—734no—
534C23—734no—
535C23—734no—
536C23—734no—
537C23—1462no—
538C23—1462no—
539C23—1464no—
540C23—1462no—
541C23—1462no—
542C23—1464no—
543C23—1464no—
544C23—1464no—
545C23—1464no—
546C23—1464no—
547C23—1464no—
548C23—1464no—
549C23—1464no—
550C23—1464no—
551C23—1464no—
552C23—1464no—
553C23C22×C4×D7311yescharacteristic
554C23C2×D4×D7311yescharacteristic
555C23C2×D4×D7311yescharacteristic
556C23C2×Q8×D7311yescharacteristic
557C23C2×Q8×D7311yescharacteristic
558C23#41×D7311yescharacteristic
559C23#41×D7311yescharacteristic
560C23#1455663612yes—
561C23#1455663712yes—
562C23#1455663712yes—
563C23#1455663812yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—29216no—
593C2×C4—29216no—
594C2×C4—29216no—
595C2×C4—2921no—
596C2×C4—2922no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
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—2922no—
611C2×C4—2922no—
612C2×C4—734no—
613C2×C4—1464no—
614C2×C4—1464no—
615C2×C4—1464no—
616C2×C4—1464no—
617C2×C4—1464no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4—1462no—
621C2×C4—1462no—
622C2×C4—1464no—
623C2×C4—1464no—
624C2×C4C4⋊C4×D7314yes—
625C22—738no—
626C22—1464no—
627C22—1464no—
628C22—1468no—
629C22—1464no—
630C22—1464no—
631C22—1468no—
632C22—1468no—
633C22—1464no—
634C22—1464no—
635C22—1468no—
636C22C2×C4⋊C4×D7311yescharacteristic
637C22C4×D4×D7311yescharacteristic
638C22C4×Q8×D7311yescharacteristic
639C22#1798905912yes—
640C22#1798906712yes—
641C22#121×D7311yescharacteristic
642C22C22⋊Q8×D7311yescharacteristic
643C22#125×D7311yescharacteristic
644C22C4⋊Q8×D7311yescharacteristic
645C22#1798914711yescharacteristic
646C22#1798914811yescharacteristic
647C22#1798914812yes—
648C22#1798915511yescharacteristic
649C22#1798915711yescharacteristic
650C22#1798916612yes—
651C22#1798917112yes—
652C22#1798917612yes—
653C22#1798917812yes—
654C22#1798918012yes—
655C22#1798921212yes—
656C22#1798921212yes—
657C22#1798921712yes—
658C22#1798921712yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—2922no—
665C4—2924no—
666C4—2922no—
667C4—5842no—
668C4—1464no—
669C2—1468no—
670C2#389×D7311yescharacteristic
671C2#2261254211yescharacteristic
672C2#2261255611yescharacteristic
673C2#2261263512yes—
674C2#2261265812yes—
675C2#2261566611yescharacteristic
676C2#2261567211yescharacteristic
677C2#2261573511yescharacteristic
678C2#2261574511yescharacteristic
679C2#2261590212yes—
680C2#2261604612yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1442C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1442D73
hypocenterC73#14275
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. #22608196C2semidirect
6. #22608195C2semidirect
7. #22608190C2semidirect
8. #22608185C2semidirect
9. #22608182C2semidirect
11. C73×#1442C2semidirect
22. #17988446C4semidirect
70. C2×#11924007C2×C4semidirect
85. #14556498C2×C4semidirect
96. #95×C73C2×C4semidirect
151. C2×C4×D73C4⋊C4semidirect
348. #1442D73semidirect
352. C73#14275semidirect
437. #95C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884612
C2×#179884462
C2×#179884442
C2×#95×D732
#226081962
#226081952
#226081902
#226081852
#226081822
#226081522
C73×#14422
#1427573