Finite Group Databasebeta

← #33154100

Subgroups — #33154100

Order 18688.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614307C211yesmaximal, characteristic
2#22612940C211yesmaximal, characteristic
3#22611251C211yesmaximal, characteristic, complemented
4#22611131C211yesmaximal, characteristic
5#22611070C211yesmaximal, characteristic, complemented
6#22610877C211yesmaximal, characteristic, complemented
7#22610609C211yesmaximal, characteristic, complemented
8#22610575C211yesmaximal, characteristic, complemented
9#22610534C211yesmaximal, characteristic
10#22610526C211yesmaximal, characteristic, complemented
11#22610274C211yesmaximal, characteristic, complemented
12#22610239C211yesmaximal, characteristic
13#22610022C211yesmaximal, characteristic
14#103×Dic73C211yesmaximal, characteristic, complemented
15C73×#2001C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988525—21no—
28#17988499—21no—
29#17988494C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988494C2211yescharacteristic
32#17988492—21no—
33#17988486—21no—
34#17988486C2211yescharacteristic
35#17988481—21no—
36#17988473—21no—
37#17988462—21no—
38#17988447C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×Q16×C73C2211yescharacteristic
53C2×C22⋊Q8×C73C2211yescharacteristic
54C2×C22⋊Q8×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×Dic146—21no—
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556590—22no—
64#14556590—41no—
65Q8×Dic73—22no—
66Q8×Dic73—41no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—41no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73D411yescharacteristic
79C2×C22⋊Dic73D411yescharacteristic
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85#14556524—22no—
86#14556520—41no—
87#14556515—22no—
88#14556515—22no—
89#14556515—41no—
90#14556515—41no—
91C22⋊#10596831—41no—
92C8×Dic73—41no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C22×Q8×C73C2311yescharacteristic
106C22×Q8×C73C2311yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C2×Q16×C73—22no—
109C2×Q16×C73—41no—
110C2×Q16×C73—41no—
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22⋊Q8×C73—22no—
114C22⋊Q8×C73—22no—
115C22⋊Q8×C73—22no—
116C22⋊Q8×C73—41no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123#103×C73—22no—
124#103×C73—22no—
125C22⋊C8×C73—41no—
126C2×Dic146—42no—
127C2×Dic146—42no—
128C2×Dic146—81no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—81no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—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—
154C4⋊Dic73—42no—
155C4⋊Dic73—81no—
156C4⋊Dic73#4112yes—
157#11924007—42no—
158C22⋊Dic73—41no—
159C22⋊Dic73—81no—
160C22⋊Dic73Q1614yescomplemented
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—42no—
166C4×Dic73—81no—
167C4×Dic73—81no—
168C24×C73C2×D411yescharacteristic
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73#4112yes—
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—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2411yescharacteristic
193Q16×C73—44no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C4⋊C4×C73—24no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—42no—
204C4⋊C4×C73#4112yes—
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×C73C2×Q1612yes—
215C23×C73C22×D411yescharacteristic
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
220Q8×C73—41no—
221Q8×C73—41no—
222Q8×C73—41no—
223Q8×C73—41no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
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×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73H(5,𝔽2)11yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—22no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
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—41no—
271C2×Dic73—84no—
272C2×Dic73C2×Q1612yes—
273C2×Dic73C2×Q1612yes—
274Dic146—84no—
275C8×C73—41no—
276C8×C73—41no—
277#10596831—41no—
278#10596831—41no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73#52511yescharacteristic
287C22×C73#54611yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73C22×Q1611yescharacteristic
290C22×C73#57511yescharacteristic
291C22×C73#57711yescharacteristic
292C22×C73#57711yescharacteristic
293C4×C73—42no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—42no—
297C4×C73—81no—
298C4×C73#52911yescharacteristic
299C4×C73#53811yescharacteristic
300C4×C73#55511yescharacteristic
301C4×C73#55911yescharacteristic
302Dic73—22no—
303Dic73—22no—
304Dic73—41no—
305Dic73—42no—
306Dic73—41no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310#26726—731nomaximal, Sylow
311C2×C73—42no—
312C2×C73#245911yescharacteristic
313C2×C73#310411yescharacteristic
314C2×C73#312511yescharacteristic
315C2×C73#329911yescharacteristic
316C2×C73#329911yescharacteristic
317C2×C73#331111yescharacteristic
318C2×C73#332311yescharacteristic
319#2492—731no—
320#2360—731no—
321#2056—731no—
322#2036—731no—
323#2024—731no—
324#2001D7311yescharacteristic
325#1987—731no—
326#1933—731no—
327#1928—731no—
328#1923—731no—
329#1921—731no—
330#1878—731no—
331#1872—731no—
332#1824—731no—
333C4×#103—731no—
334C73#2672611yescharacteristic, complemented, Sylow
335C22×Q16C2×D7311yescharacteristic
336C2×C4⋊Q8—731no—
337C2×C22⋊Q8C2×D7311yescharacteristic
338C2×C22⋊Q8C2×D7311yescharacteristic
339C2×C4×Q8—731no—
340C22×C4⋊C4—731no—
341C2×#107—731no—
342C2×C4⋊C8—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103C2×D7311yescharacteristic
348C2×#103C2×D7311yescharacteristic
349C2×C22⋊C8—731no—
350C2×C22⋊C8C2×D7311yescharacteristic
351C2×C4×C8—731no—
352#400—1461no—
353#393—1461no—
354#391—731no—
355#391—731no—
356#391—731no—
357#391—1461no—
358#389—731no—
359#389—1461no—
360#388—1461no—
361#387—1461no—
362#385—1461no—
363#380—731no—
364C4×C4⋊C4—731no—
365C4×C4⋊C4—731no—
366C4×C22⋊C4—731no—
367C4×C22⋊C4—1461no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340C2×D7311yescharacteristic
376#336—731no—
377#336—731no—
378C22×Q8—1461no—
379C22×Q8C22×D7311yescharacteristic
380C22×Q8C22×D7311yescharacteristic
381C23×C4—1461no—
382C23×C4C22×D7311yescharacteristic
383C2×Q16—22no—
384C2×Q16—41no—
385C2×Q16—41no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8C22×D7311yescharacteristic
389C22×C8C22×D7311yescharacteristic
390C4⋊Q8—2921no—
391C4⋊Q8—1462no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—41no—
396C4×Q8—2921no—
397C4×Q8—1462no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—2921no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1462no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—1461no—
417C2×C42—731no—
418C2×C42—731no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422#107—1462no—
423C4⋊C8—2921no—
424#103—22no—
425#103—22no—
426#103—2921no—
427#103—2921no—
428#103—1462no—
429#103—1462no—
430C22⋊C8—41no—
431C22⋊C8—2921no—
432C4×C8—2921no—
433#95—731no—
434#95—731no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445C24D4×D7311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—2922no—
458C2×Q8—2922no—
459C2×Q8—5841no—
460C2×Q8#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—21no—
471C22×C4—2922no—
472C22×C4—2922no—
473C22×C4—5841no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4D4×D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484Q16—44no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—2922no—
502C4⋊C4—2922no—
503C4⋊C4—2922no—
504C4⋊C4—5841no—
505C4⋊C4—732no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4#1192402712yes—
514C4⋊C4#1192402712yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—2921no—
518C22⋊C4—5841no—
519C22⋊C4—734no—
520C42—2922no—
521C42—5841no—
522C42—5841no—
523C42—1461no—
524C42—1461no—
525C42—1461no—
526C42—1461no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23Q16×D7312yes—
534C23C2×D4×D7311yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—5844no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
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—41no—
562C2×C4—41no—
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—5844no—
575C2×C4—732no—
576C2×C4—732no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1462no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4#41×D7311yescharacteristic
589C2×C4C2×#1192402711yescharacteristic
590C2×C4C2×#1192402711yescharacteristic
591C2×C4C2×#1192402711yescharacteristic
592C2×C4#1455663611yescharacteristic
593C2×C4#1455663611yescharacteristic
594C8—41no—
595C8—41no—
596C8—2921no—
597C8—2921no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1798909311yescharacteristic
606C22#1798922411yescharacteristic
607C22#1798922611yescharacteristic
608C22C2×Q16×D7311yescharacteristic
609C22#1798938611yescharacteristic
610C22#1798939411yescharacteristic
611C22#1798939611yescharacteristic
612C4—42no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—2922no—
618C4—2921no—
619C4—2921no—
620C4—5841no—
621C4—5841no—
622C4—5841no—
623C4—1462no—
624C4—1462no—
625C4#1798911811yescharacteristic
626C4#1798917211yescharacteristic
627C4#1798930111yescharacteristic
628C4#1798932311yescharacteristic
629C2—42no—
630C2#2261386311yescharacteristic
631C2#2261996511yescharacteristic
632C2#2262020111yescharacteristic
633C2#2262180011yescharacteristic
634C2#2262180211yescharacteristic
635C2#2262196211yescharacteristic
636C2#2262214011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2001C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2001D73
hypocenterC73#26726
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #22611251C2semidirect
5. #22611070C2semidirect
6. #22610877C2semidirect
7. #22610609C2semidirect
8. #22610575C2semidirect
10. #22610526C2semidirect
11. #22610274C2semidirect
14. #103×Dic73C2semidirect
160. C22⋊Dic73Q16semidirect
334. C73#26726semidirect

Maximal subgroups

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

subgroupindex
#226143072
#226129402
#226112512
#226111312
#226110702
#226108772
#226106092
#226105752
#226105342
#226105262
#226102742
#226102392
#226100222
#103×Dic732
C73×#20012
#2672673