Finite Group Databasebeta

← #33086757

Subgroups — #33086757

Order 18688.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989068C211yesmaximal, characteristic
2C2×#17988623C211yesmaximal, characteristic, complemented
3C2×#17988555C211yesmaximal, characteristic
4#22612519C211yesmaximal, characteristic
5#22610129C212yesmaximal, complemented
6#22610127C212yesmaximal, complemented
7#22610122C212yesmaximal, complemented
8#22610117C212yesmaximal, complemented
9#22609651C211yesmaximal, characteristic, complemented
10C2×#17988349C211yesmaximal, characteristic, complemented
11C2×#428×C73C211yesmaximal, characteristic, complemented
12C22×#11924028C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14#17989068—24no—
15#17989068—24no—
16C2×#14556556—21no—
17C2×#14556553—21no—
18C2×C4×C22⋊D73—22no—
19C2×#14556550C2211yescharacteristic
20C2×#14556549C2212yes—
21C22×C4⋊Dic73C2211yescharacteristic
22C2×C4⋊C4×D73C2212yes—
23#17988623—24no—
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28#17988555C418yescomplemented
29C2×#14556503C2212yes—
30C2×#14556503C2212yes—
31#17988493—21no—
32#17988489—21no—
33#17988464—22no—
34#17988444C2212yes—
35#17988444C2212yes—
36#17988350C2212yes—
37#17988350C2212yes—
38#17988349C2212yes—
39#17988349C2212yes—
40C22×M16×C73C2211yescharacteristic
41C2×#117×C73C2211yescharacteristic
42C22×C4⋊C4×C73C2211yescharacteristic
43#428×C73—24no—
44C2×#107×C73C2212yes—
45C2×#106×C73C2212yes—
46C2×#11924028—22no—
47C2×#11924028—22no—
48C2×#11924028—22no—
49C2×#11924028—22no—
50C2×#11924028C2×C414yes—
51C2×#11924028Q814yes—
52C22×Dic146D411yescharacteristic
53C22×D292D411yescharacteristic
54C22×C22⋊D73D412yes—
55C23×Dic73—21no—
56C22×C4×D73—22no—
57C22×C4×D73C2312yes—
58C22×#10596831—22no—
59C22×#10596831—22no—
60C22×#10596831C2×C414yescomplemented
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556556—28no—
64#14556553—28no—
65C4×C22⋊D73—216no—
66#14556550—22no—
67#14556550D414yes—
68#14556549—24no—
69#14556549—24no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73C2312yes—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—22no—
77C4⋊C4×D73—24no—
78C4⋊C4×D73—24no—
79C2×C22⋊Dic73—21no—
80C2×#11924005—22no—
81C2×#11924005—22no—
82C2×#11924005—22no—
83C2×C4×Dic73—22no—
84C2×C4×Dic73—21no—
85#14556524—22no—
86#14556524—22no—
87#14556522—22no—
88#14556522—22no—
89#14556503C2×C418yescomplemented
90#14556503C2×C418yescomplemented
91#14556498—22no—
92#14556498—22no—
93C23×C4×C73—21no—
94C23×C4×C73C2311yescharacteristic
95C2×M16×C73—22no—
96C2×M16×C73—22no—
97C2×M16×C73C2×C414yescomplemented
98C22×C8×C73C2312yes—
99#117×C73—22no—
100#117×C73—22no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—22no—
103C2×C4⋊C4×C73D414yes—
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2312yes—
107C2×C22⋊C4×C73—21no—
108C2×C42×C73—21no—
109#107×C73—24no—
110#106×C73—24no—
111C23×D73C2×D412yes—
112#11924028—28no—
113#11924028—28no—
114#11924028—28no—
115#11924028—28no—
116C2×Dic146—22no—
117C2×Dic146—22no—
118C2×Dic146C4⋊C414yes—
119C2×D292—22no—
120C2×D292—22no—
121C2×D292C4⋊C414yes—
122C2×C22⋊D73—24no—
123C2×C22⋊D73—24no—
124C2×C22⋊D73C4⋊C418yes—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—22no—
128C22×Dic73—42no—
129C22×Dic73—81no—
130C22×Dic73—81no—
131C22×Dic73C2×D412yes—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—22no—
136C2×C4×D73—28no—
137C2×C4×D73—82no—
138C2×C4×D73—82no—
139C2×C4×D73C22×C414yes—
140C2×C4×D73C2×Q814yes—
141C2×#10596831—28no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831C22×C412yes—
147C2×#10596831C22×C412yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—22no—
151C4⋊Dic73—22no—
152C4⋊Dic73—42no—
153C4⋊Dic73C2×D412yes—
154C4⋊Dic73C2×D412yes—
155#11924007—28no—
156C22⋊Dic73—24no—
157#11924005—28no—
158#11924005—82no—
159#11924005—82no—
160C4×Dic73—28no—
161C4×Dic73—24no—
162C24×C73C2×D411yescharacteristic
163C22×C4×C73—22no—
164C22×C4×C73—24no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—81no—
168C22×C4×C73—81no—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×Q812yes—
174C22×C4×C73#4112yes—
175C22×C4×C73#4112yes—
176C22×C4×C73C2411yescharacteristic
177M16×C73—28no—
178C2×C8×C73—22no—
179C2×C8×C73—22no—
180C2×C8×C73C22×C414yes—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73C2×D414yes—
188C22⋊C4×C73—42no—
189C42×C73—42no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73C2×C4⋊C412yes—
195C23×C73#12012yes—
196C23×C73#12312yes—
197C23×C73C22×D411yescharacteristic
198C22×D73—22no—
199C22×D73—22no—
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—22no—
204C22×D73C2×C4⋊C414yes—
205C2×C4×C73—24no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—84no—
215C2×C4×C73C2×C4⋊C412yes—
216C2×C4×C73C4×D414yes—
217C2×C4×C73C4×D414yes—
218C2×C4×C73#12012yes—
219C2×C4×C73C22⋊Q814yes—
220C2×C4×C73C22⋊Q814yes—
221C2×C4×C73#12312yes—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C22×D411yescharacteristic
225C2×C4×C73C22×Q811yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C22⋊D73—216no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—84no—
241C2×Dic73C2×C4⋊C414yes—
242D292—28no—
243C4×D73—24no—
244C4×D73—24no—
245C4×D73—24no—
246C4×D73—24no—
247C4×D73—88no—
248Dic146—28no—
249C8×C73—24no—
250#10596831—22no—
251#10596831—22no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73#38514yes—
259C22×C73C22×C4⋊C411yescharacteristic
260C22×C73C2×#12011yescharacteristic
261C22×C73C2×#12311yescharacteristic
262C22×C73#57611yescharacteristic
263C22×C73#57711yescharacteristic
264C22×C73#57711yescharacteristic
265C22×C73#57811yescharacteristic
266C2×D73—24no—
267C2×D73—24no—
268C2×D73—24no—
269C2×D73—42no—
270C2×D73—42no—
271C2×D73—42no—
272C2×D73—42no—
273C4×C73—44no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73#38514yes—
277C4×C73C2×C4×D411yescharacteristic
278C4×C73C2×C4×D411yescharacteristic
279C4×C73C2×C22⋊Q811yescharacteristic
280C4×C73C2×C22⋊Q811yescharacteristic
281Dic73—24no—
282Dic73—42no—
283Dic73—42no—
284Dic73—82no—
285#20849—731nomaximal, Sylow
286C2×C73—24no—
287C2×C73C2×#38511yescharacteristic
288C2×C73#291211yescharacteristic
289C2×C73#291311yescharacteristic
290C2×C73#301211yescharacteristic
291C2×C73#301311yescharacteristic
292C2×C73#309611yescharacteristic
293C2×C73#309711yescharacteristic
294D73—44no—
295C2×#520—731no—
296C2×#428D7311yescharacteristic, complemented
297C2×#427—731no—
298C2×#408—731no—
299#2311—731no—
300#1854—732no—
301#1854—732no—
302#1853—732no—
303#1852—732no—
304#1736—731no—
305C2×#340—731no—
306C73#2084911yescharacteristic, complemented, Sylow
307C22×#41—731no—
308C22×M16C2×D7311yescharacteristic
309C23×C8—731no—
310#520—1464no—
311#520—1464no—
312C2×C4×Q8—1461no—
313C2×C4×D4—1461no—
314C2×C4×D4—1462no—
315C2×#117—731no—
316C2×#117—732no—
317C2×#117C2×D7311yescharacteristic
318C22×C4⋊C4—731no—
319C22×C4⋊C4—732no—
320C22×C4⋊C4C2×D7311yescharacteristic
321#428—24no—
322#427—1464no—
323C2×#107—731no—
324C2×#107—731no—
325C2×#107C2×D7312yes—
326C2×#106—731no—
327C2×#106—731no—
328C2×#106C2×D7312yes—
329#408—738no—
330C2×C22⋊C8—732no—
331C2×C22⋊C8—732no—
332#391—1461no—
333#390—1461no—
334#386—1462no—
335#380—732no—
336#380—732no—
337#340—732no—
338#340—732no—
339#340—732no—
340#340—732no—
341C2×#41—734no—
342C2×#41—734no—
343C2×#41—1462no—
344C2×#41—1462no—
345C2×#41—1462no—
346C2×#41—1462no—
347C22×Q8—731no—
348C22×D4—732no—
349C22×D4—731no—
350C23×C4—21no—
351C23×C4—732no—
352C23×C4—1462no—
353C23×C4—1461no—
354C23×C4C22×D7311yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D7314yescomplemented
358C22×C8—734no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—1462no—
362C22×C8—1462no—
363C22×C8C22×D7312yes—
364C4×Q8—1468no—
365C4×D4—1468no—
366C4×D4—14616no—
367#117—22no—
368#117—22no—
369#117—734no—
370#117—1464no—
371#117—1462no—
372#117—1464no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—732no—
378C2×C4⋊C4—1464no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1462no—
381C2×C4⋊C4—1462no—
382C2×C4⋊C4—1462no—
383C2×C4⋊C4—1464no—
384C2×C4⋊C4C22⋊D7314yes—
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7312yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1462no—
390C2×C22⋊C4—1462no—
391C2×C22⋊C4—1462no—
392C2×C22⋊C4—1461no—
393C2×C42—21no—
394C2×C42—1462no—
395C2×C42—1461no—
396#107—24no—
397#107—1462no—
398#107—1462no—
399#106—24no—
400#106—1462no—
401#106—1462no—
402C22⋊C8—738no—
403C22⋊C8—738no—
404#95—1462no—
405#95—1462no—
406C24—732no—
407C24D4×D7311yescharacteristic
408#41—1468no—
409#41—1468no—
410#41—1468no—
411#41—1468no—
412C2×Q8—734no—
413C2×Q8—1462no—
414C2×Q8—1462no—
415C2×D4—738no—
416C2×D4—734no—
417C2×D4—1464no—
418C2×D4—1464no—
419C2×D4—1462no—
420C2×D4—1462no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—2922no—
428C22×C4—5842no—
429C22×C4—5842no—
430C22×C4—5841no—
431C22×C4—5841no—
432C22×C4—732no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—1468no—
436C22×C4—1462no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4C2×C4×D7312yes—
444C22×C4D4×D7311yescharacteristic
445C22×C4D4×D7312yes—
446C22×C4C2×C22⋊D7312yes—
447C22×C4Q8×D7312yes—
448C22×C4#1192402812yes—
449C22×C4#1192402912yes—
450C22×C4C23×D7311yescharacteristic
451M16—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—732no—
455C2×C8—732no—
456C2×C8—1468no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8C2×C4×D7314yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—2922no—
469C4⋊C4—732no—
470C4⋊C4—732no—
471C4⋊C4—1468no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4C2×C22⋊D7314yes—
477C22⋊C4—42no—
478C22⋊C4—5842no—
479C22⋊C4—5842no—
480C22⋊C4—1468no—
481C22⋊C4—1464no—
482C42—42no—
483C42—1468no—
484C42—1464no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—734no—
490C23—1462no—
491C23—1462no—
492C23—1462no—
493C23—1462no—
494C23—1462no—
495C23—1462no—
496C23C4⋊C4×D7312yes—
497C23#1455655912yes—
498C23#1455657612yes—
499C23C2×D4×D7311yescharacteristic
500Q8—1468no—
501D4—14616no—
502D4—1468no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—84no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2922no—
518C2×C4—2922no—
519C2×C4—2922no—
520C2×C4—5848no—
521C2×C4—5844no—
522C2×C4—734no—
523C2×C4—1462no—
524C2×C4—1462no—
525C2×C4—1462no—
526C2×C4—1462no—
527C2×C4—1464no—
528C2×C4—1464no—
529C2×C4—1464no—
530C2×C4—1464no—
531C2×C4C4⋊C4×D7312yes—
532C2×C4C4×C22⋊D7314yes—
533C2×C4#1455655314yes—
534C2×C4#1455655912yes—
535C2×C4#1455656714yes—
536C2×C4#1455656914yes—
537C2×C4#1455657612yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4C2×D4×D7311yescharacteristic
540C2×C4C22×C22⋊D7311yescharacteristic
541C2×C4C2×Q8×D7311yescharacteristic
542C2×C4C2×#1192402811yescharacteristic
543C2×C4C2×#1192402911yescharacteristic
544C8—24no—
545C8—1462no—
546C8—1462no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—2922no—
554C22—2922no—
555C22—2922no—
556C22—2922no—
557C22—1464no—
558C22—1464no—
559C22—1464no—
560C22#1798845814yes—
561C22C2×C4⋊C4×D7311yescharacteristic
562C22C2×#1455655911yescharacteristic
563C22C2×#1455657611yescharacteristic
564C22#1798939111yescharacteristic
565C22#1798939611yescharacteristic
566C22#1798939711yescharacteristic
567C22#1798940311yescharacteristic
568C4—44no—
569C4—81no—
570C4—81no—
571C4—2922no—
572C4—2922no—
573C4—5842no—
574C4—1464no—
575C4#1798845814yes—
576C4C2×C4×C22⋊D7311yescharacteristic
577C4C2×#1455655311yescharacteristic
578C4C2×#1455656711yescharacteristic
579C4C2×#1455656911yescharacteristic
580C2—24no—
581C2—2924no—
582C2C2×#1798845811yescharacteristic
583C2#2261852511yescharacteristic
584C2#2261853011yescharacteristic
585C2#2261923411yescharacteristic
586C2#2261924511yescharacteristic
587C2#2261989111yescharacteristic
588C2#2261990211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#428D73
hypocenterC73#20849
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#17988623C2semidirect
5. #22610129C2semidirect
6. #22610127C2semidirect
7. #22610122C2semidirect
8. #22610117C2semidirect
9. #22609651C2semidirect
10. C2×#17988349C2semidirect
11. C2×#428×C73C2semidirect
28. #17988555C4semidirect
60. C22×#10596831C2×C4semidirect
89. #14556503C2×C4semidirect
90. #14556503C2×C4semidirect
97. C2×M16×C73C2×C4semidirect
296. C2×#428D73semidirect
306. C73#20849semidirect
357. C2×M16C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890682
C2×#179886232
C2×#179885552
#226125192
#226101292
#226101272
#226101222
#226101172
#226096512
C2×#179883492
C2×#428×C732
#2084973