Finite Group Databasebeta

← #32019470

Subgroups — #32019470

Order 18176.

850 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×#17986276C211yesmaximal, characteristic
2#22591251C211yesmaximal, characteristic
3#22589314C211yesmaximal, characteristic, complemented
4#22589108C211yesmaximal, characteristic, complemented
5#22589075C211yesmaximal, characteristic, complemented
6#22589070C211yesmaximal, characteristic, complemented
7#22588932C211yesmaximal, characteristic, complemented
8#22588911C211yesmaximal, characteristic, complemented
9#22588600C211yesmaximal, characteristic
10#22588586C211yesmaximal, characteristic
11#22588215C211yesmaximal, characteristic, complemented
12#22588168C211yesmaximal, characteristic
13#22588168C211yesmaximal, characteristic
14C4×#14556048C211yesmaximal, characteristic
15C71×#2369C211yesmaximal, characteristic, complemented
16C2×C4×D284C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19#17986276C418yescomplemented
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556048C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985870—21no—
35#17985870—21no—
36#17985865C2211yescharacteristic
37#17985858—21no—
38#95×D71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4⋊Q8×C71C2211yescharacteristic
50#391×C71—21no—
51#391×C71—21no—
52#389×C71—21no—
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55C4×C4⋊C4×C71—21no—
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×D284—22no—
69C4×D284—22no—
70C4×D284—22no—
71C4×D284—41no—
72C4×D284—41no—
73C4×D284C2×C414yes—
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×(C2×D71)⋊C4—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82#14556061—41no—
83#14556061C2×C414yescomplemented
84#14556061C2×C414yescomplemented
85#14556059—41no—
86C4⋊(C71⋊C8)C2×C414yescomplemented
87#14556048—22no—
88#14556048C2×C414yes—
89#14556048C2×C414yes—
90#14556048Q814yescomplemented
91#14556039—22no—
92#14556039—41no—
93C4×C71⋊C8—41no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C22×Q8×C71—21no—
99C4⋊Q8×C71—22no—
100C4⋊Q8×C71C2×C414yescomplemented
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71D411yescharacteristic
105C2×C4⋊C4×C71D411yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C42×C71—21no—
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113C2×C42×C71C2311yescharacteristic
114C2×C42×C71C2311yescharacteristic
115C2×C42×C71C2311yescharacteristic
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123C23×D71—21no—
124C2×D284—21no—
125C2×D284—21no—
126C2×D284—21no—
127C2×D284C22×C412yes—
128C2×D284C2×Q812yes—
129C2×D284#4112yes—
130C22×Dic71—21no—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C2×C4×D71—24no—
134C2×C4×D71—42no—
135C2×C4×D71—42no—
136C2×C4×D71—42no—
137C2×C4×D71—42no—
138C2×C4×D71—81no—
139C2×C4×D71—81no—
140C2×C4×D71—81no—
141C2×C4×D71—81no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8C22×C412yes—
152C2×C71⋊C8C22×C412yes—
153C2×C71⋊C8C2×Q812yes—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71C22×C412yes—
160Dic71⋊C4—81no—
161Dic71⋊C4—81no—
162(C2×D71)⋊C4—24no—
163(C2×D71)⋊C4—42no—
164(C2×D71)⋊C4—42no—
165C2×Q8×C71—24no—
166C2×Q8×C71—81no—
167C2×Q8×C71—81no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2411yescharacteristic
179C4⋊C4×C71—24no—
180C4⋊C4×C71—22no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—41no—
187C4⋊C4×C71—81no—
188C4⋊C4×C71—81no—
189C4⋊C4×C71C22×C412yes—
190C4⋊C4×C71C22×C412yes—
191C4⋊C4×C71C2×Q812yes—
192C42×C71—21no—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—21no—
196C42×C71—42no—
197C42×C71—81no—
198C42×C71—81no—
199C42×C71C22×C412yes—
200C42×C71#4112yes—
201C23×C71C22×D411yescharacteristic
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—41no—
206C22×D71—41no—
207C22×D71—41no—
208C22×D71—41no—
209Q8×C71—84no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—22no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71C4×D412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71#13712yes—
240C2×C4×C71C23×C411yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71H(5,𝔽2)11yescharacteristic
245C2×C4×C712−1+411yescharacteristic
246C2×Dic71—22no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255D284—22no—
256D284—22no—
257D284—22no—
258D284C4×Q814yes—
259C4×D71—84no—
260C4×D71—84no—
261C4×D71—84no—
262C71⋊C8—22no—
263C71⋊C8—22no—
264C71⋊C8—22no—
265C71⋊C8—41no—
266C22×C71C2×C4×D411yescharacteristic
267C22×C71#54611yescharacteristic
268C22×C71D4×Q811yescharacteristic
269C22×C71C2×#13711yescharacteristic
270C22×C71#57511yescharacteristic
271C22×C71#57611yescharacteristic
272C22×C71#57611yescharacteristic
273C2×D71—24no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C2×D71—42no—
279C2×D71—42no—
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—22no—
283C4×C71—42no—
284C4×C71—42no—
285C4×C71—42no—
286C4×C71—41no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—81no—
290C4×C71C2×C4×Q811yescharacteristic
291C4×C71#52011yescharacteristic
292C4×C71#55211yescharacteristic
293C4×C71#55411yescharacteristic
294Dic71—42no—
295Dic71—81no—
296Dic71—81no—
297#22979—711nomaximal, Sylow
298C2×C71#236011yescharacteristic
299C2×C71C4×#13711yescharacteristic
300C2×C71#294811yescharacteristic
301C2×C71#331411yescharacteristic
302C2×C71#331911yescharacteristic
303C2×C71#338511yescharacteristic
304C2×C71#338911yescharacteristic
305D71—44no—
306C2×#474—711no—
307#2369D7111yescharacteristic, complemented
308#2349—711no—
309#1993—711no—
310#1949—711no—
311#1943—711no—
312#1942—711no—
313#1921—711no—
314#1918—711no—
315#1864—711no—
316#1862—711no—
317#1771—711no—
318#1762—711no—
319#1762—711no—
320C4×#102—711no—
321C71#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8C2×D7111yescharacteristic
323C2×C4×D4—711no—
324C2×C4×D4—711no—
325C2×C4×D4—711no—
326#474—718no—
327C2×#107—711no—
328C2×#107—711no—
329C2×#107—711no—
330C2×#106—711no—
331C2×C4⋊C8—711no—
332C2×#102—711no—
333C2×#102—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#97—711no—
337C2×#97—711no—
338C2×C4×C8—711no—
339#391—21no—
340#391—21no—
341#389—21no—
342#389—711no—
343#389—711no—
344#389C2×D7111yescharacteristic
345#389C2×D7111yescharacteristic
346#385—1421no—
347#385—1421no—
348#382—711no—
349#380—1421no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4C2×D7111yescharacteristic
352C4×C4⋊C4C2×D7111yescharacteristic
353C4×C4⋊C4C2×D7111yescharacteristic
354C4×C4⋊C4C2×D7111yescharacteristic
355C2×#95—1421no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#336—711no—
365#336—711no—
366C22×Q8—21no—
367C22×D4—711no—
368C23×C4—1421no—
369C23×C4—1421no—
370C23×C4—1421no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8—711no—
374C22×C8—711no—
375C4⋊Q8—22no—
376C4⋊Q8C4×D7114yescomplemented
377C4×D4—2841no—
378C4×D4—2841no—
379C4×D4—714no—
380C4×D4—1422no—
381C4×D4—1422no—
382C4×D4—1422no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4C22⋊D7111yescharacteristic
392C2×C4⋊C4C22⋊D7111yescharacteristic
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C22⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22×D7111yescharacteristic
404C2×C42C22×D7111yescharacteristic
405C2×C42C22×D7111yescharacteristic
406#107—2841no—
407#107—714no—
408#107—714no—
409#106—2841no—
410C4⋊C8—714no—
411#102—714no—
412#102—714no—
413#102—714no—
414#102—1422no—
415#97—2841no—
416#97—1422no—
417C4×C8—2841no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1421no—
424#95—1421no—
425#95—1421no—
426#95—1421no—
427#95C22⋊D7111yescharacteristic
428#95C22⋊D7111yescharacteristic
429C24—1421no—
430C2×Q8—24no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×D4—712no—
434C2×D4—712no—
435C2×D4—712no—
436C2×D4—1421no—
437C2×D4—1421no—
438C2×D4—1421no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—2842no—
444C22×C4—2842no—
445C22×C4—2842no—
446C22×C4—2842no—
447C22×C4—5681no—
448C22×C4—5681no—
449C22×C4—5681no—
450C22×C4—5681no—
451C22×C4—1421no—
452C22×C4—1421no—
453C22×C4—1421no—
454C22×C4—1424no—
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C23×D7111yescharacteristic
462C2×C8—712no—
463C2×C8—712no—
464C2×C8—712no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—5681no—
485C4⋊C4—5681no—
486C4⋊C4—712no—
487C4⋊C4—1421no—
488C4⋊C4—1421no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4C2×C4×D7112yes—
493C4⋊C4C2×C4×D7112yes—
494C4⋊C4Q8×D7112yes—
495C22⋊C4—2842no—
496C22⋊C4—2842no—
497C22⋊C4—1424no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—81no—
504C42—81no—
505C42C2×C4×D7112yes—
506C42#1176586512yes—
507C23—2841no—
508C23—2841no—
509C23—2841no—
510C23—2841no—
511C23—1422no—
512C23—1422no—
513C23—1422no—
514C23C22×C22⋊D7111yescharacteristic
515Q8—84no—
516D4—714no—
517D4—1422no—
518D4—1422no—
519D4—1422no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—5684no—
556C2×C4—5684no—
557C2×C4—5684no—
558C2×C4—1422no—
559C2×C4C4×C22⋊D7112yes—
560C2×C4C4×C22⋊D7112yes—
561C2×C4#1455615912yes—
562C2×C4C22×C4×D7111yescharacteristic
563C2×C4C2×Q8×D7111yescharacteristic
564C2×C4#41×D7111yescharacteristic
565C2×C4C2×#1176586511yescharacteristic
566C2×C4#1455617411yescharacteristic
567C2×C4#1455617611yescharacteristic
568C8—2841no—
569C8—1422no—
570C8—1422no—
571C8—1422no—
572C22—2842no—
573C22—2842no—
574C22—2842no—
575C22—2842no—
576C22—2842no—
577C22—2842no—
578C22—1424no—
579C22C2×C4×C22⋊D7111yescharacteristic
580C22#1798663711yescharacteristic
581C22Q8×C22⋊D7111yescharacteristic
582C22C2×#1455615911yescharacteristic
583C22#1798679711yescharacteristic
584C22#1798680211yescharacteristic
585C22#1798680211yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—2842no—
597C4—5681no—
598C4—5681no—
599C4C4×Q8×D7111yescharacteristic
600C4#1798648111yescharacteristic
601C4#1798669011yescharacteristic
602C4#1798670911yescharacteristic
603C2—2844no—
604C2#2259134111yescharacteristic
605C2C4×#1455615911yescharacteristic
606C2#2259719011yescharacteristic
607C2#2260041511yescharacteristic
608C2#2260049011yescharacteristic
609C2#2260124611yescharacteristic
610C2#2260129411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2369C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2369D71
hypocenterC71#22979
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. #22589314C2semidirect
4. #22589108C2semidirect
5. #22589075C2semidirect
6. #22589070C2semidirect
7. #22588932C2semidirect
8. #22588911C2semidirect
11. #22588215C2semidirect
15. C71×#2369C2semidirect
19. #17986276C4semidirect
83. #14556061C2×C4semidirect
84. #14556061C2×C4semidirect
86. C4⋊(C71⋊C8)C2×C4semidirect
90. #14556048Q8semidirect
100. C4⋊Q8×C71C2×C4semidirect
307. #2369D71semidirect
321. C71#22979semidirect
376. C4⋊Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862762
#225912512
#225893142
#225891082
#225890752
#225890702
#225889322
#225889112
#225886002
#225885862
#225882152
#225881682
C4×#145560482
C71×#23692
#2297971