Finite Group Databasebeta

← #33154039

Subgroups — #33154039

Order 18688.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614305C211yesmaximal, characteristic, complemented
2#22611248C211yesmaximal, characteristic, complemented
3#22611117C211yesmaximal, characteristic
4#22611067C211yesmaximal, characteristic, complemented
5#22610934C211yesmaximal, characteristic
6#22610879C211yesmaximal, characteristic, complemented
7#22610606C211yesmaximal, characteristic, complemented
8#22610565C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610506C211yesmaximal, characteristic, complemented
11#22610265C211yesmaximal, characteristic, complemented
12#22610238C211yesmaximal, characteristic, complemented
13#22610017C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2361C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
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#17988529—21no—
31#17988499—21no—
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988491—21no—
35#17988480—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×Dic146—21no—
60C22×Dic146C2311yescharacteristic
61C2×#11924016—22no—
62C2×#11924016—41no—
63C2×#11924016—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556591—22no—
69#14556591—41no—
70#14556570—22no—
71#14556570—41no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80#14556522—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—
133C2×Dic146—42no—
134C2×Dic146—42no—
135C2×Dic146—81no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140#11924016—44no—
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—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—42no—
159C4⋊Dic73#4112yes—
160#11924007—42no—
161C22⋊Dic73—42no—
162C22⋊Dic73—81no—
163C4×Dic73—81no—
164C24×C73—21no—
165C24×C73C2×D411yescharacteristic
166C2×D4×C73—24no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—42no—
173C2×D4×C73—42no—
174C2×D4×C73#4112yes—
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—21no—
183C22×C4×C73—21no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—42no—
187C22×C4×C73—42no—
188C22×C4×C73—81no—
189C22×C4×C73—81no—
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2411yescharacteristic
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73#4112yes—
203C4⋊C4×C73#4112yes—
204C22⋊C4×C73—24no—
205C22⋊C4×C73—41no—
206C22⋊C4×C73—42no—
207C22⋊C4×C73—42no—
208C22⋊C4×C73—81no—
209C22⋊C4×C73SD1614yescomplemented
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—42no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—22no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73C2×SD1612yes—
229C23×C73C22×D411yescharacteristic
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—48no—
233D4×C73—41no—
234D4×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
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—22no—
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—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—41no—
261C2×C4×C73—84no—
262C2×C4×C73—84no—
263C2×C4×C73C2×SD1612yes—
264C2×C4×C73C2×SD1612yes—
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×Dic73—22no—
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—
281C2×Dic73—41no—
282C2×Dic73—41no—
283Dic146—41no—
284Dic146—41no—
285Dic146—41no—
286Dic146—41no—
287Dic146—84no—
288#10596831—41no—
289#10596831—41no—
290#10596831—41no—
291#10596831—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—42no—
300C22×C73—42no—
301C22×C73—42no—
302C22×C73—81no—
303C22×C73—81no—
304C22×C73—81no—
305C22×C73—81no—
306C22×C73#52511yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73#54611yescharacteristic
309C22×C73C22×SD1611yescharacteristic
310C22×C73#57511yescharacteristic
311C22×C73#57611yescharacteristic
312C22×C73#57811yescharacteristic
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—42no—
316C4×C73—41no—
317C4×C73—42no—
318C4×C73—41no—
319C4×C73—41no—
320C4×C73—81no—
321C4×C73#52911yescharacteristic
322C4×C73#53811yescharacteristic
323C4×C73#55511yescharacteristic
324C4×C73#55911yescharacteristic
325Dic73—42no—
326Dic73—81no—
327Dic73—81no—
328Dic73—81no—
329#26722—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73—82no—
332C2×C73#245911yescharacteristic
333C2×C73#310111yescharacteristic
334C2×C73#312411yescharacteristic
335C2×C73#329511yescharacteristic
336C2×C73#329811yescharacteristic
337C2×C73#331311yescharacteristic
338C2×C73#332211yescharacteristic
339#2492—731no—
340#2361D7311yescharacteristic
341#2055—731no—
342#2034—731no—
343#2023—731no—
344#2000—731no—
345#1987—731no—
346#1933—731no—
347#1927—731no—
348#1923—731no—
349#1919—731no—
350#1877—731no—
351#1872—731no—
352#1823—731no—
353C4×#102—731no—
354C73#2672211yescharacteristic, complemented, Sylow
355C22×SD16—731no—
356C2×C4⋊Q8—731no—
357C2×C22⋊Q8—731no—
358C2×#121C2×D7311yescharacteristic
359C2×C4×D4—21no—
360C2×C4×D4C2×D7311yescharacteristic
361C22×C4⋊C4C2×D7311yescharacteristic
362C2×#106—731no—
363C2×C4⋊C8—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#102—731no—
368C2×#102—731no—
369C2×#102—731no—
370C2×C22⋊C8—731no—
371C2×C22⋊C8—731no—
372C2×C4×C8—731no—
373#400—1461no—
374#393—1461no—
375#391—731no—
376#391—731no—
377#391—1461no—
378#390C2×D7311yescharacteristic
379#389C2×D7311yescharacteristic
380#388—21no—
381#388—1461no—
382#386—21no—
383#385—21no—
384#380—731no—
385C4×C4⋊C4—731no—
386C4×C4⋊C4C2×D7311yescharacteristic
387C4×C22⋊C4C2×D7311yescharacteristic
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#336—731no—
397#336—731no—
398C22×Q8—731no—
399C22×Q8—1461no—
400C22×D4—21no—
401C22×D4C22×D7311yescharacteristic
402C23×C4—21no—
403C23×C4—21no—
404C23×C4C22×D7311yescharacteristic
405C2×SD16—2921no—
406C2×SD16—2921no—
407C2×SD16—1462no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C4⋊Q8—2921no—
413C4⋊Q8—1462no—
414C22⋊Q8—2921no—
415C22⋊Q8—1462no—
416#121—22no—
417#121—22no—
418C4×D4—22no—
419C4×D4—41no—
420C4×D4—44no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—41no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4C22⋊D7311yescharacteristic
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C42—21no—
442C2×C42—1461no—
443C2×C42C22×D7311yescharacteristic
444C2×C42C22×D7311yescharacteristic
445#106—1462no—
446C4⋊C8—2921no—
447#103—2921no—
448#103—1462no—
449#103—1462no—
450#102—2921no—
451#102—1462no—
452#102—1462no—
453C22⋊C8—2921no—
454C22⋊C8—2921no—
455C4×C8—2921no—
456#95—21no—
457#95—21no—
458#95—1461no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462#95—1461no—
463#95—1461no—
464#95C22⋊D7311yescharacteristic
465#95C22⋊D7311yescharacteristic
466C24—21no—
467C24C2×C22⋊D7311yescharacteristic
468C2×Q8—2922no—
469C2×Q8—2922no—
470C2×Q8—5841no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×D4—24no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4#1192402812yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
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—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4C2×C22⋊D7311yescharacteristic
509C22×C4C2×C22⋊D7311yescharacteristic
510C22×C4C23×D7311yescharacteristic
511SD16—2924no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—2922no—
531C4⋊C4—2922no—
532C4⋊C4—732no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4#1192402712yes—
539C4⋊C4#1192402812yes—
540C22⋊C4—24no—
541C22⋊C4—41no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—2922no—
546C22⋊C4—5841no—
547C22⋊C4#1192401614yescomplemented
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—42no—
553C42—5841no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C2×#1192401612yes—
568C23C22×C22⋊D7311yescharacteristic
569Q8—2921no—
570Q8—2921no—
571Q8—2921no—
572Q8—2921no—
573Q8—5844no—
574D4—22no—
575D4—22no—
576D4—48no—
577D4—41no—
578D4—41no—
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—22no—
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—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—1462no—
619C2×C4C2×#1192401612yes—
620C2×C4C2×#1192401612yes—
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402711yescharacteristic
623C2×C4C2×#1192402811yescharacteristic
624C2×C4C2×#1192402811yescharacteristic
625C2×C4#1455663611yescharacteristic
626C2×C4#1455663711yescharacteristic
627C8—2921no—
628C8—2921no—
629C8—2921no—
630C8—2921no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#1798909811yescharacteristic
646C22#1798922511yescharacteristic
647C22#1798923111yescharacteristic
648C22C22×#1192401611yescharacteristic
649C22#1798938511yescharacteristic
650C22#1798939011yescharacteristic
651C22#1798940211yescharacteristic
652C4—22no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—2922no—
661C4—5841no—
662C4—5841no—
663C4—5841no—
664C4#1798911711yescharacteristic
665C4#1798917011yescharacteristic
666C4#1798930211yescharacteristic
667C4#1798932011yescharacteristic
668C2—42no—
669C2—82no—
670C2#2261386211yescharacteristic
671C2#2261993911yescharacteristic
672C2#2262019311yescharacteristic
673C2#2262175511yescharacteristic
674C2#2262179011yescharacteristic
675C2#2262199711yescharacteristic
676C2#2262212611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2361C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2361D73
hypocenterC73#26722
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. #22614305C2semidirect
2. #22611248C2semidirect
4. #22611067C2semidirect
6. #22610879C2semidirect
7. #22610606C2semidirect
8. #22610565C2semidirect
9. #22610539C2semidirect
10. #22610506C2semidirect
11. #22610265C2semidirect
12. #22610238C2semidirect
13. #22610017C2semidirect
14. C4×#14556512C2semidirect
209. C22⋊C4×C73SD16semidirect
354. C73#26722semidirect
547. C22⋊C4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226143052
#226112482
#226111172
#226110672
#226109342
#226108792
#226106062
#226105652
#226105392
#226105062
#226102652
#226102382
#226100172
C4×#145565122
C73×#23612
#2672273