Finite Group Databasebeta

← #33092507

Subgroups — #33092507

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988814C211yesmaximal, characteristic
2C2×#17988678C211yesmaximal, characteristic, complemented
3C2×#17988545C211yesmaximal, characteristic, complemented
4#22612626C211yesmaximal, characteristic
5#22612609C211yesmaximal, characteristic, complemented
6#22610786C211yesmaximal, characteristic, complemented
7#22610658C211yesmaximal, characteristic, complemented
8#22610625C211yesmaximal, characteristic, complemented
9#22610622C211yesmaximal, characteristic, complemented
10#22610364C211yesmaximal, characteristic, complemented
11#22609954C211yesmaximal, characteristic, complemented
12#22609819C211yesmaximal, characteristic, complemented
13#22609760C211yesmaximal, characteristic, complemented
14#22609758C211yesmaximal, characteristic, complemented
15C73×#1895C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21C2×#14556550C2211yescharacteristic
22#17988814C418yescomplemented
23#17988678—24no—
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31#17988545—24no—
32C2×#14556501C2211yescharacteristic
33C2×#14556501C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36#17988492C2211yescharacteristic
37#17988490—21no—
38#17988490C2211yescharacteristic
39#17988486C2211yescharacteristic
40#17988484—21no—
41#17988483C2211yescharacteristic
42#17988481—21no—
43#17988480—21no—
44#17988475—21no—
45#17988474—21no—
46#17988467—21no—
47#17988462—21no—
48#17988447C2211yescharacteristic
49C4⋊C4×Dic73C2211yescharacteristic
50C4⋊C4×Dic73C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C22×M16×C73C2211yescharacteristic
56C2×#121×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C2×#11924016—22no—
65C2×#11924016—22no—
66C2×#11924016C2×C414yes—
67C22×#10596831—22no—
68C22×#10596831—22no—
69C22×#10596831C2×C414yescomplemented
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570C2×C414yescomplemented
74#14556556—22no—
75#14556556—22no—
76D4×Dic73—22no—
77D4×Dic73—22no—
78#14556550—22no—
79#14556550—22no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73D411yescharacteristic
82C2×C4⋊Dic73D411yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C22⋊Dic73—21no—
93C2×C22⋊Dic73—21no—
94C2×C22⋊Dic73D411yescharacteristic
95C2×C22⋊Dic73D411yescharacteristic
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101#14556522—22no—
102#14556522—22no—
103#14556522C2×C414yescomplemented
104#14556516—22no—
105#14556516—22no—
106#14556514C2×C414yescomplemented
107#14556512C2×C414yes—
108#14556501—22no—
109#14556501—22no—
110C8×Dic73—22no—
111C8×Dic73D414yescomplemented
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119C22×D4×C73—21no—
120C22×D4×C73C2311yescharacteristic
121C23×C4×C73C2311yescharacteristic
122C2×M16×C73—22no—
123C2×M16×C73—22no—
124C2×M16×C73—22no—
125C22×C8×C73C2311yescharacteristic
126C22×C8×C73C2311yescharacteristic
127#121×C73—22no—
128#121×C73C2×C414yes—
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C22⋊C4×C73—21no—
133C2×C22⋊C4×C73—21no—
134#102×C73—22no—
135#102×C73—22no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—42no—
148C22×Dic73—81no—
149C22×Dic73—81no—
150C22×Dic73C2×D411yescharacteristic
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154#11924016—28no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—28no—
158C2×#10596831C22×C412yes—
159C2×#10596831C22×C412yes—
160C2×#10596831C2×D412yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—22no—
164C4⋊Dic73—22no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—81no—
170C4⋊Dic73C22×C412yes—
171#11924007—24no—
172#11924007—42no—
173#11924007—81no—
174C22⋊Dic73—24no—
175C22⋊Dic73—22no—
176C22⋊Dic73—22no—
177C22⋊Dic73—42no—
178C22⋊Dic73—81no—
179C22⋊Dic73—81no—
180C22⋊Dic73—81no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—42no—
184C4×Dic73—81no—
185C4×Dic73—81no—
186C4×Dic73C2×D412yes—
187C4×Dic73#4112yes—
188C24×C73—21no—
189C24×C73C2×D411yescharacteristic
190C2×D4×C73—24no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73—81no—
197C2×D4×C73—81no—
198C2×D4×C73C22×C412yes—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73C22×C412yes—
207C22×C4×C73C2×D411yescharacteristic
208C22×C4×C73#4112yes—
209C22×C4×C73C2411yescharacteristic
210M16×C73—28no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73C2×D412yes—
216C2×C8×C73#4112yes—
217C4⋊C4×C73—21no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73C22×C412yes—
223C22⋊C4×C73—24no—
224C22⋊C4×C73—81no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—22no—
232C23×C73—41no—
233C23×C73—41no—
234C23×C73—41no—
235C23×C73—41no—
236C23×C73—41no—
237C23×C73—41no—
238C23×C73C4×D412yes—
239C23×C73C22×D411yescharacteristic
240D4×C73—22no—
241D4×C73—22no—
242D4×C73—41no—
243D4×C73—41no—
244D4×C73—84no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73C4×D412yes—
261C2×C4×C73C23×C411yescharacteristic
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—84no—
290C2×Dic73#12112yes—
291C2×Dic73#12112yes—
292Dic146—22no—
293Dic146—22no—
294Dic146—41no—
295Dic146—41no—
296C8×C73—22no—
297C8×C73—22no—
298#10596831—22no—
299#10596831C4×D414yes—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73—22no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73—41no—
307C22×C73—42no—
308C22×C73—42no—
309C22×C73—42no—
310C22×C73—81no—
311C22×C73—81no—
312C22×C73—81no—
313C22×C73—81no—
314C22×C73C2×C4×D411yescharacteristic
315C22×C73C2×#12111yescharacteristic
316C22×C73#53211yescharacteristic
317C22×C73#57711yescharacteristic
318C22×C73#57711yescharacteristic
319C22×C73#57711yescharacteristic
320C22×C73#57711yescharacteristic
321C4×C73—22no—
322C4×C73—42no—
323C4×C73—81no—
324C4×C73—81no—
325C4×C73C2×C4×D411yescharacteristic
326C4×C73C4×#4111yescharacteristic
327C4×C73#52511yescharacteristic
328C4×C73#52911yescharacteristic
329Dic73—22no—
330Dic73—22no—
331Dic73—42no—
332Dic73—41no—
333Dic73—41no—
334Dic73—42no—
335Dic73—81no—
336Dic73—81no—
337#21523—731nomaximal, Sylow
338C2×C73—42no—
339C2×C73—82no—
340C2×C73C4×#12111yescharacteristic
341C2×C73#294911yescharacteristic
342C2×C73#295211yescharacteristic
343C2×C73#305711yescharacteristic
344C2×C73#306011yescharacteristic
345C2×C73#317311yescharacteristic
346C2×C73#317411yescharacteristic
347C2×#465—731no—
348C2×C4×SD16—731no—
349C2×#405—731no—
350#2324—731no—
351#2323—731no—
352#1968—731no—
353#1941—731no—
354#1936—731no—
355#1935—731no—
356#1895D7311yescharacteristic
357#1894—731no—
358#1807—731no—
359C4×#106—731no—
360#1763—731no—
361#1762—731no—
362C73#2152311yescharacteristic, complemented, Sylow
363C22×SD16—731no—
364C22×M16C2×D7311yescharacteristic
365C23×C8—731no—
366C2×C22⋊Q8—731no—
367C2×#121C2×D7311yescharacteristic
368C2×C4×Q8—731no—
369C2×C4×D4—731no—
370C2×#117—731no—
371#465—738no—
372C4×SD16—1464no—
373C2×#106—731no—
374C2×#106—731no—
375C2×#106—731no—
376C2×#103—731no—
377C2×#103—731no—
378C2×#103—731no—
379C2×#102—731no—
380C2×#102C2×D7311yescharacteristic
381C2×#102C2×D7311yescharacteristic
382#405—1464no—
383C2×#97—731no—
384C2×#97—731no—
385C2×C4×C8—731no—
386C2×C4×C8—731no—
387#391—731no—
388#390—731no—
389#390—1461no—
390#389—731no—
391#389—731no—
392#389—1461no—
393#388—1461no—
394#388—1461no—
395#388—1461no—
396#387—1461no—
397#386—1461no—
398#385—1461no—
399#380—731no—
400#380C2×D7311yescharacteristic
401C4×C4⋊C4—731no—
402C4×C4⋊C4—731no—
403#340—731no—
404#340—731no—
405#340—731no—
406#340—731no—
407#340C2×D7311yescharacteristic
408#340C2×D7311yescharacteristic
409C22×Q8—731no—
410C22×D4—21no—
411C22×D4C22×D7311yescharacteristic
412C23×C4—1461no—
413C23×C4C22×D7311yescharacteristic
414C2×SD16—734no—
415C2×SD16—1462no—
416C2×SD16—1462no—
417C2×M16—22no—
418C2×M16—22no—
419C2×M16—22no—
420C22×C8—731no—
421C22×C8—734no—
422C22×C8—731no—
423C22×C8—1462no—
424C22×C8—1462no—
425C22×C8C22×D7311yescharacteristic
426C22×C8C22×D7311yescharacteristic
427C22⋊Q8—734no—
428C22⋊Q8—1462no—
429#121—22no—
430#121C4×D7314yes—
431C4×Q8—1462no—
432C4×Q8—1462no—
433C4×D4—1462no—
434C4×D4—1462no—
435#117—1462no—
436#117—1462no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—731no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4—1461no—
446C2×C4⋊C4C22×D7311yescharacteristic
447C2×C4⋊C4C22×D7311yescharacteristic
448C2×C4⋊C4C22×D7311yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—731no—
452C2×C22⋊C4—731no—
453C2×C22⋊C4—1461no—
454C2×C22⋊C4—1461no—
455C2×C22⋊C4—1461no—
456C2×C22⋊C4—1461no—
457C2×C22⋊C4—1461no—
458C2×C42—731no—
459C2×C42—731no—
460C2×C42—1461no—
461C2×C42—1461no—
462C2×C42—1461no—
463#106—734no—
464#106—1462no—
465#106—1462no—
466#103—734no—
467#103—1462no—
468#103—1462no—
469#102—22no—
470#102—22no—
471#102—734no—
472#97—1462no—
473#97—1462no—
474C4×C8—734no—
475C4×C8—1462no—
476#95—21no—
477#95—1461no—
478#95—1461no—
479#95—1461no—
480#95—1461no—
481#95—1461no—
482#95—1461no—
483#95—1461no—
484C24—21no—
485C24D4×D7311yescharacteristic
486C2×Q8—732no—
487C2×Q8—1461no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×Q8—1461no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—81no—
499C2×D4—81no—
500C2×D4C2×C4×D7312yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—2922no—
509C22×C4—5841no—
510C22×C4—5841no—
511C22×C4—731no—
512C22×C4—731no—
513C22×C4—731no—
514C22×C4—731no—
515C22×C4—1461no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4C2×C4×D7312yes—
520C22×C4D4×D7311yescharacteristic
521C22×C4#1192402712yes—
522C22×C4C23×D7311yescharacteristic
523SD16—1468no—
524M16—28no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—732no—
530C2×C8—732no—
531C2×C8—732no—
532C2×C8—732no—
533C2×C8—1468no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8D4×D7312yes—
537C2×C8#1192402712yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—2922no—
544C4⋊C4—5841no—
545C4⋊C4—5841no—
546C4⋊C4—732no—
547C4⋊C4—1462no—
548C4⋊C4—1462no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1464no—
554C4⋊C4—1461no—
555C4⋊C4C2×C4×D7312yes—
556C22⋊C4—24no—
557C22⋊C4—81no—
558C22⋊C4—2922no—
559C22⋊C4—5841no—
560C22⋊C4—5841no—
561C22⋊C4—5841no—
562C22⋊C4—1462no—
563C22⋊C4—1462no—
564C22⋊C4—1464no—
565C42—2922no—
566C42—5841no—
567C42—5841no—
568C42—732no—
569C42—732no—
570C42—1461no—
571C42—1461no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23#1455655112yes—
586C23C2×D4×D7311yescharacteristic
587Q8—2921no—
588Q8—2921no—
589Q8—1462no—
590Q8—1462no—
591D4—22no—
592D4—22no—
593D4—41no—
594D4—41no—
595D4—84no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—5844no—
622C2×C4—732no—
623C2×C4—732no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1461no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1461no—
631C2×C4—1461no—
632C2×C4—1461no—
633C2×C4—1461no—
634C2×C4—1461no—
635C2×C4—1461no—
636C2×C4#1455655112yes—
637C2×C4C22×C4×D7311yescharacteristic
638C2×C4C2×D4×D7311yescharacteristic
639C2×C4#41×D7311yescharacteristic
640C2×C4#41×D7311yescharacteristic
641C2×C4C2×#1192402711yescharacteristic
642C2×C4C2×#1192402711yescharacteristic
643C8—22no—
644C8—22no—
645C8—734no—
646C8—1462no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#1455655111yescharacteristic
662C22#121×D7311yescharacteristic
663C22#1798913211yescharacteristic
664C22#1798939611yescharacteristic
665C22#1798939711yescharacteristic
666C22#1798939711yescharacteristic
667C22#1798939711yescharacteristic
668C4—22no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—2921no—
673C4—2922no—
674C4—2921no—
675C4—2922no—
676C4—5841no—
677C4—5841no—
678C4—1462no—
679C4—1462no—
680C4C4×D4×D7311yescharacteristic
681C4C4×#1192402711yescharacteristic
682C4#1798909311yescharacteristic
683C4#1798911411yescharacteristic
684C2—42no—
685C2—82no—
686C2#2261244111yescharacteristic
687C2#2261879311yescharacteristic
688C2#2261881011yescharacteristic
689C2#2261956811yescharacteristic
690C2#2261959111yescharacteristic
691C2#2262054311yescharacteristic
692C2#2262055411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1895C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
cosocleC22×C4×C73C24
hypercenter#1895D73
hypocenterC73#21523
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×#17988678C2semidirect
3. C2×#17988545C2semidirect
5. #22612609C2semidirect
6. #22610786C2semidirect
7. #22610658C2semidirect
8. #22610625C2semidirect
9. #22610622C2semidirect
10. #22610364C2semidirect
11. #22609954C2semidirect
12. #22609819C2semidirect
13. #22609760C2semidirect
14. #22609758C2semidirect
22. #17988814C4semidirect
69. C22×#10596831C2×C4semidirect
73. #14556570C2×C4semidirect
103. #14556522C2×C4semidirect
106. #14556514C2×C4semidirect
111. C8×Dic73D4semidirect
362. C73#21523semidirect

Maximal subgroups

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

subgroupindex
C2×#179888142
C2×#179886782
C2×#179885452
#226126262
#226126092
#226107862
#226106582
#226106252
#226106222
#226103642
#226099542
#226098192
#226097602
#226097582
C73×#18952
#2152373