Finite Group Databasebeta

← #33154024

Subgroups — #33154024

Order 18688.

832 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614272C211yesmaximal, characteristic, complemented
2#22611248C211yesmaximal, characteristic, complemented
3#22611110C211yesmaximal, characteristic
4#22611077C211yesmaximal, characteristic, complemented
5#22610934C211yesmaximal, characteristic
6#22610888C211yesmaximal, characteristic, complemented
7#22610606C211yesmaximal, characteristic, complemented
8#22610572C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610499C211yesmaximal, characteristic, complemented
11#22610265C211yesmaximal, characteristic, complemented
12#22610238C211yesmaximal, characteristic, complemented
13#22610021C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2361C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556514C2211yescharacteristic
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988528—21no—
31#17988515—21no—
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988475—21no—
35#17988451—21no—
36#17988447C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#121×C73C2211yescharacteristic
49C2×C4×D4×C73—21no—
50C2×C4×D4×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#386×C73—21no—
56#385×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924016—22no—
61C2×#11924016—41no—
62C2×#11924016—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556583—22no—
68#14556583—41no—
69#14556570—22no—
70#14556570—41no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—22no—
81C4⋊#10596831—41no—
82#14556514—22no—
83#14556514—22no—
84#14556514—41no—
85#14556512—22no—
86#14556512—22no—
87#14556512—41no—
88C22⋊#10596831—41no—
89C22⋊#10596831—41no—
90C4×#10596831—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×D4×C73—21no—
98C22×D4×C73C2311yescharacteristic
99C23×C4×C73—21no—
100C23×C4×C73—21no—
101C23×C4×C73C2311yescharacteristic
102#121×C73—22no—
103#121×C73—22no—
104C4×D4×C73—22no—
105C4×D4×C73—41no—
106C4×D4×C73—44no—
107C2×C4⋊C4×C73—22no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—41no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137#11924016—44no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73#4112yes—
156#11924007—42no—
157#11924007—42no—
158#11924007—42no—
159C22⋊Dic73—42no—
160C22⋊Dic73—42no—
161C4×Dic73—42no—
162C24×C73—21no—
163C24×C73C2×D411yescharacteristic
164C2×D4×C73—24no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—42no—
171C2×D4×C73—42no—
172C2×D4×C73#4112yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—81no—
187C22×C4×C73—81no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2411yescharacteristic
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—42no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73#4112yes—
201C4⋊C4×C73#4112yes—
202C22⋊C4×C73—24no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—42no—
206C22⋊C4×C73—42no—
207C22⋊C4×C73—81no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—42no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73#13512yes—
227C23×C73C22×D411yescharacteristic
228D4×C73—22no—
229D4×C73—22no—
230D4×C73—48no—
231D4×C73—41no—
232D4×C73—41no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
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×C73—21no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
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—41no—
260C2×C4×C73—84no—
261C2×C4×C73—84no—
262C2×C4×C73AGL(1,ℤ/8)12yes—
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73H(5,𝔽2)11yescharacteristic
268C2×C4×C73H(5,𝔽2)11yescharacteristic
269C2×Dic73—22no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281Dic146—41no—
282Dic146—41no—
283Dic146—41no—
284Dic146—41no—
285#10596831—41no—
286#10596831—41no—
287#10596831—41no—
288#10596831—41no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—42no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73—81no—
302C22×C73—81no—
303C22×C73#52511yescharacteristic
304C22×C73#54611yescharacteristic
305C22×C73#54611yescharacteristic
306C22×C73C2×#13511yescharacteristic
307C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C73#57711yescharacteristic
309C22×C73#57811yescharacteristic
310C4×C73—22no—
311C4×C73—41no—
312C4×C73—42no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—42no—
317C4×C73—81no—
318C4×C73#52911yescharacteristic
319C4×C73#53811yescharacteristic
320C4×C73#55511yescharacteristic
321C4×C73#55911yescharacteristic
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326#26721—731nomaximal, Sylow
327C2×C73—42no—
328C2×C73—82no—
329C2×C73#245911yescharacteristic
330C2×C73#310511yescharacteristic
331C2×C73#312011yescharacteristic
332C2×C73#330011yescharacteristic
333C2×C73#330111yescharacteristic
334C2×C73#331011yescharacteristic
335C2×C73#333011yescharacteristic
336#2489—731no—
337#2361D7311yescharacteristic
338#2055—731no—
339#2033—731no—
340#2025—731no—
341#2000—731no—
342#1989—731no—
343#1933—731no—
344#1928—731no—
345#1923—731no—
346#1918—731no—
347#1877—731no—
348#1872—731no—
349#1824—731no—
350C4×#102—731no—
351C73#2672111yescharacteristic, complemented, Sylow
352C22×SD16—731no—
353C2×#125—731no—
354C2×C22⋊Q8—731no—
355C2×#121C2×D7311yescharacteristic
356C2×C4×D4—21no—
357C2×C4×D4C2×D7311yescharacteristic
358C22×C4⋊C4C2×D7311yescharacteristic
359C2×#107—731no—
360C2×C4⋊C8—731no—
361C2×#103—731no—
362C2×#103—731no—
363C2×#103—731no—
364C2×#102—731no—
365C2×#102—731no—
366C2×#102—731no—
367C2×C22⋊C8—731no—
368C2×C22⋊C8—731no—
369C2×C4×C8—731no—
370#400—1461no—
371#397—1461no—
372#391—731no—
373#391—731no—
374#390C2×D7311yescharacteristic
375#389C2×D7311yescharacteristic
376#388—21no—
377#388—1461no—
378#386—21no—
379#385—21no—
380#382—1461no—
381#380—731no—
382C4×C4⋊C4—731no—
383C4×C4⋊C4C2×D7311yescharacteristic
384C4×C22⋊C4C2×D7311yescharacteristic
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#336—731no—
394#336—731no—
395C22×Q8—731no—
396C22×D4—21no—
397C22×D4C22×D7311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4C22×D7311yescharacteristic
401C2×SD16—2921no—
402C2×SD16—2921no—
403C2×SD16—1462no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408#125—2921no—
409#125—1462no—
410C22⋊Q8—2921no—
411C22⋊Q8—1462no—
412#121—22no—
413#121—22no—
414C4×D4—22no—
415C4×D4—41no—
416C4×D4—44no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1461no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4C22⋊D7311yescharacteristic
437C2×C22⋊C4C22⋊D7311yescharacteristic
438C2×C42—21no—
439C2×C42—1461no—
440C2×C42C22×D7311yescharacteristic
441C2×C42C22×D7311yescharacteristic
442#107—1462no—
443C4⋊C8—2921no—
444#103—2921no—
445#103—1462no—
446#103—1462no—
447#102—2921no—
448#102—1462no—
449#102—1462no—
450C22⋊C8—2921no—
451C22⋊C8—2921no—
452C4×C8—2921no—
453#95—21no—
454#95—21no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95—1461no—
461#95C22⋊D7311yescharacteristic
462#95C22⋊D7311yescharacteristic
463C24—21no—
464C24C2×C22⋊D7311yescharacteristic
465C2×Q8—1461no—
466C2×Q8—1461no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4#1192402812yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C2×C22⋊D7311yescharacteristic
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C23×D7311yescharacteristic
505SD16—2924no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—2922no—
527C4⋊C4—732no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4#1192402712yes—
534C4⋊C4#1192402812yes—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—2922no—
542C22⋊C4—2922no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—2922no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1455661112yes—
563C23C22×C22⋊D7311yescharacteristic
564Q8—2921no—
565Q8—2921no—
566Q8—2921no—
567Q8—2921no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
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—1462no—
614C2×C4#1455661712yes—
615C2×C4#41×D7311yescharacteristic
616C2×C4C2×#1192402711yescharacteristic
617C2×C4C2×#1192402811yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C2×C4#1455663611yescharacteristic
620C2×C4#1455663711yescharacteristic
621C8—2921no—
622C8—2921no—
623C8—2921no—
624C8—2921no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22#1798909811yescharacteristic
640C22#1798922511yescharacteristic
641C22#1798923111yescharacteristic
642C22C2×#1455661111yescharacteristic
643C22C2×#1455661711yescharacteristic
644C22#1798939811yescharacteristic
645C22#1798940211yescharacteristic
646C4—22no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—2922no—
655C4—5841no—
656C4—5841no—
657C4—5841no—
658C4#1798911711yescharacteristic
659C4#1798917011yescharacteristic
660C4#1798930211yescharacteristic
661C4#1798932011yescharacteristic
662C2—42no—
663C2—82no—
664C2#2261386211yescharacteristic
665C2#2261997811yescharacteristic
666C2#2262014511yescharacteristic
667C2#2262181811yescharacteristic
668C2#2262182711yescharacteristic
669C2#2262195211yescharacteristic
670C2#2262223011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2361C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2361D73
hypocenterC73#26721
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #22614272C2semidirect
2. #22611248C2semidirect
4. #22611077C2semidirect
6. #22610888C2semidirect
7. #22610606C2semidirect
8. #22610572C2semidirect
9. #22610539C2semidirect
10. #22610499C2semidirect
11. #22610265C2semidirect
12. #22610238C2semidirect
13. #22610021C2semidirect
14. C4×#14556512C2semidirect
351. C73#26721semidirect

Maximal subgroups

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

subgroupindex
#226142722
#226112482
#226111102
#226110772
#226109342
#226108882
#226106062
#226105722
#226105392
#226104992
#226102652
#226102382
#226100212
C4×#145565122
C73×#23612
#2672173