Finite Group Databasebeta

← #33088681

Subgroups — #33088681

Order 18688.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988586C211yesmaximal, characteristic
2C2×#17988584C211yesmaximal, characteristic
3C2×#17988542C211yesmaximal, characteristic, complemented
4#22612549C211yesmaximal, characteristic, complemented
5#22612524C211yesmaximal, characteristic
6#22610629C212yesmaximal, complemented
7#22610622C212yesmaximal, complemented
8#22610615C212yesmaximal, complemented
9#22610608C212yesmaximal, complemented
10#22609650C211yesmaximal, characteristic, complemented
11C2×#340×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14C2×#14556550C2211yescharacteristic
15C2×#14556550C2211yescharacteristic
16#17988586C418yescomplemented
17#17988584—24no—
18C2×#14556516C2211yescharacteristic
19C2×#14556516C2211yescharacteristic
20C2×#14556514C2212yes—
21C2×#14556513C2211yescharacteristic
22C2×#14556513C2211yescharacteristic
23C2×#14556511C2212yes—
24#17988542—24no—
25C2×#14556501C2212yes—
26C2×C8×Dic73C2212yes—
27#17988492C2212yes—
28#17988488C2212yes—
29#17988486—21no—
30#17988486—21no—
31#17988486C2212yes—
32#17988486C2212yes—
33#17988466—22no—
34#17988447—22no—
35#17988444—21no—
36#17988444—21no—
37#17988349C2212yes—
38#17988349C2212yes—
39C23×C8×C73C2211yescharacteristic
40C22×C4⋊C4×C73C2211yescharacteristic
41C22×C4⋊C4×C73C2211yescharacteristic
42#340×C73C2212yes—
43#340×C73C2212yes—
44C2×#11924028—22no—
45C2×#11924028—22no—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028C2×C414yes—
50C22×Dic146C2311yescharacteristic
51C22×D292C2311yescharacteristic
52C22×C22⋊D73—21no—
53C22×C4×D73—21no—
54C2×#11924012—22no—
55C2×#11924012—22no—
56C2×#11924012C2×C414yes—
57C22×#10596831C2312yes—
58#14556550—22no—
59#14556550—22no—
60#14556550—22no—
61#14556550—22no—
62C2×C4⋊Dic73—22no—
63C2×C4⋊Dic73—22no—
64C2×C4⋊Dic73D411yescharacteristic
65C2×C4⋊Dic73D411yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C22⋊Dic73—21no—
69C2×C22⋊Dic73D412yes—
70C2×#11924005—22no—
71C2×#11924005—22no—
72C2×C4×Dic73—21no—
73C2×C4×Dic73C2312yes—
74#14556516—22no—
75#14556516—22no—
76#14556514C2×C418yescomplemented
77#14556513—22no—
78#14556513—22no—
79#14556511C2×C418yescomplemented
80#14556501—24no—
81C8×Dic73—24no—
82#14556498—22no—
83#14556498—22no—
84#14556498—22no—
85#14556498—22no—
86C23×C4×C73—21no—
87C23×C4×C73—21no—
88C23×C4×C73C2311yescharacteristic
89C22×C8×C73—22no—
90C22×C8×C73—22no—
91C22×C8×C73D414yes—
92C22×C8×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—22no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73C2×C414yes—
100C2×C4⋊C4×C73C2312yes—
101C2×C4⋊C4×C73C2312yes—
102C23×D73—21no—
103#11924028—28no—
104#11924028—28no—
105#11924028—44no—
106#11924028—44no—
107C2×Dic146—21no—
108C2×Dic146—21no—
109C2×Dic146—21no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146C22×C412yes—
113C2×D292—21no—
114C2×D292—21no—
115C2×D292—21no—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292C22×C412yes—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—42no—
121C2×C22⋊D73—42no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73C2×D412yes—
126C22×Dic73C2×D412yes—
127C2×C4×D73—24no—
128C2×C4×D73—42no—
129C2×C4×D73—42no—
130#11924012—28no—
131C2×#10596831—22no—
132C2×#10596831—22no—
133C2×#10596831C22×C414yes—
134C4⋊Dic73—22no—
135C4⋊Dic73—22no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—82no—
141C4⋊Dic73—82no—
142C22⋊Dic73—24no—
143C22⋊Dic73—42no—
144#11924005—82no—
145#11924005—82no—
146C4×Dic73—22no—
147C4×Dic73—22no—
148C4×Dic73—42no—
149C24×C73C2×D411yescharacteristic
150C22×C4×C73—22no—
151C22×C4×C73—24no—
152C22×C4×C73—22no—
153C22×C4×C73—42no—
154C22×C4×C73—81no—
155C22×C4×C73—81no—
156C22×C4×C73—81no—
157C22×C4×C73—81no—
158C22×C4×C73C22×C412yes—
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D412yes—
161C22×C4×C73#4112yes—
162C22×C4×C73#4112yes—
163C22×C4×C73#4112yes—
164C22×C4×C73#4112yes—
165C22×C4×C73C2411yescharacteristic
166C2×C8×C73—28no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73—21no—
170C2×C8×C73—21no—
171C2×C8×C73C2×D412yes—
172C2×C8×C73C2×D412yes—
173C4⋊C4×C73—24no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73C22×C414yes—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—21no—
183C23×C73C4×D412yes—
184C23×C73#12112yes—
185C23×C73#12412yes—
186C23×C73C22×D411yescharacteristic
187C22×D73—22no—
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—41no—
191C22×D73—41no—
192C22×D73—41no—
193C22×D73—41no—
194C2×C4×C73—24no—
195C2×C4×C73—42no—
196C2×C4×C73—42no—
197C2×C4×C73—42no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—84no—
201C2×C4×C73—84no—
202C2×C4×C73#11714yes—
203C2×C4×C73C4×D412yes—
204C2×C4×C73C4×D414yes—
205C2×C4×C73#12112yes—
206C2×C4×C73#12314yes—
207C2×C4×C73#12412yes—
208C2×C4×C73#12614yes—
209C2×C4×C73C23×C411yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C2×#4111yescharacteristic
212C2×C4×C73C2×#4111yescharacteristic
213C2×C4×C73C2×#4111yescharacteristic
214C2×C4×C73C2×#4111yescharacteristic
215C22⋊D73—48no—
216C2×Dic73—22no—
217C2×Dic73—22no—
218C2×Dic73—22no—
219C2×Dic73—22no—
220C2×Dic73—22no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—41no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231D292—22no—
232D292—22no—
233D292—41no—
234D292—41no—
235C4×D73—48no—
236Dic146—22no—
237Dic146—22no—
238Dic146—41no—
239Dic146—41no—
240C8×C73—22no—
241C8×C73—22no—
242#10596831—24no—
243C22×C73—22no—
244C22×C73—22no—
245C22×C73—22no—
246C22×C73—22no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73#38814yes—
250C22×C73C2×C4×D411yescharacteristic
251C22×C73C2×#12111yescharacteristic
252C22×C73C2×#12411yescharacteristic
253C22×C73#57611yescharacteristic
254C22×C73#57711yescharacteristic
255C22×C73#57711yescharacteristic
256C22×C73#57811yescharacteristic
257C2×D73—42no—
258C2×D73—42no—
259C2×D73—42no—
260C2×D73—81no—
261C2×D73—81no—
262C2×D73—81no—
263C2×D73—81no—
264C4×C73—44no—
265C4×C73—82no—
266C4×C73#38814yes—
267C4×C73C2×#11711yescharacteristic
268C4×C73C2×C4×D411yescharacteristic
269C4×C73C2×#12311yescharacteristic
270C4×C73C2×#12611yescharacteristic
271Dic73—42no—
272Dic73—42no—
273Dic73—42no—
274Dic73—81no—
275Dic73—81no—
276#21092—731nomaximal, Sylow
277C2×C73—24no—
278C2×C73C2×#38811yescharacteristic
279C2×C73#294611yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#306011yescharacteristic
282C2×C73#306111yescharacteristic
283C2×C73#313611yescharacteristic
284C2×C73#313711yescharacteristic
285D73—82no—
286C2×#417—731no—
287C2×#416—731no—
288C2×#405—731no—
289#2316—731no—
290#2311—731no—
291#1936—732no—
292#1935—732no—
293#1934—732no—
294#1933—732no—
295#1736—731no—
296C2×#340D7311yescharacteristic, complemented
297C73#2109211yescharacteristic, complemented, Sylow
298C22×#41—731no—
299C22×M16—731no—
300C23×C8C2×D7311yescharacteristic
301C2×#117—731no—
302C2×#117—731no—
303C22×C4⋊C4C2×D7311yescharacteristic
304C22×C4⋊C4C2×D7311yescharacteristic
305#417—738no—
306#416—1464no—
307C2×#103—732no—
308C2×#103—731no—
309C2×#103—731no—
310C2×#102—732no—
311C2×#102—731no—
312C2×#102—731no—
313#405—1464no—
314C2×#97—732no—
315C2×C4×C8—732no—
316#391—732no—
317#390—732no—
318#389—732no—
319#389—732no—
320#389—1461no—
321#389—1461no—
322#386—1462no—
323#380—1462no—
324#380—1461no—
325#380—1461no—
326#340—732no—
327#340—732no—
328#340C2×D7312yes—
329#340C2×D7312yes—
330C2×#41—734no—
331C2×#41—1462no—
332C2×#41—1462no—
333C2×#41—1462no—
334C2×#41—1462no—
335C2×#41—1462no—
336C22×Q8—731no—
337C22×D4—731no—
338C22×D4—1461no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1461no—
342C23×C4C22×D7311yescharacteristic
343C2×M16—734no—
344C2×M16—1462no—
345C2×M16—1462no—
346C22×C8—22no—
347C22×C8—22no—
348C22×C8—732no—
349C22×C8D29214yes—
350C22×C8C22×D7311yescharacteristic
351C22×C8C22×D7311yescharacteristic
352#117—1462no—
353#117—1462no—
354#117—1462no—
355#117—1462no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—731no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4C4×D7314yes—
368C2×C4⋊C4C22×D7312yes—
369C2×C4⋊C4C22×D7312yes—
370C2×C22⋊C4—732no—
371C2×C22⋊C4—1461no—
372C2×C22⋊C4—1462no—
373C2×C22⋊C4—1462no—
374C2×C42—732no—
375C2×C42—1461no—
376#103—738no—
377#103—1462no—
378#103—1462no—
379#102—738no—
380#102—1462no—
381#102—1462no—
382#97—1464no—
383C4×C8—1464no—
384#95—1462no—
385#95—1462no—
386#95—1462no—
387#95—1462no—
388C24—1461no—
389C24D4×D7311yescharacteristic
390#41—2924no—
391#41—2924no—
392#41—1468no—
393#41—1468no—
394C2×Q8—732no—
395C2×Q8—1461no—
396C2×Q8—1461no—
397C2×Q8—1461no—
398C2×Q8—1461no—
399C2×Q8—1461no—
400C2×D4—2922no—
401C2×D4—2922no—
402C2×D4—732no—
403C2×D4—1464no—
404C2×D4—1461no—
405C2×D4—1461no—
406C2×D4—1461no—
407C2×D4—1461no—
408C2×D4—1461no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—2922no—
418C22×C4—2922no—
419C22×C4—732no—
420C22×C4—732no—
421C22×C4—1461no—
422C22×C4—1461no—
423C22×C4—1461no—
424C22×C4—1464no—
425C22×C4C2×C4×D7312yes—
426C22×C4D4×D7311yescharacteristic
427C22×C4C2×D29212yes—
428C22×C4#1192402712yes—
429C22×C4#1192402712yes—
430C22×C4#1192402812yes—
431C22×C4#1192402912yes—
432C22×C4C23×D7311yescharacteristic
433M16—1468no—
434C2×C8—28no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—734no—
440C2×C8—1462no—
441C2×C8—1462no—
442C2×C8C2×D29212yes—
443C2×C8C2×D29212yes—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—5842no—
450C4⋊C4—5842no—
451C4⋊C4—1461no—
452C4⋊C4—1461no—
453C4⋊C4—1461no—
454C4⋊C4—1461no—
455C4⋊C4—1462no—
456C4⋊C4—1462no—
457C4⋊C4C2×C4×D7314yes—
458C22⋊C4—2922no—
459C22⋊C4—5842no—
460C22⋊C4—5842no—
461C22⋊C4—1464no—
462C42—2922no—
463C42—1462no—
464C42—1462no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—2921no—
470C23—2921no—
471C23—2921no—
472C23—2921no—
473C23—2921no—
474C23—2921no—
475C23—1462no—
476C23#1455655112yes—
477C23#1455656612yes—
478C23#1455657712yes—
479C23C2×D4×D7311yescharacteristic
480Q8—2921no—
481Q8—2921no—
482Q8—1462no—
483Q8—1462no—
484D4—2928no—
485D4—2921no—
486D4—2921no—
487D4—1462no—
488D4—1462no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—2921no—
498C2×C4—2921no—
499C2×C4—2928no—
500C2×C4—2921no—
501C2×C4—2921no—
502C2×C4—2921no—
503C2×C4—2921no—
504C2×C4—2921no—
505C2×C4—2921no—
506C2×C4—1462no—
507C2×C4—1462no—
508C2×C4—1462no—
509C2×C4—1462no—
510C2×C4—1462no—
511C2×C4—1462no—
512C2×C4—1462no—
513C2×C4#1455654914yes—
514C2×C4#1455655112yes—
515C2×C4C4×D29214yes—
516C2×C4#1455656612yes—
517C2×C4#1455657514yes—
518C2×C4#1455657712yes—
519C2×C4#1455658614yes—
520C2×C4C22×C4×D7311yescharacteristic
521C2×C4C22×D29211yescharacteristic
522C2×C4C2×#1192402711yescharacteristic
523C2×C4C2×#1192402711yescharacteristic
524C2×C4C2×#1192402811yescharacteristic
525C2×C4C2×#1192402911yescharacteristic
526C8—22no—
527C8—22no—
528C8—1464no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—2922no—
536C22—2922no—
537C22—2922no—
538C22—5841no—
539C22—5841no—
540C22—5841no—
541C22—5841no—
542C22#1798847914yes—
543C22C2×#1455655111yescharacteristic
544C22C2×#1455656611yescharacteristic
545C22C2×#1455657711yescharacteristic
546C22#1798938811yescharacteristic
547C22#1798939311yescharacteristic
548C22#1798939311yescharacteristic
549C22#1798940011yescharacteristic
550C4—44no—
551C4—82no—
552C4—2922no—
553C4—2922no—
554C4—2922no—
555C4—5841no—
556C4—5841no—
557C4#1798847914yes—
558C4C2×#1455654911yescharacteristic
559C4C2×C4×D29211yescharacteristic
560C4C2×#1455657511yescharacteristic
561C4C2×#1455658611yescharacteristic
562C2—24no—
563C2—5842no—
564C2C2×#1798847911yescharacteristic
565C2#2261875711yescharacteristic
566C2#2261876811yescharacteristic
567C2#2261959411yescharacteristic
568C2#2261960311yescharacteristic
569C2#2262028311yescharacteristic
570C2#2262029411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#340×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#340D73
hypocenterC73#21092
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C2×#17988542C2semidirect
4. #22612549C2semidirect
6. #22610629C2semidirect
7. #22610622C2semidirect
8. #22610615C2semidirect
9. #22610608C2semidirect
10. #22609650C2semidirect
11. C2×#340×C73C2semidirect
16. #17988586C4semidirect
76. #14556514C2×C4semidirect
79. #14556511C2×C4semidirect
296. C2×#340D73semidirect
297. C73#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#179885862
C2×#179885842
C2×#179885422
#226125492
#226125242
#226106292
#226106222
#226106152
#226106082
#226096502
C2×#340×C732
#2109273