Finite Group Databasebeta

← #32055111

Subgroups — #32055111

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#22591215C211yesmaximal, characteristic
2#22589897C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589841C211yesmaximal, characteristic, complemented
5#22589670C211yesmaximal, characteristic, complemented
6#22589293C211yesmaximal, characteristic, complemented
7#22589101C211yesmaximal, characteristic, complemented
8#22588915C211yesmaximal, characteristic, complemented
9#22588915C211yesmaximal, characteristic, complemented
10#22588533C211yesmaximal, characteristic
11#22588520C211yesmaximal, characteristic
12#22588428C211yesmaximal, characteristic
13#22588363C211yesmaximal, characteristic
14C2×#17985761C211yesmaximal, characteristic
15C71×#2440C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C22×C4⋊Dic71C2211yescharacteristic
18C2×#14556059C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25#17985897C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985882—21no—
28#17985882—21no—
29#17985859C2211yescharacteristic
30#17985857—21no—
31C4×C4⋊Dic71C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C22⋊C4×Dic71—21no—
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×C4⋊Q8×C71C2211yescharacteristic
49C2×C22⋊Q8×C71—21no—
50C22×C4⋊C4×C71C2211yescharacteristic
51#398×C71—21no—
52#395×C71—21no—
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#385×C71—21no—
56#384×C71C2211yescharacteristic
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
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71—22no—
69C2×C4⋊Dic71Q814yescomplemented
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×C22⋊Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79#14556059—22no—
80#14556059—22no—
81#14556059—41no—
82#14556059—41no—
83C4⋊(C71⋊C8)—22no—
84C4⋊(C71⋊C8)—41no—
85C22⋊(C71⋊C8)—22no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C22×Q8×C71—21no—
92C23×C4×C71—21no—
93C23×C4×C71C2311yescharacteristic
94C4⋊Q8×C71—22no—
95C4⋊Q8×C71—41no—
96C22⋊Q8×C71—44no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—41no—
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C22⋊C4×C71—21no—
110C2×C22⋊C4×C71D411yescharacteristic
111C2×C22⋊C4×C71D411yescharacteristic
112C2×C42×C71C2311yescharacteristic
113C2×C42×C71C2311yescharacteristic
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—24no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—81no—
125C22×Dic71—81no—
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—81no—
149Dic71⋊C4—81no—
150C22⋊Dic71—42no—
151C4×Dic71—81no—
152C4×Dic71—81no—
153C24×C71C2×D411yescharacteristic
154C2×Q8×C71—42no—
155C2×Q8×C71—42no—
156C2×Q8×C71—81no—
157C22×C4×C71—21no—
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—21no—
165C22×C4×C71—42no—
166C22×C4×C71—42no—
167C22×C4×C71—81no—
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×Q812yes—
174C22×C4×C71#4112yes—
175C22×C4×C71C2411yescharacteristic
176C4⋊C4×C71—21no—
177C4⋊C4×C71—21no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—42no—
184C4⋊C4×C71—42no—
185C4⋊C4×C71—42no—
186C4⋊C4×C71—42no—
187C4⋊C4×C71—42no—
188C4⋊C4×C71—81no—
189C4⋊C4×C71#4112yes—
190C22⋊C4×C71—22no—
191C22⋊C4×C71—42no—
192C22⋊C4×C71SD1614yes—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—21no—
196C42×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71C22⋊Q812yes—
202C23×C71C2×SD1612yes—
203C23×C71#13712yes—
204C23×C71C22×D411yescharacteristic
205Q8×C71—84no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—21no—
217C2×C4×C71—22no—
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—41no—
234C2×C4×C71—84no—
235C2×C4×C71C22⋊Q812yes—
236C2×C4×C71C2×SD1612yes—
237C2×C4×C71C2×SD1612yes—
238C2×C4×C71C22×Q811yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C2×C4×C71H(5,𝔽2)11yescharacteristic
243C2×C4×C712−1+411yescharacteristic
244C2×Dic71—22no—
245C2×Dic71—22no—
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—41no—
256C2×Dic71—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—41no—
279C4×C71—42no—
280C4×C71—41no—
281C4×C71—81no—
282C4×C71—81no—
283C4×C71—81no—
284C4×C71#53311yescharacteristic
285C4×C71#54411yescharacteristic
286C4×C71#55111yescharacteristic
287C4×C71#55211yescharacteristic
288Dic71—42no—
289Dic71—42no—
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#2440D7111yescharacteristic
302#2345—711no—
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⋊Q8C2×D7111yescharacteristic
319C2×C22⋊Q8—21no—
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—21no—
330#395—21no—
331#389—711no—
332#389—711no—
333#389C2×D7111yescharacteristic
334#389C2×D7111yescharacteristic
335#387—1421no—
336#387—1421no—
337#385—21no—
338#384C2×D7111yescharacteristic
339#380—711no—
340#380—1421no—
341#380C2×D7111yescharacteristic
342C4×C4⋊C4—711no—
343C4×C4⋊C4—711no—
344C4×C22⋊C4—1421no—
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—21no—
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—22no—
372C4⋊Q8—41no—
373C22⋊Q8—44no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—714no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1422no—
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—1421no—
400C2×C42—1421no—
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—1421no—
414#95—1421no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95C22⋊D7111yescharacteristic
419#95C22⋊D7111yescharacteristic
420C24C2×C22⋊D7111yescharacteristic
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4—5681no—
436C22×C4—5681no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4—1424no—
442C22×C4C2×C22⋊D7111yescharacteristic
443C22×C4C2×C22⋊D7111yescharacteristic
444C22×C4C2×C22⋊D7111yescharacteristic
445C22×C4C2×C22⋊D7111yescharacteristic
446C22×C4C2×C22⋊D7111yescharacteristic
447C22×C4Q8×D7112yes—
448C22×C4#1176586612yes—
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—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—5681no—
477C4⋊C4—5681no—
478C4⋊C4—712no—
479C4⋊C4—712no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4—1424no—
487C4⋊C4#1176586412yes—
488C22⋊C4—22no—
489C22⋊C4—42no—
490C22⋊C4—2842no—
491C22⋊C4#1176585314yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—5681no—
497C42—5681no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1455610412yes—
503C23C2×#1176585312yes—
504C23#1455615912yes—
505C23C22×C22⋊D7111yescharacteristic
506Q8—84no—
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—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
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—5684no—
547C2×C4—1422no—
548C2×C4—1422no—
549C2×C4#1455610412yes—
550C2×C4C2×#1176585312yes—
551C2×C4C2×#1176585312yes—
552C2×C4C2×Q8×D7111yescharacteristic
553C2×C4#41×D7111yescharacteristic
554C2×C4C2×#1176586411yescharacteristic
555C2×C4C2×#1176586611yescharacteristic
556C2×C4#1455617311yescharacteristic
557C2×C4#1455617511yescharacteristic
558C8—2841no—
559C8—2841no—
560C8—2841no—
561C8—1422no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1798628114yes—
569C22C2×#1455610411yescharacteristic
570C22#1798664311yescharacteristic
571C22#1798664811yescharacteristic
572C22C22×#1176585311yescharacteristic
573C22C2×#1455615911yescharacteristic
574C22#1798680211yescharacteristic
575C22#1798680711yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—2842no—
586C4—2842no—
587C4—5681no—
588C4—5681no—
589C4#1798655111yescharacteristic
590C4#1798662811yescharacteristic
591C4#1798667611yescharacteristic
592C4#1798669111yescharacteristic
593C2—24no—
594C2#2259201211yescharacteristic
595C2C2×#1798628111yescharacteristic
596C2#2259816511yescharacteristic
597C2#2260016411yescharacteristic
598C2#2260051111yescharacteristic
599C2#2260069711yescharacteristic
600C2#2260081911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2440C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2440D71
hypocenterC71#25992
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #22589897C2semidirect
3. #22589841C2semidirect
4. #22589841C2semidirect
5. #22589670C2semidirect
6. #22589293C2semidirect
7. #22589101C2semidirect
8. #22588915C2semidirect
9. #22588915C2semidirect
69. C2×C4⋊Dic71Q8semidirect
316. C71#25992semidirect

Maximal subgroups

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

subgroupindex
#225912152
#225898972
#225898412
#225896702
#225892932
#225891012
#225889152
#225885332
#225885202
#225884282
#225883632
C2×#179857612
C71×#24402
#2599271