Finite Group Databasebeta

← #33088880

Subgroups — #33088880

Order 18688.

884 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988646C211yesmaximal, characteristic, complemented
2C2×#17988541C211yesmaximal, characteristic, complemented
3#22612395C211yesmaximal, characteristic
4#22610543C211yesmaximal, characteristic, complemented
5#22610536C211yesmaximal, characteristic, complemented
6#22610494C211yesmaximal, characteristic, complemented
7#22610487C211yesmaximal, characteristic, complemented
8#22610227C211yesmaximal, characteristic
9#22610222C211yesmaximal, characteristic
10#22610112C211yesmaximal, characteristic, complemented
11#22610107C211yesmaximal, characteristic, complemented
12#22609698C211yesmaximal, characteristic, complemented
13#22609698C211yesmaximal, characteristic, complemented
14#22609680C211yesmaximal, characteristic, complemented
15C4×C4⋊C8×C73C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×#14556548C2211yescharacteristic
20C2×#14556548C2211yescharacteristic
21#17988646—24no—
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×#14556503C2211yescharacteristic
30#17988541—24no—
31C2×#14556502C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×#14556501C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35#17988489C411yescharacteristic
36#17988489C411yescharacteristic
37#17988485C411yescharacteristic
38#17988485C411yescharacteristic
39#17988478C411yescharacteristic
40#17988478C411yescharacteristic
41#17988478C411yescharacteristic
42#17988478C411yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44C4×#11924005C2211yescharacteristic
45C4×#11924005C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C43×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×D292C2×C411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C22×C4×D73C2311yescharacteristic
64C2×#11924011—22no—
65C2×#11924011—22no—
66C2×#11924011—22no—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73—22no—
74C4×D292—22no—
75C4×D292—22no—
76#14556548—22no—
77#14556548—22no—
78#14556548—22no—
79#14556548—22no—
80C2×C4⋊Dic73C2×C411yescharacteristic
81C2×C4⋊Dic73C2×C411yescharacteristic
82C2×C4⋊Dic73C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924005C2×C411yescharacteristic
88C2×#11924005C2×C411yescharacteristic
89C2×#11924005C2×C411yescharacteristic
90C2×#11924005C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×#11924005C2×C411yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99#14556520—22no—
100#14556520—22no—
101#14556503—22no—
102#14556503—22no—
103#14556503—22no—
104#14556503—22no—
105#14556503—22no—
106#14556503—22no—
107#14556502—22no—
108#14556502D414yescomplemented
109#14556501—22no—
110C8×Dic73—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×C42×C73C2×C411yescharacteristic
122C2×C42×C73C2×C411yescharacteristic
123C2×C42×C73C2×C411yescharacteristic
124C2×C42×C73C2×C411yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C4⋊C8×C73—22no—
129C4⋊C8×C73—22no—
130C4×C8×C73—22no—
131C4×C8×C73—22no—
132C23×D73C22×C411yescharacteristic
133C23×D73C22×C411yescharacteristic
134C2×D292—22no—
135C2×D292—22no—
136C2×D292—41no—
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C411yescharacteristic
143C2×C4×D73—21no—
144C2×C4×D73—21no—
145C2×C4×D73—21no—
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73#4112yes—
154C2×C4×D73#4112yes—
155#11924011—28no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831C2×D412yes—
167C2×#10596831C2×D412yes—
168C8×D73—28no—
169C4⋊Dic73—22no—
170C4⋊Dic73—22no—
171C4⋊Dic73—41no—
172#11924007—22no—
173#11924007—22no—
174#11924007—22no—
175#11924007—22no—
176#11924005—22no—
177#11924005—22no—
178#11924005—22no—
179#11924005—22no—
180#11924005—22no—
181#11924005—22no—
182#11924005—22no—
183#11924005—22no—
184#11924005—41no—
185#11924005—41no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C22×C411yescharacteristic
194C22×C4×C73C22×C411yescharacteristic
195C22×C4×C73C22×C411yescharacteristic
196C22×C4×C73C2411yescharacteristic
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×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73#4112yes—
208C2×C8×C73#4112yes—
209C42×C73—22no—
210C42×C73—22no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—41no—
216C42×C73—41no—
217C42×C73C2×D412yes—
218C42×C73#4112yes—
219C23×C73C23×C411yescharacteristic
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—21no—
229C22×D73—21no—
230C22×D73—21no—
231C22×D73—21no—
232C22×D73—21no—
233C22×D73#11712yes—
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×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73C4×D412yes—
249C2×C4×C73C4×D412yes—
250C2×C4×C73#13112yes—
251C2×C4×C73#13112yes—
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73#11712yes—
276D292—44no—
277C4×D73—22no—
278C4×D73—22no—
279C4×D73—22no—
280C4×D73—22no—
281C4×D73—41no—
282C4×D73—41no—
283C4×D73—41no—
284C4×D73—41no—
285C8×C73—22no—
286C8×C73—22no—
287C8×C73—41no—
288C8×C73—41no—
289#10596831—22no—
290#10596831—22no—
291#10596831—41no—
292#10596831—41no—
293C22×C73C2×#11711yescharacteristic
294C22×C73C2×C4×D411yescharacteristic
295C22×C73C4×#4111yescharacteristic
296C22×C73C2×#13111yescharacteristic
297C22×C73C2×#13111yescharacteristic
298C22×C73#56811yescharacteristic
299C22×C73#56811yescharacteristic
300C2×D73—22no—
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—41no—
304C2×D73—41no—
305C2×D73—41no—
306C2×D73—41no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—41no—
310C2×D73—41no—
311C2×D73—41no—
312C2×D73—41no—
313C2×D73—41no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—22no—
317C4×C73—22no—
318C4×C73—41no—
319C4×C73—41no—
320C4×C73#52511yescharacteristic
321C4×C73#52511yescharacteristic
322C4×C73#52911yescharacteristic
323C4×C73#52911yescharacteristic
324Dic73—22no—
325Dic73—41no—
326Dic73—41no—
327Dic73—41no—
328Dic73—41no—
329Dic73—41no—
330#21111—731nomaximal, Sylow
331C2×C73#229011yescharacteristic
332C2×C73#292411yescharacteristic
333C2×C73#292511yescharacteristic
334C2×C73#293511yescharacteristic
335C2×C73#293511yescharacteristic
336C2×C73#296611yescharacteristic
337C2×C73#296711yescharacteristic
338D73—42no—
339D73—81no—
340C2×#433—731no—
341C2×#405—731no—
342#2293—731no—
343#1924—731no—
344#1923—731no—
345#1917—731no—
346#1916—731no—
347#1870—731no—
348#1869—731no—
349#1851—731no—
350#1850—731no—
351C4×C4⋊C8D7311yescharacteristic, complemented
352#1750—731no—
353#1750—731no—
354#1745—731no—
355C73#2111111yescharacteristic, complemented, Sylow
356C22×M16—731no—
357C23×C8—731no—
358C2×C4×D4—731no—
359C2×#117—731no—
360C2×#117—731no—
361#433—1464no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8C2×D7311yescharacteristic
365C2×C4⋊C8C2×D7311yescharacteristic
366C2×C22⋊C8—731no—
367C2×C22⋊C8—731no—
368C2×C22⋊C8—731no—
369C2×C22⋊C8—731no—
370C2×C22⋊C8—731no—
371C2×C22⋊C8—731no—
372#405—1464no—
373C2×#97—731no—
374C2×#97—731no—
375C2×#97—731no—
376C2×C4×C8—731no—
377C2×C4×C8C2×D7311yescharacteristic
378C2×C4×C8C2×D7311yescharacteristic
379#390—731no—
380#390—731no—
381#389—731no—
382#389—731no—
383#388—731no—
384#388—731no—
385#388—731no—
386#388—731no—
387C4×C4⋊C4—731no—
388C4×C22⋊C4—731no—
389C4×C22⋊C4—731no—
390C43C2×D7311yescharacteristic
391#336—731no—
392#336—731no—
393#336—731no—
394#336—731no—
395#336—731no—
396#336—731no—
397#336—731no—
398#336—731no—
399#336C2×D7311yescharacteristic
400#336C2×D7311yescharacteristic
401C22×D4—731no—
402C23×C4—731no—
403C23×C4—731no—
404C2×M16—1462no—
405C2×M16—1462no—
406C2×M16—1462no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—1462no—
412C22×C8—1462no—
413C22×C8—1462no—
414C22×C8C22×D7311yescharacteristic
415C22×C8C22×D7311yescharacteristic
416C22×C8C22×D7311yescharacteristic
417C22×C8C22×D7311yescharacteristic
418C4×D4—1462no—
419C4×D4—1462no—
420#117—1462no—
421#117—1462no—
422#117—1462no—
423#117—1462no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C22⋊C4—731no—
432C2×C22⋊C4—731no—
433C2×C22⋊C4—731no—
434C2×C22⋊C4—731no—
435C2×C22⋊C4—731no—
436C2×C22⋊C4—731no—
437C2×C22⋊C4—731no—
438C2×C22⋊C4—731no—
439C2×C22⋊C4—731no—
440C2×C22⋊C4—731no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42C4×D7311yescharacteristic
444C2×C42C4×D7311yescharacteristic
445C2×C42C4×D7311yescharacteristic
446C2×C42C4×D7311yescharacteristic
447C2×C42C22×D7311yescharacteristic
448C2×C42C22×D7311yescharacteristic
449C2×C42C22×D7311yescharacteristic
450C4⋊C8—22no—
451C4⋊C8—22no—
452C4⋊C8—1462no—
453C4⋊C8—1462no—
454C22⋊C8—1462no—
455C22⋊C8—1462no—
456C22⋊C8—1462no—
457C22⋊C8—1462no—
458C22⋊C8—1462no—
459C22⋊C8—1462no—
460#97—734no—
461#97—1462no—
462#97—1462no—
463C4×C8—22no—
464C4×C8—22no—
465C4×C8—1462no—
466#95—731no—
467#95—731no—
468#95—731no—
469#95—731no—
470#95—731no—
471#95—731no—
472C24—731no—
473C24—731no—
474C2×D4—2921no—
475C2×D4—1462no—
476C2×D4—1462no—
477C22×C4—731no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4C2×C4×D7311yescharacteristic
496C22×C4C2×C4×D7311yescharacteristic
497C22×C4C2×C4×D7311yescharacteristic
498C22×C4C2×C4×D7311yescharacteristic
499C22×C4C2×C4×D7311yescharacteristic
500C22×C4C2×C4×D7311yescharacteristic
501C22×C4C23×D7311yescharacteristic
502M16—1468no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—732no—
514C2×C8—732no—
515C2×C8—1468no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8#1192402812yes—
527C2×C8#1192402812yes—
528C4⋊C4—2921no—
529C4⋊C4—1462no—
530C4⋊C4—1462no—
531C4⋊C4—1462no—
532C4⋊C4—1462no—
533C4⋊C4—1462no—
534C4⋊C4—1462no—
535C22⋊C4—2921no—
536C22⋊C4—2921no—
537C22⋊C4—1462no—
538C22⋊C4—1462no—
539C22⋊C4—1462no—
540C22⋊C4—1462no—
541C22⋊C4—1462no—
542C22⋊C4—1462no—
543C22⋊C4—1462no—
544C22⋊C4—1462no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—41no—
552C42—41no—
553C42—1461no—
554C42—1461no—
555C42—1461no—
556C42—1461no—
557C42D4×D7312yes—
558C42#1192402912yes—
559C23—732no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23—1461no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23—1461no—
568C23—1461no—
569C23—1461no—
570C23—1461no—
571C23—1461no—
572C23—1461no—
573C23C22×C4×D7311yescharacteristic
574D4—2924no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—732no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4#1455655312yes—
616C2×C4#1455655312yes—
617C2×C4#1455659712yes—
618C2×C4#1455659712yes—
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402811yescharacteristic
623C2×C4C2×#1192402811yescharacteristic
624C2×C4C2×#1192402911yescharacteristic
625C8—22no—
626C8—22no—
627C8—41no—
628C8—41no—
629C8—2921no—
630C8—2921no—
631C8—1462no—
632C8—1462no—
633C22—2921no—
634C22—2921no—
635C22—2921no—
636C22—2921no—
637C22—2921no—
638C22—2921no—
639C22—2921no—
640C22—2921no—
641C22—2921no—
642C22—2921no—
643C22—2921no—
644C22—1462no—
645C22—1462no—
646C22—1462no—
647C22#117×D7311yescharacteristic
648C22C2×#1455655311yescharacteristic
649C22C4×#1192402811yescharacteristic
650C22C2×#1455659711yescharacteristic
651C22C2×#1455659711yescharacteristic
652C22#1798935311yescharacteristic
653C22#1798935311yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—2921no—
661C4—2921no—
662C4—2921no—
663C4—2921no—
664C4—2921no—
665C4—1462no—
666C4#1798909611yescharacteristic
667C4#1798909611yescharacteristic
668C4#1798912411yescharacteristic
669C4#1798912411yescharacteristic
670C2—2922no—
671C2—5841no—
672C2#2261237211yescharacteristic
673C2#2261858411yescharacteristic
674C2#2261859211yescharacteristic
675C2#2261867411yescharacteristic
676C2#2261867411yescharacteristic
677C2#2261890611yescharacteristic
678C2#2261891611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC4×C4⋊C8×C73C2
prime radicalC22×D292C2×C4
cosocleC22×C4×C73C24
hypercenterC4×C4⋊C8D73
hypocenterC73#21111
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#17988646C2semidirect
2. C2×#17988541C2semidirect
4. #22610543C2semidirect
5. #22610536C2semidirect
6. #22610494C2semidirect
7. #22610487C2semidirect
10. #22610112C2semidirect
11. #22610107C2semidirect
12. #22609698C2semidirect
13. #22609698C2semidirect
14. #22609680C2semidirect
15. C4×C4⋊C8×C73C2semidirect
108. #14556502D4semidirect
351. C4×C4⋊C8D73semidirect
355. C73#21111semidirect

Maximal subgroups

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

subgroupindex
C2×#179886462
C2×#179885412
#226123952
#226105432
#226105362
#226104942
#226104872
#226102272
#226102222
#226101122
#226101072
#226096982
#226096802
C4×C4⋊C8×C732
#2111173