Finite Group Databasebeta

← #33144852

Subgroups — #33144852

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988628C211yesmaximal, characteristic
2#22613651C211yesmaximal, characteristic
3#22613618C211yesmaximal, characteristic
4#22611416C211yesmaximal, characteristic, complemented
5#22611415C211yesmaximal, characteristic, complemented
6#22611266C211yesmaximal, characteristic, complemented
7#22611256C211yesmaximal, characteristic, complemented
8#22610669C211yesmaximal, characteristic, complemented
9#22610665C211yesmaximal, characteristic, complemented
10#22610129C211yesmaximal, characteristic
11#22610117C211yesmaximal, characteristic
12#22610092C211yesmaximal, characteristic, complemented
13#22609961C211yesmaximal, characteristic
14#22609716C211yesmaximal, characteristic
15C2×#501×C73C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×#14556549C2211yescharacteristic
18C2×C4⋊C4×D73C2211yescharacteristic
19#17988628—24no—
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28#17988525—21no—
29#17988520—21no—
30#17988509—21no—
31#17988498—21no—
32#17988494—21no—
33#17988486C2211yescharacteristic
34#17988486C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988468—21no—
38#17988458—21no—
39#17988452C2211yescharacteristic
40#17988452C2211yescharacteristic
41#17988444—21no—
42#17988444C2211yescharacteristic
43#17988444C2211yescharacteristic
44C4×#11924005C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4⋊Q8×C73C2211yescharacteristic
54C2×#125×C73C2211yescharacteristic
55#501×C73—24no—
56C2×#107×C73C2211yescharacteristic
57C2×#107×C73C2211yescharacteristic
58C2×#106×C73C2211yescharacteristic
59C2×#106×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C2×#11924011—22no—
64C2×#11924011—22no—
65C2×#11924011Q814yescomplemented
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556549—22no—
69#14556549—41no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C4⋊C4×D73—22no—
80C4⋊C4×D73—41no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005D411yescharacteristic
85C2×#11924005D411yescharacteristic
86C2×C4×Dic73—21no—
87C2×C4×Dic73C2311yescharacteristic
88#14556525—22no—
89#14556524—22no—
90#14556524—41no—
91#14556524—41no—
92#14556523—22no—
93#14556522—22no—
94#14556503Q814yescomplemented
95C8×Dic73—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498D411yescharacteristic
107#14556498D411yescharacteristic
108C22×Q8×C73—21no—
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C4⋊Q8×C73—22no—
112C4⋊Q8×C73—41no—
113#125×C73—22no—
114#125×C73—41no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
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×C73D414yes—
125#106×C73—22no—
126#106×C73—22no—
127#97×C73—22no—
128C23×D73C2×D411yescharacteristic
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73—81no—
143C2×C4×D73C2×Q812yes—
144C2×C4×D73C2×Q812yes—
145#11924011—28no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831C2×Q812yes—
151C2×#10596831C2×Q812yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—81no—
161C4⋊Dic73—81no—
162#11924007—42no—
163#11924007—42no—
164#11924007—81no—
165#11924005—41no—
166#11924005—41no—
167#11924005—42no—
168#11924005—42no—
169#11924005—81no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—42no—
173C2×Q8×C73—42no—
174C2×Q8×C73—42no—
175C2×Q8×C73—81no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73C2×D412yes—
186C2×C8×C73C2×Q812yes—
187C2×C8×C73C2×Q812yes—
188C2×C8×C73#4112yes—
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—42no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73C2×D412yes—
199C4⋊C4×C73C2×D412yes—
200C42×C73—21no—
201C42×C73—21no—
202C23×C73C22×D411yescharacteristic
203C22×D73—21no—
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—21no—
209C22×D73C4⋊Q812yes—
210Q8×C73—84no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×Q811yescharacteristic
231C2×C4×C73C22×Q811yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73C4⋊Q812yes—
256C4×D73—22no—
257C4×D73—22no—
258C4×D73—22no—
259C4×D73—22no—
260C4×D73—84no—
261C8×C73—22no—
262C8×C73C22⋊Q814yes—
263#10596831—22no—
264#10596831—22no—
265C22×C73C2×C4⋊Q811yescharacteristic
266C22×C73D4211yescharacteristic
267C22×C73#54711yescharacteristic
268C22×C73#57511yescharacteristic
269C22×C73#57511yescharacteristic
270C22×C73#57611yescharacteristic
271C22×C73#57811yescharacteristic
272C2×D73—22no—
273C2×D73—22no—
274C2×D73—22no—
275C2×D73—41no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73—41no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—42no—
282C4×C73—41no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73C2×C22⋊Q811yescharacteristic
286C4×C73#54411yescharacteristic
287C4×C73D4×Q811yescharacteristic
288C4×C73#55411yescharacteristic
289Dic73—22no—
290Dic73—41no—
291Dic73—41no—
292Dic73—81no—
293Dic73—81no—
294Dic73—81no—
295Dic73—81no—
296#26079—731nomaximal, Sylow
297C2×C73#244011yescharacteristic
298C2×C73#316411yescharacteristic
299C2×C73#316611yescharacteristic
300C2×C73#327311yescharacteristic
301C2×C73#327711yescharacteristic
302C2×C73#334711yescharacteristic
303C2×C73#334911yescharacteristic
304D73—42no—
305C2×#501D7311yescharacteristic, complemented
306C2×#428—731no—
307#2442—731no—
308#2439—731no—
309#2085—731no—
310#2085—731no—
311#2059—731no—
312#2057—731no—
313#1943—731no—
314#1942—731no—
315#1854—731no—
316#1852—731no—
317#1846—731no—
318#1809—731no—
319#1753—731no—
320C73#2607911yescharacteristic, complemented, Sylow
321C22×M16—731no—
322C2×C4⋊Q8C2×D7311yescharacteristic
323C2×#125C2×D7311yescharacteristic
324C2×#117—731no—
325C22×C4⋊C4—731no—
326#501—24no—
327#428—1464no—
328C2×#107—731no—
329C2×#107—731no—
330C2×#107—731no—
331C2×#107—731no—
332C2×#107C2×D7311yescharacteristic
333C2×#107C2×D7311yescharacteristic
334C2×#106—731no—
335C2×#106—731no—
336C2×#106C2×D7311yescharacteristic
337C2×#106C2×D7311yescharacteristic
338C2×C22⋊C8—731no—
339C2×#97C2×D7311yescharacteristic
340C2×C4×C8—731no—
341#400—1461no—
342#398—1461no—
343#396—1461no—
344#393—1461no—
345#391—1461no—
346#389—731no—
347#389—731no—
348#389—731no—
349#389—731no—
350#387—1461no—
351#385—1461no—
352#382—731no—
353#382—731no—
354#380—731no—
355#380—731no—
356#380—1461no—
357C4×C22⋊C4—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336—731no—
365#336—731no—
366C22×Q8—21no—
367C23×C4—731no—
368C23×C4—1461no—
369C2×M16—734no—
370C2×M16—1462no—
371C2×M16—1462no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8C22×D7311yescharacteristic
375C22×C8C22×D7311yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378#125—22no—
379#125—41no—
380#117—2921no—
381#117—1462no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—2921no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1462no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—731no—
401C2×C22⋊C4—731no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C22⋊C4—1461no—
405C2×C42—731no—
406C2×C42—1461no—
407C2×C42C22×D7311yescharacteristic
408#107—22no—
409#107—2921no—
410#107—2921no—
411#107—1462no—
412#107—1462no—
413#107C22⋊D7314yes—
414#106—22no—
415#106—22no—
416#106—1462no—
417#106—1462no—
418C22⋊C8—734no—
419#97—22no—
420C4×C8—1462no—
421#95—731no—
422#95—731no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433C24—731no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2922no—
442C22×C4—2922no—
443C22×C4—5841no—
444C22×C4—731no—
445C22×C4—732no—
446C22×C4—732no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4D4×D7311yescharacteristic
458C22×C4D4×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460M16—1468no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—732no—
464C2×C8—732no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8C2×C22⋊D7312yes—
470C2×C8Q8×D7312yes—
471C2×C8Q8×D7312yes—
472C2×C8#1192402912yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—2922no—
483C4⋊C4—2922no—
484C4⋊C4—5841no—
485C4⋊C4—5841no—
486C4⋊C4—5841no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4C2×C22⋊D7312yes—
496C4⋊C4C2×C22⋊D7312yes—
497C22⋊C4—2922no—
498C22⋊C4—2921no—
499C22⋊C4—2921no—
500C22⋊C4—2922no—
501C22⋊C4—5841no—
502C42—21no—
503C42—21no—
504C42—2922no—
505C42—1461no—
506C42—1461no—
507C23—732no—
508C23—1461no—
509C23—1461no—
510C23—1461no—
511C23—1461no—
512C23—1461no—
513C23—1461no—
514C23C2×D4×D7311yescharacteristic
515Q8—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—5844no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4C22×C22⋊D7311yescharacteristic
561C2×C4C2×Q8×D7311yescharacteristic
562C2×C4C2×Q8×D7311yescharacteristic
563C2×C4C2×#1192402911yescharacteristic
564C2×C4#1455663611yescharacteristic
565C2×C4#1455663911yescharacteristic
566C8—22no—
567C8—1462no—
568C8—1462no—
569C8#1455656714yes—
570C22—2921no—
571C22—2921no—
572C22—2921no—
573C22—2921no—
574C22—1462no—
575C22—1462no—
576C22—1462no—
577C22C4⋊Q8×D7311yescharacteristic
578C22D4×C22⋊D7311yescharacteristic
579C22#1798924011yescharacteristic
580C22#1798938311yescharacteristic
581C22#1798938611yescharacteristic
582C22#1798939111yescharacteristic
583C22#1798940311yescharacteristic
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—2921no—
591C4—2921no—
592C4—5841no—
593C4—5841no—
594C4—5841no—
595C4—5841no—
596C4—1462no—
597C4C2×#1455656711yescharacteristic
598C4#1798921411yescharacteristic
599C4Q8×C22⋊D7311yescharacteristic
600C4#1798929811yescharacteristic
601C2—2922no—
602C2#2261362911yescharacteristic
603C2#2262046611yescharacteristic
604C2#2262048211yescharacteristic
605C2#2262153611yescharacteristic
606C2#2262158211yescharacteristic
607C2#2262243011yescharacteristic
608C2#2262246011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#501×C73C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
hypercenterC2×#501D73
hypocenterC73#26079
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #22611416C2semidirect
5. #22611415C2semidirect
6. #22611266C2semidirect
7. #22611256C2semidirect
8. #22610669C2semidirect
9. #22610665C2semidirect
12. #22610092C2semidirect
15. C2×#501×C73C2semidirect
65. C2×#11924011Q8semidirect
94. #14556503Q8semidirect
305. C2×#501D73semidirect
320. C73#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
#226136512
#226136182
#226114162
#226114152
#226112662
#226112562
#226106692
#226106652
#226101292
#226101172
#226100922
#226099612
#226097162
C2×#501×C732
#2607973