Finite Group Databasebeta

← #33095092

Subgroups — #33095092

Order 18688.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989158C211yesmaximal, characteristic
2#22612523C211yesmaximal, characteristic
3#22610364C212yesmaximal, complemented
4#22610354C212yesmaximal, complemented
5#22610241C212yesmaximal, complemented
6#22610236C212yesmaximal, complemented
7#22610027C212yesmaximal, complemented
8#22609867C212yesmaximal, complemented
9C2×#408×C73C211yesmaximal, characteristic
10C23×Dic146C2211yescharacteristic
11#17989158—24no—
12#17989158C418yescomplemented
13C2×#14556589—21no—
14C2×#14556578—21no—
15C2×#14556572—21no—
16C2×#14556572—21no—
17C2×#14556570C2212yes—
18C2×#14556570C2212yes—
19C2×#14556550C2211yescharacteristic
20C2×#14556550C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22C2×#14556516C2212yes—
23C2×#14556516C2212yes—
24C2×#14556516C2212yes—
25C2×#14556516C2212yes—
26#17988494—21no—
27#17988490—21no—
28#17988467—21no—
29#17988467—21no—
30#17988447C2212yes—
31#17988447C2212yes—
32#17988350C2212yes—
33#17988350C2212yes—
34#17988350C2212yes—
35#17988350C2212yes—
36C22×#41×C73C2211yescharacteristic
37C22×M16×C73C2211yescharacteristic
38C23×C8×C73C2211yescharacteristic
39#408×C73—24no—
40C2×C22⋊C8×C73C2212yes—
41C2×C22⋊C8×C73C2212yes—
42C22×Dic146—21no—
43C22×Dic146—22no—
44C22×Dic146—41no—
45C22×Dic146—41no—
46C22×Dic146—41no—
47C22×Dic146—41no—
48C22×Dic146C2×C414yescomplemented
49C22×Dic146C2312yes—
50C23×Dic73—21no—
51C23×Dic73—21no—
52#14556589—28no—
53#14556578—28no—
54#14556572—28no—
55#14556572—28no—
56#14556570—24no—
57#14556570—24no—
58#14556570C2×C418yes—
59#14556570C2×C418yescomplemented
60#14556550—22no—
61#14556550—41no—
62#14556550—41no—
63#14556550C2×C414yescomplemented
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—41no—
66C2×C4⋊Dic73—41no—
67C2×C4⋊Dic73C2312yes—
68C2×C4⋊Dic73C2312yes—
69C2×C4⋊Dic73C2312yes—
70C2×#11924007—22no—
71C2×#11924007—22no—
72C2×C22⋊Dic73—21no—
73C2×C22⋊Dic73—22no—
74C2×C22⋊Dic73—22no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—22no—
77C2×C22⋊Dic73—22no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80#14556516—42no—
81#14556516—42no—
82#14556516—42no—
83#14556516—42no—
84#14556498—22no—
85#14556498—22no—
86C2×#41×C73—41no—
87C2×#41×C73—41no—
88C2×#41×C73—41no—
89C2×#41×C73—41no—
90C2×#41×C73C2×C414yes—
91C2×#41×C73D414yes—
92C22×Q8×C73D411yescharacteristic
93C22×D4×C73D411yescharacteristic
94C22×D4×C73D411yescharacteristic
95C22×D4×C73D411yescharacteristic
96C23×C4×C73C2311yescharacteristic
97C23×C4×C73C2311yescharacteristic
98C23×C4×C73C2311yescharacteristic
99C2×M16×C73—22no—
100C2×M16×C73—41no—
101C2×M16×C73—41no—
102C22×C8×C73—22no—
103C22×C8×C73—41no—
104C22×C8×C73—41no—
105C22×C8×C73C2312yes—
106C22×C8×C73C2312yes—
107C22⋊C8×C73—24no—
108C22⋊C8×C73—24no—
109C2×Dic146—22no—
110C2×Dic146—22no—
111C2×Dic146—22no—
112C2×Dic146—22no—
113C2×Dic146—24no—
114C2×Dic146—22no—
115C2×Dic146—44no—
116C2×Dic146—44no—
117C2×Dic146—42no—
118C2×Dic146—42no—
119C2×Dic146—44no—
120C2×Dic146—44no—
121C2×Dic146C22×C414yes—
122C22×Dic73—24no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—42no—
128C22×Dic73—42no—
129C22×Dic73—42no—
130C22×Dic73—82no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—22no—
134C4⋊Dic73—22no—
135C4⋊Dic73—22no—
136C4⋊Dic73—42no—
137C4⋊Dic73C22×C414yes—
138#11924007—28no—
139#11924007—28no—
140C22⋊Dic73—24no—
141C22⋊Dic73—28no—
142C22⋊Dic73—28no—
143C22⋊Dic73—42no—
144C22⋊Dic73—82no—
145C22⋊Dic73—82no—
146C4×Dic73—24no—
147C4×Dic73—42no—
148C24×C73C2×D411yescharacteristic
149C24×C73C2×D411yescharacteristic
150C24×C73C2×D411yescharacteristic
151#41×C73—44no—
152#41×C73—44no—
153#41×C73—44no—
154#41×C73—44no—
155C2×Q8×C73—41no—
156C2×Q8×C73—41no—
157C2×Q8×C73C22⋊C414yes—
158C2×D4×C73—41no—
159C2×D4×C73—41no—
160C2×D4×C73—41no—
161C2×D4×C73—41no—
162C2×D4×C73—41no—
163C2×D4×C73—41no—
164C2×D4×C73C22⋊C414yes—
165C2×D4×C73C22⋊C414yes—
166C2×D4×C73C22⋊C414yes—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
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—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73C22×C412yes—
180C22×C4×C73C22×C412yes—
181C22×C4×C73C22×C412yes—
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189M16×C73—44no—
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C2×C8×C73—22no—
193C2×C8×C73—22no—
194C2×C8×C73—22no—
195C2×C8×C73—22no—
196C2×C8×C73—44no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—22no—
209C23×C73C2×C22⋊C412yes—
210C23×C73C2×C22⋊C412yes—
211C23×C73C2×C22⋊C412yes—
212C23×C73C22×D411yescharacteristic
213Q8×C73—44no—
214D4×C73—44no—
215D4×C73—44no—
216D4×C73—44no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73C2×C22⋊C412yes—
236C2×C4×C73C2×C22⋊C412yes—
237C2×C4×C73C2×C22⋊C412yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73H(5,𝔽2)11yescharacteristic
241C2×C4×C73H(5,𝔽2)11yescharacteristic
242C2×C4×C73H(5,𝔽2)11yescharacteristic
243C2×C4×C73H(5,𝔽2)11yescharacteristic
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—48no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—84no—
258Dic146—24no—
259Dic146—24no—
260Dic146—48no—
261Dic146—42no—
262Dic146—42no—
263C8×C73—42no—
264C8×C73—42no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—24no—
269C22×C73—24no—
270C22×C73—24no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73C22×C22⋊C411yescharacteristic
281C22×C73#53411yescharacteristic
282C22×C73#53411yescharacteristic
283C22×C73#57712yes—
284C22×C73#57812yes—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73#51811yescharacteristic
289C4×C73#51811yescharacteristic
290C4×C73#53411yescharacteristic
291C4×C73#53411yescharacteristic
292Dic73—44no—
293Dic73—44no—
294Dic73—82no—
295Dic73—82no—
296#21771—731nomaximal, Sylow
297C2×C73—42no—
298C2×C73—42no—
299C2×C73—42no—
300C2×C73#230811yescharacteristic
301C2×C73#289812yes—
302C2×C73#320011yescharacteristic
303C2×C73#320112yes—
304C2×C73#320211yescharacteristic
305C2×#536—731no—
306C2×#408D7311yescharacteristic
307#2311—731no—
308#1894—732no—
309#1892—732no—
310#1873—732no—
311#1872—732no—
312#1825—732no—
313#1787—732no—
314C73#2177111yescharacteristic, complemented, Sylow
315C22×#41C2×D7311yescharacteristic
316C23×Q8—731no—
317C22×M16C2×D7311yescharacteristic
318C23×C8C2×D7311yescharacteristic
319#536—738no—
320#536—1464no—
321C2×C4⋊Q8—1461no—
322C2×#124—1461no—
323C2×#123—1461no—
324C2×#123—1461no—
325C2×C22⋊Q8—732no—
326C2×C22⋊Q8—732no—
327C2×#117—731no—
328C2×#117—731no—
329C22×C4⋊C4—731no—
330C2×#103—732no—
331C2×#103—732no—
332C2×#103—732no—
333C2×#103—732no—
334#408—24no—
335C2×C22⋊C8C2×D7312yes—
336C2×C22⋊C8C2×D7312yes—
337#391—1461no—
338#390—1461no—
339#386—1461no—
340#386—1461no—
341#380—732no—
342#380—732no—
343#340—732no—
344#340—732no—
345#340—732no—
346#340—732no—
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C4×D7314yes—
352C2×#41C22⋊D7314yes—
353C22×Q8—2921no—
354C22×Q8—2921no—
355C22×Q8—2921no—
356C22×Q8—2921no—
357C22×Q8—734no—
358C22×Q8—732no—
359C22×Q8—1462no—
360C22×Q8—1461no—
361C22×Q8D29211yescharacteristic
362C22×D4D29211yescharacteristic
363C22×D4D29211yescharacteristic
364C22×D4D29211yescharacteristic
365C23×C4—1461no—
366C23×C4—1461no—
367C23×C4C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C23×C4C22×D7311yescharacteristic
370C2×M16—22no—
371C2×M16—41no—
372C2×M16—41no—
373C22×C8—22no—
374C22×C8—41no—
375C22×C8—41no—
376C22×C8C22×D7312yes—
377C22×C8C22×D7312yes—
378C4⋊Q8—1468no—
379#124—1468no—
380#123—1468no—
381#123—1468no—
382C22⋊Q8—738no—
383C22⋊Q8—738no—
384C22⋊Q8—1464no—
385C22⋊Q8—1464no—
386#117—2921no—
387#117—2921no—
388#117—734no—
389#117—1462no—
390C2×C4⋊C4—2921no—
391C2×C4⋊C4—2921no—
392C2×C4⋊C4—732no—
393C2×C4⋊C4—732no—
394C2×C4⋊C4—732no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1462no—
397C2×C4⋊C4—1462no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1462no—
400C2×C22⋊C4—1462no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1462no—
403C2×C22⋊C4—1462no—
404C2×C42—1461no—
405C2×C42—1461no—
406#103—2922no—
407#103—2922no—
408#103—2922no—
409#103—2922no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412#95—1462no—
413#95—1462no—
414C24C2×D29211yescharacteristic
415C24C2×D29211yescharacteristic
416C24C2×D29211yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—2922no—
424C2×Q8—2922no—
425C2×Q8—2924no—
426C2×Q8—2924no—
427C2×Q8—2924no—
428C2×Q8—2924no—
429C2×Q8—734no—
430C2×Q8—1462no—
431C2×Q8—1462no—
432C2×Q8—1462no—
433C2×Q8—1462no—
434C2×Q8—1464no—
435C2×Q8—1462no—
436C2×Q8#1192400514yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4#1192400514yes—
444C2×D4#1192400514yes—
445C2×D4#1192400514yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—5842no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1464no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4C2×C4×D7312yes—
468C22×C4C2×C4×D7312yes—
469C22×C4C2×C4×D7312yes—
470C22×C4C2×C22⋊D7312yes—
471C22×C4C2×C22⋊D7312yes—
472C22×C4C2×C22⋊D7312yes—
473C22×C4C2×D29211yescharacteristic
474C22×C4C2×D29211yescharacteristic
475C22×C4C2×D29211yescharacteristic
476C22×C4C23×D7311yescharacteristic
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C4⋊C4—2922no—
486C4⋊C4—734no—
487C4⋊C4—1462no—
488C4⋊C4—1462no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C4⋊C4—1468no—
492C4⋊C4—1468no—
493C4⋊C4—1462no—
494C22⋊C4—2922no—
495C22⋊C4—5842no—
496C22⋊C4—5842no—
497C22⋊C4—1464no—
498C22⋊C4—1468no—
499C22⋊C4—1468no—
500C42—2922no—
501C42—1464no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C2×#1192400512yes—
515C23C2×#1192400512yes—
516C23C2×#1192400512yes—
517C23C22×D29211yescharacteristic
518Q8—44no—
519Q8—2928no—
520Q8—2922no—
521Q8—2922no—
522Q8—1464no—
523Q8—1464no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—2928no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
555C2×C4—2922no—
556C2×C4—5844no—
557C2×C4—1464no—
558C2×C4—1464no—
559C2×C4C2×#1192400512yes—
560C2×C4C2×#1192400512yes—
561C2×C4C2×#1192400512yes—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C22×C22⋊D7311yescharacteristic
564C2×C4#1455663611yescharacteristic
565C2×C4#1455663611yescharacteristic
566C2×C4#1455663611yescharacteristic
567C2×C4#1455663611yescharacteristic
568C8—42no—
569C8—42no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×#1192400511yescharacteristic
586C22#1798914911yescharacteristic
587C22#1798914911yescharacteristic
588C22#1798939912yes—
589C22#1798940412yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—2924no—
594C4—2924no—
595C4—5842no—
596C4—5842no—
597C4#1798906011yescharacteristic
598C4#1798906011yescharacteristic
599C4#1798914611yescharacteristic
600C4#1798914611yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#2261250611yescharacteristic
605C2#2261846512yes—
606C2#2262080411yescharacteristic
607C2#2262081512yes—
608C2#2262082611yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#408×C73C2
socleC23×C73C22×D4
prime radicalC22×#41×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#408D73
hypocenterC73#21771
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
3. #22610364C2semidirect
4. #22610354C2semidirect
5. #22610241C2semidirect
6. #22610236C2semidirect
7. #22610027C2semidirect
8. #22609867C2semidirect
12. #17989158C4semidirect
48. C22×Dic146C2×C4semidirect
59. #14556570C2×C4semidirect
63. #14556550C2×C4semidirect
314. C73#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#179891582
#226125232
#226103642
#226103542
#226102412
#226102362
#226100272
#226098672
C2×#408×C732
#2177173