Finite Group Databasebeta

← #32055409

Subgroups — #32055409

Order 18176.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22592053C211yesmaximal, characteristic
2#22589898C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589836C211yesmaximal, characteristic, complemented
5#22589678C211yesmaximal, characteristic, complemented
6#22589297C211yesmaximal, characteristic, complemented
7#22589098C211yesmaximal, characteristic, complemented
8#22588925C211yesmaximal, characteristic, complemented
9#22588918C211yesmaximal, characteristic, complemented
10#22588537C211yesmaximal, characteristic
11#22588525C211yesmaximal, characteristic
12#22588430C211yesmaximal, characteristic
13#22588371C211yesmaximal, characteristic
14#22588059C211yesmaximal, characteristic
15C71×#2344C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
18C2×#14556087C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C22⋊(C71⋊C8)C2211yescharacteristic
26#17985935—21no—
27#17985922—21no—
28#17985897C2211yescharacteristic
29#17985897C2211yescharacteristic
30#17985896—21no—
31#17985882—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
48C2×#117×C71—21no—
49C22×C4⋊C4×C71C2211yescharacteristic
50#389×C71C2211yescharacteristic
51#389×C71C2211yescharacteristic
52#387×C71—21no—
53#387×C71—21no—
54#381×C71—21no—
55#380×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71C2211yescharacteristic
59C2×#11765849—22no—
60C2×#11765849—22no—
61C2×#11765849—41no—
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66#14556120—22no—
67#14556120—41no—
68#14556087—22no—
69#14556087—41no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×C22⋊Dic71—21no—
81C2×C22⋊Dic71—21no—
82C2×C4×Dic71—21no—
83#14556061—41no—
84#14556061—41no—
85#14556059—22no—
86#14556059—22no—
87C4⋊(C71⋊C8)—22no—
88C4⋊(C71⋊C8)—41no—
89C22⋊(C71⋊C8)—22no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C23×C4×C71—21no—
94C23×C4×C71C2311yescharacteristic
95#117×C71—44no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71Q814yes—
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
119C22×Dic71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123#11765849—44no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
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#4112yes—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—42no—
144C4⋊Dic71—81no—
145C4⋊Dic71#4112yes—
146Dic71⋊C4—42no—
147Dic71⋊C4—42no—
148Dic71⋊C4—42no—
149Dic71⋊C4—81no—
150C22⋊Dic71—42no—
151C22⋊Dic71—81no—
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—24no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
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—24no—
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×C71C2×Q812yes—
183C4⋊C4×C71C2×Q812yes—
184C22⋊C4×C71—22no—
185C22⋊C4×C71—41no—
186C22⋊C4×C71—42no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—21no—
190C42×C71—21no—
191C42×C71—42no—
192C42×C71—42no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71C22⋊Q812yes—
200C23×C71C22×D411yescharacteristic
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
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—48no—
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×C71C22⋊Q812yes—
232C2×C4×C71AGL(1,ℤ/8)12yes—
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—41no—
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—
251C71⋊C8—22no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71C2×C22⋊Q811yescharacteristic
263C22×C71#54611yescharacteristic
264C22×C71#54711yescharacteristic
265C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C71#57511yescharacteristic
267C22×C71#57711yescharacteristic
268C22×C71#57811yescharacteristic
269C4×C71—22no—
270C4×C71—22no—
271C4×C71—41no—
272C4×C71—42no—
273C4×C71—42no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71#53311yescharacteristic
279C4×C71#54411yescharacteristic
280C4×C71#55111yescharacteristic
281C4×C71#55211yescharacteristic
282Dic71—42no—
283Dic71—81no—
284Dic71—81no—
285Dic71—81no—
286#26012—711nomaximal, Sylow
287C2×C71—42no—
288C2×C71#243811yescharacteristic
289C2×C71#307711yescharacteristic
290C2×C71#308211yescharacteristic
291C2×C71#331511yescharacteristic
292C2×C71#333211yescharacteristic
293C2×C71#334311yescharacteristic
294C2×C71#335111yescharacteristic
295#2441—711no—
296#2344D7111yescharacteristic
297#2099—711no—
298#2088—711no—
299#2087—711no—
300#2059—711no—
301#1989—711no—
302#1947—711no—
303#1920—711no—
304#1919—711no—
305#1854—711no—
306#1851—711no—
307#1824—711no—
308#1809—711no—
309#1736—711no—
310C71#2601211yescharacteristic, complemented, Sylow
311C22×M16—711no—
312C2×#125—711no—
313C2×#117—21no—
314C2×#117—711no—
315C22×C4⋊C4C2×D7111yescharacteristic
316C2×#107—711no—
317C2×#107—711no—
318C2×#106—711no—
319C2×#106—711no—
320C2×C4⋊C8—711no—
321C2×C4⋊C8—711no—
322C2×C22⋊C8—711no—
323#399—1421no—
324#396—1421no—
325#389—711no—
326#389—711no—
327#389—1421no—
328#389C2×D7111yescharacteristic
329#389C2×D7111yescharacteristic
330#387—21no—
331#387—21no—
332#387—1421no—
333#384—711no—
334#381—21no—
335#380—711no—
336#380C2×D7111yescharacteristic
337C4×C4⋊C4C2×D7111yescharacteristic
338C4×C4⋊C4C2×D7111yescharacteristic
339C4×C22⋊C4C2×D7111yescharacteristic
340#340—711no—
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#336—711no—
353#336—711no—
354C23×C4—21no—
355C23×C4C22×D7111yescharacteristic
356C2×M16—2841no—
357C2×M16—1422no—
358C2×M16—1422no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362C22×C8—711no—
363#125—2841no—
364#125—1422no—
365#117—44no—
366#117—2841no—
367#117—1422no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4Dic14214yes—
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1421no—
389C2×C22⋊C4—1421no—
390C2×C22⋊C4C22⋊D7111yescharacteristic
391C2×C22⋊C4C22⋊D7111yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—1421no—
395C2×C42C22×D7111yescharacteristic
396C2×C42C22×D7111yescharacteristic
397#107—2841no—
398#107—2841no—
399#106—1422no—
400#106—1422no—
401C4⋊C8—2841no—
402C4⋊C8—1422no—
403C22⋊C8—1422no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—1421no—
410#95—1421no—
411#95—1421no—
412#95C22⋊D7111yescharacteristic
413#95C22⋊D7111yescharacteristic
414C24C2×C22⋊D7111yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—1421no—
427C22×C4—1421no—
428C22×C4—1421no—
429C22×C4—1421no—
430C22×C4C2×C22⋊D7111yescharacteristic
431C22×C4C2×C22⋊D7111yescharacteristic
432C22×C4C2×C22⋊D7111yescharacteristic
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×Dic14212yes—
436C22×C4#1176586512yes—
437C22×C4C23×D7111yescharacteristic
438M16—2844no—
439C2×C8—712no—
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—2842no—
461C4⋊C4—2842no—
462C4⋊C4—2842no—
463C4⋊C4—2842no—
464C4⋊C4—5681no—
465C4⋊C4—5681no—
466C4⋊C4—712no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4C2×Dic14212yes—
475C4⋊C4C2×Dic14212yes—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—2842no—
480C22⋊C4—5681no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—42no—
487C42—2842no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#1455610712yes—
495C23C22×C22⋊D7111yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—48no—
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—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—1422no—
538C2×C4#1455610712yes—
539C2×C4#1455615412yes—
540C2×C4C22×Dic14211yescharacteristic
541C2×C4#41×D7111yescharacteristic
542C2×C4#41×D7111yescharacteristic
543C2×C4C2×#1176586511yescharacteristic
544C2×C4#1455617411yescharacteristic
545C2×C4#1455617511yescharacteristic
546C8—2841no—
547C8—2841no—
548C8—2841no—
549C8—1422no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#1455610711yescharacteristic
558C22#1798663711yescharacteristic
559C22#1798664811yescharacteristic
560C22C2×#1455615411yescharacteristic
561C22#1798679711yescharacteristic
562C22#1798680711yescharacteristic
563C22#1798681411yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—2842no—
574C4—5681no—
575C4—5681no—
576C4—5681no—
577C4#1798655211yescharacteristic
578C4#1798662711yescharacteristic
579C4#1798667311yescharacteristic
580C4#1798668211yescharacteristic
581C2—42no—
582C2#2259201011yescharacteristic
583C2#2259813711yescharacteristic
584C2#2259818211yescharacteristic
585C2#2260043011yescharacteristic
586C2#2260066111yescharacteristic
587C2#2260077411yescharacteristic
588C2#2260090011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2344C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2344D71
hypocenterC71#26012
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
2. #22589898C2semidirect
3. #22589841C2semidirect
4. #22589836C2semidirect
5. #22589678C2semidirect
6. #22589297C2semidirect
7. #22589098C2semidirect
8. #22588925C2semidirect
9. #22588918C2semidirect
310. C71#26012semidirect

Maximal subgroups

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

subgroupindex
#225920532
#225898982
#225898412
#225898362
#225896782
#225892972
#225890982
#225889252
#225889182
#225885372
#225885252
#225884302
#225883712
#225880592
C71×#23442
#2601271