Finite Group Databasebeta

← #32055109

Subgroups — #32055109

Order 18176.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22592042C211yesmaximal, characteristic
2#22589898C211yesmaximal, characteristic, complemented
3#22589842C211yesmaximal, characteristic, complemented
4#22589842C211yesmaximal, characteristic, complemented
5#22589673C211yesmaximal, characteristic, complemented
6#22589292C211yesmaximal, characteristic, complemented
7#22589103C211yesmaximal, characteristic, complemented
8#22588918C211yesmaximal, characteristic, complemented
9#22588918C211yesmaximal, characteristic, complemented
10#22588532C211yesmaximal, characteristic
11#22588520C211yesmaximal, characteristic
12#22588430C211yesmaximal, characteristic
13#22588363C211yesmaximal, characteristic
14C2×#17985761C211yesmaximal, characteristic
15C71×#2345C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556128C2211yescharacteristic
18C2×#14556105—21no—
19C22×C4⋊Dic71C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C4⋊(C71⋊C8)C2211yescharacteristic
26C2×C22⋊(C71⋊C8)C2211yescharacteristic
27#17985931—21no—
28#17985920—21no—
29#17985897C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985874—21no—
32#17985869C2211yescharacteristic
33#17985859C2211yescharacteristic
34#17985761C2211yescharacteristic
35#17985761C2211yescharacteristic
36#17985761C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C22×C4⋊C4×C71C2211yescharacteristic
49#389×C71C2211yescharacteristic
50#389×C71C2211yescharacteristic
51#387×C71—21no—
52#387×C71—21no—
53#380×C71—21no—
54#380×C71C2211yescharacteristic
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C22⋊C4×C71—21no—
58C4×C22⋊C4×C71C2211yescharacteristic
59C22×Dic142—21no—
60C23×Dic71—21no—
61C22×C71⋊C8—22no—
62C22×C71⋊C8—22no—
63C22×C71⋊C8—41no—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68#14556128—22no—
69#14556128—41no—
70#14556105—44no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—41no—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×Dic71⋊C4—21no—
82C2×Dic71⋊C4—21no—
83C2×C22⋊Dic71—21no—
84#14556059—22no—
85#14556059—22no—
86#14556059—41no—
87#14556059—41no—
88C4⋊(C71⋊C8)—22no—
89C4⋊(C71⋊C8)—41no—
90C22⋊(C71⋊C8)—22no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C23×C4×C71—21no—
95C23×C4×C71C2311yescharacteristic
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71Q814yescomplemented
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71D411yescharacteristic
107C2×C22⋊C4×C71D411yescharacteristic
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71D411yescharacteristic
118#95×C71D411yescharacteristic
119C2×Dic142—42no—
120C2×Dic142—42no—
121C2×Dic142—81no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—42no—
127C22×Dic71—42no—
128C22×Dic71—81no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—44no—
141C2×C71⋊C8#4112yes—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—42no—
150C4⋊Dic71—42no—
151C4⋊Dic71—42no—
152C4⋊Dic71—81no—
153C4⋊Dic71#4112yes—
154Dic71⋊C4—42no—
155Dic71⋊C4—42no—
156C22⋊Dic71—42no—
157C24×C71C2×D411yescharacteristic
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—24no—
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—81no—
168C22×C4×C71—81no—
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×Q812yes—
175C22×C4×C71#4112yes—
176C22×C4×C71C2411yescharacteristic
177C4⋊C4×C71—24no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—81no—
185C4⋊C4×C71—81no—
186C4⋊C4×C71C2×Q812yes—
187C4⋊C4×C71C2×Q812yes—
188C22⋊C4×C71—22no—
189C22⋊C4×C71—42no—
190C22⋊C4×C71SD1614yes—
191C42×C71—21no—
192C42×C71—21no—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—81no—
196C42×C71—81no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71C22⋊Q812yes—
202C23×C71C2×SD1612yes—
203C23×C71#13712yes—
204C23×C71C22×D411yescharacteristic
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
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—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—84no—
234C2×C4×C71C22⋊Q812yes—
235C2×C4×C71C2×SD1612yes—
236C2×C4×C71C2×SD1612yes—
237C2×C4×C71C22×Q811yescharacteristic
238C2×C4×C71C2×#4111yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71H(5,𝔽2)11yescharacteristic
242C2×C4×C712−1+411yescharacteristic
243C2×Dic71—22no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—84no—
256Dic142—84no—
257C71⋊C8—22no—
258C71⋊C8—41no—
259C71⋊C8—41no—
260C71⋊C8—41no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71#47514yes—
268C22×C71C2×C22⋊Q811yescharacteristic
269C22×C71#54611yescharacteristic
270C22×C71#54711yescharacteristic
271C22×C71C22×SD1611yescharacteristic
272C22×C71C2×#13711yescharacteristic
273C22×C71#57611yescharacteristic
274C22×C71#57711yescharacteristic
275C4×C71—22no—
276C4×C71—22no—
277C4×C71—22no—
278C4×C71—42no—
279C4×C71—42no—
280C4×C71—41no—
281C4×C71—41no—
282C4×C71—81no—
283C4×C71—81no—
284C4×C71#53311yescharacteristic
285C4×C71#54411yescharacteristic
286C4×C71#55111yescharacteristic
287C4×C71#55211yescharacteristic
288Dic71—42no—
289Dic71—81no—
290Dic71—81no—
291Dic71—81no—
292#25992—711nomaximal, Sylow
293C2×C71—24no—
294C2×C71#243811yescharacteristic
295C2×C71C2×#47511yescharacteristic
296C2×C71#308111yescharacteristic
297C2×C71#329511yescharacteristic
298C2×C71#332011yescharacteristic
299C2×C71#333511yescharacteristic
300C2×C71#334611yescharacteristic
301#2440—711no—
302#2345D7111yescharacteristic
303#2099—711no—
304#2088—711no—
305#2088—711no—
306#2058—711no—
307#1988—711no—
308#1948—711no—
309#1919—711no—
310#1919—711no—
311#1853—711no—
312#1850—711no—
313#1824—711no—
314#1807—711no—
315C2×#340—711no—
316C71#2599211yescharacteristic, complemented, Sylow
317C23×C8—711no—
318C2×C4⋊Q8—711no—
319C2×C22⋊Q8—1421no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4C2×D7111yescharacteristic
322C2×#106—711no—
323C2×#106—711no—
324C2×#106—711no—
325C2×#106—711no—
326C2×C4⋊C8—711no—
327C2×C4⋊C8—711no—
328C2×C22⋊C8—711no—
329#398—1421no—
330#395—1421no—
331#389—711no—
332#389—711no—
333#389C2×D7111yescharacteristic
334#389C2×D7111yescharacteristic
335#387—21no—
336#387—21no—
337#385—1421no—
338#384—711no—
339#380—21no—
340#380—711no—
341#380C2×D7111yescharacteristic
342C4×C4⋊C4C2×D7111yescharacteristic
343C4×C4⋊C4C2×D7111yescharacteristic
344C4×C22⋊C4—21no—
345C4×C22⋊C4C2×D7111yescharacteristic
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#336—711no—
359#336—711no—
360C22×Q8—1421no—
361C23×C4—21no—
362C23×C4—1421no—
363C23×C4C22×D7111yescharacteristic
364C22×C8—2841no—
365C22×C8—711no—
366C22×C8—711no—
367C22×C8—711no—
368C22×C8—711no—
369C22×C8—1422no—
370C22×C8—1422no—
371C4⋊Q8—2841no—
372C4⋊Q8—1422no—
373C22⋊Q8—2844no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—2841no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—711no—
381C2×C4⋊C4—711no—
382C2×C4⋊C4—711no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1421no—
385C2×C4⋊C4—1421no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4Dic14214yescomplemented
391C2×C4⋊C4C22×D7111yescharacteristic
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1421no—
397C2×C22⋊C4C22⋊D7111yescharacteristic
398C2×C22⋊C4C22⋊D7111yescharacteristic
399C2×C42—21no—
400C2×C42—21no—
401C2×C42C22×D7111yescharacteristic
402C2×C42C22×D7111yescharacteristic
403#106—2841no—
404#106—2841no—
405#106—1422no—
406#106—1422no—
407C4⋊C8—2841no—
408C4⋊C8—1422no—
409C22⋊C8—1422no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95C22⋊D7111yescharacteristic
419#95C22⋊D7111yescharacteristic
420C24C2×C22⋊D7111yescharacteristic
421C2×Q8—2842no—
422C2×Q8—2842no—
423C2×Q8—5681no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—2842no—
436C22×C4—2842no—
437C22×C4—5681no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4C2×C22⋊D7111yescharacteristic
443C22×C4C2×C22⋊D7111yescharacteristic
444C22×C4C2×C22⋊D7111yescharacteristic
445C22×C4C2×C22⋊D7111yescharacteristic
446C22×C4C2×C22⋊D7111yescharacteristic
447C22×C4C2×Dic14212yes—
448C22×C4#1176586512yes—
449C22×C4C23×D7111yescharacteristic
450C2×C8—2844no—
451C2×C8—712no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8—1421no—
456C2×C8—1421no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8—1421no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—2842no—
473C4⋊C4—2842no—
474C4⋊C4—2842no—
475C4⋊C4—2842no—
476C4⋊C4—2842no—
477C4⋊C4—5681no—
478C4⋊C4—712no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4C2×Dic14212yes—
487C4⋊C4C2×Dic14212yes—
488C22⋊C4—22no—
489C22⋊C4—42no—
490C22⋊C4—2842no—
491C22⋊C4#1176585214yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—81no—
497C42—81no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1455610712yes—
503C23C2×#1176585212yes—
504C23#1455615812yes—
505C23C22×C22⋊D7111yescharacteristic
506Q8—5684no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
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—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—5684no—
548C2×C4—1422no—
549C2×C4#1455610712yes—
550C2×C4C2×#1176585212yes—
551C2×C4C2×#1176585212yes—
552C2×C4C22×Dic14211yescharacteristic
553C2×C4#41×D7111yescharacteristic
554C2×C4#41×D7111yescharacteristic
555C2×C4C2×#1176586511yescharacteristic
556C2×C4#1455617411yescharacteristic
557C2×C4#1455617511yescharacteristic
558C8—2841no—
559C8—2841no—
560C8—2841no—
561C8—1422no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1798628414yes—
569C22C2×#1455610711yescharacteristic
570C22#1798663711yescharacteristic
571C22#1798664811yescharacteristic
572C22C22×#1176585211yescharacteristic
573C22C2×#1455615811yescharacteristic
574C22#1798680211yescharacteristic
575C22#1798681011yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—2842no—
586C4—5681no—
587C4—5681no—
588C4—5681no—
589C4#1798655211yescharacteristic
590C4#1798662711yescharacteristic
591C4#1798667311yescharacteristic
592C4#1798668211yescharacteristic
593C2—24no—
594C2#2259201011yescharacteristic
595C2C2×#1798628411yescharacteristic
596C2#2259817111yescharacteristic
597C2#2260015811yescharacteristic
598C2#2260050511yescharacteristic
599C2#2260069111yescharacteristic
600C2#2260082511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2345C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2345D71
hypocenterC71#25992
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #22589898C2semidirect
3. #22589842C2semidirect
4. #22589842C2semidirect
5. #22589673C2semidirect
6. #22589292C2semidirect
7. #22589103C2semidirect
8. #22588918C2semidirect
9. #22588918C2semidirect
100. C2×C4⋊C4×C71Q8semidirect
316. C71#25992semidirect
390. C2×C4⋊C4Dic142semidirect

Maximal subgroups

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

subgroupindex
#225920422
#225898982
#225898422
#225896732
#225892922
#225891032
#225889182
#225885322
#225885202
#225884302
#225883632
C2×#179857612
C71×#23452
#2599271