Finite Group Databasebeta

← #32055229

Subgroups — #32055229

Order 18176.

730 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#22592064C211yesmaximal, characteristic
2#22589908C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589831C211yesmaximal, characteristic, complemented
5#22589678C211yesmaximal, characteristic, complemented
6#22589297C211yesmaximal, characteristic, complemented
7#22589108C211yesmaximal, characteristic, complemented
8#22588925C211yesmaximal, characteristic, complemented
9#22588911C211yesmaximal, characteristic, complemented
10#22588527C211yesmaximal, characteristic
11#22588520C211yesmaximal, characteristic
12#22588430C211yesmaximal, characteristic
13#22588363C211yesmaximal, characteristic
14C2×#17985761C211yesmaximal, characteristic
15C71×#2345C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
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#17985941—21no—
27#17985933—21no—
28#17985897C2211yescharacteristic
29#17985897C2211yescharacteristic
30#17985871—21no—
31#17985865C2211yescharacteristic
32#17985859C2211yescharacteristic
33#17985857—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
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
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#14556120—22no—
68#14556120—41no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—41no—
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—
81#14556061—22no—
82#14556061—41no—
83#14556059—22no—
84#14556059—41no—
85C4⋊(C71⋊C8)—22no—
86C4⋊(C71⋊C8)—41no—
87C22⋊(C71⋊C8)—22no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C23×C4×C71—21no—
94C23×C4×C71C2311yescharacteristic
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71Q814yes—
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C22⋊C4×C71—21no—
105C2×C22⋊C4×C71D411yescharacteristic
106C2×C22⋊C4×C71D411yescharacteristic
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110C2×C42×C71C2311yescharacteristic
111#95×C71—21no—
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71D411yescharacteristic
117#95×C71D411yescharacteristic
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—42no—
123C22×Dic71—42no—
124C22×Dic71—81no—
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—21no—
136C2×C71⋊C8—44no—
137C2×C71⋊C8#4112yes—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—42no—
146C4⋊Dic71#4112yes—
147Dic71⋊C4—42no—
148Dic71⋊C4—42no—
149Dic71⋊C4—81no—
150Dic71⋊C4—81no—
151C22⋊Dic71—42no—
152C24×C71C2×D411yescharacteristic
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—24no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—81no—
163C22×C4×C71—81no—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×Q812yes—
170C22×C4×C71#4112yes—
171C22×C4×C71C2411yescharacteristic
172C4⋊C4×C71—24no—
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—81no—
180C4⋊C4×C71—81no—
181C4⋊C4×C71C2×Q812yes—
182C4⋊C4×C71C2×Q812yes—
183C22⋊C4×C71—22no—
184C22⋊C4×C71—22no—
185C22⋊C4×C71—42no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—21no—
190C42×C71—81no—
191C42×C71—81no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71C22⋊Q812yes—
197C23×C71#13512yes—
198C23×C71#13712yes—
199C23×C71C22×D411yescharacteristic
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
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—84no—
231C2×C4×C71C22⋊Q812yes—
232C2×C4×C71C22×Q811yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71H(5,𝔽2)11yescharacteristic
237C2×C4×C712−1+411yescharacteristic
238C2×Dic71—22no—
239C2×Dic71—41no—
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—84no—
251C71⋊C8—22no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71#47914yes—
262C22×C71C2×C22⋊Q811yescharacteristic
263C22×C71#54611yescharacteristic
264C22×C71#54711yescharacteristic
265C22×C71C2×#13511yescharacteristic
266C22×C71C2×#13711yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57711yescharacteristic
269C4×C71—22no—
270C4×C71—42no—
271C4×C71—42no—
272C4×C71—41no—
273C4×C71—41no—
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#26000—711nomaximal, Sylow
287C2×C71—24no—
288C2×C71#243811yescharacteristic
289C2×C71C2×#47911yescharacteristic
290C2×C71#308111yescharacteristic
291C2×C71#329711yescharacteristic
292C2×C71#331611yescharacteristic
293C2×C71#333711yescharacteristic
294C2×C71#334611yescharacteristic
295#2442—711no—
296#2345D7111yescharacteristic
297#2101—711no—
298#2088—711no—
299#2086—711no—
300#2059—711no—
301#1989—711no—
302#1949—711no—
303#1920—711no—
304#1918—711no—
305#1852—711no—
306#1850—711no—
307#1824—711no—
308#1807—711no—
309C2×#340—711no—
310C71#2600011yescharacteristic, complemented, Sylow
311C23×C8—711no—
312C2×#125—711no—
313C22×C4⋊C4—711no—
314C22×C4⋊C4C2×D7111yescharacteristic
315C2×#107—711no—
316C2×#107—711no—
317C2×#106—711no—
318C2×#106—711no—
319C2×C4⋊C8—711no—
320C2×C4⋊C8—711no—
321C2×C22⋊C8—711no—
322#400—1421no—
323#398—1421no—
324#389—711no—
325#389—711no—
326#389C2×D7111yescharacteristic
327#389C2×D7111yescharacteristic
328#387—21no—
329#387—21no—
330#385—1421no—
331#382—711no—
332#380—21no—
333#380—711no—
334#380—1421no—
335#380C2×D7111yescharacteristic
336C4×C4⋊C4C2×D7111yescharacteristic
337C4×C4⋊C4C2×D7111yescharacteristic
338C4×C22⋊C4—21no—
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×C4—1421no—
356C23×C4C22×D7111yescharacteristic
357C22×C8—2841no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362C22×C8—1422no—
363C22×C8—1422no—
364#125—2841no—
365#125—1422no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—2841no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1422no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4Dic14214yes—
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1421no—
388C2×C22⋊C4C22⋊D7111yescharacteristic
389C2×C22⋊C4C22⋊D7111yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42C22×D7111yescharacteristic
393C2×C42C22×D7111yescharacteristic
394#107—2841no—
395#107—1422no—
396#106—2841no—
397#106—1422no—
398C4⋊C8—2841no—
399C4⋊C8—1422no—
400C22⋊C8—1422no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95C22⋊D7111yescharacteristic
412#95C22⋊D7111yescharacteristic
413C24C2×C22⋊D7111yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—2842no—
426C22×C4—2842no—
427C22×C4—5681no—
428C22×C4—1421no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4C2×C22⋊D7111yescharacteristic
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×Dic14212yes—
438C22×C4#1176586512yes—
439C22×C4C23×D7111yescharacteristic
440C2×C8—2844no—
441C2×C8—712no—
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—
451C2×C8—1421no—
452C2×C8—1421no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—2842no—
463C4⋊C4—2842no—
464C4⋊C4—2842no—
465C4⋊C4—5681no—
466C4⋊C4—5681no—
467C4⋊C4—712no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4C2×Dic14212yes—
476C4⋊C4C2×Dic14212yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—42no—
480C22⋊C4—2842no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#1455610712yes—
492C23#1455614812yes—
493C23#1455615912yes—
494C23C22×C22⋊D7111yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
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—41no—
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—84no—
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—5684no—
538C2×C4—1422no—
539C2×C4#1455610712yes—
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—22no—
554C22—22no—
555C22—22no—
556C22#1798630614yes—
557C22C2×#1455610711yescharacteristic
558C22#1798663711yescharacteristic
559C22#1798664811yescharacteristic
560C22C2×#1455614811yescharacteristic
561C22C2×#1455615911yescharacteristic
562C22#1798680211yescharacteristic
563C22#1798681011yescharacteristic
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—2842no—
574C4—5681no—
575C4—5681no—
576C4—5681no—
577C4#1798655211yescharacteristic
578C4#1798662711yescharacteristic
579C4#1798667311yescharacteristic
580C4#1798668211yescharacteristic
581C2—24no—
582C2#2259201011yescharacteristic
583C2C2×#1798630611yescharacteristic
584C2#2259817111yescharacteristic
585C2#2260018411yescharacteristic
586C2#2260044511yescharacteristic
587C2#2260071311yescharacteristic
588C2#2260081911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2345C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2345D71
hypocenterC71#26000
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. #22589908C2semidirect
3. #22589841C2semidirect
4. #22589831C2semidirect
5. #22589678C2semidirect
6. #22589297C2semidirect
7. #22589108C2semidirect
8. #22588925C2semidirect
9. #22588911C2semidirect
310. C71#26000semidirect

Maximal subgroups

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

subgroupindex
#225920642
#225899082
#225898412
#225898312
#225896782
#225892972
#225891082
#225889252
#225889112
#225885272
#225885202
#225884302
#225883632
C2×#179857612
C71×#23452
#2600071