Finite Group Databasebeta

← #33088046

Subgroups — #33088046

Order 18688.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988612C211yesmaximal, characteristic
2C2×#97×D73C211yesmaximal, characteristic
3#22612411C211yesmaximal, characteristic
4#22610498C211yesmaximal, characteristic, complemented
5#22610497C211yesmaximal, characteristic, complemented
6#22610496C211yesmaximal, characteristic, complemented
7#22610495C211yesmaximal, characteristic, complemented
8#22610494C211yesmaximal, characteristic, complemented
9#22610493C211yesmaximal, characteristic, complemented
10#22610492C211yesmaximal, characteristic, complemented
11#22609626C211yesmaximal, characteristic
12#22609626C211yesmaximal, characteristic
13#22609626C211yesmaximal, characteristic
14#22609623C211yesmaximal, characteristic
15C73×#1917C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×#11924011C2211yescharacteristic
18C22×C8×D73C2211yescharacteristic
19C22×C8×D73C2211yescharacteristic
20C2×#14556548C2211yescharacteristic
21C2×#14556548C2211yescharacteristic
22C2×C42×D73C2211yescharacteristic
23#17988612—24no—
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556520C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30#97×D73—24no—
31#17988484C411yescharacteristic
32#17988484C411yescharacteristic
33#17988482C411yescharacteristic
34#17988482C411yescharacteristic
35#17988461C411yescharacteristic
36#17988461C411yescharacteristic
37#17988458C411yescharacteristic
38#17988458C411yescharacteristic
39C4×#11924007C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988342C2211yescharacteristic
43#17988342C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×C4×D73C2×C411yescharacteristic
62C22×C4×D73C2×C411yescharacteristic
63C22×C4×D73C2311yescharacteristic
64C2×#11924011—22no—
65C2×#11924011—22no—
66C2×#11924011—22no—
67C2×#11924011—22no—
68C2×#11924011—22no—
69C2×#11924011—22no—
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74C2×C8×D73—22no—
75C2×C8×D73—22no—
76C2×C8×D73—22no—
77C2×C8×D73—22no—
78C2×C8×D73D414yes—
79C2×C8×D73Q814yes—
80#14556548—22no—
81#14556548—22no—
82#14556548—22no—
83#14556548—22no—
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×C4⋊Dic73C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×#11924007C2×C411yescharacteristic
89C2×#11924007C2×C411yescharacteristic
90C2×#11924007C2×C411yescharacteristic
91C2×#11924007C2×C411yescharacteristic
92C2×#11924007C2×C411yescharacteristic
93C2×#11924007C2×C411yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×C4×Dic73C2×C411yescharacteristic
99C2×C4×Dic73C2×C411yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C42×D73—22no—
104C42×D73—22no—
105C4⋊#10596831—22no—
106#14556520—22no—
107#14556520—22no—
108#14556502—22no—
109#14556501—22no—
110#14556501—22no—
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113#14556498C2×C411yescharacteristic
114#14556498C2×C411yescharacteristic
115#14556498C2×C411yescharacteristic
116#14556498C2×C411yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2×C411yescharacteristic
122C2×C4⋊C4×C73C2×C411yescharacteristic
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C4⋊C8×C73—22no—
127#97×C73—22no—
128#95×C73C2×C411yescharacteristic
129#95×C73C2×C411yescharacteristic
130C23×D73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144C2×C4×D73C2×D412yes—
145C2×C4×D73C2×Q812yes—
146C2×C4×D73#4112yes—
147C2×C4×D73#4112yes—
148C2×C4×D73#4112yes—
149C2×C4×D73#4112yes—
150#11924011—44no—
151#11924011—44no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C2×D412yes—
163C2×#10596831C2×Q812yes—
164C8×D73—28no—
165C8×D73—28no—
166C4⋊Dic73—41no—
167C4⋊Dic73—41no—
168#11924007—22no—
169#11924007—22no—
170#11924007—22no—
171#11924007—22no—
172#11924007—41no—
173#11924007—41no—
174#11924007—41no—
175#11924007—41no—
176#11924005—22no—
177#11924005—22no—
178#11924005—22no—
179#11924005—22no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—22no—
187C4×Dic73—22no—
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C22×C411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C2×D412yes—
206C2×C8×C73C2×Q812yes—
207C4⋊C4×C73—41no—
208C4⋊C4×C73—41no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C23×C73C23×C411yescharacteristic
216C22×D73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73#11712yes—
221C22×D73C4×D412yes—
222C22×D73C4×Q812yes—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C22×Q811yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73#11712yes—
266C2×Dic73C4×D412yes—
267C2×Dic73C4×Q812yes—
268C4×D73—44no—
269C4×D73—44no—
270C4×D73C22⋊Q814yes—
271C4×D73#12314yes—
272C4×D73#12514yes—
273C4×D73#12614yes—
274C8×C73—22no—
275C8×C73—22no—
276C8×C73—41no—
277C8×C73—41no—
278#10596831—22no—
279#10596831—22no—
280#10596831—41no—
281#10596831—41no—
282C22×C73C2×#11711yescharacteristic
283C22×C73C2×C4×D411yescharacteristic
284C22×C73C2×C4×Q811yescharacteristic
285C22×C73#56811yescharacteristic
286C22×C73#56811yescharacteristic
287C22×C73#56811yescharacteristic
288C22×C73#56811yescharacteristic
289C2×D73—22no—
290C2×D73—22no—
291C2×D73—22no—
292C2×D73—22no—
293C2×D73—22no—
294C2×D73—22no—
295C2×D73#38714yes—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73C2×C22⋊Q811yescharacteristic
303C4×C73C2×#12311yescharacteristic
304C4×C73C2×#12511yescharacteristic
305C4×C73C2×#12611yescharacteristic
306Dic73—41no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312Dic73#38714yes—
313#21021—731nomaximal, Sylow
314C2×C73C2×#38711yescharacteristic
315C2×C73#292311yescharacteristic
316C2×C73#292311yescharacteristic
317C2×C73#293011yescharacteristic
318C2×C73#293011yescharacteristic
319C2×C73#296111yescharacteristic
320C2×C73#296111yescharacteristic
321D73—24no—
322C2×#424—731no—
323C22×#97—731no—
324#2296—731no—
325#1917—731no—
326#1917—731no—
327#1917—731no—
328#1917—731no—
329#1917—731no—
330#1917—731no—
331#1917—731no—
332#1917D7311yescharacteristic, complemented
333#1729—731no—
334#1729—731no—
335#1729—731no—
336#1728—731no—
337C73#2102111yescharacteristic, complemented, Sylow
338C22×M16—731no—
339C22×M16—731no—
340C23×C8—731no—
341C23×C8—731no—
342C2×#117—731no—
343C2×#117—731no—
344C22×C42—731no—
345#424—1464no—
346C2×C4⋊C8—731no—
347C2×C4⋊C8—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8C2×D7311yescharacteristic
350C2×#97—731no—
351C2×#97—731no—
352C2×#97—731no—
353C2×#97—1464no—
354C2×#97C2×D7311yescharacteristic
355#389—731no—
356#389—731no—
357#389—731no—
358#389—731no—
359#385—731no—
360#385—731no—
361#385—731no—
362#385—731no—
363C4×C4⋊C4—731no—
364C4×C4⋊C4—731no—
365C4×C4⋊C4—731no—
366C4×C4⋊C4C2×D7311yescharacteristic
367#336—731no—
368#336—731no—
369#336—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377#336—731no—
378#336—731no—
379#336C2×D7311yescharacteristic
380#336C2×D7311yescharacteristic
381#336C2×D7311yescharacteristic
382#336C2×D7311yescharacteristic
383C23×C4—731no—
384C23×C4—731no—
385C23×C4—731no—
386C2×M16—1462no—
387C2×M16—1462no—
388C2×M16—1462no—
389C2×M16—1462no—
390C2×M16—1462no—
391C2×M16—1462no—
392C22×C8—731no—
393C22×C8—731no—
394C22×C8—734no—
395C22×C8—734no—
396C22×C8—731no—
397C22×C8—731no—
398C22×C8—1462no—
399C22×C8—1462no—
400C22×C8—1462no—
401C22×C8—1462no—
402C22×C8C22×D7311yescharacteristic
403C22×C8C22×D7311yescharacteristic
404C22×C8C22×D7311yescharacteristic
405C22×C8C22×D7311yescharacteristic
406#117—1462no—
407#117—1462no—
408#117—1462no—
409#117—1462no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4C4×D7311yescharacteristic
421C2×C4⋊C4C4×D7311yescharacteristic
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C42—731no—
427C2×C42—731no—
428C2×C42—731no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—1462no—
432C2×C42—1462no—
433C2×C42C22×D7311yescharacteristic
434C2×C42C22×D7311yescharacteristic
435C2×C42C22×D7311yescharacteristic
436C4⋊C8—22no—
437C4⋊C8—1462no—
438C4⋊C8—1462no—
439C4⋊C8—1462no—
440#97—22no—
441#97—1462no—
442#97—1462no—
443#97—1462no—
444#95—731no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—731no—
449#95—731no—
450#95C4×D7311yescharacteristic
451#95C4×D7311yescharacteristic
452C24—731no—
453C22×C4—2921no—
454C22×C4—2921no—
455C22×C4—2921no—
456C22×C4—2921no—
457C22×C4—731no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—732no—
462C22×C4—732no—
463C22×C4—732no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4C2×C4×D7311yescharacteristic
473C22×C4C2×C4×D7311yescharacteristic
474C22×C4C2×C4×D7311yescharacteristic
475C22×C4C2×C4×D7311yescharacteristic
476C22×C4C2×C4×D7311yescharacteristic
477C22×C4C2×C4×D7311yescharacteristic
478C22×C4C23×D7311yescharacteristic
479M16—2924no—
480M16—2924no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—732no—
492C2×C8—732no—
493C2×C8—1468no—
494C2×C8—1468no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8D4×D7312yes—
506C2×C8Q8×D7312yes—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—2921no—
510C4⋊C4—2921no—
511C4⋊C4—2921no—
512C4⋊C4—2921no—
513C4⋊C4—2921no—
514C4⋊C4—2921no—
515C4⋊C4—1462no—
516C4⋊C4—1462no—
517C4⋊C4—1462no—
518C4⋊C4—1462no—
519C22⋊C4—1462no—
520C22⋊C4—1462no—
521C22⋊C4—1462no—
522C22⋊C4—1462no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—1462no—
530C42—1462no—
531C42—1461no—
532C42—1461no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C42—1461no—
537C23—732no—
538C23—732no—
539C23—732no—
540C23—1461no—
541C23—1461no—
542C23—1461no—
543C23—1461no—
544C23C22×C4×D7311yescharacteristic
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—2924no—
564C2×C4—2924no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—732no—
568C2×C4—734no—
569C2×C4—734no—
570C2×C4—734no—
571C2×C4—734no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4C2×Q8×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4#41×D7311yescharacteristic
594C2×C4#41×D7311yescharacteristic
595C2×C4#41×D7311yescharacteristic
596C8—22no—
597C8—22no—
598C8—41no—
599C8—41no—
600C8—2921no—
601C8—2921no—
602C8—1462no—
603C8—1462no—
604C22—734no—
605C22—1462no—
606C22—1462no—
607C22—1462no—
608C22—1462no—
609C22—1462no—
610C22—1462no—
611C22#117×D7311yescharacteristic
612C22C4×D4×D7311yescharacteristic
613C22C4×Q8×D7311yescharacteristic
614C22#1798935211yescharacteristic
615C22#1798935211yescharacteristic
616C22#1798935311yescharacteristic
617C22#1798935311yescharacteristic
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—2921no—
625C4—2921no—
626C4—2921no—
627C4—2921no—
628C4—2921no—
629C4—2921no—
630C4—734no—
631C4C22⋊Q8×D7311yescharacteristic
632C4#123×D7311yescharacteristic
633C4#125×D7311yescharacteristic
634C4#126×D7311yescharacteristic
635C2—1464no—
636C2#387×D7311yescharacteristic
637C2#2261857711yescharacteristic
638C2#2261857911yescharacteristic
639C2#2261862911yescharacteristic
640C2#2261863111yescharacteristic
641C2#2261886711yescharacteristic
642C2#2261886711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1917C2
prime radicalC23×D73C22×C4
cosocleC22×C4×C73C24
hypercenter#1917D73
hypocenterC73#21021
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
4. #22610498C2semidirect
5. #22610497C2semidirect
6. #22610496C2semidirect
7. #22610495C2semidirect
8. #22610494C2semidirect
9. #22610493C2semidirect
10. #22610492C2semidirect
15. C73×#1917C2semidirect
332. #1917D73semidirect
337. C73#21021semidirect

Maximal subgroups

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

subgroupindex
C2×#179886122
C2×#97×D732
#226124112
#226104982
#226104972
#226104962
#226104952
#226104942
#226104932
#226104922
#226096262
#226096232
C73×#19172
#2102173