Finite Group Databasebeta

← #33115971

Subgroups — #33115971

Order 18688.

880 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic73C211yesmaximal, characteristic
2#22612333C211yesmaximal, characteristic, complemented
3#22611492C211yesmaximal, characteristic, complemented
4#22611457C211yesmaximal, characteristic, complemented
5#22611405C211yesmaximal, characteristic, complemented
6#22611357C211yesmaximal, characteristic, complemented
7#22611354C211yesmaximal, characteristic, complemented
8#22611337C211yesmaximal, characteristic, complemented
9#22611110C211yesmaximal, characteristic
10#107×Dic73C211yesmaximal, characteristic, complemented
11#22609823C211yesmaximal, characteristic, complemented
12#102×Dic73C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic73C211yesmaximal, characteristic, complemented
15C73×#2033C211yesmaximal, characteristic
16C2×D4×Dic73—21no—
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556547—21no—
19C22×C4×Dic73C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28#17988490C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988480—21no—
31#17988467—21no—
32#17988456C2211yescharacteristic
33#17988456C2211yescharacteristic
34#17988451—21no—
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37#17988445—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41#17988438—21no—
42C22⋊C4×Dic73—21no—
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#121×C73C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C23×Dic73—21no—
60C23×Dic73D411yescharacteristic
61C23×Dic73D411yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64D4×Dic73—22no—
65D4×Dic73—22no—
66D4×Dic73—44no—
67#14556547—44no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—22no—
83C2×C4×Dic73—22no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556525—22no—
92#14556524—22no—
93#14556524—22no—
94#14556512—22no—
95#14556512—22no—
96C22⋊#10596831—22no—
97C8×Dic73—22no—
98C8×Dic73—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×D4×C73—21no—
106C22×D4×C73C2311yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110#121×C73—22no—
111#121×C73—22no—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117#107×C73—22no—
118#102×C73—22no—
119#102×C73—22no—
120C22⋊C8×C73—22no—
121#95×C73—21no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—41no—
131C22×Dic73—42no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73D814yes—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831#4112yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154#11924007—42no—
155#11924007—42no—
156#11924007—42no—
157#11924007—42no—
158C22⋊Dic73—42no—
159C22⋊Dic73—42no—
160C22⋊Dic73—42no—
161C22⋊Dic73—42no—
162C22⋊Dic73—42no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—22no—
166C4×Dic73—22no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C4×Dic73#4112yes—
171C4×Dic73#4112yes—
172C24×C73—21no—
173C24×C73C2×D411yescharacteristic
174C2×D4×C73—24no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—42no—
181C2×D4×C73—42no—
182C2×D4×C73#4112yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2411yescharacteristic
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73#4112yes—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73#4112yes—
206C22⋊C4×C73—24no—
207C22⋊C4×C73—42no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—22no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73C2×D812yes—
222C23×C73C22×D411yescharacteristic
223D4×C73—22no—
224D4×C73—22no—
225D4×C73—48no—
226D4×C73—41no—
227D4×C73—41no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
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×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—28no—
257C2×Dic73—44no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—84no—
271C2×Dic73#12312yes—
272C2×Dic73#12312yes—
273C2×Dic73C2×D812yes—
274C2×Dic73C2×D812yes—
275C2×Dic73#13512yes—
276C2×Dic73#13712yes—
277C8×C73—22no—
278C8×C73—41no—
279#10596831—22no—
280#10596831—41no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—81no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73—81no—
295C22×C73C2×#12311yescharacteristic
296C22×C73#52511yescharacteristic
297C22×C73#53211yescharacteristic
298C22×C73C22×D811yescharacteristic
299C22×C73C2×#13511yescharacteristic
300C22×C73C2×#13711yescharacteristic
301C22×C73#57511yescharacteristic
302C4×C73—42no—
303C4×C73—41no—
304C4×C73—81no—
305C4×C73—81no—
306C4×C73#52911yescharacteristic
307C4×C73#52911yescharacteristic
308C4×C73#52911yescharacteristic
309C4×C73#52911yescharacteristic
310Dic73—22no—
311Dic73—22no—
312Dic73—22no—
313Dic73—81no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317Dic73#48014yes—
318#23754—731nomaximal, Sylow
319C2×C73—42no—
320C2×C73—82no—
321C2×C73#239511yescharacteristic
322C2×C73C2×#48011yescharacteristic
323C2×C73#309111yescharacteristic
324C2×C73#310011yescharacteristic
325C2×C73#310711yescharacteristic
326C2×C73#316911yescharacteristic
327C2×C73#317211yescharacteristic
328C4×#121—731no—
329#2285—731no—
330#2099—731no—
331#2093—731no—
332#2083—731no—
333#2075—731no—
334#2075—731no—
335#2071—731no—
336#2033—731no—
337#2033D7311yescharacteristic
338C4×#107—731no—
339C4×#107—731no—
340C4×#102—731no—
341C4×#102—731no—
342C4×C22⋊C8—731no—
343C73#2375411yescharacteristic, complemented, Sylow
344C2×#121C2×D7311yescharacteristic
345C2×C4×D4—731no—
346C2×C4×D4—1461no—
347C2×#117—1461no—
348C22×C42—731no—
349C2×#107—731no—
350C2×#107—731no—
351C2×#107—731no—
352C2×#107C2×D7311yescharacteristic
353C2×#102—731no—
354C2×#102—731no—
355C2×#102C2×D7311yescharacteristic
356C2×#102C2×D7311yescharacteristic
357C2×C22⋊C8—731no—
358C2×C22⋊C8C2×D7311yescharacteristic
359C2×C4×C8—731no—
360C2×C4×C8—731no—
361#390—731no—
362#389—731no—
363#388—1461no—
364#386—1461no—
365#384—731no—
366#384—731no—
367#382—1461no—
368#380—731no—
369#380—731no—
370#380—1461no—
371#380C2×D7311yescharacteristic
372C4×C4⋊C4—731no—
373C4×C4⋊C4—731no—
374C4×C4⋊C4—731no—
375C4×C4⋊C4—1461no—
376C4×C22⋊C4—1461no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340C2×D7311yescharacteristic
385#336—731no—
386#336—731no—
387C22×D4—21no—
388C22×D4C22×D7311yescharacteristic
389C23×C4—731no—
390C23×C4—731no—
391C23×C4—1461no—
392C23×C4C22×D7311yescharacteristic
393C22×C8—731no—
394C22×C8—731no—
395C22×C8C22×D7311yescharacteristic
396C22×C8C22×D7311yescharacteristic
397#121—22no—
398#121—22no—
399C4×D4—2924no—
400C4×D4—1462no—
401C4×D4—1462no—
402#117—2924no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4C22×D7311yescharacteristic
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C4⋊C4C22×D7311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1461no—
418C2×C22⋊C4—1461no—
419C2×C22⋊C4—1461no—
420C2×C22⋊C4—1461no—
421C2×C22⋊C4—1461no—
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—731no—
425C2×C42—731no—
426C2×C42—1461no—
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42—1462no—
430C2×C42—1462no—
431#107—22no—
432#107—1462no—
433#107—1462no—
434#107—1462no—
435#102—22no—
436#102—22no—
437#102—1462no—
438#102—1462no—
439C22⋊C8—22no—
440C22⋊C8—1462no—
441C4×C8—1462no—
442C4×C8—1462no—
443#95—21no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450C24—21no—
451C24D4×D7311yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4#1192402712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—2921no—
471C22×C4—2922no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—731no—
475C22×C4—734no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4D4×D7311yescharacteristic
488C22×C4C23×D7311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—732no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8#1192402712yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—2922no—
507C4⋊C4—2922no—
508C4⋊C4—2922no—
509C4⋊C4—2922no—
510C4⋊C4—2922no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4#1192402712yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—2922no—
523C22⋊C4—2922no—
524C22⋊C4—2922no—
525C22⋊C4—2922no—
526C22⋊C4—2922no—
527C42—2922no—
528C42—2922no—
529C42—2922no—
530C42—732no—
531C42—732no—
532C42—1462no—
533C42—1462no—
534C42—1461no—
535C42—1461no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23D8×D7312yes—
550C23C2×D4×D7311yescharacteristic
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2924no—
585C2×C4—5844no—
586C2×C4—732no—
587C2×C4—732no—
588C2×C4—732no—
589C2×C4—732no—
590C2×C4—732no—
591C2×C4—732no—
592C2×C4—1468no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4#41×D7311yescharacteristic
600C2×C4#41×D7311yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4C2×#1192402711yescharacteristic
604C2×C4C2×#1192402711yescharacteristic
605C8—22no—
606C8—41no—
607C8—2921no—
608C8—1462no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#123×D7311yescharacteristic
624C22#1798909311yescharacteristic
625C22#1798913211yescharacteristic
626C22C2×D8×D7311yescharacteristic
627C22#135×D7311yescharacteristic
628C22#137×D7311yescharacteristic
629C22#1798938311yescharacteristic
630C4—42no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—5841no—
635C4—5841no—
636C4—5841no—
637C4—5841no—
638C4—734no—
639C4—1462no—
640C4—1462no—
641C4—1462no—
642C4#1798911411yescharacteristic
643C4#1798911611yescharacteristic
644C4#1798911811yescharacteristic
645C4#1798911811yescharacteristic
646C2—42no—
647C2—82no—
648C2#2261318711yescharacteristic
649C2#480×D7311yescharacteristic
650C2#2261984311yescharacteristic
651C2#2261992611yescharacteristic
652C2#2261999511yescharacteristic
653C2#2262050511yescharacteristic
654C2#2262053011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2033C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2033D73
hypocenterC73#23754
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #22612333C2semidirect
3. #22611492C2semidirect
4. #22611457C2semidirect
5. #22611405C2semidirect
6. #22611357C2semidirect
7. #22611354C2semidirect
8. #22611337C2semidirect
10. #107×Dic73C2semidirect
11. #22609823C2semidirect
12. #102×Dic73C2semidirect
13. #102×Dic73C2semidirect
14. C22⋊C8×Dic73C2semidirect
343. C73#23754semidirect

Maximal subgroups

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

subgroupindex
#121×Dic732
#226123332
#226114922
#226114572
#226114052
#226113572
#226113542
#226113372
#226111102
#107×Dic732
#226098232
#102×Dic732
C22⋊C8×Dic732
C73×#20332
#2375473