Finite Group Databasebeta

← #32001785

Subgroups — #32001785

Order 18176.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986099C211yesmaximal, characteristic
2#22590817C211yesmaximal, characteristic, complemented
3#22589220C211yesmaximal, characteristic
4#22589193C211yesmaximal, characteristic
5#22589075C211yesmaximal, characteristic, complemented
6#22589070C211yesmaximal, characteristic, complemented
7#22589040C211yesmaximal, characteristic
8#22589029C211yesmaximal, characteristic, complemented
9#22589026C211yesmaximal, characteristic
10#22589015C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12#22588173C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic
14C4×#14556039C211yesmaximal, characteristic, complemented
15C71×#2293C211yesmaximal, characteristic
16C22×#11765853C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18#17986099C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556051C2211yescharacteristic
23C2×#14556051C2211yescharacteristic
24C2×#14556051C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×#14556039C2211yescharacteristic
32C2×C4×C71⋊C8C2211yescharacteristic
33C2×C4×C71⋊C8C2211yescharacteristic
34#17985905C2211yescharacteristic
35#17985905C2211yescharacteristic
36#17985897C2211yescharacteristic
37#17985897C2211yescharacteristic
38#17985883—21no—
39#17985883—21no—
40C4×Dic71⋊C4—21no—
41C4×C4⋊Dic71C2211yescharacteristic
42#17985845—21no—
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×#117×C71—21no—
51#390×C71C2211yescharacteristic
52#390×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#388×C71—21no—
56#388×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C43×C71C2211yescharacteristic
60C22×Dic142C2311yescharacteristic
61C2×#11765853—22no—
62C2×#11765853—22no—
63C2×#11765853C2×C414yes—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×Dic142—22no—
69C4×Dic142C2×C414yes—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78#14556061—22no—
79#14556061C2×C414yes—
80#14556059—22no—
81#14556051—22no—
82#14556051—22no—
83#14556051C2×C414yes—
84#14556049—22no—
85#14556049—22no—
86#14556049C2×C414yes—
87#14556039—22no—
88#14556039—22no—
89#14556039—22no—
90#14556039C2×C414yes—
91C4×C71⋊C8—22no—
92C4×C71⋊C8—22no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C22×D4×C71C2311yescharacteristic
101C23×C4×C71—21no—
102C4×D4×C71—22no—
103C4×D4×C71C2×C414yes—
104#117×C71—44no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C22⋊C4×C71—21no—
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71D411yescharacteristic
119C2×C42×C71D411yescharacteristic
120C2×C42×C71C2311yescharacteristic
121C2×C42×C71C2311yescharacteristic
122C2×C42×C71C2311yescharacteristic
123#95×C71—21no—
124#95×C71—21no—
125#95×C71—21no—
126C2×Dic142—21no—
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142—21no—
130C2×Dic142—21no—
131C2×Dic142C22×C412yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C22×Dic71—21no—
136#11765853—28no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8C22×C412yes—
144C2×C71⋊C8C22×C412yes—
145C2×C71⋊C8#4112yes—
146C2×C71⋊C8#4112yes—
147C2×C71⋊C8#4112yes—
148C2×C71⋊C8#4112yes—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71C22×C412yes—
155Dic71⋊C4—24no—
156Dic71⋊C4—42no—
157Dic71⋊C4—42no—
158C4×Dic71—24no—
159C4×Dic71—42no—
160C24×C71—21no—
161C2×D4×C71—21no—
162C2×D4×C71—21no—
163C2×D4×C71—21no—
164C2×D4×C71—21no—
165C2×D4×C71—21no—
166C2×D4×C71C22×C412yes—
167C22×C4×C71—21no—
168C22×C4×C71—24no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—42no—
172C22×C4×C71—42no—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—42no—
186C4⋊C4×C71—42no—
187C4⋊C4×C71C22×C412yes—
188C22⋊C4×C71—24no—
189C22⋊C4×C71—42no—
190C22⋊C4×C71—42no—
191C22⋊C4×C71—42no—
192C22⋊C4×C71—42no—
193C42×C71—22no—
194C42×C71—22no—
195C42×C71—21no—
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—41no—
200C42×C71—41no—
201C42×C71—42no—
202C42×C71C22×C412yes—
203C42×C71#4112yes—
204C23×C71—22no—
205C23×C71—41no—
206C23×C71—41no—
207C23×C71—41no—
208C23×C71—41no—
209C23×C71—41no—
210C23×C71—41no—
211C23×C71C22×D411yescharacteristic
212D4×C71—22no—
213D4×C71—22no—
214D4×C71—41no—
215D4×C71—41no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—22no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—48no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71C4×D412yes—
241C2×C4×C71C4×D412yes—
242C2×C4×C71AGL(1,ℤ/8)12yes—
243C2×C4×C71#13712yes—
244C2×C4×C71C23×C411yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262Dic142—22no—
263Dic142—22no—
264Dic142—41no—
265Dic142—41no—
266C71⋊C8—22no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C71⋊C8#11714yescomplemented
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—42no—
273C22×C71—81no—
274C22×C71—81no—
275C22×C71—81no—
276C22×C71—81no—
277C22×C71C2×C4×D411yescharacteristic
278C22×C71#53211yescharacteristic
279C22×C71#53211yescharacteristic
280C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C71C2×#13711yescharacteristic
282C22×C71#57511yescharacteristic
283C22×C71#57511yescharacteristic
284C4×C71—22no—
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—41no—
289C4×C71—41no—
290C4×C71—42no—
291C4×C71—81no—
292C4×C71—81no—
293C4×C71C2×#11711yescharacteristic
294C4×C71C4×#4111yescharacteristic
295C4×C71#52911yescharacteristic
296C4×C71#52911yescharacteristic
297Dic71—42no—
298Dic71—42no—
299Dic71—81no—
300Dic71—81no—
301#21288—711nomaximal, Sylow
302C2×C71—82no—
303C2×C71#229311yescharacteristic
304C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
305C2×C71C4×#13711yescharacteristic
306C2×C71#318011yescharacteristic
307C2×C71#318111yescharacteristic
308C2×C71#318211yescharacteristic
309C2×C71#318311yescharacteristic
310C2×#440—711no—
311#2295—711no—
312#2293D7111yescharacteristic
313#1973—711no—
314#1968—711no—
315#1943—711no—
316#1942—711no—
317#1936—711no—
318#1935—711no—
319#1934—711no—
320#1933—711no—
321#1776—711no—
322#1763—711no—
323#1762—711no—
324C4×#97—711no—
325C71#2128811yescharacteristic, complemented, Sylow
326C22×SD16—711no—
327C2×C4×Q8—711no—
328C2×C4×D4C2×D7111yescharacteristic
329C2×#117—21no—
330#440—718no—
331C2×#107—711no—
332C2×#107—711no—
333C2×#106—711no—
334C2×#103—711no—
335C2×#103—711no—
336C2×#103—711no—
337C2×#102—711no—
338C2×#102—711no—
339C2×#102—711no—
340C2×#97—711no—
341C2×#97—711no—
342C2×#97—711no—
343C2×#97—711no—
344C2×C4×C8—711no—
345C2×C4×C8—711no—
346#391—711no—
347#391—711no—
348#390C2×D7111yescharacteristic
349#390C2×D7111yescharacteristic
350#389—711no—
351#389—711no—
352#389C2×D7111yescharacteristic
353#389C2×D7111yescharacteristic
354#388—21no—
355#388—21no—
356#387—1421no—
357#387—1421no—
358C4×C4⋊C4—711no—
359C4×C4⋊C4—1421no—
360C4×C4⋊C4C2×D7111yescharacteristic
361C4×C22⋊C4—21no—
362#376—1421no—
363C43C2×D7111yescharacteristic
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370C22×Q8—711no—
371C22×D4C22×D7111yescharacteristic
372C23×C4—21no—
373C2×SD16—714no—
374C2×SD16—1422no—
375C2×SD16—1422no—
376C22×C8—711no—
377C22×C8—711no—
378C22×C8—711no—
379C22×C8—711no—
380C4×Q8—714no—
381C4×Q8—1422no—
382C4×D4—22no—
383C4×D4C4×D7114yes—
384#117—44no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—21no—
402C2×C42—1421no—
403C2×C42—1421no—
404C2×C42C22⋊D7111yescharacteristic
405C2×C42C22⋊D7111yescharacteristic
406C2×C42C22⋊D7111yescharacteristic
407C2×C42C22⋊D7111yescharacteristic
408C2×C42C22×D7111yescharacteristic
409C2×C42C22×D7111yescharacteristic
410C2×C42C22×D7111yescharacteristic
411#107—714no—
412#107—1422no—
413#106—1422no—
414#103—714no—
415#103—1422no—
416#103—1422no—
417#102—714no—
418#102—1422no—
419#102—1422no—
420#97—714no—
421#97—1422no—
422#97—1422no—
423#97—1422no—
424C4×C8—1422no—
425C4×C8—1422no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1421no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436C24—21no—
437C2×Q8—712no—
438C2×Q8—1421no—
439C2×Q8—1421no—
440C2×Q8—1421no—
441C2×Q8—1421no—
442C2×Q8—1421no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D7112yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C23×D7111yescharacteristic
466SD16—1428no—
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—712no—
470C2×C8—712no—
471C2×C8—712no—
472C2×C8—712no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—2842no—
487C4⋊C4—2842no—
488C4⋊C4—712no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1424no—
495C4⋊C4C2×C4×D7112yes—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—2842no—
511C42—1424no—
512C42C2×C4×D7112yes—
513C42#1176586512yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊D7111yescharacteristic
522Q8—2841no—
523Q8—2841no—
524Q8—1422no—
525Q8—1422no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—48no—
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—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—1422no—
565C2×C4—1422no—
566C2×C4C4×C22⋊D7112yes—
567C2×C4C4×C22⋊D7112yes—
568C2×C4#1455615412yes—
569C2×C4#1455615812yes—
570C2×C4C22×C4×D7111yescharacteristic
571C2×C4#41×D7111yescharacteristic
572C2×C4#41×D7111yescharacteristic
573C2×C4#41×D7111yescharacteristic
574C2×C4#41×D7111yescharacteristic
575C2×C4C2×#1176586511yescharacteristic
576C8—714no—
577C8—1422no—
578C8—1422no—
579C8—1422no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×C4×C22⋊D7111yescharacteristic
588C22#1798654511yescharacteristic
589C22#1798654511yescharacteristic
590C22C2×#1455615411yescharacteristic
591C22C2×#1455615811yescharacteristic
592C22#1798679711yescharacteristic
593C22#1798679711yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—2842no—
604C4—2842no—
605C4—5681no—
606C4—5681no—
607C4#117×D7111yescharacteristic
608C4#1798646711yescharacteristic
609C4#1798652711yescharacteristic
610C4#1798652711yescharacteristic
611C2—82no—
612C2#2259079911yescharacteristic
613C2C4×#1455615411yescharacteristic
614C2C4×#1455615811yescharacteristic
615C2#2259903211yescharacteristic
616C2#2259905011yescharacteristic
617C2#2259905411yescharacteristic
618C2#2259906311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2293C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2293D71
hypocenterC71#21288
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #22590817C2semidirect
5. #22589075C2semidirect
6. #22589070C2semidirect
8. #22589029C2semidirect
10. #22589015C2semidirect
11. #22588238C2semidirect
12. #22588173C2semidirect
14. C4×#14556039C2semidirect
18. #17986099C4semidirect
269. C71⋊C8#117semidirect
325. C71#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#179860992
#225908172
#225892202
#225891932
#225890752
#225890702
#225890402
#225890292
#225890262
#225890152
#225882382
#225881732
#225881652
C4×#145560392
C71×#22932
#2128871