Finite Group Databasebeta

← #32002354

Subgroups — #32002354

Order 18176.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986059C211yesmaximal, characteristic
2#22590950C211yesmaximal, characteristic
3#22590694C211yesmaximal, characteristic
4#22589106C211yesmaximal, characteristic, complemented
5#22589103C211yesmaximal, characteristic, complemented
6#22589099C211yesmaximal, characteristic, complemented
7#22588542C211yesmaximal, characteristic
8#22588542C211yesmaximal, characteristic
9#22588531C211yesmaximal, characteristic
10#22588526C211yesmaximal, characteristic
11#22588240C211yesmaximal, characteristic, complemented
12#22588236C211yesmaximal, characteristic, complemented
13#107×Dic71C211yesmaximal, characteristic, complemented
14#22588228C211yesmaximal, characteristic, complemented
15C71×#1949C211yesmaximal, characteristic, complemented
16C2×#14556086C2211yescharacteristic
17C2×#14556085C2211yescharacteristic
18C2×C4⋊C4×D71C2211yescharacteristic
19#17986059C418yescomplemented
20C2×#14556062C2211yescharacteristic
21C2×#14556062C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×C4⋊(C71⋊C8)C2211yescharacteristic
27C2×(C2×D71)⋊C8C2211yescharacteristic
28C2×(C2×D71)⋊C8C2211yescharacteristic
29C2×#14556038C2211yescharacteristic
30C2×C8×Dic71C2211yescharacteristic
31#17985891—21no—
32#17985891—21no—
33#17985886—21no—
34#17985886—21no—
35#17985869C2211yescharacteristic
36#17985856C2211yescharacteristic
37#17985856C2211yescharacteristic
38C4⋊C4×Dic71C2211yescharacteristic
39C4⋊C4×Dic71C2211yescharacteristic
40C4⋊C4×Dic71C2211yescharacteristic
41C4⋊C4×Dic71C2211yescharacteristic
42C4×Dic71⋊C4—21no—
43C4×C4⋊Dic71C2211yescharacteristic
44#17985850—21no—
45C4×(C2×D71)⋊C4—21no—
46#17985846—21no—
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54C2×#106×C71C2211yescharacteristic
55C2×C4⋊C8×C71C2211yescharacteristic
56#382×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C4×D71—21no—
61C22×C4×D71C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556086—22no—
65#14556086—22no—
66#14556085C2×C414yes—
67#14556085Q814yes—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4D411yescharacteristic
76C2×Dic71⋊C4D411yescharacteristic
77C4⋊C4×D71—22no—
78C4⋊C4×D71—22no—
79C2×(C2×D71)⋊C4—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×(C2×D71)⋊C4D411yescharacteristic
83C2×(C2×D71)⋊C4D411yescharacteristic
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71—21no—
89C2×C4×Dic71—21no—
90C2×C4×Dic71C2311yescharacteristic
91#14556062—22no—
92#14556062—22no—
93#14556061—22no—
94#14556059—22no—
95#14556059—22no—
96#14556059—22no—
97C4⋊(C71⋊C8)C2×C414yescomplemented
98(C2×D71)⋊C8C2×C414yes—
99(C2×D71)⋊C8C2×C414yes—
100#14556038C2×C414yescomplemented
101C8×Dic71C2×C414yescomplemented
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C22×C8×C71C2311yescharacteristic
110C22×C8×C71C2311yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C42×C71—21no—
116C2×C42×C71C2311yescharacteristic
117#107×C71—22no—
118#106×C71—22no—
119C4⋊C8×C71C2×C414yescomplemented
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123C23×D71C2×D411yescharacteristic
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71C2×D411yescharacteristic
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C2×C4×D71—21no—
132C2×C4×D71—21no—
133C2×C4×D71—21no—
134C2×C4×D71—21no—
135C2×C4×D71—24no—
136C2×C4×D71—81no—
137C2×C4×D71—81no—
138C2×C4×D71C22×C412yes—
139C2×C4×D71C2×Q812yes—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8C22×C412yes—
145C2×C71⋊C8C22×C412yes—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—81no—
153C4⋊Dic71—81no—
154C4⋊Dic71#4112yes—
155C4⋊Dic71#4112yes—
156Dic71⋊C4—24no—
157Dic71⋊C4C4⋊C414yes—
158Dic71⋊C4C4⋊C414yes—
159(C2×D71)⋊C4—24no—
160(C2×D71)⋊C4—81no—
161(C2×D71)⋊C4—81no—
162(C2×D71)⋊C4C4⋊C414yes—
163(C2×D71)⋊C4C4⋊C414yes—
164C4×Dic71—24no—
165C4×Dic71—81no—
166C4×Dic71—81no—
167C4×Dic71—81no—
168C4×Dic71—81no—
169C4×Dic71—81no—
170C4×Dic71C22×C412yes—
171C4×Dic71C2×Q812yes—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2411yescharacteristic
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71C22×C412yes—
184C2×C8×C71C22×C412yes—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71#4112yes—
192C4⋊C4×C71#4112yes—
193C42×C71—81no—
194C42×C71C22×C412yes—
195C42×C71C2×Q812yes—
196C23×C71C22×D411yescharacteristic
197C22×D71—21no—
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71—21no—
202C22×D71—21no—
203C22×D71C2×C4⋊C412yes—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—41no—
211C2×C4×C71—41no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71C2×C4⋊C412yes—
221C2×C4×C71C2×C4⋊C412yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×Q811yescharacteristic
224C2×C4×C71C2×#4111yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71C2×C4⋊C412yes—
247C2×Dic71C2×C4⋊C412yes—
248C2×Dic71C2×C4⋊C412yes—
249C4×D71—22no—
250C4×D71—22no—
251C4×D71—22no—
252C4×D71—22no—
253C4×D71—84no—
254C8×C71—22no—
255C8×C71—22no—
256C71⋊C8—22no—
257C71⋊C8—22no—
258C22×C71C22×C4⋊C411yescharacteristic
259C22×C71#52511yescharacteristic
260C22×C71#52511yescharacteristic
261C22×C71#57611yescharacteristic
262C22×C71#57711yescharacteristic
263C22×C71#57711yescharacteristic
264C22×C71#57811yescharacteristic
265C2×D71—22no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—41no—
269C2×D71—41no—
270C2×D71—41no—
271C2×D71—41no—
272C4×C71—22no—
273C4×C71—22no—
274C4×C71—42no—
275C4×C71—42no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71C4×#4111yescharacteristic
279C4×C71C4×#4111yescharacteristic
280C4×C71#53311yescharacteristic
281C4×C71#53311yescharacteristic
282Dic71—22no—
283Dic71—22no—
284Dic71—22no—
285Dic71—42no—
286Dic71—42no—
287Dic71—81no—
288Dic71—81no—
289#21367—711nomaximal, Sylow
290C2×C71#229611yescharacteristic
291C2×C71#291211yescharacteristic
292C2×C71#291311yescharacteristic
293C2×C71#311811yescharacteristic
294C2×C71#311911yescharacteristic
295C2×C71#312011yescharacteristic
296C2×C71#312111yescharacteristic
297D71—42no—
298C2×#433—711no—
299#2314—711no—
300#2274—711no—
301#1949—711no—
302#1949D7111yescharacteristic, complemented
303#1948—711no—
304#1947—711no—
305#1854—711no—
306#1854—711no—
307#1853—711no—
308#1852—711no—
309#1776—711no—
310#1776—711no—
311C4×#107—711no—
312C4×#106—711no—
313C71#2136711yescharacteristic, complemented, Sylow
314C2×#117—711no—
315C2×#117—711no—
316C22×C4⋊C4—711no—
317#433—718no—
318C2×#107—711no—
319C2×#107—711no—
320C2×#107—711no—
321C2×#107C2×D7111yescharacteristic
322C2×#106—711no—
323C2×#106—711no—
324C2×#106—711no—
325C2×#106C2×D7111yescharacteristic
326C2×C4⋊C8—711no—
327C2×C4⋊C8C2×D7111yescharacteristic
328C2×C22⋊C8—711no—
329C2×C22⋊C8—711no—
330C2×#97—711no—
331C2×C4×C8—711no—
332#388—1421no—
333#388—1421no—
334#387—1421no—
335#387—1421no—
336#384—711no—
337#382C2×D7111yescharacteristic
338#380—711no—
339#380—711no—
340C4×C4⋊C4—711no—
341C4×C4⋊C4—711no—
342C4×C4⋊C4—711no—
343C4×C4⋊C4—711no—
344C4×C4⋊C4—711no—
345C4×C4⋊C4—1421no—
346C4×C4⋊C4—1421no—
347C4×C4⋊C4C2×D7111yescharacteristic
348C4×C22⋊C4—1421no—
349C4×C22⋊C4—1421no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340C2×D7111yescharacteristic
357#340C2×D7111yescharacteristic
358C23×C4—711no—
359C23×C4—1421no—
360C22×C8—711no—
361C22×C8—711no—
362C22×C8C22×D7111yescharacteristic
363C22×C8C22×D7111yescharacteristic
364#117—714no—
365#117—714no—
366#117—1422no—
367#117—1422no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—1422no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1422no—
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C22⋊C4—711no—
384C2×C22⋊C4—711no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C42—21no—
389C2×C42—711no—
390C2×C42—1421no—
391C2×C42—1421no—
392C2×C42—1421no—
393C2×C42—1421no—
394C2×C42—1421no—
395C2×C42—1421no—
396C2×C42C22×D7111yescharacteristic
397#107—22no—
398#107—1422no—
399#107—1422no—
400#107—1422no—
401#106—22no—
402#106—1422no—
403#106—1422no—
404#106—1422no—
405C4⋊C8—714no—
406C4⋊C8C4×D7114yescomplemented
407C22⋊C8—714no—
408C22⋊C8—714no—
409#97—714no—
410C4×C8—714no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1421no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421C24—711no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—5681no—
427C22×C4—5681no—
428C22×C4—711no—
429C22×C4—711no—
430C22×C4—711no—
431C22×C4—712no—
432C22×C4—712no—
433C22×C4—1424no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4D4×D7111yescharacteristic
443C22×C4D4×D7111yescharacteristic
444C22×C4C23×D7111yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—712no—
450C2×C8—712no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C4×D7112yes—
456C2×C8C2×C4×D7112yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—5681no—
464C4⋊C4—5681no—
465C4⋊C4—714no—
466C4⋊C4—714no—
467C4⋊C4—712no—
468C4⋊C4—712no—
469C4⋊C4—1424no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4#1176586412yes—
477C4⋊C4#1176586612yes—
478C22⋊C4—5681no—
479C22⋊C4—5681no—
480C22⋊C4—714no—
481C22⋊C4—714no—
482C22⋊C4—1424no—
483C42—81no—
484C42—5681no—
485C42—5681no—
486C42—5681no—
487C42—5681no—
488C42—5681no—
489C42—712no—
490C42—712no—
491C42—1424no—
492C42C2×C4×D7112yes—
493C42Q8×D7112yes—
494C23—712no—
495C23—1421no—
496C23—1421no—
497C23—1421no—
498C23—1421no—
499C23—1421no—
500C23—1421no—
501C23C2×D4×D7111yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—5684no—
529C2×C4—712no—
530C2×C4—712no—
531C2×C4—712no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1421no—
535C2×C4—1421no—
536C2×C4—1421no—
537C2×C4—1421no—
538C2×C4—1421no—
539C2×C4—1421no—
540C2×C4—1422no—
541C2×C4—1422no—
542C2×C4—1422no—
543C2×C4—1422no—
544C2×C4C4⋊C4×D7112yes—
545C2×C4C4⋊C4×D7112yes—
546C2×C4C22×C4×D7111yescharacteristic
547C2×C4C2×Q8×D7111yescharacteristic
548C2×C4#41×D7111yescharacteristic
549C2×C4#41×D7111yescharacteristic
550C2×C4C2×#1176586411yescharacteristic
551C2×C4C2×#1176586611yescharacteristic
552C8—22no—
553C8—22no—
554C8—1422no—
555C8—1422no—
556C22—2841no—
557C22—2841no—
558C22—2841no—
559C22—2841no—
560C22—1422no—
561C22—1422no—
562C22—1422no—
563C22C2×C4⋊C4×D7111yescharacteristic
564C22#1798650911yescharacteristic
565C22#1798651211yescharacteristic
566C22#1798680111yescharacteristic
567C22#1798680811yescharacteristic
568C22#1798680911yescharacteristic
569C22#1798681311yescharacteristic
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—2842no—
577C4—2842no—
578C4—5681no—
579C4—5681no—
580C4—1422no—
581C4—1422no—
582C4—1422no—
583C4C4×#1176586411yescharacteristic
584C4C4×#1176586611yescharacteristic
585C4#1798655411yescharacteristic
586C4#1798655511yescharacteristic
587C2—2842no—
588C2#2259082111yescharacteristic
589C2#2259693411yescharacteristic
590C2#2259693711yescharacteristic
591C2#2259852111yescharacteristic
592C2#2259854611yescharacteristic
593C2#2259855911yescharacteristic
594C2#2259856811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1949C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1949D71
hypocenterC71#21367
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #22589106C2semidirect
5. #22589103C2semidirect
6. #22589099C2semidirect
11. #22588240C2semidirect
12. #22588236C2semidirect
13. #107×Dic71C2semidirect
14. #22588228C2semidirect
15. C71×#1949C2semidirect
19. #17986059C4semidirect
97. C4⋊(C71⋊C8)C2×C4semidirect
100. #14556038C2×C4semidirect
101. C8×Dic71C2×C4semidirect
119. C4⋊C8×C71C2×C4semidirect
302. #1949D71semidirect
313. C71#21367semidirect
406. C4⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860592
#225909502
#225906942
#225891062
#225891032
#225890992
#225885422
#225885312
#225885262
#225882402
#225882362
#107×Dic712
#225882282
C71×#19492
#2136771