Finite Group Databasebeta

← #33086433

Subgroups — #33086433

Order 18688.

1070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988628C211yesmaximal, characteristic
2C2×#17988626C211yesmaximal, characteristic
3C2×#17988622C211yesmaximal, characteristic
4C2×#107×D73C211yesmaximal, characteristic
5C2×#17988544C211yesmaximal, characteristic
6#22612556C211yesmaximal, characteristic
7#22612552C211yesmaximal, characteristic
8#22610099C211yesmaximal, characteristic, complemented
9#22610098C211yesmaximal, characteristic, complemented
10#22610096C211yesmaximal, characteristic, complemented
11#22610093C211yesmaximal, characteristic, complemented
12#22610092C211yesmaximal, characteristic, complemented
13#22610090C211yesmaximal, characteristic, complemented
14#22610089C211yesmaximal, characteristic, complemented
15C73×#1847C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×#14556549C2211yescharacteristic
20C2×#14556548C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22C2×C4⋊C4×D73C2211yescharacteristic
23#17988628—24no—
24#17988626—24no—
25#17988622—24no—
26C2×#14556525C2211yescharacteristic
27C2×#14556525C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556524C2211yescharacteristic
30C2×#14556524C2211yescharacteristic
31C2×#14556524C2211yescharacteristic
32C2×#14556524C2211yescharacteristic
33C2×#14556524C2211yescharacteristic
34#107×D73—24no—
35C2×#14556523C2211yescharacteristic
36C2×#14556523C2211yescharacteristic
37C2×#14556522C2211yescharacteristic
38C2×#14556522C2211yescharacteristic
39#17988544C418yescomplemented
40C2×#14556501C2211yescharacteristic
41C2×C4×#10596831C2211yescharacteristic
42C2×C8×Dic73C2211yescharacteristic
43#17988484—22no—
44#17988482—22no—
45#17988461—22no—
46#17988458—22no—
47#17988457C2211yescharacteristic
48#17988456C2211yescharacteristic
49#17988456C2211yescharacteristic
50#17988453C2211yescharacteristic
51#17988452C2211yescharacteristic
52#17988452C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#106×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×#97×C73C2211yescharacteristic
58#384×C73C2211yescharacteristic
59#382×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C2×#11924011C2×C414yes—
64C2×#11924011D414yes—
65C2×#11924011Q814yes—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C2×C8×D73C2×C414yes—
69C2×C8×D73D414yes—
70C2×C8×D73Q814yes—
71#14556549—22no—
72#14556549—22no—
73#14556549—22no—
74#14556549—22no—
75#14556548C2×C414yes—
76#14556548Q814yes—
77C2×C4⋊Dic73—22no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—22no—
85C2×#11924007—22no—
86C2×#11924007D412yes—
87C4⋊C4×D73—22no—
88C4⋊C4×D73—22no—
89C4⋊C4×D73—22no—
90C4⋊C4×D73—22no—
91C2×#11924005—21no—
92C2×#11924005—21no—
93C2×#11924005D412yes—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73C2311yescharacteristic
97#14556525—22no—
98#14556525—22no—
99#14556524—22no—
100#14556524—22no—
101#14556524—22no—
102#14556524—22no—
103#14556524—22no—
104#14556524—22no—
105#14556523—22no—
106#14556523—22no—
107#14556522—22no—
108#14556522—22no—
109#14556501C2×C414yescomplemented
110C4×#10596831C2×C414yescomplemented
111C8×Dic73C2×C414yescomplemented
112#14556498—22no—
113#14556498—22no—
114#14556498—22no—
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C2×C4⋊C4×C73—22no—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#107×C73—22no—
124#107×C73—22no—
125#106×C73—22no—
126#106×C73—22no—
127#97×C73C2×C414yescomplemented
128#95×C73—22no—
129C23×D73C2×D411yescharacteristic
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D411yescharacteristic
136C2×C4×D73—42no—
137C2×C4×D73—42no—
138C2×C4×D73—82no—
139C2×C4×D73—82no—
140C2×C4×D73C22×C412yes—
141C2×C4×D73C2×D412yes—
142C2×C4×D73C2×D412yes—
143C2×C4×D73C2×Q812yes—
144C2×C4×D73C2×Q812yes—
145C2×C4×D73C2×Q812yes—
146#11924011C4⋊C4116yescomplemented
147C2×#10596831C22×C412yes—
148C2×#10596831C22×C412yes—
149C2×#10596831C2×D412yes—
150C2×#10596831C2×D412yes—
151C2×#10596831C2×Q812yes—
152C2×#10596831C2×Q812yes—
153C8×D73C4⋊C4116yescomplemented
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—82no—
163#11924007—42no—
164#11924007—42no—
165#11924007—82no—
166#11924007—82no—
167#11924007C4⋊C418yes—
168#11924005—42no—
169#11924005—42no—
170#11924005C4⋊C418yes—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73C22×C412yes—
174C4×Dic73C2×Q812yes—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73C2×D412yes—
180C22×C4×C73C2411yescharacteristic
181C2×C8×C73C22×C412yes—
182C2×C8×C73C22×C412yes—
183C2×C8×C73C2×D412yes—
184C2×C8×C73C2×D412yes—
185C2×C8×C73C2×Q812yes—
186C2×C8×C73C2×Q812yes—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—82no—
196C42×C73C22×C412yes—
197C42×C73C2×Q812yes—
198C23×C73C22×D411yescharacteristic
199C22×D73—22no—
200C22×D73—22no—
201C22×D73C2×C4⋊C412yes—
202C22×D73#12712yes—
203C22×D73C4⋊Q812yes—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73C2×C4⋊C414yes—
215C2×C4×C73C23×C411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73C22×D411yescharacteristic
218C2×C4×C73C22×Q811yescharacteristic
219C2×C4×C73C22×Q811yescharacteristic
220C2×C4×C73C22×Q811yescharacteristic
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—41no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73C2×C4⋊C414yes—
232C2×Dic73C2×C4⋊C412yes—
233C2×Dic73#12712yes—
234C2×Dic73C4⋊Q812yes—
235C4×D73—84no—
236C4×D73—84no—
237C4×D73C2×C4⋊C414yes—
238C4×D73C2×C4⋊C414yes—
239C4×D73C4⋊Q814yes—
240C4×D73C4⋊Q814yes—
241C8×C73C2×C4⋊C414yes—
242C8×C73C2×C4⋊C414yes—
243#10596831C2×C4⋊C414yes—
244#10596831C2×C4⋊C414yes—
245C22×C73C22×C4⋊C411yescharacteristic
246C22×C73C2×#12711yescharacteristic
247C22×C73C2×C4⋊Q811yescharacteristic
248C22×C73#57612yes—
249C22×C73#57812yes—
250C2×D73—22no—
251C2×D73—22no—
252C2×D73—24no—
253C2×D73—24no—
254C2×D73#38414yes—
255C4×C73—24no—
256C4×C73—81no—
257C4×C73—81no—
258C4×C73—81no—
259C4×C73—81no—
260C4×C73C22×C4⋊C411yescharacteristic
261C4×C73C22×C4⋊C411yescharacteristic
262C4×C73C2×C4⋊Q811yescharacteristic
263C4×C73C2×C4⋊Q811yescharacteristic
264Dic73—24no—
265Dic73—81no—
266Dic73—81no—
267Dic73—81no—
268Dic73—81no—
269Dic73#38414yes—
270#20803—731nomaximal, Sylow
271C2×C73C2×#38411yescharacteristic
272C2×C73#291312yes—
273C2×C73#315511yescharacteristic
274C2×C73#315611yescharacteristic
275C2×C73#316412yes—
276D73—24no—
277C2×#428—731no—
278C2×#428—731no—
279C2×#427—731no—
280C22×#107—731no—
281C2×#405—731no—
282#2317—731no—
283#2317—731no—
284#1847—731no—
285#1847—731no—
286#1847—731no—
287#1847D7311yescharacteristic, complemented
288#1846—731no—
289#1846—731no—
290#1845—731no—
291#1845—731no—
292C73#2080311yescharacteristic, complemented, Sylow
293C22×M16—731no—
294C23×C8—731no—
295C2×#117—731no—
296C2×#117—731no—
297C2×#117—731no—
298C22×C4⋊C4—731no—
299C22×C4⋊C4—731no—
300#428—1464no—
301#428—1464no—
302#427—1464no—
303C2×#107—731no—
304C2×#107—731no—
305C2×#107—731no—
306C2×#107—731no—
307C2×#107—731no—
308C2×#107—731no—
309C2×#107—731no—
310C2×#107—731no—
311C2×#107—1464no—
312C2×#107C2×D7311yescharacteristic
313C2×#107C2×D7311yescharacteristic
314C2×#106—731no—
315C2×#106—731no—
316C2×#106—731no—
317C2×#106—731no—
318C2×#106C2×D7311yescharacteristic
319C2×#106C2×D7311yescharacteristic
320#405—738no—
321C2×#97—731no—
322C2×#97C2×D7311yescharacteristic
323C2×C4×C8—731no—
324C2×C4×C8—731no—
325#389—1462no—
326#389—1462no—
327#385—1462no—
328#385—1462no—
329#384—731no—
330#384—731no—
331#384—731no—
332#384C2×D7311yescharacteristic
333#382—731no—
334#382—731no—
335#382—731no—
336#382C2×D7311yescharacteristic
337C23×C4—731no—
338C23×C4—1461no—
339C23×C4—1461no—
340C2×M16—734no—
341C2×M16—734no—
342C2×M16—734no—
343C22×C8—731no—
344C22×C8—731no—
345C22×C8—734no—
346C22×C8—734no—
347C22×C8—734no—
348C22×C8C22×D7311yescharacteristic
349C22×C8C22×D7311yescharacteristic
350#117—734no—
351#117—734no—
352#117—1462no—
353#117—1462no—
354#117—1462no—
355#117—1462no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—732no—
358C2×C4⋊C4—731no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—731no—
361C2×C4⋊C4—731no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1462no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1461no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4C22×D7311yescharacteristic
372C2×C4⋊C4C22×D7311yescharacteristic
373C2×C4⋊C4C22×D7311yescharacteristic
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C22⋊C4—732no—
376C2×C22⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C42—731no—
379C2×C42—1461no—
380C2×C42—1461no—
381C2×C42C22×D7311yescharacteristic
382#107—22no—
383#107—22no—
384#107—1462no—
385#107—1462no—
386#107—1462no—
387#107—1462no—
388#107—1462no—
389#107—1462no—
390#107—1462no—
391#107—1462no—
392#106—22no—
393#106—22no—
394#106—1462no—
395#106—1462no—
396#106—1462no—
397#106—1462no—
398#97—734no—
399#97C4×D7314yescomplemented
400C4×C8—734no—
401C4×C8—734no—
402#95—22no—
403#95—1462no—
404#95—1462no—
405#95—1462no—
406C24—731no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—2922no—
412C22×C4—2922no—
413C22×C4—5842no—
414C22×C4—5842no—
415C22×C4—731no—
416C22×C4—732no—
417C22×C4—732no—
418C22×C4—732no—
419C22×C4—732no—
420C22×C4—732no—
421C22×C4—732no—
422C22×C4—732no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4D4×D7312yes—
428C22×C4C23×D7311yescharacteristic
429M16—7316no—
430C2×C8—732no—
431C2×C8—732no—
432C2×C8—732no—
433C2×C8—732no—
434C2×C8—732no—
435C2×C8—732no—
436C2×C8—7316no—
437C2×C8C2×C4×D7312yes—
438C2×C8C2×C4×D7312yes—
439C2×C8D4×D7312yes—
440C2×C8D4×D7312yes—
441C2×C8Q8×D7312yes—
442C2×C8Q8×D7312yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—82no—
452C4⋊C4—2922no—
453C4⋊C4—2922no—
454C4⋊C4—5842no—
455C4⋊C4—5842no—
456C4⋊C4—5842no—
457C4⋊C4—738no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C22⋊C4—2922no—
467C22⋊C4—2922no—
468C22⋊C4—738no—
469C42—2922no—
470C42—2922no—
471C42—732no—
472C42—732no—
473C42C2×C4×D7312yes—
474C42Q8×D7312yes—
475C23—732no—
476C23—732no—
477C23—732no—
478C23—1462no—
479C23—1462no—
480C23C2×D4×D7311yescharacteristic
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—2921no—
492C2×C4—2921no—
493C2×C4—2921no—
494C2×C4—2921no—
495C2×C4—2922no—
496C2×C4—2922no—
497C2×C4—2922no—
498C2×C4—2922no—
499C2×C4—5844no—
500C2×C4—5844no—
501C2×C4—732no—
502C2×C4—732no—
503C2×C4—732no—
504C2×C4—734no—
505C2×C4—734no—
506C2×C4—734no—
507C2×C4—734no—
508C2×C4—734no—
509C2×C4—1462no—
510C2×C4—1462no—
511C2×C4C4⋊C4×D7314yes—
512C2×C4C22×C4×D7311yescharacteristic
513C2×C4C2×D4×D7311yescharacteristic
514C2×C4C2×D4×D7311yescharacteristic
515C2×C4C2×Q8×D7311yescharacteristic
516C2×C4C2×Q8×D7311yescharacteristic
517C2×C4C2×Q8×D7311yescharacteristic
518C8—734no—
519C8—734no—
520C8C4⋊C4×D7314yes—
521C8C4⋊C4×D7314yes—
522C22—734no—
523C22—1462no—
524C22—1464no—
525C22—1464no—
526C22—1462no—
527C22C2×C4⋊C4×D7311yescharacteristic
528C22#127×D7311yescharacteristic
529C22C4⋊Q8×D7311yescharacteristic
530C22#1798939112yes—
531C22#1798940312yes—
532C4—24no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—5841no—
538C4—5841no—
539C4—5841no—
540C4—5841no—
541C4—734no—
542C4—1464no—
543C4C2×C4⋊C4×D7311yescharacteristic
544C4C2×C4⋊C4×D7311yescharacteristic
545C4C4⋊Q8×D7311yescharacteristic
546C4C4⋊Q8×D7311yescharacteristic
547C2—1464no—
548C2#384×D7311yescharacteristic
549C2#2261853012yes—
550C2#2262041511yescharacteristic
551C2#2262042311yescharacteristic
552C2#2262046612yes—
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1847C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1847D73
hypocenterC73#20803
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
8. #22610099C2semidirect
9. #22610098C2semidirect
10. #22610096C2semidirect
11. #22610093C2semidirect
12. #22610092C2semidirect
13. #22610090C2semidirect
14. #22610089C2semidirect
15. C73×#1847C2semidirect
39. #17988544C4semidirect
109. #14556501C2×C4semidirect
110. C4×#10596831C2×C4semidirect
111. C8×Dic73C2×C4semidirect
127. #97×C73C2×C4semidirect
146. #11924011C4⋊C4semidirect
153. C8×D73C4⋊C4semidirect
287. #1847D73semidirect
292. C73#20803semidirect
399. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
C2×#179886262
C2×#179886222
C2×#107×D732
C2×#179885442
#226125562
#226125522
#226100992
#226100982
#226100962
#226100932
#226100922
#226100902
#226100892
C73×#18472
#2080373