Finite Group Databasebeta

← #33016266

Subgroups — #33016266

Order 18688.

1650 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988480C212yesmaximal, complemented
2#22608184C211yesmaximal, characteristic, complemented
3#22608176C212yesmaximal, complemented
4#22608172C212yesmaximal
5#22608168C212yesmaximal, complemented
6#22608168C212yesmaximal, complemented
7#22608160C211yesmaximal, characteristic, complemented
8#22608156C212yesmaximal, complemented
9C2×#394×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2211yescharacteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C22×C22⋊Dic73C2212yes—
13C22×C22⋊Dic73C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2212yes—
16C22×C4×Dic73C2212yes—
17C22×C4×Dic73C2211yescharacteristic
18#17988480—28no—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2211yescharacteristic
27C23×D4×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2212yes—
30C22×C22⋊C4×C73C2212yes—
31#394×C73C418yescomplemented
32C2×#95×C73C2211yescharacteristic
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C2×C4⋊Dic73—41no—
38C2×C4⋊Dic73—41no—
39C2×C4⋊Dic73—42no—
40C2×C4⋊Dic73—41no—
41C2×C4⋊Dic73D414yescomplemented
42C2×C22⋊Dic73—24no—
43C2×C22⋊Dic73—24no—
44C2×C22⋊Dic73—24no—
45C2×C22⋊Dic73—24no—
46C2×C22⋊Dic73—24no—
47C2×C22⋊Dic73—24no—
48C2×C22⋊Dic73—42no—
49C2×C22⋊Dic73—42no—
50C2×C22⋊Dic73—42no—
51C2×C22⋊Dic73—42no—
52C2×C22⋊Dic73—42no—
53C2×C22⋊Dic73—42no—
54C2×C4×Dic73—24no—
55C2×C4×Dic73—24no—
56C2×C4×Dic73—22no—
57C2×C4×Dic73—41no—
58C2×C4×Dic73—41no—
59C2×C4×Dic73—42no—
60#14556498—24no—
61#14556498—24no—
62#14556498—24no—
63#14556498—42no—
64#14556498—42no—
65#14556498—41no—
66#14556498—42no—
67#14556498—42no—
68C25×C73C2312yes—
69C22×D4×C73—41no—
70C22×D4×C73—41no—
71C22×D4×C73—42no—
72C22×D4×C73—42no—
73C22×D4×C73C2×C414yes—
74C23×C4×C73C2311yescharacteristic
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C2×C4⋊C4×C73—41no—
79C2×C4⋊C4×C73—41no—
80C2×C4⋊C4×C73C2×C414yes—
81C2×C22⋊C4×C73—24no—
82C2×C22⋊C4×C73—24no—
83C2×C22⋊C4×C73—42no—
84C2×C22⋊C4×C73—42no—
85C2×C22⋊C4×C73C2×C418yes—
86C2×C22⋊C4×C73C2×C418yes—
87#95×C73C2×C414yes—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
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—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73C2×D414yes—
114C22×Dic73#4114yes—
115C22×Dic73#4114yes—
116C4⋊Dic73—44no—
117C4⋊Dic73—44no—
118C22⋊Dic73—48no—
119C22⋊Dic73—48no—
120C22⋊Dic73—48no—
121C22⋊Dic73—48no—
122C4×Dic73—48no—
123C4×Dic73—44no—
124C24×C73—22no—
125C24×C73—22no—
126C24×C73—22no—
127C24×C73—22no—
128C24×C73—22no—
129C24×C73—22no—
130C24×C73—22no—
131C24×C73—22no—
132C24×C73—22no—
133C24×C73—22no—
134C24×C73—22no—
135C24×C73—22no—
136C24×C73—22no—
137C24×C73—22no—
138C24×C73C22×C414yes—
139C24×C73C2411yescharacteristic
140C2×D4×C73—44no—
141C2×D4×C73—48no—
142C2×D4×C73—44no—
143C2×D4×C73—48no—
144C2×D4×C73—44no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73C22×C414yes—
164C22×C4×C73C22×C412yes—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C22×C412yes—
167C22×C4×C73C2×D412yes—
168C22×C4×C73#4112yes—
169C4⋊C4×C73—44no—
170C22⋊C4×C73—216no—
171C22⋊C4×C73—48no—
172C23×C73—24no—
173C23×C73—24no—
174C23×C73—24no—
175C23×C73—24no—
176C23×C73—24no—
177C23×C73—24no—
178C23×C73—24no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73C23×C411yescharacteristic
208C23×C73C22×D411yescharacteristic
209C23×C73C2×#4112yes—
210C23×C73C2×#4112yes—
211C23×C73C2×#4111yescharacteristic
212C23×C73H(5,𝔽2)11yescharacteristic
213C23×C73H(5,𝔽2)11yescharacteristic
214C23×C73H(5,𝔽2)12yes—
215C23×C73H(5,𝔽2)12yes—
216C23×C73H(5,𝔽2)11yescharacteristic
217C23×C732−1+411yescharacteristic
218D4×C73—416no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—24no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—41no—
237C2×C4×C73—42no—
238C2×C4×C73—41no—
239C2×C4×C73—42no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73C4×D414yes—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
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×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C22×C73—44no—
272C22×C73—44no—
273C22×C73—44no—
274C22×C73—44no—
275C22×C73—44no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—82no—
279C22×C73—82no—
280C22×C73—82no—
281C22×C73—82no—
282C22×C73—82no—
283C22×C73—82no—
284C22×C73—82no—
285C22×C73—82no—
286C22×C73C2×C4×D411yescharacteristic
287C22×C73C4×#4112yes—
288C22×C73#51811yescharacteristic
289C22×C73#51812yes—
290C22×C73#52011yescharacteristic
291C22×C73#53411yescharacteristic
292C22×C73#53711yescharacteristic
293C22×C73#53812yes—
294C22×C73#53811yescharacteristic
295C22×C73#53912yes—
296C22×C73#53911yescharacteristic
297C22×C73#54012yes—
298C22×C73#54212yes—
299C22×C73#54612yes—
300C22×C73#55012yes—
301C22×C73#55112yes—
302C22×C73#55312yes—
303C22×C73#55511yescharacteristic
304C22×C73#55511yescharacteristic
305C22×C73#55512yes—
306C22×C73#56011yescharacteristic
307C22×C73#56012yes—
308C22×C73H(3,𝔽4)11yescharacteristic
309C4×C73—24no—
310C4×C73—44no—
311C4×C73—42no—
312C4×C73—42no—
313Dic73—44no—
314Dic73—44no—
315Dic73—44no—
316Dic73—82no—
317#14589—731nomaximal, Sylow
318C2×C73—84no—
319C2×C73#232511yescharacteristic
320C2×C73#233512yes—
321C2×C73#236312yes—
322C2×C73#237211yescharacteristic
323C2×C73#237511yescharacteristic
324C2×C73#269911yescharacteristic
325C2×C73#270512yes—
326C2×C73#273311yescharacteristic
327C2×C73#275612yes—
328C2×C73#278511yescharacteristic
329C2×C73#279511yescharacteristic
330C2×#394D7311yescharacteristic
331C2×#388—732no—
332#1441—731no—
333#1439—732no—
334#1438—732no—
335#1437—732no—
336#1437—732no—
337C4⋊#95—731no—
338#1434—732no—
339C73#1458911yescharacteristic, complemented, Sylow
340C23×D4C2×D7311yescharacteristic
341C22×C4⋊C4—731no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4C2×D7311yescharacteristic
344C22×C22⋊C4—732no—
345C22×C22⋊C4—732no—
346C22×C22⋊C4—732no—
347C22×C22⋊C4—732no—
348C22×C22⋊C4C2×D7312yes—
349C22×C22⋊C4C2×D7312yes—
350C22×C42—731no—
351C22×C42—732no—
352#394Dic7318yescomplemented
353#388—1468no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—731no—
359C2×#95—732no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95C2×D7311yescharacteristic
363C25C22×D7312yes—
364C22×D4—42no—
365C22×D4—42no—
366C22×D4—41no—
367C22×D4—41no—
368C22×D4C2×Dic7314yes—
369C23×C4—732no—
370C23×C4—732no—
371C23×C4—732no—
372C23×C4—732no—
373C23×C4C22×D7311yescharacteristic
374C23×C4C22×D7312yes—
375C23×C4C22×D7311yescharacteristic
376C23×C4C22×D7311yescharacteristic
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—2921no—
380C2×C4⋊C4—2921no—
381C2×C4⋊C4—2922no—
382C2×C4⋊C4—2921no—
383C2×C4⋊C4—734no—
384C2×C4⋊C4C2×Dic7314yes—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—2922no—
390C2×C22⋊C4—2922no—
391C2×C22⋊C4—2922no—
392C2×C22⋊C4—2922no—
393C2×C22⋊C4—2922no—
394C2×C22⋊C4—2922no—
395C2×C22⋊C4—1464no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1464no—
398C2×C22⋊C4—1464no—
399C2×C22⋊C4—1464no—
400C2×C22⋊C4—1464no—
401C2×C22⋊C4C2×Dic7318yes—
402C2×C22⋊C4C2×Dic7318yes—
403C2×C42—2921no—
404C2×C42—2921no—
405C2×C42—2922no—
406C2×C42—1464no—
407C2×C42—1464no—
408C2×C42—1462no—
409#95—2922no—
410#95—2922no—
411#95—2921no—
412#95—2922no—
413#95—2922no—
414#95—1464no—
415#95—1464no—
416#95—1464no—
417#95C2×Dic7314yes—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×Dic7314yes—
433C24C23×D7311yescharacteristic
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—734no—
458C22×C4—734no—
459C22×C4—734no—
460C22×C4—1462no—
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—1462no—
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—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4C22×Dic7314yes—
486C22×C4C22×Dic7312yes—
487C22×C4C22×Dic7312yes—
488C22×C4C22×Dic7312yes—
489C22×C4D4×D7312yes—
490C22×C4#1192402712yes—
491C4⋊C4—44no—
492C4⋊C4—2924no—
493C4⋊C4—2924no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—2928no—
497C22⋊C4—2928no—
498C22⋊C4—2928no—
499C22⋊C4—2928no—
500C42—2924no—
501C42—2928no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×Dic7311yescharacteristic
538C23C2×D4×D7311yescharacteristic
539C23#41×D7312yes—
540C23#41×D7312yes—
541C23C2×#1192402711yescharacteristic
542C23#1455663611yescharacteristic
543C23#1455663611yescharacteristic
544C23#1455663612yes—
545C23#1455663612yes—
546C23#1455663711yescharacteristic
547C23#1455663811yescharacteristic
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4D4×Dic7314yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×D4×Dic7311yescharacteristic
617C22#41×Dic7312yes—
618C22#1798906211yescharacteristic
619C22#1798906212yes—
620C22#1798907011yescharacteristic
621C22#1798914511yescharacteristic
622C22#1798916411yescharacteristic
623C22#1798917312yes—
624C22#1798917411yescharacteristic
625C22#1798918012yes—
626C22#1798918211yescharacteristic
627C22#1798918912yes—
628C22#1798920512yes—
629C22#1798923412yes—
630C22#1798925512yes—
631C22#1798926712yes—
632C22#1798928412yes—
633C22#1798929911yescharacteristic
634C22#1798929911yescharacteristic
635C22#1798930112yes—
636C22#1798932612yes—
637C22#1798932611yescharacteristic
638C22#1798933011yescharacteristic
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—2924no—
644C4—2924no—
645C4—2924no—
646C4—5842no—
647C2—84no—
648C2#2261263611yescharacteristic
649C2#2261272512yes—
650C2#2261297312yes—
651C2#2261304311yescharacteristic
652C2#2261307211yescharacteristic
653C2#2261651311yescharacteristic
654C2#2261659712yes—
655C2#2261695411yescharacteristic
656C2#2261725212yes—
657C2#2261763811yescharacteristic
658C2#2261776411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#394×C73C2
socle, cosocleC24×C73C24
prime radicalC23×D4×C73C22
hypercenterC2×#394D73
hypocenterC73#14589
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#17988480C2semidirect
2. #22608184C2semidirect
3. #22608176C2semidirect
5. #22608168C2semidirect
6. #22608168C2semidirect
7. #22608160C2semidirect
8. #22608156C2semidirect
31. #394×C73C4semidirect
41. C2×C4⋊Dic73D4semidirect
339. C73#14589semidirect
352. #394Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
#226081842
#226081762
#226081722
#226081682
#226081602
#226081562
C2×#394×C732
#1458973