Finite Group Databasebeta

← #33116018

Subgroups — #33116018

Order 18688.

836 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612329C211yesmaximal, characteristic, complemented
2#22611499C211yesmaximal, characteristic, complemented
3#22611452C211yesmaximal, characteristic, complemented
4#22611412C211yesmaximal, characteristic, complemented
5#22611368C211yesmaximal, characteristic, complemented
6#22611362C211yesmaximal, characteristic, complemented
7#22611344C211yesmaximal, characteristic, complemented
8#22611117C211yesmaximal, characteristic
9#22611110C211yesmaximal, characteristic
10C4×#14556524C211yesmaximal, characteristic, complemented
11C4×#14556522C211yesmaximal, characteristic, complemented
12C4×#14556512C211yesmaximal, characteristic, complemented
13C4×#14556512C211yesmaximal, characteristic, complemented
14C4×C22⋊#10596831C211yesmaximal, characteristic, complemented
15C4×#121×C73C211yesmaximal, characteristic
16C2×#14556524C2211yescharacteristic
17C2×#14556524C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556512C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C4×#10596831C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988455—21no—
29#17988453C2211yescharacteristic
30#17988453C2211yescharacteristic
31#17988450—21no—
32#17988447C2211yescharacteristic
33#17988447C2211yescharacteristic
34C4×C4⋊Dic73C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36C4×C22⋊Dic73—21no—
37#17988433—21no—
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×C73—21no—
50C2×C4×D4×C73C2211yescharacteristic
51C22×C42×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#386×C73—21no—
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C22⋊Dic73—21no—
68C2×C22⋊Dic73—21no—
69C2×C4×Dic73—21no—
70C2×C4×Dic73—21no—
71#14556524—22no—
72#14556524—22no—
73#14556522—22no—
74#14556522—22no—
75#14556512—22no—
76#14556512—22no—
77#14556512—22no—
78#14556512—22no—
79C22⋊#10596831—22no—
80C22⋊#10596831—22no—
81C4×#10596831—22no—
82C4×#10596831—22no—
83#14556498—21no—
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×C73D411yescharacteristic
95C23×C4×C73D411yescharacteristic
96C23×C4×C73C2311yescharacteristic
97#121×C73—22no—
98#121×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—44no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C42×C73—22no—
112C2×C42×C73—22no—
113C2×C42×C73—21no—
114C2×C42×C73D411yescharacteristic
115C2×C42×C73D411yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—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—
136C2×#10596831#4112yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C22⋊Dic73—42no—
146C22⋊Dic73—42no—
147C4×Dic73—42no—
148C4×Dic73—42no—
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—42no—
158C2×D4×C73—42no—
159C2×D4×C73#4112yes—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
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—41no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73#4112yes—
191C22⋊C4×C73—24no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—42no—
196C42×C73—22no—
197C42×C73—22no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—42no—
201C42×C73#4112yes—
202C42×C73#4112yes—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—22no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73#13512yes—
217C23×C73C22×D411yescharacteristic
218D4×C73—22no—
219D4×C73—22no—
220D4×C73—48no—
221D4×C73—41no—
222D4×C73—41no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—44no—
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—44no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—84no—
250C2×C4×C73#12312yes—
251C2×C4×C73#12312yes—
252C2×C4×C73#13512yes—
253C2×C4×C73AGL(1,ℤ/8)12yes—
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
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—41no—
272#10596831—22no—
273#10596831—22no—
274#10596831—41no—
275#10596831—41no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—81no—
290C22×C73C2×#12311yescharacteristic
291C22×C73#52511yescharacteristic
292C22×C73#53211yescharacteristic
293C22×C73C2×#13511yescharacteristic
294C22×C73C2×#13511yescharacteristic
295C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C73#57511yescharacteristic
297C4×C73—22no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—42no—
301C4×C73—41no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73#48214yes—
305C4×C73#52911yescharacteristic
306C4×C73#52911yescharacteristic
307C4×C73#52911yescharacteristic
308C4×C73#52911yescharacteristic
309Dic73—81no—
310Dic73—81no—
311Dic73—81no—
312Dic73—81no—
313#23758—731nomaximal, Sylow
314C2×C73—42no—
315C2×C73—82no—
316C2×C73#239511yescharacteristic
317C2×C73C2×#48211yescharacteristic
318C2×C73#309111yescharacteristic
319C2×C73#310211yescharacteristic
320C2×C73#310511yescharacteristic
321C2×C73#316811yescharacteristic
322C2×C73#317111yescharacteristic
323C4×#121D7311yescharacteristic
324#2283—731no—
325#2101—731no—
326#2092—731no—
327#2085—731no—
328#2077—731no—
329#2076—731no—
330#2073—731no—
331#2034—731no—
332#2033—731no—
333C4×#107—731no—
334C4×#106—731no—
335C4×#102—731no—
336C4×#102—731no—
337C4×C22⋊C8—731no—
338C73#2375811yescharacteristic, complemented, Sylow
339C2×#121C2×D7311yescharacteristic
340C2×C4×D4—21no—
341C2×C4×D4C2×D7311yescharacteristic
342C22×C42C2×D7311yescharacteristic
343C2×#107—731no—
344C2×#107—731no—
345C2×#106—731no—
346C2×#106—731no—
347C2×#102—731no—
348C2×#102—731no—
349C2×#102—731no—
350C2×#102—731no—
351C2×C22⋊C8—731no—
352C2×C22⋊C8—731no—
353C2×C4×C8—731no—
354C2×C4×C8—731no—
355#390C2×D7311yescharacteristic
356#389C2×D7311yescharacteristic
357#388—21no—
358#386—21no—
359#383—1461no—
360#382—731no—
361#382—731no—
362#381—1461no—
363#380—731no—
364#380—731no—
365#380C2×D7311yescharacteristic
366C4×C4⋊C4—731no—
367C4×C4⋊C4—731no—
368C4×C4⋊C4C2×D7311yescharacteristic
369C4×C22⋊C4—21no—
370C4×C22⋊C4—1461no—
371#376—1461no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#336—731no—
381#336—731no—
382C22×D4—21no—
383C22×D4C22×D7311yescharacteristic
384C23×C4—21no—
385C23×C4C22⋊D7311yescharacteristic
386C23×C4C22⋊D7311yescharacteristic
387C23×C4C22×D7311yescharacteristic
388C22×C8—731no—
389C22×C8—731no—
390C22×C8—731no—
391C22×C8—731no—
392#121—22no—
393#121—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—44no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42C22⋊D7311yescharacteristic
418C2×C42C22⋊D7311yescharacteristic
419C2×C42C22×D7311yescharacteristic
420C2×C42C22×D7311yescharacteristic
421#107—1462no—
422#107—1462no—
423#106—1462no—
424#106—1462no—
425#102—1462no—
426#102—1462no—
427#102—1462no—
428#102—1462no—
429C22⋊C8—1462no—
430C22⋊C8—1462no—
431C4×C8—1462no—
432C4×C8—1462no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95—1461no—
444C24—21no—
445C24C2×C22⋊D7311yescharacteristic
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4#1192402812yes—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
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—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C23×D7311yescharacteristic
483C2×C8—732no—
484C2×C8—732no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
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#1192402812yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—2922no—
516C22⋊C4—2922no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—42no—
522C42—2922no—
523C42—2922no—
524C42#1192402712yes—
525C42#1192402712yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23#1455661112yes—
540C23C22×C22⋊D7311yescharacteristic
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
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—84no—
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—2921no—
585C2×C4#1455657212yes—
586C2×C4#1455657212yes—
587C2×C4#1455661112yes—
588C2×C4#1455661712yes—
589C2×C4#41×D7311yescharacteristic
590C2×C4#41×D7311yescharacteristic
591C2×C4C2×#1192402711yescharacteristic
592C2×C4C2×#1192402711yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C2×C4C2×#1192402811yescharacteristic
595C8—2921no—
596C8—2921no—
597C8—1462no—
598C8—1462no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22C2×#1455657211yescharacteristic
614C22#1798909811yescharacteristic
615C22#1798913311yescharacteristic
616C22C2×#1455661111yescharacteristic
617C22C2×#1455661111yescharacteristic
618C22C2×#1455661711yescharacteristic
619C22#1798938511yescharacteristic
620C4—22no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—5841no—
628C4—5841no—
629C4—5841no—
630C4—5841no—
631C4#1798890814yes—
632C4#1798911711yescharacteristic
633C4#1798911711yescharacteristic
634C4#1798911911yescharacteristic
635C4#1798911911yescharacteristic
636C2—42no—
637C2—82no—
638C2#2261319011yescharacteristic
639C2C2×#1798890811yescharacteristic
640C2#2261984011yescharacteristic
641C2#2261994911yescharacteristic
642C2#2261997811yescharacteristic
643C2#2262049511yescharacteristic
644C2#2262052011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#121×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC4×#121D73
hypocenterC73#23758
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #22612329C2semidirect
2. #22611499C2semidirect
3. #22611452C2semidirect
4. #22611412C2semidirect
5. #22611368C2semidirect
6. #22611362C2semidirect
7. #22611344C2semidirect
10. C4×#14556524C2semidirect
11. C4×#14556522C2semidirect
12. C4×#14556512C2semidirect
13. C4×#14556512C2semidirect
14. C4×C22⋊#10596831C2semidirect
338. C73#23758semidirect

Maximal subgroups

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

subgroupindex
#226123292
#226114992
#226114522
#226114122
#226113682
#226113622
#226113442
#226111172
#226111102
C4×#145565242
C4×#145565222
C4×#145565122
C4×C22⋊#105968312
C4×#121×C732
#2375873