Finite Group Databasebeta

← #33113330

Subgroups — #33113330

Order 18688.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988586C211yesmaximal, characteristic
2C2×#17988581C211yesmaximal, characteristic
3#22612519C211yesmaximal, characteristic
4#22611079C212yesmaximal, complemented
5#22611074C212yesmaximal, complemented
6#22611069C212yesmaximal, complemented
7#22611064C212yesmaximal, complemented
8#22609652C211yesmaximal, characteristic, complemented
9C2×#17988349C211yesmaximal, characteristic, complemented
10#22609628C211yesmaximal, characteristic, complemented
11C2×#544×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14C23×#10596831C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16#17988586—24no—
17#17988581—24no—
18C2×#14556514C2211yescharacteristic
19C2×#14556514C2211yescharacteristic
20C2×#14556514C2212yes—
21C2×#14556511C2211yescharacteristic
22C2×#14556511C2211yescharacteristic
23C2×#14556511C2212yes—
24#17988493C2212yes—
25#17988489C2212yes—
26#17988464—22no—
27#17988444—21no—
28#17988444—21no—
29#17988349C2212yes—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#17988342C2212yes—
35#544×C73—24no—
36#544×C73—24no—
37C2×C4⋊Q8×C73—21no—
38C2×C4⋊Q8×C73C2212yes—
39C2×#125×C73—21no—
40C2×#125×C73C2212yes—
41C2×C22⋊Q8×C73—22no—
42C2×#117×C73C2211yescharacteristic
43C2×#117×C73C2211yescharacteristic
44C22×C4⋊C4×C73C2211yescharacteristic
45C2×#11924028—22no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028Q814yescomplemented
51C22×Dic146C2311yescharacteristic
52C22×D292C2311yescharacteristic
53C22×C22⋊D73—21no—
54C23×Dic73—21no—
55C22×C4×D73—21no—
56C2×#11924012—22no—
57C2×#11924012—22no—
58C2×#11924012—22no—
59C22×#10596831—22no—
60C22×#10596831—22no—
61C22×#10596831—22no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2312yes—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—22no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73C2312yes—
69C2×#11924005—22no—
70C2×#11924005—22no—
71#14556514—22no—
72#14556514—22no—
73#14556514—24no—
74#14556511—22no—
75#14556511—22no—
76#14556511—24no—
77#14556498—22no—
78#14556498—22no—
79C22×Q8×C73—22no—
80C23×C4×C73—21no—
81C23×C4×C73C2311yescharacteristic
82C4⋊Q8×C73—28no—
83C4⋊Q8×C73—24no—
84C4⋊Q8×C73—24no—
85#125×C73—28no—
86#125×C73—24no—
87#125×C73—24no—
88C22⋊Q8×C73—216no—
89#117×C73—22no—
90#117×C73—22no—
91#117×C73—22no—
92#117×C73D414yes—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—22no—
95C2×C4⋊C4×C73—22no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73D411yescharacteristic
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C4⋊C4×C73D414yes—
101C2×C4⋊C4×C73C2312yes—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73D412yes—
106C2×C42×C73—21no—
107C2×C42×C73C2312yes—
108C23×D73—21no—
109#11924028—28no—
110#11924028—28no—
111#11924028—44no—
112#11924028—44no—
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146C2×Q812yes—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292C2×Q812yes—
125C2×C22⋊D73—24no—
126C2×C22⋊D73—42no—
127C2×C22⋊D73—42no—
128C22×Dic73—21no—
129C22×Dic73—22no—
130C22×Dic73—42no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C2×C4×D73—24no—
134C2×C4×D73—42no—
135C2×C4×D73—42no—
136#11924012—44no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—22no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—22no—
145C2×#10596831—22no—
146C2×#10596831—44no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73—42no—
150#11924005—82no—
151#11924005—82no—
152C24×C73C2×D411yescharacteristic
153C2×Q8×C73—28no—
154C2×Q8×C73—82no—
155C2×Q8×C73—82no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—22no—
159C22×C4×C73—24no—
160C22×C4×C73—81no—
161C22×C4×C73—81no—
162C22×C4×C73C2×D412yes—
163C22×C4×C73C2×D412yes—
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D412yes—
167C22×C4×C73C2×Q812yes—
168C22×C4×C73#4112yes—
169C22×C4×C73#4112yes—
170C22×C4×C73#4112yes—
171C22×C4×C73C2411yescharacteristic
172C4⋊C4×C73—28no—
173C4⋊C4×C73—28no—
174C4⋊C4×C73—28no—
175C4⋊C4×C73—24no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73C2×D414yes—
182C4⋊C4×C73C2×D412yes—
183C4⋊C4×C73C2×D412yes—
184C22⋊C4×C73—24no—
185C22⋊C4×C73—24no—
186C42×C73—24no—
187C42×C73—22no—
188C42×C73—22no—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73#12012yes—
194C23×C73C22⋊Q812yes—
195C23×C73#12412yes—
196C23×C73C22×D411yescharacteristic
197C22×D73—22no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204Q8×C73—88no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—42no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—84no—
221C2×C4×C73#12012yes—
222C2×C4×C73C22⋊Q814yes—
223C2×C4×C73C22⋊Q814yes—
224C2×C4×C73C22⋊Q812yes—
225C2×C4×C73#12414yes—
226C2×C4×C73#12414yes—
227C2×C4×C73#12412yes—
228C2×C4×C73C22×D411yescharacteristic
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×Q811yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C22⋊D73—48no—
235C2×Dic73—22no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—84no—
242D292—22no—
243D292—22no—
244D292—41no—
245D292—41no—
246C4×D73—48no—
247Dic146—22no—
248Dic146—22no—
249Dic146—41no—
250Dic146—41no—
251#10596831—41no—
252#10596831—41no—
253#10596831—42no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73#39314yes—
261C22×C73C2×#12011yescharacteristic
262C22×C73C2×C22⋊Q811yescharacteristic
263C22×C73C2×#12411yescharacteristic
264C22×C73#57611yescharacteristic
265C22×C73#57711yescharacteristic
266C22×C73#57711yescharacteristic
267C22×C73#57811yescharacteristic
268C2×D73—42no—
269C2×D73—42no—
270C2×D73—42no—
271C2×D73—81no—
272C2×D73—81no—
273C2×D73—81no—
274C2×D73—81no—
275C4×C73—42no—
276C4×C73—44no—
277C4×C73—42no—
278C4×C73—42no—
279C4×C73—42no—
280C4×C73#39314yes—
281C4×C73C2×C22⋊Q811yescharacteristic
282C4×C73C2×C22⋊Q811yescharacteristic
283C4×C73C2×#12411yescharacteristic
284C4×C73C2×#12411yescharacteristic
285Dic73—42no—
286Dic73—82no—
287#23458—731nomaximal, Sylow
288C2×C73—24no—
289C2×C73C2×#39311yescharacteristic
290C2×C73#301211yescharacteristic
291C2×C73#301311yescharacteristic
292C2×C73#308311yescharacteristic
293C2×C73#308411yescharacteristic
294C2×C73#313611yescharacteristic
295C2×C73#313711yescharacteristic
296D73—82no—
297C2×#544D7311yescharacteristic, complemented
298C2×#417—731no—
299C2×#416—731no—
300#2311—731no—
301#2026—732no—
302#2025—732no—
303#2024—732no—
304#2023—732no—
305#1736—731no—
306C2×#340—731no—
307#1729—731no—
308C73#2345811yescharacteristic, complemented, Sylow
309C22×#41—731no—
310C22×M16—731no—
311C23×C8—731no—
312#544—24no—
313#544—24no—
314C2×C4⋊Q8—21no—
315C2×C4⋊Q8C2×D7312yes—
316C2×#125—21no—
317C2×#125C2×D7312yes—
318C2×C22⋊Q8—22no—
319C2×#117C2×D7311yescharacteristic
320C2×#117C2×D7311yescharacteristic
321C22×C4⋊C4—731no—
322C22×C4⋊C4C2×D7311yescharacteristic
323#417—1464no—
324#416—1464no—
325C2×#103—731no—
326C2×#103—731no—
327C2×#103—732no—
328C2×#102—731no—
329C2×#102—731no—
330C2×#102—732no—
331#391—732no—
332#390—732no—
333#386—1462no—
334#380—1461no—
335#380—1461no—
336#340—732no—
337#340—732no—
338#340—732no—
339#340—732no—
340#336—732no—
341#336—732no—
342C2×#41—734no—
343C2×#41—1462no—
344C2×#41—1462no—
345C2×#41—1462no—
346C2×#41—1462no—
347C2×#41—1462no—
348C22×Q8—22no—
349C22×Q8—731no—
350C22×D4—731no—
351C22×D4—1461no—
352C23×C4—21no—
353C23×C4—1461no—
354C23×C4—1461no—
355C23×C4C22×D7311yescharacteristic
356C2×M16—1462no—
357C2×M16—1462no—
358C2×M16—1462no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—732no—
362C22×C8—1462no—
363C22×C8—1462no—
364C22×C8—1462no—
365C4⋊Q8—28no—
366C4⋊Q8—24no—
367C4⋊Q8—24no—
368#125—28no—
369#125—24no—
370#125—24no—
371C22⋊Q8—216no—
372#117—22no—
373#117—22no—
374#117—22no—
375#117C22⋊D7314yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—732no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1462no—
384C2×C4⋊C4—1462no—
385C2×C4⋊C4C22⋊D7311yescharacteristic
386C2×C4⋊C4C22⋊D7311yescharacteristic
387C2×C4⋊C4C22⋊D7314yes—
388C2×C4⋊C4C22×D7312yes—
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1462no—
393C2×C22⋊C4—1462no—
394C2×C22⋊C4C22⋊D7312yes—
395C2×C42—21no—
396C2×C42C22×D7312yes—
397#103—1462no—
398#103—1462no—
399#103—1464no—
400#102—1462no—
401#102—1462no—
402#102—1464no—
403#95—1462no—
404#95—1462no—
405C24—1461no—
406C24C2×C22⋊D7311yescharacteristic
407#41—2924no—
408#41—2924no—
409#41—1468no—
410#41—1468no—
411C2×Q8—28no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—732no—
415C2×Q8—1461no—
416C2×Q8—1461no—
417C2×Q8—1461no—
418C2×Q8—1461no—
419C2×Q8—1461no—
420C2×D4—2922no—
421C2×D4—2922no—
422C2×D4—732no—
423C2×D4—1464no—
424C2×D4—1461no—
425C2×D4—1461no—
426C2×D4—1461no—
427C2×D4—1461no—
428C2×D4—1461no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—2922no—
436C22×C4—2922no—
437C22×C4—2922no—
438C22×C4—5841no—
439C22×C4—5841no—
440C22×C4—1462no—
441C22×C4—1461no—
442C22×C4—1464no—
443C22×C4C2×C22⋊D7312yes—
444C22×C4C2×C22⋊D7312yes—
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7312yes—
447C22×C4C2×C22⋊D7312yes—
448C22×C4Q8×D7312yes—
449C22×C4#1192402912yes—
450C22×C4#1192402912yes—
451C22×C4#1192402912yes—
452C22×C4C23×D7311yescharacteristic
453M16—2924no—
454C2×C8—2924no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1462no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1462no—
463C2×C8—1462no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—2922no—
474C4⋊C4—1462no—
475C4⋊C4—1462no—
476C4⋊C4C2×C22⋊D7314yes—
477C4⋊C4C2×C22⋊D7312yes—
478C4⋊C4C2×C22⋊D7312yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—5842no—
482C22⋊C4—5842no—
483C42—24no—
484C42—22no—
485C42—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—2921no—
491C23—2921no—
492C23—2921no—
493C23—2921no—
494C23—2921no—
495C23—2921no—
496C23—1462no—
497C23#1455656012yes—
498C23#1455656712yes—
499C23#1455658012yes—
500C23C22×C22⋊D7311yescharacteristic
501Q8—88no—
502Q8—2921no—
503Q8—2921no—
504Q8—1462no—
505Q8—1462no—
506D4—2928no—
507D4—2921no—
508D4—2921no—
509D4—1462no—
510D4—1462no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2928no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—5844no—
534C2×C4—1462no—
535C2×C4#1455656012yes—
536C2×C4#1455656712yes—
537C2×C4#1455656714yes—
538C2×C4#1455656714yes—
539C2×C4#1455658012yes—
540C2×C4#1455658014yes—
541C2×C4#1455658014yes—
542C2×C4C22×C22⋊D7311yescharacteristic
543C2×C4C22×C22⋊D7311yescharacteristic
544C2×C4C2×Q8×D7311yescharacteristic
545C2×C4C2×#1192402911yescharacteristic
546C2×C4C2×#1192402911yescharacteristic
547C2×C4C2×#1192402911yescharacteristic
548C8—2921no—
549C8—2921no—
550C8—2922no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—2922no—
558C22—2922no—
559C22—2922no—
560C22—5841no—
561C22—5841no—
562C22—5841no—
563C22—5841no—
564C22#1798849814yes—
565C22C2×#1455656011yescharacteristic
566C22C2×#1455656711yescharacteristic
567C22C2×#1455658011yescharacteristic
568C22#1798939011yescharacteristic
569C22#1798939511yescharacteristic
570C22#1798939811yescharacteristic
571C22#1798940211yescharacteristic
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—2922no—
578C4—5842no—
579C4#1798849814yes—
580C4C2×#1455656711yescharacteristic
581C4C2×#1455656711yescharacteristic
582C4C2×#1455658011yescharacteristic
583C4C2×#1455658011yescharacteristic
584C2—24no—
585C2—5842no—
586C2C2×#1798849811yescharacteristic
587C2#2261923511yescharacteristic
588C2#2261924611yescharacteristic
589C2#2261977811yescharacteristic
590C2#2261978911yescharacteristic
591C2#2262029011yescharacteristic
592C2#2262030111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
hypercenterC2×#544D73
hypocenterC73#23458
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #22611079C2semidirect
5. #22611074C2semidirect
6. #22611069C2semidirect
7. #22611064C2semidirect
8. #22609652C2semidirect
9. C2×#17988349C2semidirect
10. #22609628C2semidirect
11. C2×#544×C73C2semidirect
50. C2×#11924028Q8semidirect
297. C2×#544D73semidirect
308. C73#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#179885862
C2×#179885812
#226125192
#226110792
#226110742
#226110692
#226110642
#226096522
C2×#179883492
#226096282
C2×#544×C732
#2345873