Finite Group Databasebeta

← #33108447

Subgroups — #33108447

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988792C211yesmaximal, characteristic
2C2×#17988687C211yesmaximal, characteristic
3C2×C4×#11924016C211yesmaximal, characteristic
4#22612987C211yesmaximal, characteristic, complemented
5#22610750C211yesmaximal, characteristic, complemented
6#22610697C211yesmaximal, characteristic, complemented
7#22610620C211yesmaximal, characteristic, complemented
8#22610606C211yesmaximal, characteristic, complemented
9#22610265C211yesmaximal, characteristic
10#22610265C211yesmaximal, characteristic
11#22610198C211yesmaximal, characteristic, complemented
12#22610184C211yesmaximal, characteristic, complemented
13#22609806C211yesmaximal, characteristic, complemented
14#22609756C211yesmaximal, characteristic
15C73×#2349C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556591C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988792C418yescomplemented
22#17988687—24no—
23C4×#11924016—24no—
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×#14556512C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556502C2211yescharacteristic
37C2×C4×#10596831C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988493C2211yescharacteristic
40#17988483—21no—
41#17988470—21no—
42#17988470—21no—
43C4⋊C4×Dic73—21no—
44C4×C4⋊Dic73C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988342C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#385×C73—21no—
58#382×C73C2211yescharacteristic
59#380×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C2×#95×C73—21no—
62C22×Dic146C2311yescharacteristic
63C22×Dic146C2311yescharacteristic
64C2×#11924016—22no—
65C2×#11924016—22no—
66C2×#11924016—22no—
67C2×#11924016—22no—
68C2×#11924016C2×C414yes—
69C2×#11924016C2×C414yes—
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74#14556591—22no—
75#14556591C2×C414yescomplemented
76C4×Dic146—22no—
77C4×Dic146—22no—
78C4×Dic146—41no—
79C4×Dic146—41no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556524—22no—
90#14556522—22no—
91C4⋊#10596831C2×C414yescomplemented
92#14556514—22no—
93#14556514—22no—
94#14556514—41no—
95#14556514—41no—
96#14556514C2×C414yescomplemented
97#14556514C2×C414yescomplemented
98#14556512—22no—
99#14556512D414yes—
100#14556502—22no—
101#14556502—22no—
102C4×#10596831—22no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×D4×C73C2311yescharacteristic
109C23×C4×C73—21no—
110C23×C4×C73—21no—
111C23×C4×C73—21no—
112C4×D4×C73—22no—
113C4×D4×C73—22no—
114C4×D4×C73—22no—
115C4×D4×C73—41no—
116C4×D4×C73—41no—
117C4×D4×C73C2×C414yes—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73D411yescharacteristic
121C2×C4⋊C4×C73D411yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C22⋊C4×C73—21no—
126C2×C22⋊C4×C73—21no—
127C2×C22⋊C4×C73—21no—
128C2×C42×C73C2311yescharacteristic
129C2×C42×C73C2311yescharacteristic
130C2×C42×C73C2311yescharacteristic
131#95×C73—21no—
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135#95×C73D411yescharacteristic
136#95×C73D411yescharacteristic
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146—21no—
147C2×Dic146C22×C412yes—
148C2×Dic146C22×C412yes—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153#11924016—28no—
154#11924016—28no—
155C2×#10596831—21no—
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×#10596831C22×C412yes—
165C2×#10596831C22×C412yes—
166C2×#10596831C2×D412yes—
167C4⋊Dic73—24no—
168C4⋊Dic73—24no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73C2×D412yes—
173#11924007—42no—
174#11924007—42no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C24×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73C22×C412yes—
183C2×D4×C73C2×D412yes—
184C2×D4×C73#4112yes—
185C22×C4×C73—21no—
186C22×C4×C73—24no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—42no—
190C22×C4×C73—42no—
191C22×C4×C73—42no—
192C22×C4×C73—42no—
193C22×C4×C73—81no—
194C22×C4×C73—81no—
195C22×C4×C73—81no—
196C22×C4×C73—81no—
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2411yescharacteristic
204C4⋊C4×C73—22no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—41no—
211C4⋊C4×C73—81no—
212C4⋊C4×C73—81no—
213C4⋊C4×C73C22×C412yes—
214C22⋊C4×C73—24no—
215C22⋊C4×C73—42no—
216C22⋊C4×C73—42no—
217C42×C73—21no—
218C42×C73—21no—
219C42×C73—21no—
220C42×C73—21no—
221C42×C73C22×C412yes—
222C42×C73#4112yes—
223C23×C73—22no—
224C23×C73—22no—
225C23×C73—22no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73—41no—
230C23×C73C22×D411yescharacteristic
231D4×C73—22no—
232D4×C73—22no—
233D4×C73—22no—
234D4×C73C4×D414yes—
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—22no—
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×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—84no—
260C2×C4×C73—84no—
261C2×C4×C73—84no—
262C2×C4×C73C4×D412yes—
263C2×C4×C73C4×D412yes—
264C2×C4×C73#13712yes—
265C2×C4×C73C23×C411yescharacteristic
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×C4×C732−1+411yescharacteristic
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283Dic146—22no—
284Dic146—22no—
285Dic146—22no—
286Dic146—22no—
287Dic146—41no—
288Dic146—41no—
289Dic146—41no—
290Dic146—41no—
291#10596831—22no—
292#10596831—22no—
293#10596831—22no—
294#10596831—41no—
295C22×C73—24no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—42no—
299C22×C73—42no—
300C22×C73—42no—
301C22×C73—42no—
302C22×C73C2×C4×D411yescharacteristic
303C22×C73D4211yescharacteristic
304C22×C73#54811yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57511yescharacteristic
307C22×C73#57711yescharacteristic
308C22×C73#57711yescharacteristic
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—42no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—81no—
317C4×C73—81no—
318C4×C73C2×C4×D411yescharacteristic
319C4×C73#52011yescharacteristic
320C4×C73#54711yescharacteristic
321C4×C73#55311yescharacteristic
322Dic73—42no—
323Dic73—42no—
324Dic73—42no—
325Dic73—81no—
326#23012—731nomaximal, Sylow
327C2×C73—44no—
328C2×C73#236111yescharacteristic
329C2×C73C4×#13711yescharacteristic
330C2×C73#294911yescharacteristic
331C2×C73#326611yescharacteristic
332C2×C73#326711yescharacteristic
333C2×C73#337211yescharacteristic
334C2×C73#337611yescharacteristic
335C2×#461—731no—
336C2×#440—731no—
337C2×C4×SD16—731no—
338#2365—731no—
339#2349D7311yescharacteristic
340#1960—731no—
341#1949—731no—
342#1935—731no—
343#1933—731no—
344#1877—731no—
345#1877—731no—
346#1865—731no—
347#1863—731no—
348#1771—731no—
349#1762—731no—
350C73#2301211yescharacteristic, complemented, Sylow
351C22×SD16—731no—
352C22×SD16—731no—
353C2×C4⋊Q8—731no—
354C2×C4×Q8—731no—
355C2×C4×Q8—731no—
356C2×C4×D4C2×D7311yescharacteristic
357C2×C4×D4C2×D7311yescharacteristic
358C2×C4×D4C2×D7311yescharacteristic
359#461—738no—
360#440—1464no—
361C4×SD16—1464no—
362C2×#107—731no—
363C2×#106—731no—
364C2×C4⋊C8—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#102—731no—
372C2×#102—731no—
373C2×#97—731no—
374C2×#97—731no—
375C2×C4×C8—731no—
376#391—731no—
377#391—731no—
378#389—1461no—
379#389C2×D7311yescharacteristic
380#389C2×D7311yescharacteristic
381#387—1461no—
382#387—1461no—
383#385—21no—
384#385—21no—
385#382C2×D7311yescharacteristic
386#380—21no—
387C4×C4⋊C4—731no—
388C4×C4⋊C4—1461no—
389C4×C4⋊C4C2×D7311yescharacteristic
390C2×#95—21no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—731no—
398C22×Q8—731no—
399C22×D4C22×D7311yescharacteristic
400C23×C4—21no—
401C23×C4—21no—
402C23×C4—21no—
403C2×SD16—734no—
404C2×SD16—734no—
405C2×SD16—1462no—
406C2×SD16—1462no—
407C2×SD16—1462no—
408C2×SD16—1462no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—731no—
413C4⋊Q8—734no—
414C4⋊Q8—1462no—
415C4×Q8—2921no—
416C4×Q8—2921no—
417C4×Q8—1462no—
418C4×Q8—1462no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—41no—
423C4×D4—41no—
424C4×D4C4×D7314yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4C22⋊D7311yescharacteristic
434C2×C4⋊C4C22⋊D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C42—1461no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446C2×C42C22×D7311yescharacteristic
447#107—1462no—
448#106—1462no—
449C4⋊C8—734no—
450#103—2921no—
451#103—2921no—
452#103—734no—
453#103—734no—
454#103—1462no—
455#103—1462no—
456#102—734no—
457#102—1462no—
458#97—1462no—
459#97—1462no—
460C4×C8—1462no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95C22⋊D7311yescharacteristic
471#95C22⋊D7311yescharacteristic
472C24—21no—
473C2×Q8—732no—
474C2×Q8—732no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×Q8—1461no—
484C2×Q8—1461no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D7312yes—
489C2×D4D4×D7312yes—
490C2×D4#1192402912yes—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4C2×C22⋊D7311yescharacteristic
509C22×C4C2×C22⋊D7311yescharacteristic
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C2×C22⋊D7311yescharacteristic
513C22×C4C23×D7311yescharacteristic
514SD16—1468no—
515SD16—1468no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—732no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C2×C8—1461no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—41no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—2922no—
538C4⋊C4—2922no—
539C4⋊C4—732no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1464no—
543C4⋊C4—1464no—
544C4⋊C4—1461no—
545C4⋊C4C2×C4×D7312yes—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—2922no—
554C42—2922no—
555C42—2922no—
556C42C2×C4×D7312yes—
557C42#1192402812yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C22×C22⋊D7311yescharacteristic
566Q8—2921no—
567Q8—2921no—
568Q8—2921no—
569Q8—2921no—
570Q8—1462no—
571Q8—1462no—
572Q8—1462no—
573Q8—1462no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4#1455655314yes—
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—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4C4×C22⋊D7312yes—
618C2×C4C4×C22⋊D7312yes—
619C2×C4#1455662112yes—
620C2×C4C22×C4×D7311yescharacteristic
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4C2×#1192402811yescharacteristic
623C2×C4C2×#1192402911yescharacteristic
624C2×C4#1455663611yescharacteristic
625C2×C4#1455663911yescharacteristic
626C8—2921no—
627C8—1462no—
628C8—1462no—
629C8—1462no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×C4×C22⋊D7311yescharacteristic
638C22D4×C22⋊D7311yescharacteristic
639C22#1798924811yescharacteristic
640C22C2×#1455662111yescharacteristic
641C22#1798938511yescharacteristic
642C22#1798939811yescharacteristic
643C22#1798939811yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—2922no—
654C4—2922no—
655C4—2922no—
656C4—5841no—
657C4C2×#1455655311yescharacteristic
658C4#1798906811yescharacteristic
659C4#1798923811yescharacteristic
660C4#1798928511yescharacteristic
661C2—44no—
662C2#2261294711yescharacteristic
663C2C4×#1455662111yescharacteristic
664C2#2261878811yescharacteristic
665C2#2262143711yescharacteristic
666C2#2262145211yescharacteristic
667C2#2262271211yescharacteristic
668C2#2262275811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2349C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#2349D73
hypocenterC73#23012
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
4. #22612987C2semidirect
5. #22610750C2semidirect
6. #22610697C2semidirect
7. #22610620C2semidirect
8. #22610606C2semidirect
11. #22610198C2semidirect
12. #22610184C2semidirect
13. #22609806C2semidirect
21. #17988792C4semidirect
75. #14556591C2×C4semidirect
91. C4⋊#10596831C2×C4semidirect
96. #14556514C2×C4semidirect
97. #14556514C2×C4semidirect
350. C73#23012semidirect

Maximal subgroups

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

subgroupindex
C2×#179887922
C2×#179886872
C2×C4×#119240162
#226129872
#226107502
#226106972
#226106202
#226106062
#226102652
#226101982
#226101842
#226098062
#226097562
C73×#23492
#2301273