Finite Group Databasebeta

← #32170505

Subgroups — #32170505

Order 18176.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596650C212yesmaximal, complemented
2#22596627C212yesmaximal
3#22594454C211yesmaximal, characteristic, complemented
4#22594454C211yesmaximal, characteristic, complemented
5#22594145C212yesmaximal
6#22593614C214yesmaximal, complemented
7#22593424C214yesmaximal, complemented
8#22593058C214yesmaximal, complemented
9#22592361C214yesmaximal, complemented
10#22591073C212yesmaximal, complemented
11#22590938C212yesmaximal
12C4×#14556107C212yesmaximal, complemented
13C2×C4×C22⋊C4×C71C211yesmaximal, characteristic
14C2×#14556120C2212yescomplemented
15C2×#14556107C2212yes—
16C2×#14556107C2212yes—
17C2×#14556107C2212yes—
18C2×#14556107C2212yes—
19C2×C4×Dic142C2212yescomplemented
20#17985935C2214yes—
21#17985931C2214yescomplemented
22#17985931C2214yescomplemented
23#17985922C2214yes—
24#17985922C2214yes—
25#17985920C2214yescomplemented
26#17985920C2214yescomplemented
27#17985911C2214yes—
28#17985905C2212yescomplemented
29#17985903C2214yescomplemented
30#17985897C2212yescomplemented
31#17985897C2212yescomplemented
32#17985897C2212yescomplemented
33#17985896C2214yescomplemented
34#17985892C2214yes—
35#17985892C2214yes—
36#17985892C2214yes—
37#17985892C2214yes—
38#17985886C2214yescomplemented
39#17985883C2214yescomplemented
40#17985883C2214yescomplemented
41#17985883C2214yescomplemented
42#17985882C2214yescomplemented
43#17985872C2214yes—
44#17985872C2214yes—
45#17985872C2214yes—
46#17985872C2214yes—
47#17985865C2212yescomplemented
48#17985859C2212yes—
49#17985859C2212yes—
50#17985859C2212yes—
51#17985859C2212yes—
52#17985859C2212yes—
53#17985859C2212yes—
54#17985855C2214yes—
55C4×Dic71⋊C4C2214yescomplemented
56C4×C4⋊Dic71C2212yescomplemented
57C4×C22⋊Dic71C2214yes—
58C24×C4×C71C2211yescharacteristic
59C22×C22⋊C4×C71C2211yescharacteristic
60C22×C22⋊C4×C71C2211yescharacteristic
61C22×C42×C71C2211yescharacteristic
62C22×C42×C71C2211yescharacteristic
63C4×C22⋊C4×C71C2214yes—
64C4×C22⋊C4×C71C2214yes—
65C2×#95×C71C2211yescharacteristic
66C2×#95×C71C2211yescharacteristic
67C22×Dic142C2312yes—
68C22×Dic142C2312yes—
69#14556120—44no—
70#14556107—28no—
71#14556107—28no—
72#14556107—44no—
73#14556107—44no—
74C4×Dic142—44no—
75C2×C4⋊Dic71C2312yes—
76C2×C4⋊Dic71C2312yes—
77C2×C4⋊Dic71C2312yes—
78C2×C4⋊Dic71C2312yes—
79C2×C4⋊Dic71C2312yes—
80C2×C4⋊Dic71C2312yes—
81C2×C4⋊Dic71C2312yes—
82C2×C4⋊Dic71C2312yes—
83C2×C4⋊Dic71C2312yes—
84C2×C4⋊Dic71C2312yes—
85C2×Dic71⋊C4C2314yes—
86C2×Dic71⋊C4C2314yes—
87C2×Dic71⋊C4C2314yes—
88C2×Dic71⋊C4C2314yes—
89C2×Dic71⋊C4C2314yes—
90C2×Dic71⋊C4C2314yes—
91C2×C22⋊Dic71C2314yes—
92C2×C22⋊Dic71C2314yes—
93C2×C22⋊Dic71C2314yes—
94C2×C22⋊Dic71C2314yes—
95C2×C22⋊Dic71C2314yes—
96C2×C22⋊Dic71C2314yes—
97C2×C4×Dic71C2314yes—
98C2×C4×Dic71C2314yes—
99C71⋊#95C2314yes—
100C71⋊#95C2314yes—
101C71⋊#95C2314yes—
102C71⋊#95C2314yes—
103C71⋊#95C2314yes—
104C71⋊#95C2314yes—
105C71⋊#95C2314yes—
106C71⋊#95C2314yes—
107C71⋊#95C2314yes—
108C71⋊#95C2314yes—
109C25×C71C2311yescharacteristic
110C23×C4×C71—24no—
111C23×C4×C71—42no—
112C23×C4×C71—42no—
113C23×C4×C71C2311yescharacteristic
114C23×C4×C71C2312yes—
115C23×C4×C71C2312yes—
116C23×C4×C71C2311yescharacteristic
117C23×C4×C71C2311yescharacteristic
118C23×C4×C71C2311yescharacteristic
119C23×C4×C71C2311yescharacteristic
120C23×C4×C71C2311yescharacteristic
121C2×C22⋊C4×C71—24no—
122C2×C22⋊C4×C71Q818yescomplemented
123C2×C22⋊C4×C71C2314yes—
124C2×C22⋊C4×C71C2314yes—
125C2×C42×C71—42no—
126C2×C42×C71—42no—
127C2×C42×C71C2312yes—
128C2×C42×C71C2312yes—
129C2×C42×C71C2312yes—
130C2×C42×C71C2312yes—
131#95×C71C2314yes—
132#95×C71C2314yes—
133C2×Dic142—24no—
134C2×Dic142—24no—
135C2×Dic142—42no—
136C2×Dic142—42no—
137C2×Dic142—42no—
138C2×Dic142—42no—
139C22×Dic71C2414yes—
140C22×Dic71C2414yes—
141C22×Dic71C2414yes—
142C22×Dic71C2414yes—
143C4⋊Dic71—24no—
144C4⋊Dic71—24no—
145C4⋊Dic71—42no—
146C4⋊Dic71—42no—
147C4⋊Dic71—42no—
148C4⋊Dic71—42no—
149C4⋊Dic71—42no—
150C4⋊Dic71—42no—
151C4⋊Dic71—42no—
152C4⋊Dic71—42no—
153Dic71⋊C4—28no—
154Dic71⋊C4—28no—
155Dic71⋊C4—28no—
156Dic71⋊C4—28no—
157Dic71⋊C4—44no—
158Dic71⋊C4—44no—
159C22⋊Dic71—28no—
160C22⋊Dic71—28no—
161C22⋊Dic71—44no—
162C22⋊Dic71—44no—
163C22⋊Dic71—44no—
164C22⋊Dic71—44no—
165C4×Dic71—44no—
166C4×Dic71—44no—
167C24×C71—22no—
168C24×C71—22no—
169C24×C71—41no—
170C24×C71—41no—
171C24×C71—41no—
172C24×C71—41no—
173C24×C71C2×Q814yes—
174C24×C71C2412yes—
175C24×C71C2411yescharacteristic
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—24no—
181C22×C4×C71—24no—
182C22×C4×C71—24no—
183C22×C4×C71—24no—
184C22×C4×C71—41no—
185C22×C4×C71—41no—
186C22×C4×C71—41no—
187C22×C4×C71—41no—
188C22×C4×C71—42no—
189C22×C4×C71—42no—
190C22×C4×C71—41no—
191C22×C4×C71—41no—
192C22×C4×C71—41no—
193C22×C4×C71—41no—
194C22×C4×C71—41no—
195C22×C4×C71—41no—
196C22×C4×C71—416no—
197C22×C4×C71C2×Q814yes—
198C22×C4×C71C2×Q814yes—
199C22×C4×C71#4114yes—
200C22×C4×C71#4114yes—
201C22×C4×C71C2412yes—
202C22×C4×C71C2412yes—
203C22×C4×C71C2412yes—
204C22×C4×C71C2412yes—
205C22×C4×C71C2412yes—
206C22×C4×C71C2411yescharacteristic
207C22×C4×C71C2411yescharacteristic
208C22⋊C4×C71—28no—
209C22⋊C4×C71C2×Q8116yes—
210C42×C71—24no—
211C42×C71—24no—
212C42×C71—42no—
213C42×C71—42no—
214C23×C71—21no—
215C23×C71—21no—
216C23×C71—21no—
217C23×C71—21no—
218C23×C71—28no—
219C23×C71—22no—
220C23×C71—22no—
221C23×C71—22no—
222C23×C71—22no—
223C23×C71—22no—
224C23×C71—22no—
225C23×C71—44no—
226C23×C71—44no—
227C23×C71—44no—
228C23×C71—44no—
229C23×C71—44no—
230C23×C71—44no—
231C23×C71C22×Q814yes—
232C23×C71C22×Q812yes—
233C23×C71C2×#4112yes—
234C23×C71C2×#4112yes—
235C23×C71C2511yescharacteristic
236C2×C4×C71—28no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—22no—
240C2×C4×C71—22no—
241C2×C4×C71—22no—
242C2×C4×C71—21no—
243C2×C4×C71—21no—
244C2×C4×C71—22no—
245C2×C4×C71—22no—
246C2×C4×C71—22no—
247C2×C4×C71—22no—
248C2×C4×C71—22no—
249C2×C4×C71—22no—
250C2×C4×C71—22no—
251C2×C4×C71—22no—
252C2×C4×C71—44no—
253C2×C4×C71—44no—
254C2×C4×C71—44no—
255C2×C4×C71—44no—
256C2×C4×C71—48no—
257C2×C4×C71—48no—
258C2×C4×C71—44no—
259C2×C4×C71C22×Q814yes—
260C2×C4×C71C22×Q814yes—
261C2×C4×C71C2×#4114yes—
262C2×C4×C71C2×#4114yes—
263C2×C4×C71H(5,𝔽2)12yes—
264C2×C4×C71H(5,𝔽2)12yes—
265C2×Dic71—24no—
266C2×Dic71—24no—
267C2×Dic71—24no—
268C2×Dic71—24no—
269C2×Dic71—24no—
270C2×Dic71—24no—
271C2×Dic71—24no—
272C2×Dic71—24no—
273C2×Dic71—24no—
274C2×Dic71—24no—
275C2×Dic71—24no—
276C2×Dic71—24no—
277Dic142—48no—
278Dic142—84no—
279C22×C71—22no—
280C22×C71—22no—
281C22×C71—22no—
282C22×C71—22no—
283C22×C71—22no—
284C22×C71—22no—
285C22×C71—24no—
286C22×C71—24no—
287C22×C71—24no—
288C22×C71—416no—
289C22×C71—42no—
290C22×C71—42no—
291C22×C71—42no—
292C22×C71—42no—
293C22×C71#53314yes—
294C22×C71C23×Q811yescharacteristic
295C22×C71C22×#4111yescharacteristic
296C22×C71C22×#4111yescharacteristic
297C22×C71C2×H(5,𝔽2)11yescharacteristic
298C22×C71C2×H(5,𝔽2)11yescharacteristic
299C22×C71#58511yescharacteristic
300C22×C71#58511yescharacteristic
301C4×C71—24no—
302C4×C71—24no—
303C4×C71—22no—
304C4×C71—22no—
305C4×C71—24no—
306C4×C71—24no—
307C4×C71—42no—
308Dic71—44no—
309Dic71—44no—
310Dic71—44no—
311Dic71—44no—
312#35012—711nomaximal, Sylow
313C2×C71—24no—
314C2×C71—44no—
315C2×C71C2×#53311yescharacteristic
316C2×C71#345911yescharacteristic
317C2×C71#346011yescharacteristic
318C2×C71#349011yescharacteristic
319C2×C71#349311yescharacteristic
320C2×C71#350111yescharacteristic
321C2×C71#350111yescharacteristic
322#2851—712no—
323#2848—712no—
324#2659—711no—
325#2659—711no—
326#2626—712no—
327#2574—714no—
328#2556—714no—
329#2522—714no—
330#2465—714no—
331#2329—712no—
332#2312—712no—
333C4×C22⋊Q8—712no—
334C2×C4×C22⋊C4D7111yescharacteristic
335C71#3501211yescharacteristic, complemented, Sylow
336C24×C4C2×D7111yescharacteristic
337C2×#125—712no—
338C2×C22⋊Q8—712no—
339C2×C22⋊Q8—712no—
340C2×C22⋊Q8—712no—
341C2×C22⋊Q8—712no—
342C2×C4×Q8—712no—
343C22×C22⋊C4C2×D7111yescharacteristic
344C22×C22⋊C4C2×D7111yescharacteristic
345C22×C42C2×D7111yescharacteristic
346C22×C42C2×D7111yescharacteristic
347#399—714no—
348#398—714no—
349#398—714no—
350#396—714no—
351#396—714no—
352#395—714no—
353#395—714no—
354#393—714no—
355#391—712no—
356#391—714no—
357#389—712no—
358#389—712no—
359#389—712no—
360#389—714no—
361#388—714no—
362#388—714no—
363#388—714no—
364#388—714no—
365#387—714no—
366#387—714no—
367#387—714no—
368#387—714no—
369#387—714no—
370#385—714no—
371#385—714no—
372#385—714no—
373#385—714no—
374#382—712no—
375#380—712no—
376#380—712no—
377#380—712no—
378#380—712no—
379#380—712no—
380#380—712no—
381#379—714no—
382C4×C4⋊C4—714no—
383C4×C4⋊C4—712no—
384C4×C22⋊C4—714no—
385C4×C22⋊C4C2×D7114yes—
386C4×C22⋊C4C2×D7114yes—
387C2×#95C2×D7111yescharacteristic
388C2×#95C2×D7111yescharacteristic
389C25C22×D7111yescharacteristic
390C22×Q8—712no—
391C22×Q8—712no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4C22×D7111yescharacteristic
396C23×C4C22×D7112yes—
397C23×C4C22×D7112yes—
398C23×C4C22×D7111yescharacteristic
399C23×C4C22×D7111yescharacteristic
400C23×C4C22×D7111yescharacteristic
401C23×C4C22×D7111yescharacteristic
402C23×C4C22×D7111yescharacteristic
403#125—2844no—
404C22⋊Q8—2844no—
405C22⋊Q8—2844no—
406C22⋊Q8—1428no—
407C22⋊Q8—1428no—
408C4×Q8—2844no—
409C2×C4⋊C4—712no—
410C2×C4⋊C4—712no—
411C2×C4⋊C4—712no—
412C2×C4⋊C4—712no—
413C2×C4⋊C4—712no—
414C2×C4⋊C4—714no—
415C2×C4⋊C4—712no—
416C2×C4⋊C4—712no—
417C2×C4⋊C4—712no—
418C2×C4⋊C4—714no—
419C2×C4⋊C4—714no—
420C2×C4⋊C4—714no—
421C2×C4⋊C4—714no—
422C2×C4⋊C4—712no—
423C2×C4⋊C4—712no—
424C2×C4⋊C4—714no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—714no—
427C2×C22⋊C4—714no—
428C2×C22⋊C4—714no—
429C2×C22⋊C4—714no—
430C2×C22⋊C4—714no—
431C2×C22⋊C4—714no—
432C2×C22⋊C4Dic14218yescomplemented
433C2×C22⋊C4C22×D7114yes—
434C2×C22⋊C4C22×D7114yes—
435C2×C42—42no—
436C2×C42—42no—
437C2×C42—714no—
438C2×C42—714no—
439C2×C42C22×D7112yes—
440C2×C42C22×D7112yes—
441C2×C42C22×D7112yes—
442C2×C42C22×D7112yes—
443#95—714no—
444#95—714no—
445#95—714no—
446#95—714no—
447#95—714no—
448#95—714no—
449#95—714no—
450#95—714no—
451#95—714no—
452#95—714no—
453#95C22×D7114yes—
454#95C22×D7114yes—
455C24—22no—
456C24—22no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24C2×Dic14214yes—
462C24C23×D7112yes—
463C24C23×D7111yescharacteristic
464C2×Q8—2842no—
465C2×Q8—2842no—
466C2×Q8—2842no—
467C2×Q8—2842no—
468C2×Q8—1424no—
469C2×Q8—1424no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—416no—
491C22×C4—714no—
492C22×C4—714no—
493C22×C4—714no—
494C22×C4—714no—
495C22×C4C2×Dic14214yes—
496C22×C4C2×Dic14214yes—
497C22×C4#1176586514yes—
498C22×C4#1176586514yes—
499C22×C4C23×D7112yes—
500C22×C4C23×D7112yes—
501C22×C4C23×D7112yes—
502C22×C4C23×D7112yes—
503C22×C4C23×D7112yes—
504C22×C4C23×D7111yescharacteristic
505C22×C4C23×D7111yescharacteristic
506C4⋊C4—2844no—
507C4⋊C4—2844no—
508C4⋊C4—2842no—
509C4⋊C4—2842no—
510C4⋊C4—2842no—
511C4⋊C4—2842no—
512C4⋊C4—2842no—
513C4⋊C4—2842no—
514C4⋊C4—2842no—
515C4⋊C4—2842no—
516C4⋊C4—1428no—
517C4⋊C4—1428no—
518C4⋊C4—1424no—
519C4⋊C4—1424no—
520C4⋊C4—1428no—
521C4⋊C4—1428no—
522C22⋊C4—28no—
523C22⋊C4—2844no—
524C22⋊C4—2844no—
525C22⋊C4—2844no—
526C22⋊C4—2844no—
527C22⋊C4—1428no—
528C22⋊C4—1428no—
529C22⋊C4C2×Dic142116yes—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—2844no—
535C42—2844no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C22×Dic14214yes—
554C23C22×Dic14212yes—
555C23C2×#1176586512yes—
556C23C2×#1176586512yes—
557C23C24×D7111yescharacteristic
558Q8—2848no—
559Q8—5684no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—1424no—
584C2×C4—1424no—
585C2×C4—1424no—
586C2×C4—1424no—
587C2×C4—1424no—
588C2×C4—1424no—
589C2×C4—1424no—
590C2×C4—1424no—
591C2×C4—1424no—
592C2×C4—1424no—
593C2×C4—1424no—
594C2×C4—1424no—
595C2×C4C22×Dic14214yes—
596C2×C4C22×Dic14214yes—
597C2×C4C2×#1176586514yes—
598C2×C4C2×#1176586514yes—
599C2×C4#1455617312yes—
600C2×C4#1455617312yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1798655614yes—
616C22C23×Dic14211yescharacteristic
617C22C22×#1176586511yescharacteristic
618C22C22×#1176586511yescharacteristic
619C22C2×#1455617311yescharacteristic
620C22C2×#1455617311yescharacteristic
621C22#1798683511yescharacteristic
622C22#1798683511yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—24no—
628C4—24no—
629C4—42no—
630C4—2844no—
631C4—2844no—
632C4—2844no—
633C4—2844no—
634C2—24no—
635C2—44no—
636C2C2×#1798655611yescharacteristic
637C2#2260174911yescharacteristic
638C2#2260176211yescharacteristic
639C2#2260202911yescharacteristic
640C2#2260206011yescharacteristic
641C2#2260214311yescharacteristic
642C2#2260214311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC2×C4×C22⋊C4×C71C2
prime radicalC25×C71C23
hypercenterC2×C4×C22⋊C4D71
hypocenterC71#35012
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22596650C2semidirect
3. #22594454C2semidirect
4. #22594454C2semidirect
6. #22593614C2semidirect
7. #22593424C2semidirect
8. #22593058C2semidirect
9. #22592361C2semidirect
10. #22591073C2semidirect
12. C4×#14556107C2semidirect
14. C2×#14556120C22semidirect
19. C2×C4×Dic142C22semidirect
21. #17985931C22semidirect
22. #17985931C22semidirect
25. #17985920C22semidirect
26. #17985920C22semidirect
28. #17985905C22semidirect
29. #17985903C22semidirect
30. #17985897C22semidirect
31. #17985897C22semidirect
32. #17985897C22semidirect
33. #17985896C22semidirect
38. #17985886C22semidirect
39. #17985883C22semidirect
40. #17985883C22semidirect
41. #17985883C22semidirect
42. #17985882C22semidirect
47. #17985865C22semidirect
55. C4×Dic71⋊C4C22semidirect
56. C4×C4⋊Dic71C22semidirect
122. C2×C22⋊C4×C71Q8semidirect
335. C71#35012semidirect
432. C2×C22⋊C4Dic142semidirect

Maximal subgroups

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

subgroupindex
#225966502
#225966272
#225944542
#225941452
#225936142
#225934242
#225930582
#225923612
#225910732
#225909382
C4×#145561072
C2×C4×C22⋊C4×C712
#3501271