Finite Group Databasebeta

← #33096377

Subgroups — #33096377

Order 18688.

830 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988610C211yesmaximal, characteristic
2#22612333C211yesmaximal, characteristic
3#22612326C211yesmaximal, characteristic
4#22610702C211yesmaximal, characteristic, complemented
5#22610700C211yesmaximal, characteristic, complemented
6#22610698C211yesmaximal, characteristic, complemented
7#22610696C211yesmaximal, characteristic, complemented
8#22610693C211yesmaximal, characteristic, complemented
9#22610691C211yesmaximal, characteristic, complemented
10#22610688C211yesmaximal, characteristic, complemented
11#22610686C211yesmaximal, characteristic, complemented
12#22609964C211yesmaximal, characteristic
13#22609962C211yesmaximal, characteristic
14#22609962C211yesmaximal, characteristic
15C73×#1809C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556547—21no—
18C22×C4×Dic73C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25#17988610C418yescomplemented
26C2×#14556520C2211yescharacteristic
27C2×#14556520C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30#17988456C2211yescharacteristic
31#17988456C2211yescharacteristic
32#17988454—21no—
33#17988452C2211yescharacteristic
34#17988452C2211yescharacteristic
35#17988451—21no—
36#17988448—21no—
37#17988447C2211yescharacteristic
38#17988447C2211yescharacteristic
39#17988445—21no—
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44#17988438—21no—
45C22⋊C4×Dic73—21no—
46#17988432—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#106×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C23×Dic73D411yescharacteristic
61C23×Dic73D411yescharacteristic
62C2×#11924012—41no—
63C2×#11924012—41no—
64C2×#11924012C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556547—44no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73C2×C414yes—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73Q814yes—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556525—41no—
89#14556524—41no—
90#14556524—41no—
91#14556523—41no—
92#14556522—41no—
93#14556522—41no—
94#14556520C2×C414yes—
95#14556520C2×C414yes—
96#14556520C2×C414yes—
97#14556520C2×C414yes—
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#14556498—21no—
107#14556498—21no—
108C23×C4×C73C2311yescharacteristic
109C2×M16×C73—41no—
110C2×M16×C73—41no—
111C2×M16×C73C2×C414yes—
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120#107×C73—41no—
121#106×C73—41no—
122#95×C73—21no—
123#95×C73—21no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—41no—
131C22×Dic73—41no—
132C22×Dic73C4⋊C414yes—
133C22×Dic73C4⋊C414yes—
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138#11924012—44no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C22×C412yes—
144C2×#10596831C22×C412yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153#11924007—42no—
154#11924007—42no—
155#11924007—42no—
156#11924007—42no—
157C22⋊Dic73—42no—
158C22⋊Dic73—42no—
159C4×Dic73—42no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—42no—
163C4×Dic73C4⋊C414yes—
164C4×Dic73C4⋊C414yes—
165C4×Dic73C22×C412yes—
166C4×Dic73C22×C412yes—
167C4×Dic73C2×Q812yes—
168C4×Dic73C2×Q812yes—
169C24×C73C2×D411yescharacteristic
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73C22×C412yes—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×Q812yes—
181C22×C4×C73C2411yescharacteristic
182M16×C73—44no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73C22×C412yes—
188C2×C8×C73C22×C412yes—
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—21no—
196C4⋊C4×C73—21no—
197C22⋊C4×C73—42no—
198C22⋊C4×C73—42no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73C2×C4⋊C412yes—
206C23×C73C22×D411yescharacteristic
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
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×C73C2×C4⋊C412yes—
226C2×C4×C73C23×C411yescharacteristic
227C2×C4×C73C22×Q811yescharacteristic
228C2×C4×C73H(5,𝔽2)11yescharacteristic
229C2×C4×C73H(5,𝔽2)11yescharacteristic
230C2×C4×C732−1+411yescharacteristic
231C2×C4×C732−1+411yescharacteristic
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—28no—
239C2×Dic73—44no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
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×Dic73C2×C4⋊C412yes—
253C2×Dic73C2×C4⋊C412yes—
254C2×Dic73C2×C4⋊C412yes—
255C2×Dic73C2×C4⋊C412yes—
256C2×Dic73AGL(1,ℤ/8)12yes—
257C2×Dic73#13712yes—
258C8×C73—22no—
259C8×C73—22no—
260#10596831—22no—
261#10596831—22no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73C22×C4⋊C411yescharacteristic
270C22×C73#53711yescharacteristic
271C22×C73#53711yescharacteristic
272C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C73C2×#13711yescharacteristic
274C22×C73#57511yescharacteristic
275C22×C73#57511yescharacteristic
276C4×C73—22no—
277C4×C73—81no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73#51811yescharacteristic
282C4×C73#51911yescharacteristic
283C4×C73#54411yescharacteristic
284C4×C73#54411yescharacteristic
285Dic73—22no—
286Dic73—22no—
287Dic73—22no—
288Dic73—81no—
289Dic73—81no—
290Dic73—81no—
291Dic73—81no—
292Dic73#42814yes—
293#21963—731nomaximal, Sylow
294C2×C73—42no—
295C2×C73#231611yescharacteristic
296C2×C73C2×#42811yescharacteristic
297C2×C73#291111yescharacteristic
298C2×C73#325111yescharacteristic
299C2×C73#325111yescharacteristic
300C2×C73#325211yescharacteristic
301C2×C73#325211yescharacteristic
302C2×#423—731no—
303#2285—731no—
304#2283—731no—
305#1949—731no—
306#1949—731no—
307#1949—731no—
308#1949—731no—
309#1948—731no—
310#1948—731no—
311#1947—731no—
312#1947—731no—
313#1809—731no—
314#1809—731no—
315#1809—731no—
316#1809D7311yescharacteristic
317C73#2196311yescharacteristic, complemented, Sylow
318C22×M16—731no—
319C22×M16C2×D7311yescharacteristic
320C2×#117—1461no—
321C22×C42—731no—
322C2×#107—731no—
323C2×#107—731no—
324C2×#107—731no—
325C2×#107C2×D7311yescharacteristic
326C2×#106—731no—
327C2×#106—731no—
328C2×#106—731no—
329C2×#106C2×D7311yescharacteristic
330#423—738no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335#384—731no—
336#384—731no—
337#383—1461no—
338#382—731no—
339#382—731no—
340#382—1461no—
341#381—1461no—
342#380—731no—
343#380—731no—
344#380—1461no—
345#380C2×D7311yescharacteristic
346#380C2×D7311yescharacteristic
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—1461no—
352C4×C22⋊C4—1461no—
353#376—1461no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340C2×D7311yescharacteristic
361#340C2×D7311yescharacteristic
362C23×C4—731no—
363C23×C4—731no—
364C23×C4C22×D7311yescharacteristic
365C2×M16—41no—
366C2×M16—41no—
367C2×M16—2921no—
368C2×M16—2921no—
369C2×M16—734no—
370C2×M16C4×D7314yes—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375#117—2924no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C42—734no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—734no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42—1461no—
401C2×C42—1461no—
402#107—41no—
403#107—2921no—
404#107—2921no—
405#107—2921no—
406#106—41no—
407#106—2921no—
408#106—2921no—
409#106—2921no—
410C4⋊C8—734no—
411C4⋊C8—734no—
412C4⋊C8—734no—
413C4⋊C8—734no—
414#95—21no—
415#95—21no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426C24D4×D7311yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—2921no—
436C22×C4—2921no—
437C22×C4—734no—
438C22×C4—731no—
439C22×C4—734no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4C2×C4×D7312yes—
450C22×C4D4×D7311yescharacteristic
451C22×C4Q8×D7312yes—
452C22×C4C23×D7311yescharacteristic
453M16—44no—
454M16—2924no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—732no—
460C2×C8—732no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8C2×C4×D7312yes—
466C2×C8C2×C4×D7312yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—2922no—
478C4⋊C4—2922no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—2922no—
490C22⋊C4—2922no—
491C42—2922no—
492C42—2922no—
493C42—2922no—
494C42—2922no—
495C42—734no—
496C42—734no—
497C42—732no—
498C42—732no—
499C42—732no—
500C42—732no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C4⋊C4×D7312yes—
508C23C2×D4×D7311yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2924no—
540C2×C4—732no—
541C2×C4—732no—
542C2×C4—732no—
543C2×C4—732no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—1468no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4C4⋊C4×D7312yes—
554C2×C4C22×C4×D7311yescharacteristic
555C2×C4C2×Q8×D7311yescharacteristic
556C2×C4#1455663611yescharacteristic
557C2×C4#1455663611yescharacteristic
558C2×C4#1455663811yescharacteristic
559C2×C4#1455663811yescharacteristic
560C8—22no—
561C8—22no—
562C8—1462no—
563C8—1462no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C2×C4⋊C4×D7311yescharacteristic
572C22#1798916411yescharacteristic
573C22#1798916411yescharacteristic
574C22AGL(1,ℤ/8)×D7311yescharacteristic
575C22#137×D7311yescharacteristic
576C22#1798938311yescharacteristic
577C22#1798938611yescharacteristic
578C4—22no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—5841no—
584C4—5841no—
585C4—5841no—
586C4—5841no—
587C4—734no—
588C4—1462no—
589C4—1462no—
590C4—1462no—
591C4#1798906011yescharacteristic
592C4#1798906411yescharacteristic
593C4#1798921611yescharacteristic
594C4#1798921611yescharacteristic
595C2—42no—
596C2#2261254811yescharacteristic
597C2#428×D7311yescharacteristic
598C2#2261851811yescharacteristic
599C2#2262128611yescharacteristic
600C2#2262128811yescharacteristic
601C2#2262129711yescharacteristic
602C2#2262129911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1809C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1809D73
hypocenterC73#21963
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #22610702C2semidirect
5. #22610700C2semidirect
6. #22610698C2semidirect
7. #22610696C2semidirect
8. #22610693C2semidirect
9. #22610691C2semidirect
10. #22610688C2semidirect
11. #22610686C2semidirect
25. #17988610C4semidirect
317. C73#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#179886102
#226123332
#226123262
#226107022
#226107002
#226106982
#226106962
#226106932
#226106912
#226106882
#226106862
#226099642
#226099622
C73×#18092
#2196373