Finite Group Databasebeta

← #32002325

Subgroups — #32002325

Order 18176.

938 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986054C211yesmaximal, characteristic
2#22590736C211yesmaximal, characteristic
3C4×C4⋊C4×D71C211yesmaximal, characteristic
4#22589111C211yesmaximal, characteristic, complemented
5#22589109C211yesmaximal, characteristic, complemented
6#22589107C211yesmaximal, characteristic, complemented
7#22589105C211yesmaximal, characteristic, complemented
8#22588534C211yesmaximal, characteristic
9#22588531C211yesmaximal, characteristic
10#22588529C211yesmaximal, characteristic
11#22588526C211yesmaximal, characteristic
12#22588238C211yesmaximal, characteristic, complemented
13#22588238C211yesmaximal, characteristic, complemented
14#22588237C211yesmaximal, characteristic, complemented
15C71×#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D71C2211yescharacteristic
17C2×C4⋊C4×D71C2211yescharacteristic
18C2×C42×D71—21no—
19C2×C42×D71C2211yescharacteristic
20#17986054C418yescomplemented
21C2×#14556062C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556061C2211yescharacteristic
24C2×#14556060C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×Dic71⋊C8C2211yescharacteristic
28C2×Dic71⋊C8C2211yescharacteristic
29C2×(C2×D71)⋊C8C2211yescharacteristic
30C2×(C2×D71)⋊C8C2211yescharacteristic
31C2×#14556039C2211yescharacteristic
32#17985866—21no—
33#17985864C2211yescharacteristic
34#17985864C2211yescharacteristic
35#17985861—21no—
36#17985856C2211yescharacteristic
37#17985856C2211yescharacteristic
38C4⋊C4×Dic71C2211yescharacteristic
39C4⋊C4×Dic71C2211yescharacteristic
40C4×Dic71⋊C4—21no—
41C4×C4⋊Dic71C2211yescharacteristic
42C4×C4⋊Dic71C2211yescharacteristic
43#17985850—21no—
44C4×(C2×D71)⋊C4—21no—
45#17985845—21no—
46#95×D71—21no—
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54C2×#106×C71C2211yescharacteristic
55C2×#97×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C4×D71—21no—
61C22×C4×D71—21no—
62C22×C4×D71D411yescharacteristic
63C22×C4×D71D411yescharacteristic
64C22×C4×D71C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C4⋊C4×D71—22no—
74C4⋊C4×D71—22no—
75C4⋊C4×D71—22no—
76C4⋊C4×D71—22no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71D411yescharacteristic
84C2×C4×Dic71D411yescharacteristic
85C2×C4×Dic71C2311yescharacteristic
86C42×D71—44no—
87C42×D71C2×C414yes—
88C42×D71Q814yes—
89#14556062—22no—
90#14556061—22no—
91#14556061—22no—
92#14556060—22no—
93#14556059—22no—
94#14556059—22no—
95Dic71⋊C8C2×C414yescomplemented
96Dic71⋊C8C2×C414yescomplemented
97(C2×D71)⋊C8C2×C414yes—
98(C2×D71)⋊C8C2×C414yes—
99#14556039C2×C414yescomplemented
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C22×C8×C71C2311yescharacteristic
111C22×C8×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C42×C71—21no—
117C2×C42×C71—21no—
118C2×C42×C71C2311yescharacteristic
119#107×C71—22no—
120#106×C71—22no—
121#97×C71C2×C414yescomplemented
122#95×C71—21no—
123#95×C71—21no—
124C23×D71C2×D411yescharacteristic
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C22×Dic71C2×D411yescharacteristic
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—21no—
135C2×C4×D71—21no—
136C2×C4×D71—21no—
137C2×C4×D71—21no—
138C2×C4×D71—24no—
139C2×C4×D71—24no—
140C2×C4×D71—41no—
141C2×C4×D71—41no—
142C2×C4×D71—42no—
143C2×C4×D71—42no—
144C2×C4×D71—42no—
145C2×C4×D71—42no—
146C2×C4×D71C4⋊C414yes—
147C2×C4×D71C4⋊C414yes—
148C2×C4×D71C22×C412yes—
149C2×C4×D71C2×Q812yes—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8C22×C412yes—
155C2×C71⋊C8C22×C412yes—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71#4112yes—
163C4⋊Dic71#4112yes—
164Dic71⋊C4—24no—
165Dic71⋊C4—24no—
166(C2×D71)⋊C4—42no—
167(C2×D71)⋊C4—42no—
168C4×Dic71—42no—
169C4×Dic71—42no—
170C4×Dic71—42no—
171C4×Dic71—42no—
172C4×Dic71C4⋊C414yes—
173C4×Dic71C4⋊C414yes—
174C4×Dic71C22×C412yes—
175C4×Dic71C2×Q812yes—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2411yescharacteristic
183C2×C8×C71—21no—
184C2×C8×C71—21no—
185C2×C8×C71—21no—
186C2×C8×C71—21no—
187C2×C8×C71C22×C412yes—
188C2×C8×C71C22×C412yes—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71#4112yes—
196C4⋊C4×C71#4112yes—
197C42×C71—42no—
198C42×C71—42no—
199C42×C71C22×C412yes—
200C42×C71C2×Q812yes—
201C23×C71C22×D411yescharacteristic
202C22×D71—21no—
203C22×D71—21no—
204C22×D71—21no—
205C22×D71—21no—
206C22×D71C2×C4⋊C412yes—
207C22×D71AGL(1,ℤ/8)12yes—
208C22×D71#13712yes—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71C2×C4⋊C412yes—
226C2×C4×C71C2×C4⋊C412yes—
227C2×C4×C71C23×C411yescharacteristic
228C2×C4×C71C22×Q811yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—21no—
238C2×Dic71—21no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71C2×C4⋊C412yes—
252C2×Dic71C2×C4⋊C412yes—
253C2×Dic71C2×C4⋊C412yes—
254C4×D71—28no—
255C4×D71—22no—
256C4×D71—22no—
257C4×D71—22no—
258C4×D71—22no—
259C4×D71—44no—
260C4×D71—48no—
261C4×D71—48no—
262C8×C71—22no—
263C8×C71—22no—
264C71⋊C8—22no—
265C71⋊C8—22no—
266C22×C71C22×C4⋊C411yescharacteristic
267C22×C71#52511yescharacteristic
268C22×C71#52511yescharacteristic
269C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C71C2×#13711yescharacteristic
271C22×C71#57511yescharacteristic
272C22×C71#57511yescharacteristic
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—22no—
279C2×D71#42814yes—
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—42no—
283C4×C71—42no—
284C4×C71—81no—
285C4×C71—81no—
286C4×C71C4×#4111yescharacteristic
287C4×C71C4×#4111yescharacteristic
288C4×C71#53311yescharacteristic
289C4×C71#53311yescharacteristic
290Dic71—22no—
291Dic71—22no—
292Dic71—22no—
293Dic71—42no—
294Dic71—42no—
295Dic71—81no—
296Dic71—81no—
297#21363—711nomaximal, Sylow
298C2×C71#229611yescharacteristic
299C2×C71C2×#42811yescharacteristic
300C2×C71#291111yescharacteristic
301C2×C71#311411yescharacteristic
302C2×C71#311411yescharacteristic
303C2×C71#311711yescharacteristic
304C2×C71#311711yescharacteristic
305D71—24no—
306C2×#432—711no—
307#2283—711no—
308C2×C4×C4⋊C4—711no—
309#1949—711no—
310#1949—711no—
311#1949—711no—
312#1949—711no—
313#1853—711no—
314#1853—711no—
315#1852—711no—
316#1852—711no—
317#1776—711no—
318#1776—711no—
319#1776—711no—
320#1776D7111yescharacteristic, complemented
321C71#2136311yescharacteristic, complemented, Sylow
322C22×C4⋊C4—711no—
323C22×C4⋊C4—711no—
324C22×C42—711no—
325C22×C42—1421no—
326#432—718no—
327C2×#107—711no—
328C2×#107—711no—
329C2×#107—711no—
330C2×#107C2×D7111yescharacteristic
331C2×#106—711no—
332C2×#106—711no—
333C2×#106—711no—
334C2×#106C2×D7111yescharacteristic
335C2×C4⋊C8—711no—
336C2×C4⋊C8—711no—
337C2×C22⋊C8—711no—
338C2×C22⋊C8—711no—
339C2×#97—711no—
340C2×#97C2×D7111yescharacteristic
341#383—1421no—
342#382—711no—
343#382—711no—
344#381—1421no—
345#380—711no—
346#380—711no—
347C4×C4⋊C4—711no—
348C4×C4⋊C4—711no—
349C4×C4⋊C4—711no—
350C4×C4⋊C4—711no—
351C4×C4⋊C4—1421no—
352C4×C4⋊C4—1421no—
353C4×C4⋊C4C2×D7111yescharacteristic
354C4×C4⋊C4C2×D7111yescharacteristic
355C4×C22⋊C4—1421no—
356#376—1421no—
357C2×#95—1421no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340C2×D7111yescharacteristic
365#340C2×D7111yescharacteristic
366C23×C4—711no—
367C23×C4—711no—
368C23×C4—711no—
369C23×C4—1421no—
370C23×C4—1421no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8C22×D7111yescharacteristic
374C22×C8C22×D7111yescharacteristic
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—1422no—
380C2×C4⋊C4—1422no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4—1422no—
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C4⋊C4C22×D7111yescharacteristic
389C2×C22⋊C4—1421no—
390C2×C22⋊C4—1421no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—2844no—
394C2×C42—711no—
395C2×C42—711no—
396C2×C42—711no—
397C2×C42—714no—
398C2×C42—714no—
399C2×C42—1421no—
400C2×C42—1421no—
401C2×C42—1421no—
402C2×C42—1421no—
403C2×C42C22×D7111yescharacteristic
404#107—22no—
405#107—1422no—
406#107—1422no—
407#107—1422no—
408#106—22no—
409#106—1422no—
410#106—1422no—
411#106—1422no—
412C4⋊C8—714no—
413C4⋊C8—714no—
414C22⋊C8—714no—
415C22⋊C8—714no—
416#97—714no—
417#97C4×D7114yescomplemented
418#95—21no—
419#95—21no—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423#95—1421no—
424#95—1421no—
425#95—1421no—
426#95—1421no—
427#95—1421no—
428#95—1421no—
429#95—1421no—
430C24—711no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—2841no—
436C22×C4—2841no—
437C22×C4—2842no—
438C22×C4—2842no—
439C22×C4—2842no—
440C22×C4—2842no—
441C22×C4—714no—
442C22×C4—711no—
443C22×C4—711no—
444C22×C4—711no—
445C22×C4—712no—
446C22×C4—712no—
447C22×C4—714no—
448C22×C4—1424no—
449C22×C4—1424no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1421no—
453C22×C4—1421no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4D4×D7111yescharacteristic
461C22×C4D4×D7111yescharacteristic
462C22×C4C23×D7111yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8C2×C4×D7112yes—
474C2×C8C2×C4×D7112yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—712no—
482C4⋊C4—712no—
483C4⋊C4—1424no—
484C4⋊C4—1424no—
485C4⋊C4—1421no—
486C4⋊C4—1421no—
487C4⋊C4—1421no—
488C4⋊C4—1421no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4#1176586512yes—
492C4⋊C4#1176586512yes—
493C22⋊C4—2842no—
494C22⋊C4—2842no—
495C42—42no—
496C42—42no—
497C42—2842no—
498C42—2842no—
499C42—2842no—
500C42—2842no—
501C42—712no—
502C42—712no—
503C42—714no—
504C42—714no—
505C42C2×C4×D7112yes—
506C42Q8×D7112yes—
507C23—712no—
508C23—712no—
509C23—712no—
510C23—1421no—
511C23—1421no—
512C23—1421no—
513C23—1421no—
514C23C2×D4×D7111yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2844no—
532C2×C4—2848no—
533C2×C4—2848no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2841no—
541C2×C4—2841no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—712no—
545C2×C4—712no—
546C2×C4—712no—
547C2×C4—1428no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1421no—
551C2×C4—1421no—
552C2×C4—1421no—
553C2×C4—1421no—
554C2×C4—1421no—
555C2×C4—1421no—
556C2×C4—1422no—
557C2×C4—1422no—
558C2×C4—1422no—
559C2×C4—1422no—
560C2×C4C4⋊C4×D7112yes—
561C2×C4C4⋊C4×D7112yes—
562C2×C4C22×C4×D7111yescharacteristic
563C2×C4C2×Q8×D7111yescharacteristic
564C2×C4#41×D7111yescharacteristic
565C2×C4#41×D7111yescharacteristic
566C2×C4C2×#1176586511yescharacteristic
567C2×C4C2×#1176586511yescharacteristic
568C8—22no—
569C8—22no—
570C8—1422no—
571C8—1422no—
572C22—714no—
573C22—1422no—
574C22—1422no—
575C22—1422no—
576C22—1422no—
577C22—1422no—
578C22—1422no—
579C22C2×C4⋊C4×D7111yescharacteristic
580C22#1798650811yescharacteristic
581C22#1798650811yescharacteristic
582C22AGL(1,ℤ/8)×D7111yescharacteristic
583C22#137×D7111yescharacteristic
584C22#1798679511yescharacteristic
585C22#1798679811yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—2842no—
593C4—2842no—
594C4—5681no—
595C4—5681no—
596C4—1422no—
597C4—1422no—
598C4—1422no—
599C4#1798646711yescharacteristic
600C4C4×#1176586511yescharacteristic
601C4#1798655311yescharacteristic
602C4#1798655311yescharacteristic
603C2—1424no—
604C2#2259082011yescharacteristic
605C2#428×D7111yescharacteristic
606C2#2259692711yescharacteristic
607C2#2259847611yescharacteristic
608C2#2259848211yescharacteristic
609C2#2259850911yescharacteristic
610C2#2259851511yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1776C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1776D71
hypocenterC71#21363
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #22589111C2semidirect
5. #22589109C2semidirect
6. #22589107C2semidirect
7. #22589105C2semidirect
12. #22588238C2semidirect
13. #22588238C2semidirect
14. #22588237C2semidirect
15. C71×#1776C2semidirect
20. #17986054C4semidirect
95. Dic71⋊C8C2×C4semidirect
96. Dic71⋊C8C2×C4semidirect
99. #14556039C2×C4semidirect
121. #97×C71C2×C4semidirect
320. #1776D71semidirect
321. C71#21363semidirect
417. #97C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860542
#225907362
C4×C4⋊C4×D712
#225891112
#225891092
#225891072
#225891052
#225885342
#225885312
#225885292
#225885262
#225882382
#225882372
C71×#17762
#2136371