Finite Group Databasebeta

← #33154069

Subgroups — #33154069

Order 18688.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614294C211yesmaximal, characteristic
2#22611246C211yesmaximal, characteristic, complemented
3#22611124C211yesmaximal, characteristic, complemented
4#22611072C211yesmaximal, characteristic, complemented
5#22610932C211yesmaximal, characteristic
6#22610883C211yesmaximal, characteristic, complemented
7#22610613C211yesmaximal, characteristic, complemented
8#22610558C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610513C211yesmaximal, characteristic, complemented
11#22610286C211yesmaximal, characteristic, complemented
12#22610248C211yesmaximal, characteristic
13#22610021C211yesmaximal, characteristic, complemented
14C4×#14556515C211yesmaximal, characteristic, complemented
15C73×#2360C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556577—21no—
19C2×#14556565C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988503—21no—
32#17988500—21no—
33#17988489C2211yescharacteristic
34#17988489C2211yescharacteristic
35#17988479—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×C22⋊Q8×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C22×C4⋊C4×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73—21no—
53#389×C73C2211yescharacteristic
54#387×C73—21no—
55#385×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73—21no—
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×D292C2311yescharacteristic
61C22×C22⋊D73—21no—
62C2×#11924015—22no—
63C2×#11924015—41no—
64C2×#11924015—41no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556591—22no—
70#14556591—41no—
71#14556577—44no—
72#14556565—22no—
73#14556565—41no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C22⋊Dic73—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×C4×Dic73—21no—
85#14556522—22no—
86C4⋊#10596831—41no—
87#14556515—22no—
88#14556515—22no—
89#14556515—41no—
90#14556511—22no—
91#14556511—22no—
92#14556511—41no—
93C22⋊#10596831—41no—
94C22⋊#10596831—41no—
95C4×#10596831—41no—
96#14556498—21no—
97#14556498—21no—
98C22×Q8×C73C2311yescharacteristic
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101C22⋊Q8×C73—22no—
102C22⋊Q8×C73—22no—
103C4×Q8×C73—22no—
104C4×Q8×C73—41no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73—21no—
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×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C42×C73—21no—
118C2×C42×C73—21no—
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C23×D73—21no—
128C2×Dic146—42no—
129C2×Dic146—42no—
130C2×Dic146—81no—
131C2×D292—21no—
132C2×D292—21no—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×C22⋊D73—42no—
138C2×C22⋊D73—42no—
139C2×C22⋊D73—81no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143#11924015—44no—
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—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—42no—
162C4⋊Dic73#4112yes—
163C22⋊Dic73—42no—
164#11924005—42no—
165#11924005—42no—
166#11924005—42no—
167#11924005—42no—
168#11924005—81no—
169C4×Dic73—42no—
170C24×C73C2×D411yescharacteristic
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73#4112yes—
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—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
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—24no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73—81no—
204C4⋊C4×C73#4112yes—
205C4⋊C4×C73#4112yes—
206C22⋊C4×C73—24no—
207C22⋊C4×C73—41no—
208C22⋊C4×C73SD1614yescomplemented
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—42no—
214C42×C73—81no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73C2×SD1612yes—
222C23×C73C22×D411yescharacteristic
223C22×D73—41no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230Q8×C73—22no—
231Q8×C73—22no—
232Q8×C73—41no—
233Q8×C73—41no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
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—41no—
262C2×C4×C73—41no—
263C2×C4×C73—84no—
264C2×C4×C73C2×SD1612yes—
265C2×C4×C73C2×SD1612yes—
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C73H(5,𝔽2)11yescharacteristic
272C22⋊D73—84no—
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—
282D292—41no—
283D292—41no—
284D292—41no—
285D292—41no—
286Dic146—84no—
287#10596831—41no—
288#10596831—41no—
289#10596831—41no—
290#10596831—41no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73#52511yescharacteristic
299C22×C73#54611yescharacteristic
300C22×C73#54611yescharacteristic
301C22×C73C22×SD1611yescharacteristic
302C22×C73#57511yescharacteristic
303C22×C73#57611yescharacteristic
304C22×C73#57811yescharacteristic
305C2×D73—81no—
306C2×D73—81no—
307C2×D73—81no—
308C2×D73—81no—
309C2×D73—81no—
310C2×D73—81no—
311C2×D73—81no—
312C4×C73—22no—
313C4×C73—22no—
314C4×C73—42no—
315C4×C73—41no—
316C4×C73—42no—
317C4×C73—41no—
318C4×C73—42no—
319C4×C73—41no—
320C4×C73—81no—
321C4×C73#52911yescharacteristic
322C4×C73#53811yescharacteristic
323C4×C73#55511yescharacteristic
324C4×C73#55911yescharacteristic
325Dic73—42no—
326Dic73—81no—
327Dic73—81no—
328#26724—731nomaximal, Sylow
329C2×C73—42no—
330C2×C73#245911yescharacteristic
331C2×C73#310311yescharacteristic
332C2×C73#312411yescharacteristic
333C2×C73#329511yescharacteristic
334C2×C73#329811yescharacteristic
335C2×C73#331911yescharacteristic
336C2×C73#332711yescharacteristic
337D73—161no—
338#2491—731no—
339#2360D7311yescharacteristic, complemented
340#2055—731no—
341#2035—731no—
342#2024—731no—
343#2000—731no—
344#1988—731no—
345#1934—731no—
346#1926—731no—
347#1923—731no—
348#1920—731no—
349#1880—731no—
350#1874—731no—
351#1824—731no—
352C4×#103—731no—
353C73#2672411yescharacteristic, complemented, Sylow
354C22×SD16—731no—
355C2×C4⋊Q8—731no—
356C2×#124—1461no—
357C2×C22⋊Q8C2×D7311yescharacteristic
358C2×#121—731no—
359C2×C4×Q8C2×D7311yescharacteristic
360C22×C4⋊C4C2×D7311yescharacteristic
361C2×#106—731no—
362C2×C4⋊C8—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×#102—731no—
369C2×C22⋊C8—731no—
370C2×C22⋊C8—731no—
371C2×C4×C8—731no—
372#394—1461no—
373#393—1461no—
374#391C2×D7311yescharacteristic
375#390—731no—
376#390—731no—
377#389—21no—
378#389C2×D7311yescharacteristic
379#388—1461no—
380#387—21no—
381#385—21no—
382#380—731no—
383C4×C4⋊C4—731no—
384C4×C4⋊C4C2×D7311yescharacteristic
385C4×C22⋊C4—21no—
386C4×C22⋊C4C2×D7311yescharacteristic
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—1461no—
398C22×Q8C22×D7311yescharacteristic
399C22×D4—731no—
400C22×D4—1461no—
401C23×C4—21no—
402C23×C4C22×D7311yescharacteristic
403C2×SD16—2921no—
404C2×SD16—2921no—
405C2×SD16—1462no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C4⋊Q8—2921no—
411C4⋊Q8—1462no—
412#124—2924no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415#121—2921no—
416#121—1462no—
417C4×Q8—22no—
418C4×Q8—41no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—21no—
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⋊C4C22×D7311yescharacteristic
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4—1461no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4C22⋊D7311yescharacteristic
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446#106—1462no—
447C4⋊C8—2921no—
448#103—2921no—
449#103—1462no—
450#103—1462no—
451#102—2921no—
452#102—1462no—
453#102—1462no—
454C22⋊C8—2921no—
455C22⋊C8—2921no—
456C4×C8—2921no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—1461no—
462#95—1461no—
463#95C22⋊D7311yescharacteristic
464#95C22⋊D7311yescharacteristic
465C24—1461no—
466C24C2×C22⋊D7311yescharacteristic
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—2922no—
473C2×Q8—2922no—
474C2×Q8—5841no—
475C2×Q8#1192402812yes—
476C2×D4—2922no—
477C2×D4—2922no—
478C2×D4—5841no—
479C2×D4—1461no—
480C2×D4—1461no—
481C2×D4—1461no—
482C2×D4—1461no—
483C2×D4—1461no—
484C2×D4—1461no—
485C22×C4—21no—
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—81no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C23×D7311yescharacteristic
507SD16—2924no—
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—
518C2×C8—1461no—
519C2×C8—1461no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—2922no—
529C4⋊C4—732no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4#1192402712yes—
536C4⋊C4#1192402812yes—
537C22⋊C4—24no—
538C22⋊C4—41no—
539C22⋊C4—2922no—
540C22⋊C4—2922no—
541C22⋊C4—2922no—
542C22⋊C4—2922no—
543C22⋊C4—2922no—
544C22⋊C4—5841no—
545C22⋊C4#1192401514yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—81no—
552C42—2922no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—2921no—
560C23—2921no—
561C23—2921no—
562C23—2921no—
563C23—2921no—
564C23—2921no—
565C23—2921no—
566C23C2×#1192401512yes—
567C23C22×C22⋊D7311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—5844no—
573D4—2921no—
574D4—2921no—
575D4—2921no—
576D4—2921no—
577D4—5844no—
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—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
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—41no—
606C2×C4—41no—
607C2×C4—84no—
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—1462no—
617C2×C4C2×#1192401512yes—
618C2×C4C2×#1192401512yes—
619C2×C4#41×D7311yescharacteristic
620C2×C4C2×#1192402711yescharacteristic
621C2×C4C2×#1192402811yescharacteristic
622C2×C4C2×#1192402811yescharacteristic
623C2×C4#1455663611yescharacteristic
624C2×C4#1455663711yescharacteristic
625C8—2921no—
626C8—2921no—
627C8—2921no—
628C8—2921no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—5841no—
637C22—5841no—
638C22—5841no—
639C22—5841no—
640C22—5841no—
641C22—5841no—
642C22—5841no—
643C22#1798909811yescharacteristic
644C22#1798922511yescharacteristic
645C22#1798923111yescharacteristic
646C22C22×#1192401511yescharacteristic
647C22#1798938511yescharacteristic
648C22#1798939011yescharacteristic
649C22#1798940211yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—81no—
659C4—2922no—
660C4—5841no—
661C4—5841no—
662C4#1798911711yescharacteristic
663C4#1798917011yescharacteristic
664C4#1798930211yescharacteristic
665C4#1798932011yescharacteristic
666C2—42no—
667C2—11681no—
668C2#2261386211yescharacteristic
669C2#2261996211yescharacteristic
670C2#2262019311yescharacteristic
671C2#2262174911yescharacteristic
672C2#2262179611yescharacteristic
673C2#2262208711yescharacteristic
674C2#2262219311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2360C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenter#2360D73
hypocenterC73#26724
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #22611246C2semidirect
3. #22611124C2semidirect
4. #22611072C2semidirect
6. #22610883C2semidirect
7. #22610613C2semidirect
8. #22610558C2semidirect
9. #22610539C2semidirect
10. #22610513C2semidirect
11. #22610286C2semidirect
13. #22610021C2semidirect
14. C4×#14556515C2semidirect
15. C73×#2360C2semidirect
208. C22⋊C4×C73SD16semidirect
339. #2360D73semidirect
353. C73#26724semidirect
545. C22⋊C4#11924015semidirect

Maximal subgroups

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

subgroupindex
#226142942
#226112462
#226111242
#226110722
#226109322
#226108832
#226106132
#226105582
#226105392
#226105132
#226102862
#226102482
#226100212
C4×#145565152
C73×#23602
#2672473