Finite Group Databasebeta

← #32055171

Subgroups — #32055171

Order 18176.

758 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591207C211yesmaximal, characteristic
2#22589907C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589836C211yesmaximal, characteristic, complemented
5#22589675C211yesmaximal, characteristic, complemented
6#22589298C211yesmaximal, characteristic, complemented
7#22589106C211yesmaximal, characteristic, complemented
8#22588929C211yesmaximal, characteristic, complemented
9#22588915C211yesmaximal, characteristic, complemented
10#22588533C211yesmaximal, characteristic
11#22588520C211yesmaximal, characteristic
12#22588424C211yesmaximal, characteristic
13#22588367C211yesmaximal, characteristic
14C2×#17985761C211yesmaximal, characteristic
15C71×#2442C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556084—21no—
18C22×C4⋊Dic71C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C22⋊(C71⋊C8)C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985897C2211yescharacteristic
28#17985884—21no—
29#17985884—21no—
30#17985860—21no—
31#17985859C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C4×C4⋊Dic71C2211yescharacteristic
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
48C2×#125×C71C2211yescharacteristic
49C22×C4⋊C4×C71C2211yescharacteristic
50#400×C71—21no—
51#398×C71—21no—
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#385×C71—21no—
55#382×C71C2211yescharacteristic
56#380×C71—21no—
57#380×C71C2211yescharacteristic
58C4×C22⋊C4×C71C2211yescharacteristic
59C23×Dic71—21no—
60C22×C71⋊C8—22no—
61C22×C71⋊C8—22no—
62C22×C71⋊C8—41no—
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556084—44no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71Q814yes—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×C22⋊Dic71—21no—
78C2×C4×Dic71—21no—
79C2×C4×Dic71—21no—
80#14556061—22no—
81#14556061—41no—
82#14556059—22no—
83#14556059—41no—
84C4⋊(C71⋊C8)—22no—
85C4⋊(C71⋊C8)—41no—
86C22⋊(C71⋊C8)—22no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C23×C4×C71—21no—
93C23×C4×C71C2311yescharacteristic
94#125×C71—22no—
95#125×C71—41no—
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—41no—
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71D411yescharacteristic
109C2×C22⋊C4×C71D411yescharacteristic
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
119C22×Dic71—24no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—42no—
125C22×Dic71—42no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
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—44no—
138C2×C71⋊C8#4112yes—
139C4⋊Dic71—24no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71C2×Q812yes—
147C4⋊Dic71C2×Q812yes—
148Dic71⋊C4—42no—
149Dic71⋊C4—42no—
150C22⋊Dic71—42no—
151C4×Dic71—42no—
152C4×Dic71—42no—
153C24×C71C2×D411yescharacteristic
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—42no—
163C22×C4×C71—42no—
164C22×C4×C71—81no—
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×Q812yes—
171C22×C4×C71#4112yes—
172C22×C4×C71C2411yescharacteristic
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—21no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—42no—
181C4⋊C4×C71—42no—
182C4⋊C4×C71—42no—
183C4⋊C4×C71—81no—
184C4⋊C4×C71—81no—
185C4⋊C4×C71#4112yes—
186C22⋊C4×C71—22no—
187C22⋊C4×C71—22no—
188C22⋊C4×C71—42no—
189C42×C71—21no—
190C42×C71—21no—
191C42×C71—21no—
192C42×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71C22⋊Q812yes—
198C23×C71#13512yes—
199C23×C71AGL(1,ℤ/8)12yes—
200C23×C71C22×D411yescharacteristic
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—22no—
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—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
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—84no—
232C2×C4×C71C22⋊Q812yes—
233C2×C4×C71C22×Q811yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71H(5,𝔽2)11yescharacteristic
238C2×C4×C712−1+411yescharacteristic
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—48no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C71⋊C8—22no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8—41no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71#47814yes—
263C22×C71C2×C22⋊Q811yescharacteristic
264C22×C71#54611yescharacteristic
265C22×C71#54711yescharacteristic
266C22×C71C2×#13511yescharacteristic
267C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C71#57711yescharacteristic
269C22×C71#57811yescharacteristic
270C4×C71—22no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71—42no—
275C4×C71—41no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71—81no—
279C4×C71#53311yescharacteristic
280C4×C71#54411yescharacteristic
281C4×C71#55111yescharacteristic
282C4×C71#55211yescharacteristic
283Dic71—42no—
284Dic71—42no—
285Dic71—81no—
286Dic71—81no—
287#25996—711nomaximal, Sylow
288C2×C71—24no—
289C2×C71#243811yescharacteristic
290C2×C71C2×#47811yescharacteristic
291C2×C71#308211yescharacteristic
292C2×C71#329611yescharacteristic
293C2×C71#330811yescharacteristic
294C2×C71#333611yescharacteristic
295C2×C71#334511yescharacteristic
296#2442D7111yescharacteristic
297#2344—711no—
298#2101—711no—
299#2088—711no—
300#2087—711no—
301#2059—711no—
302#1989—711no—
303#1949—711no—
304#1921—711no—
305#1919—711no—
306#1853—711no—
307#1850—711no—
308#1823—711no—
309#1808—711no—
310C2×#340—711no—
311C71#2599611yescharacteristic, complemented, Sylow
312C23×C8—711no—
313C2×#125C2×D7111yescharacteristic
314C2×#117—1421no—
315C22×C4⋊C4—711no—
316C22×C4⋊C4C2×D7111yescharacteristic
317C2×#107—711no—
318C2×#107—711no—
319C2×#106—711no—
320C2×#106—711no—
321C2×C4⋊C8—711no—
322C2×C4⋊C8—711no—
323C2×C22⋊C8—711no—
324#400—21no—
325#398—21no—
326#389—711no—
327#389—711no—
328#389C2×D7111yescharacteristic
329#389C2×D7111yescharacteristic
330#387—1421no—
331#387—1421no—
332#385—21no—
333#382C2×D7111yescharacteristic
334#381—1421no—
335#380—21no—
336#380—711no—
337#380C2×D7111yescharacteristic
338C4×C4⋊C4—711no—
339C4×C4⋊C4—711no—
340C4×C22⋊C4C2×D7111yescharacteristic
341#340—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#336—711no—
354#336—711no—
355C23×C4—21no—
356C23×C4—1421no—
357C23×C4C22×D7111yescharacteristic
358C22×C8—2841no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362C22×C8—711no—
363C22×C8—1422no—
364C22×C8—1422no—
365#125—22no—
366#125—41no—
367#117—2844no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—714no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1421no—
390C2×C22⋊C4C22⋊D7111yescharacteristic
391C2×C22⋊C4C22⋊D7111yescharacteristic
392C2×C42—1421no—
393C2×C42—1421no—
394C2×C42C22×D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396#107—2841no—
397#107—1422no—
398#106—2841no—
399#106—1422no—
400C4⋊C8—2841no—
401C4⋊C8—1422no—
402C22⋊C8—1422no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95—1421no—
412#95—1421no—
413#95C22⋊D7111yescharacteristic
414#95C22⋊D7111yescharacteristic
415C24C2×C22⋊D7111yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—2842no—
428C22×C4—2842no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4—1424no—
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4Q8×D7112yes—
440C22×C4#1176586612yes—
441C22×C4C23×D7111yescharacteristic
442C2×C8—2844no—
443C2×C8—712no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—2842no—
468C4⋊C4—2842no—
469C4⋊C4—712no—
470C4⋊C4—712no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4—1424no—
478C4⋊C4#1176586412yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—42no—
482C22⋊C4—2842no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—2842no—
488C42—2842no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#1455610412yes—
494C23#1455614812yes—
495C23#1455615312yes—
496C23C22×C22⋊D7111yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2848no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—1422no—
540C2×C4—1422no—
541C2×C4#1455610412yes—
542C2×C4C2×Q8×D7111yescharacteristic
543C2×C4#41×D7111yescharacteristic
544C2×C4C2×#1176586411yescharacteristic
545C2×C4C2×#1176586611yescharacteristic
546C2×C4#1455617311yescharacteristic
547C2×C4#1455617511yescharacteristic
548C8—2841no—
549C8—2841no—
550C8—2841no—
551C8—1422no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22#1798629614yes—
559C22C2×#1455610411yescharacteristic
560C22#1798664311yescharacteristic
561C22#1798664811yescharacteristic
562C22C2×#1455614811yescharacteristic
563C22C2×#1455615311yescharacteristic
564C22#1798681011yescharacteristic
565C22#1798681411yescharacteristic
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—2842no—
576C4—2842no—
577C4—5681no—
578C4—5681no—
579C4#1798655111yescharacteristic
580C4#1798662811yescharacteristic
581C4#1798667611yescharacteristic
582C4#1798669111yescharacteristic
583C2—24no—
584C2#2259201211yescharacteristic
585C2C2×#1798629611yescharacteristic
586C2#2259817611yescharacteristic
587C2#2260017511yescharacteristic
588C2#2260033111yescharacteristic
589C2#2260070211yescharacteristic
590C2#2260080411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2442C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2442D71
hypocenterC71#25996
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. #22589907C2semidirect
3. #22589841C2semidirect
4. #22589836C2semidirect
5. #22589675C2semidirect
6. #22589298C2semidirect
7. #22589106C2semidirect
8. #22588929C2semidirect
9. #22588915C2semidirect
311. C71#25996semidirect

Maximal subgroups

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

subgroupindex
#225912072
#225899072
#225898412
#225898362
#225896752
#225892982
#225891062
#225889292
#225889152
#225885332
#225885202
#225884242
#225883672
C2×#179857612
C71×#24422
#2599671