Finite Group Databasebeta

← #33158785

Subgroups — #33158785

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614331C211yesmaximal, characteristic, complemented
2#22611344C211yesmaximal, characteristic, complemented
3#22611305C211yesmaximal, characteristic, complemented
4#22611117C211yesmaximal, characteristic
5#22611110C211yesmaximal, characteristic
6#22610699C211yesmaximal, characteristic, complemented
7#22610656C211yesmaximal, characteristic, complemented
8#22610539C211yesmaximal, characteristic, complemented
9#22610520C211yesmaximal, characteristic, complemented
10#22610502C211yesmaximal, characteristic, complemented
11#22610178C211yesmaximal, characteristic, complemented
12#22610130C211yesmaximal, characteristic, complemented
13#22610026C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2363C211yesmaximal, characteristic
16C2×#14556524C2211yescharacteristic
17C2×#14556524C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28C73⋊#401—21no—
29#17988515—21no—
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988481—21no—
33#17988469—21no—
34#17988453C2211yescharacteristic
35#17988447C2211yescharacteristic
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×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51#390×C73—21no—
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#386×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73—21no—
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×#11924007—21no—
68C2×#11924007—21no—
69C2×C22⋊Dic73—21no—
70C2×C22⋊Dic73—21no—
71C2×C4×Dic73—21no—
72#14556524—22no—
73#14556524—22no—
74#14556522—22no—
75#14556522—41no—
76C4⋊#10596831—22no—
77#14556512—22no—
78#14556512—22no—
79#14556512—22no—
80#14556512—41no—
81C22⋊#10596831—22no—
82C22⋊#10596831—41no—
83C4×#10596831—41no—
84#14556498—21no—
85#14556498—21no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73—21no—
92C22×D4×C73C2311yescharacteristic
93C23×C4×C73—21no—
94C23×C4×C73C2311yescharacteristic
95#121×C73—22no—
96#121×C73—22no—
97C4×D4×C73—22no—
98C4×D4×C73—41no—
99#117×C73—22no—
100#117×C73—41no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73D411yescharacteristic
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C42×C73—21no—
113C2×C42×C73—21no—
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116#95×C73—21no—
117#95×C73—21no—
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C2×#10596831—21no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831#4112yes—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144#11924007—42no—
145#11924007—81no—
146C22⋊Dic73—42no—
147C22⋊Dic73—81no—
148C4×Dic73—81no—
149C24×C73—21no—
150C24×C73C2×D411yescharacteristic
151C2×D4×C73—24no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—81no—
158C2×D4×C73—81no—
159C2×D4×C73#4112yes—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—42no—
170C22×C4×C73—42no—
171C22×C4×C73—81no—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—81no—
183C4⋊C4×C73#4112yes—
184C4⋊C4×C73#4112yes—
185C22⋊C4×C73—24no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—41no—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—42no—
190C22⋊C4×C73—42no—
191C22⋊C4×C73—81no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—42no—
197C42×C73—81no—
198C23×C73—22no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73C22×D411yescharacteristic
213D4×C73—22no—
214D4×C73—22no—
215D4×C73—41no—
216D4×C73—41no—
217D4×C73—84no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—84no—
246C2×C4×C73#13512yes—
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C732−1+411yescharacteristic
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265#10596831—22no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—42no—
273C22×C73—41no—
274C22×C73—42no—
275C22×C73—41no—
276C22×C73—42no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—81no—
283C22×C73#52511yescharacteristic
284C22×C73#54611yescharacteristic
285C22×C73#54711yescharacteristic
286C22×C73C2×#13511yescharacteristic
287C22×C73#57511yescharacteristic
288C22×C73#57611yescharacteristic
289C22×C73#57711yescharacteristic
290C4×C73—22no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—42no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—41no—
297C4×C73—81no—
298C4×C73#52911yescharacteristic
299C4×C73#53911yescharacteristic
300C4×C73#55311yescharacteristic
301C4×C73#55611yescharacteristic
302Dic73—81no—
303Dic73—81no—
304Dic73—81no—
305Dic73—81no—
306#27040—731nomaximal, Sylow
307C2×C73—81no—
308C2×C73—82no—
309C2×C73#246611yescharacteristic
310C2×C73#310211yescharacteristic
311C2×C73#311811yescharacteristic
312C2×C73#329711yescharacteristic
313C2×C73#331711yescharacteristic
314C2×C73#334011yescharacteristic
315C2×C73#335511yescharacteristic
316#2494—731no—
317#2363D7311yescharacteristic
318#2073—731no—
319#2065—731no—
320#2034—731no—
321#2033—731no—
322#1949—731no—
323#1941—731no—
324#1923—731no—
325#1921—731no—
326#1918—731no—
327#1862—731no—
328#1854—731no—
329#1825—731no—
330C4×#102—731no—
331C73#2704011yescharacteristic, complemented, Sylow
332C2×#121C2×D7311yescharacteristic
333C2×C4×D4C2×D7311yescharacteristic
334C2×#117C2×D7311yescharacteristic
335C2×#107—731no—
336C2×#107—731no—
337C2×#106—731no—
338C2×#106—731no—
339C2×C4⋊C8—731no—
340C2×#102—731no—
341C2×#102—731no—
342C2×#102—731no—
343C2×#102—731no—
344C2×C22⋊C8—731no—
345C2×C22⋊C8—731no—
346C2×C4×C8—731no—
347#401—1461no—
348#397—1461no—
349#390—21no—
350#390C2×D7311yescharacteristic
351#389—731no—
352#389—731no—
353#389C2×D7311yescharacteristic
354#388—21no—
355#388—1461no—
356#387—1461no—
357#386—21no—
358#382—731no—
359#380—731no—
360#380—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4C2×D7311yescharacteristic
363C4×C22⋊C4—21no—
364C4×C22⋊C4C2×D7311yescharacteristic
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#336—731no—
374#336—731no—
375C22×D4—21no—
376C22×D4C22×D7311yescharacteristic
377C23×C4—21no—
378C23×C4C22×D7311yescharacteristic
379C22×C8—731no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—731no—
383#121—22no—
384#121—22no—
385C4×D4—22no—
386C4×D4—41no—
387#117—22no—
388#117—41no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4C22⋊D7311yescharacteristic
407C2×C22⋊C4C22⋊D7311yescharacteristic
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1461no—
411C2×C42C22×D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413#107—1462no—
414#107—1462no—
415#106—2921no—
416#106—1462no—
417C4⋊C8—1462no—
418#102—2921no—
419#102—1462no—
420#102—1462no—
421#102—1462no—
422C22⋊C8—2921no—
423C22⋊C8—1462no—
424C4×C8—2921no—
425#95—21no—
426#95—21no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95C22⋊D7311yescharacteristic
435#95C22⋊D7311yescharacteristic
436C24—21no—
437C24C2×C22⋊D7311yescharacteristic
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—81no—
445C2×D4—81no—
446C2×D4#1192402812yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468C22×C4C23×D7311yescharacteristic
469C2×C8—732no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—81no—
486C4⋊C4—2922no—
487C4⋊C4—5841no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4#1192402712yes—
497C4⋊C4#1192402812yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C22⋊C4—2922no—
506C22⋊C4—5841no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—81no—
513C42—5841no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C22×C22⋊D7311yescharacteristic
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4#1455661112yes—
575C2×C4#41×D7311yescharacteristic
576C2×C4C2×#1192402711yescharacteristic
577C2×C4C2×#1192402811yescharacteristic
578C2×C4C2×#1192402811yescharacteristic
579C2×C4#1455663611yescharacteristic
580C2×C4#1455663811yescharacteristic
581C8—2921no—
582C8—2921no—
583C8—2921no—
584C8—1462no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1798909811yescharacteristic
600C22#1798923111yescharacteristic
601C22#1798923611yescharacteristic
602C22C2×#1455661111yescharacteristic
603C22#1798938511yescharacteristic
604C22#1798939011yescharacteristic
605C22#1798939811yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—5841no—
615C4—5841no—
616C4—5841no—
617C4—5841no—
618C4#1798911711yescharacteristic
619C4#1798917911yescharacteristic
620C4#1798928711yescharacteristic
621C4#1798930711yescharacteristic
622C2—81no—
623C2—82no—
624C2#2261396111yescharacteristic
625C2#2261994911yescharacteristic
626C2#2262011511yescharacteristic
627C2#2262178111yescharacteristic
628C2#2262205711yescharacteristic
629C2#2262234411yescharacteristic
630C2#2262253111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2363C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2363D73
hypocenterC73#27040
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #22614331C2semidirect
2. #22611344C2semidirect
3. #22611305C2semidirect
6. #22610699C2semidirect
7. #22610656C2semidirect
8. #22610539C2semidirect
9. #22610520C2semidirect
10. #22610502C2semidirect
11. #22610178C2semidirect
12. #22610130C2semidirect
13. #22610026C2semidirect
14. C4×#14556512C2semidirect
331. C73#27040semidirect

Maximal subgroups

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

subgroupindex
#226143312
#226113442
#226113052
#226111172
#226111102
#226106992
#226106562
#226105392
#226105202
#226105022
#226101782
#226101302
#226100262
C4×#145565122
C73×#23632
#2704073