Finite Group Databasebeta

← #31997657

Subgroups — #31997657

Order 18176.

894 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985956C211yesmaximal, characteristic
2#22590965C211yesmaximal, characteristic
3#22590961C211yesmaximal, characteristic
4#22588474C211yesmaximal, characteristic, complemented
5#22588472C211yesmaximal, characteristic, complemented
6#22588472C211yesmaximal, characteristic, complemented
7#22588470C211yesmaximal, characteristic, complemented
8#22588469C211yesmaximal, characteristic, complemented
9#22588467C211yesmaximal, characteristic, complemented
10#22588466C211yesmaximal, characteristic, complemented
11#22588058C211yesmaximal, characteristic
12#22588058C211yesmaximal, characteristic
13#22588055C211yesmaximal, characteristic
14#340×D71C211yesmaximal, characteristic
15C71×#1836C211yesmaximal, characteristic, complemented
16C22×#11765848C2211yescharacteristic
17C22×C8×D71C2211yescharacteristic
18C2×#14556086C2211yescharacteristic
19C2×#14556086C2211yescharacteristic
20C2×#14556085C2211yescharacteristic
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C4⋊C4×D71C2211yescharacteristic
23#17985956C418yescomplemented
24C2×#14556038C2211yescharacteristic
25C2×C4×C71⋊C8C2211yescharacteristic
26C2×C8×Dic71C2211yescharacteristic
27#17985896—21no—
28#17985896—21no—
29#17985894—21no—
30#17985894—21no—
31#17985873—21no—
32#17985873—21no—
33#17985870—21no—
34#17985870—21no—
35#17985869C2211yescharacteristic
36#17985868C2211yescharacteristic
37#17985868C2211yescharacteristic
38#17985865C2211yescharacteristic
39#17985864C2211yescharacteristic
40#17985864C2211yescharacteristic
41#17985762C2211yescharacteristic
42#17985762C2211yescharacteristic
43#17985762C2211yescharacteristic
44#17985762C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#97×C71C2211yescharacteristic
54#384×C71C2211yescharacteristic
55#382×C71C2211yescharacteristic
56#340×C71C2211yescharacteristic
57#340×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C4×D71—21no—
61C22×C4×D71—21no—
62C22×C4×D71C2311yescharacteristic
63C2×#11765848—22no—
64C2×#11765848—22no—
65C2×#11765848C2×C414yes—
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C2×C8×D71—22no—
69C2×C8×D71—22no—
70C2×C8×D71C2×C414yes—
71#14556086—22no—
72#14556086—22no—
73#14556086—22no—
74#14556086—22no—
75#14556085C2×C414yes—
76#14556085Q814yes—
77C2×C4⋊Dic71—21no—
78C2×C4⋊Dic71—21no—
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C2×Dic71⋊C4—21no—
87C2×Dic71⋊C4—21no—
88C2×Dic71⋊C4—21no—
89C2×Dic71⋊C4D411yescharacteristic
90C2×Dic71⋊C4D411yescharacteristic
91C4⋊C4×D71—22no—
92C4⋊C4×D71—22no—
93C4⋊C4×D71—22no—
94C4⋊C4×D71—22no—
95C2×(C2×D71)⋊C4—21no—
96C2×(C2×D71)⋊C4—21no—
97C2×(C2×D71)⋊C4D411yescharacteristic
98C2×(C2×D71)⋊C4D411yescharacteristic
99C2×C4×Dic71—21no—
100C2×C4×Dic71—21no—
101C2×C4×Dic71C2311yescharacteristic
102#14556038C2×C414yescomplemented
103C4×C71⋊C8C2×C414yescomplemented
104C8×Dic71C2×C414yescomplemented
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C71⋊#95—21no—
111C22×C8×C71C2311yescharacteristic
112C22×C8×C71C2311yescharacteristic
113C2×C4⋊C4×C71—21no—
114C2×C4⋊C4×C71—21no—
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120#97×C71C2×C414yescomplemented
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—42no—
132C2×C4×D71—42no—
133C2×C4×D71—81no—
134C2×C4×D71—81no—
135C2×C4×D71—81no—
136C2×C4×D71—81no—
137C2×C4×D71C22×C412yes—
138C2×C4×D71C2×Q812yes—
139C2×C4×D71#4112yes—
140C2×C4×D71#4112yes—
141C2×C4×D71#4112yes—
142C2×C4×D71#4112yes—
143#11765848—28no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8C22×C412yes—
149C2×C71⋊C8C22×C412yes—
150C8×D71—28no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—81no—
160C4⋊Dic71—81no—
161Dic71⋊C4—42no—
162Dic71⋊C4—42no—
163Dic71⋊C4—81no—
164Dic71⋊C4—81no—
165Dic71⋊C4—81no—
166Dic71⋊C4—81no—
167Dic71⋊C4C4⋊C414yes—
168Dic71⋊C4C4⋊C414yes—
169(C2×D71)⋊C4—42no—
170(C2×D71)⋊C4—42no—
171(C2×D71)⋊C4C4⋊C414yes—
172(C2×D71)⋊C4C4⋊C414yes—
173C4×Dic71—42no—
174C4×Dic71—42no—
175C4×Dic71C22×C412yes—
176C4×Dic71C2×Q812yes—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2411yescharacteristic
184C2×C8×C71—21no—
185C2×C8×C71—21no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71C22×C412yes—
189C2×C8×C71C22×C412yes—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—81no—
199C4⋊C4×C71—81no—
200C42×C71C22×C412yes—
201C42×C71C2×Q812yes—
202C23×C71C22×D411yescharacteristic
203C22×D71—21no—
204C22×D71—21no—
205C22×D71—21no—
206C22×D71—21no—
207C22×D71C2×C4⋊C412yes—
208C22×D71#12412yes—
209C22×D71#12412yes—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
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×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71C2×C4⋊C412yes—
227C2×C4×C71C2×C4⋊C412yes—
228C2×C4×C71C23×C411yescharacteristic
229C2×C4×C71C22×Q811yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—21no—
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×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71C2×C4⋊C412yes—
251C2×Dic71C2×C4⋊C412yes—
252C2×Dic71C2×C4⋊C412yes—
253C2×Dic71#12412yes—
254C2×Dic71#12412yes—
255C4×D71—84no—
256C4×D71—84no—
257C4×D71#11714yes—
258C4×D71#11714yes—
259C4×D71#12514yes—
260C4×D71#12514yes—
261C8×C71—22no—
262C8×C71—22no—
263C71⋊C8—22no—
264C71⋊C8—22no—
265C22×C71C22×C4⋊C411yescharacteristic
266C22×C71C2×#12411yescharacteristic
267C22×C71C2×#12411yescharacteristic
268C22×C71#57611yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57711yescharacteristic
271C22×C71#57811yescharacteristic
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71#38214yes—
279C4×C71—22no—
280C4×C71—22no—
281C4×C71—81no—
282C4×C71—81no—
283C4×C71—81no—
284C4×C71—81no—
285C4×C71C2×#11711yescharacteristic
286C4×C71C2×#11711yescharacteristic
287C4×C71C2×#12511yescharacteristic
288C4×C71C2×#12511yescharacteristic
289Dic71—22no—
290Dic71—22no—
291Dic71—81no—
292Dic71—81no—
293Dic71—81no—
294Dic71—81no—
295Dic71#38214yes—
296#20763—711nomaximal, Sylow
297C2×C71C2×#38211yescharacteristic
298C2×C71#291211yescharacteristic
299C2×C71#291311yescharacteristic
300C2×C71#313611yescharacteristic
301C2×C71#313611yescharacteristic
302C2×C71#313711yescharacteristic
303C2×C71#313711yescharacteristic
304D71—24no—
305C2×#405—711no—
306#2317—711no—
307#2317—711no—
308#1836—711no—
309#1836—711no—
310#1836—711no—
311#1836D7111yescharacteristic, complemented
312#1835—711no—
313#1835—711no—
314#1834—711no—
315#1834—711no—
316#1736—711no—
317#1736—711no—
318#1735—711no—
319C2×#340—711no—
320C71#2076311yescharacteristic, complemented, Sylow
321C22×M16—711no—
322C23×C8—711no—
323C2×#117—711no—
324C2×#117—711no—
325C2×#117—711no—
326C22×C4⋊C4—711no—
327C22×C4⋊C4—711no—
328#405—718no—
329C2×#97—711no—
330C2×#97C2×D7111yescharacteristic
331C2×C4×C8—711no—
332C2×C4×C8—711no—
333#389—1421no—
334#389—1421no—
335#389—1421no—
336#389—1421no—
337#385—1421no—
338#385—1421no—
339#385—1421no—
340#385—1421no—
341#384—711no—
342#384—711no—
343#384—711no—
344#384C2×D7111yescharacteristic
345#382—711no—
346#382—711no—
347#382—711no—
348#382C2×D7111yescharacteristic
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#340—711no—
359#340—711no—
360#340—711no—
361#340C2×D7111yescharacteristic
362#340C2×D7111yescharacteristic
363#340C2×D7111yescharacteristic
364#340C2×D7111yescharacteristic
365C23×C4—711no—
366C23×C4—1421no—
367C23×C4—1421no—
368C2×M16—714no—
369C2×M16—1422no—
370C2×M16—1422no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8—714no—
374C22×C8—1422no—
375C22×C8—1422no—
376C22×C8C22×D7111yescharacteristic
377C22×C8C22×D7111yescharacteristic
378#117—714no—
379#117—714no—
380#117—1422no—
381#117—1422no—
382#117—1422no—
383#117—1422no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—711no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4—1422no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4—1421no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4—1421no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4—1421no—
404C2×C4⋊C4C22×D7111yescharacteristic
405C2×C4⋊C4C22×D7111yescharacteristic
406C2×C4⋊C4C22×D7111yescharacteristic
407C2×C4⋊C4C22×D7111yescharacteristic
408C2×C22⋊C4—711no—
409C2×C22⋊C4—711no—
410C2×C22⋊C4—1421no—
411C2×C22⋊C4—1421no—
412C2×C42—711no—
413C2×C42—1421no—
414C2×C42—1421no—
415C2×C42C22×D7111yescharacteristic
416#97—714no—
417#97C4×D7114yescomplemented
418C4×C8—714no—
419C4×C8—714no—
420#95—21no—
421#95—21no—
422#95—1421no—
423#95—1421no—
424#95—1421no—
425#95—1421no—
426#95—1421no—
427#95—1421no—
428C24—711no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—2842no—
434C22×C4—2842no—
435C22×C4—5681no—
436C22×C4—5681no—
437C22×C4—5681no—
438C22×C4—5681no—
439C22×C4—711no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—712no—
445C22×C4—712no—
446C22×C4—711no—
447C22×C4—711no—
448C22×C4—1421no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4D4×D7111yescharacteristic
453C22×C4D4×D7111yescharacteristic
454C22×C4C23×D7111yescharacteristic
455M16—1428no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—712no—
461C2×C8—712no—
462C2×C8—1428no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8C2×C4×D7112yes—
468C2×C8C2×C4×D7112yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—2842no—
480C4⋊C4—2842no—
481C4⋊C4—5681no—
482C4⋊C4—5681no—
483C4⋊C4—5681no—
484C4⋊C4—5681no—
485C4⋊C4—5681no—
486C4⋊C4—5681no—
487C4⋊C4—714no—
488C4⋊C4—714no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C22⋊C4—2842no—
498C22⋊C4—2842no—
499C22⋊C4—714no—
500C22⋊C4—714no—
501C42—2842no—
502C42—2842no—
503C42—712no—
504C42—712no—
505C42C2×C4×D7112yes—
506C42Q8×D7112yes—
507C23—712no—
508C23—712no—
509C23—712no—
510C23—1421no—
511C23—1421no—
512C23—1421no—
513C23—1421no—
514C23C2×D4×D7111yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
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—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2841no—
541C2×C4—2841no—
542C2×C4—2841no—
543C2×C4—5684no—
544C2×C4—5684no—
545C2×C4—712no—
546C2×C4—712no—
547C2×C4—712no—
548C2×C4—714no—
549C2×C4—714no—
550C2×C4—714no—
551C2×C4—714no—
552C2×C4—712no—
553C2×C4—712no—
554C2×C4—1421no—
555C2×C4—1421no—
556C2×C4—1421no—
557C2×C4—1421no—
558C2×C4C4⋊C4×D7112yes—
559C2×C4C4⋊C4×D7112yes—
560C2×C4C22×C4×D7111yescharacteristic
561C2×C4C2×Q8×D7111yescharacteristic
562C2×C4#41×D7111yescharacteristic
563C2×C4#41×D7111yescharacteristic
564C2×C4#41×D7111yescharacteristic
565C2×C4#41×D7111yescharacteristic
566C8—22no—
567C8—22no—
568C8—1422no—
569C8—1422no—
570C22—714no—
571C22—1422no—
572C22—1422no—
573C22—1422no—
574C22—1422no—
575C22—1422no—
576C22—1422no—
577C22C2×C4⋊C4×D7111yescharacteristic
578C22#124×D7111yescharacteristic
579C22#124×D7111yescharacteristic
580C22#1798680311yescharacteristic
581C22#1798680811yescharacteristic
582C22#1798680911yescharacteristic
583C22#1798681511yescharacteristic
584C4—22no—
585C4—22no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—5681no—
591C4—5681no—
592C4—5681no—
593C4—5681no—
594C4—714no—
595C4—1422no—
596C4—1422no—
597C4#117×D7111yescharacteristic
598C4#117×D7111yescharacteristic
599C4#125×D7111yescharacteristic
600C4#125×D7111yescharacteristic
601C2—1424no—
602C2#382×D7111yescharacteristic
603C2#2259693411yescharacteristic
604C2#2259693911yescharacteristic
605C2#2259869111yescharacteristic
606C2#2259869511yescharacteristic
607C2#2259870211yescharacteristic
608C2#2259870611yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1836C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1836D71
hypocenterC71#20763
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. #22588474C2semidirect
5. #22588472C2semidirect
6. #22588472C2semidirect
7. #22588470C2semidirect
8. #22588469C2semidirect
9. #22588467C2semidirect
10. #22588466C2semidirect
15. C71×#1836C2semidirect
23. #17985956C4semidirect
102. #14556038C2×C4semidirect
103. C4×C71⋊C8C2×C4semidirect
104. C8×Dic71C2×C4semidirect
120. #97×C71C2×C4semidirect
311. #1836D71semidirect
320. C71#20763semidirect
417. #97C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859562
#225909652
#225909612
#225884742
#225884722
#225884702
#225884692
#225884672
#225884662
#225880582
#225880552
#340×D712
C71×#18362
#2076371