Finite Group Databasebeta

← #33130474

Subgroups — #33130474

Order 18688.

776 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614242C211yesmaximal, characteristic
2#22612856C211yesmaximal, characteristic
3#22611501C211yesmaximal, characteristic, complemented
4#22611417C211yesmaximal, characteristic, complemented
5#22611366C211yesmaximal, characteristic, complemented
6#22611358C211yesmaximal, characteristic, complemented
7#22610508C211yesmaximal, characteristic, complemented
8#22610500C211yesmaximal, characteristic, complemented
9#22610223C211yesmaximal, characteristic
10#22610131C211yesmaximal, characteristic, complemented
11#22610119C211yesmaximal, characteristic, complemented
12#22610024C211yesmaximal, characteristic, complemented
13#22610021C211yesmaximal, characteristic, complemented
14#22609956C211yesmaximal, characteristic, complemented
15C2×#340×C73C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D73C2211yescharacteristic
17C2×#14556549C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22C2×C22⋊#10596831C2211yescharacteristic
23C2×#14556503C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25#17988525—21no—
26#17988522—21no—
27#17988483C2211yescharacteristic
28#17988483C2211yescharacteristic
29#17988477—21no—
30#17988468—21no—
31#17988466—21no—
32#17988466—21no—
33#17988460—21no—
34#17988452C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37#17988444C2211yescharacteristic
38#17988444C2211yescharacteristic
39#17988444C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×C22⋊Dic73—21no—
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C22×C4⋊C4×C73C2211yescharacteristic
54C22×C4⋊C4×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×C22⋊D73D411yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C4×C22⋊D73—22no—
64C4×C22⋊D73—22no—
65#14556549—22no—
66#14556549—41no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007D411yescharacteristic
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73D411yescharacteristic
77C2×#11924005—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005D411yescharacteristic
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85#14556524—22no—
86#14556524—41no—
87#14556523—22no—
88#14556523—41no—
89C22⋊#10596831—22no—
90#14556503—22no—
91#14556503—41no—
92#14556498—21no—
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—
102C23×C4×C73—21no—
103C23×C4×C73—21no—
104C23×C4×C73C2311yescharacteristic
105C22×C8×C73—22no—
106C22×C8×C73—22no—
107C22×C8×C73—41no—
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C2×C4⋊C4×C73—22no—
111C2×C4⋊C4×C73—22no—
112C2×C4⋊C4×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—41no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C23×D73C2×D411yescharacteristic
121C2×C22⋊D73—22no—
122C2×C22⋊D73—22no—
123C2×C22⋊D73—22no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151#11924007—22no—
152#11924007—81no—
153#11924007—81no—
154C22⋊Dic73—22no—
155C22⋊Dic73—42no—
156C22⋊Dic73—42no—
157#11924005—22no—
158#11924005—42no—
159#11924005—81no—
160#11924005—81no—
161#11924005—81no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—42no—
165C4×Dic73—81no—
166C24×C73C2×D411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—81no—
179C22×C4×C73—81no—
180C22×C4×C73—81no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73#4112yes—
183C22×C4×C73#4112yes—
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—44no—
191C2×C8×C73#4112yes—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73#4112yes—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73#12312yes—
207C23×C73#13512yes—
208C23×C73#13712yes—
209C23×C73C22×D411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—21no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—84no—
236C2×C4×C73—84no—
237C2×C4×C73#12312yes—
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73H(5,𝔽2)11yescharacteristic
243C2×C4×C732−1+411yescharacteristic
244C22⋊D73—44no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
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—
265C2×Dic73—41no—
266C4×D73—41no—
267C4×D73—41no—
268C4×D73—41no—
269C4×D73—41no—
270C8×C73—22no—
271C8×C73—41no—
272#10596831—41no—
273#10596831—41no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73#48514yes—
281C22×C73C2×#12311yescharacteristic
282C22×C73#54611yescharacteristic
283C22×C73#54711yescharacteristic
284C22×C73C2×#13511yescharacteristic
285C22×C73C2×#13711yescharacteristic
286C22×C73#57611yescharacteristic
287C22×C73#57711yescharacteristic
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—41no—
293C2×D73—41no—
294C2×D73—41no—
295C4×C73—22no—
296C4×C73—42no—
297C4×C73—81no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73#52911yescharacteristic
301C4×C73#53911yescharacteristic
302C4×C73#55111yescharacteristic
303C4×C73#55211yescharacteristic
304Dic73—41no—
305Dic73—41no—
306Dic73—41no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310Dic73—81no—
311#24943—731nomaximal, Sylow
312C2×C73—24no—
313C2×C73#241911yescharacteristic
314C2×C73C2×#48511yescharacteristic
315C2×C73#309611yescharacteristic
316C2×C73#329711yescharacteristic
317C2×C73#331611yescharacteristic
318C2×C73#333711yescharacteristic
319C2×C73#334611yescharacteristic
320D73—81no—
321#2487—731no—
322#2352—731no—
323#2101—731no—
324#2085—731no—
325#2077—731no—
326#2075—731no—
327#1919—731no—
328#1918—731no—
329#1869—731no—
330#1854—731no—
331#1852—731no—
332#1825—731no—
333#1824—731no—
334#1807—731no—
335C2×#340D7311yescharacteristic, complemented
336C73#2494311yescharacteristic, complemented, Sylow
337C23×C8C2×D7311yescharacteristic
338C2×C4×D4—731no—
339C2×#117—731no—
340C22×C4⋊C4C2×D7311yescharacteristic
341C22×C4⋊C4C2×D7311yescharacteristic
342C2×#107—731no—
343C2×#107—731no—
344C2×#106—731no—
345C2×#106—731no—
346C2×C22⋊C8—731no—
347C2×C22⋊C8—731no—
348C2×C22⋊C8—731no—
349#400—1461no—
350#399—1461no—
351#389—731no—
352#389—731no—
353#388—1461no—
354#387—1461no—
355#386—1461no—
356#386—1461no—
357#385—1461no—
358#382—731no—
359#380—731no—
360#380—731no—
361#380—731no—
362#380—731no—
363#380—731no—
364C4×C4⋊C4—731no—
365C4×C22⋊C4—1461no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340C2×D7311yescharacteristic
375#340C2×D7311yescharacteristic
376#340C2×D7311yescharacteristic
377#340C2×D7311yescharacteristic
378#336—731no—
379#336—731no—
380C22×D4—731no—
381C23×C4—21no—
382C23×C4—21no—
383C23×C4—731no—
384C23×C4C22×D7311yescharacteristic
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—41no—
388C22×C8—731no—
389C22×C8—731no—
390C22×C8C22×D7311yescharacteristic
391C22×C8C22×D7311yescharacteristic
392C4×D4—1462no—
393C4×D4—1462no—
394#117—2921no—
395#117—1462no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—41no—
402C2×C4⋊C4—731no—
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⋊C4C22×D7311yescharacteristic
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C4⋊C4C22×D7311yescharacteristic
412C2×C4⋊C4C22×D7311yescharacteristic
413C2×C22⋊C4—731no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—1461no—
416C2×C22⋊C4—1461no—
417C2×C22⋊C4—1461no—
418C2×C22⋊C4—1461no—
419C2×C22⋊C4—1461no—
420C2×C22⋊C4—1461no—
421C2×C42—731no—
422C2×C42—1461no—
423C2×C42—1461no—
424#107—2921no—
425#107—1462no—
426#106—2921no—
427#106—1462no—
428C22⋊C8—2921no—
429C22⋊C8—1462no—
430C22⋊C8—1462no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441C24—731no—
442C24D4×D7311yescharacteristic
443C2×D4—1462no—
444C2×D4—1462no—
445C2×D4—1462no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4D4×D7311yescharacteristic
474C22×C4#1192402712yes—
475C22×C4#1192402812yes—
476C22×C4C23×D7311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—44no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8#1192402812yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—5841no—
500C4⋊C4—5841no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1462no—
510C4⋊C4#1192402812yes—
511C22⋊C4—2922no—
512C22⋊C4—2922no—
513C22⋊C4—2922no—
514C22⋊C4—5841no—
515C22⋊C4—5841no—
516C22⋊C4—5841no—
517C22⋊C4—1462no—
518C22⋊C4—1462no—
519C42—2922no—
520C42—5841no—
521C42—1461no—
522C42—1461no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—1461no—
528C23—1461no—
529C23—1461no—
530C23—1461no—
531C23—1461no—
532C23—1461no—
533C23—1461no—
534C23#1455657312yes—
535C23#1455661212yes—
536C23#1455662312yes—
537C23C2×D4×D7311yescharacteristic
538D4—2924no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
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—2921no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4#1455657312yes—
585C2×C4C2×#1192402711yescharacteristic
586C2×C4C2×#1192402811yescharacteristic
587C2×C4C2×#1192402811yescharacteristic
588C2×C4C2×#1192402811yescharacteristic
589C2×C4#1455663611yescharacteristic
590C2×C4#1455663911yescharacteristic
591C8—22no—
592C8—41no—
593C8—2921no—
594C8—2921no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—2921no—
602C22—2921no—
603C22—2921no—
604C22—2921no—
605C22—2921no—
606C22—2921no—
607C22—2921no—
608C22#1798892614yes—
609C22C2×#1455657311yescharacteristic
610C22#1798922711yescharacteristic
611C22#1798923811yescharacteristic
612C22C2×#1455661211yescharacteristic
613C22C2×#1455662311yescharacteristic
614C22#1798938811yescharacteristic
615C22#1798939311yescharacteristic
616C4—22no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—2921no—
622C4—2921no—
623C4—2921no—
624C4—5841no—
625C4—5841no—
626C4—5841no—
627C4—5841no—
628C4#1798912111yescharacteristic
629C4#1798918411yescharacteristic
630C4#1798926311yescharacteristic
631C4#1798927211yescharacteristic
632C2—24no—
633C2—5841no—
634C2#2261340811yescharacteristic
635C2C2×#1798892611yescharacteristic
636C2#2261988511yescharacteristic
637C2#2262177711yescharacteristic
638C2#2262203811yescharacteristic
639C2#2262230611yescharacteristic
640C2#2262241211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#340×C73C2
socleC23×C73C22×D4
prime radicalC22×C22⋊D73D4
hypercenterC2×#340D73
hypocenterC73#24943
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #22611501C2semidirect
4. #22611417C2semidirect
5. #22611366C2semidirect
6. #22611358C2semidirect
7. #22610508C2semidirect
8. #22610500C2semidirect
10. #22610131C2semidirect
11. #22610119C2semidirect
12. #22610024C2semidirect
13. #22610021C2semidirect
14. #22609956C2semidirect
15. C2×#340×C73C2semidirect
335. C2×#340D73semidirect
336. C73#24943semidirect

Maximal subgroups

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

subgroupindex
#226142422
#226128562
#226115012
#226114172
#226113662
#226113582
#226105082
#226105002
#226102232
#226101312
#226101192
#226100242
#226100212
#226099562
C2×#340×C732
#2494373