Finite Group Databasebeta

← #31999608

Subgroups — #31999608

Order 18176.

926 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×#17986036C211yesmaximal, characteristic
2#22590821C211yesmaximal, characteristic
3C4×#14556086C211yesmaximal, characteristic
4#22588933C211yesmaximal, characteristic, complemented
5#22588931C211yesmaximal, characteristic, complemented
6#22588925C211yesmaximal, characteristic, complemented
7#22588923C211yesmaximal, characteristic, complemented
8#22588921C211yesmaximal, characteristic, complemented
9#22588915C211yesmaximal, characteristic, complemented
10#22588913C211yesmaximal, characteristic, complemented
11#22588055C211yesmaximal, characteristic
12#22588055C211yesmaximal, characteristic
13#22588055C211yesmaximal, characteristic
14#336×D71C211yesmaximal, characteristic
15C71×#1918C211yesmaximal, characteristic, complemented
16C22×C8×D71C2211yescharacteristic
17C22×C8×D71C2211yescharacteristic
18C2×#14556086C2211yescharacteristic
19C2×#14556086C2211yescharacteristic
20C2×#14556086C2211yescharacteristic
21C2×#14556086C2211yescharacteristic
22C2×C42×D71C2211yescharacteristic
23#17986036C418yescomplemented
24C2×#14556062C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27#17985898C2211yescharacteristic
28#17985897C2211yescharacteristic
29#17985895C2211yescharacteristic
30#17985873—21no—
31#17985870—21no—
32C4⋊C4×Dic71—21no—
33C4⋊C4×Dic71C2211yescharacteristic
34C4⋊C4×Dic71C2211yescharacteristic
35C4×C4⋊Dic71C2211yescharacteristic
36#17985850—21no—
37C4×(C2×D71)⋊C4—21no—
38#17985846—21no—
39#17985844—21no—
40C42×Dic71—21no—
41#17985762C2211yescharacteristic
42#17985762C2211yescharacteristic
43#17985762C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985754C2211yescharacteristic
51#17985753C2211yescharacteristic
52#17985753C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55C4×C4⋊C4×C71C2211yescharacteristic
56#340×C71C2211yescharacteristic
57#340×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#336×C71C2211yescharacteristic
60C22×C4×D71D411yescharacteristic
61C22×C4×D71D411yescharacteristic
62C22×C4×D71C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C2×C8×D71—22no—
66C2×C8×D71—22no—
67C2×C8×D71—22no—
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#14556086—22no—
76#14556086C2×C414yes—
77#14556086C2×C414yes—
78#14556086Q814yes—
79C2×C4⋊Dic71—21no—
80C2×C4⋊Dic71C2311yescharacteristic
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×C4⋊Dic71C2311yescharacteristic
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C2×(C2×D71)⋊C4—21no—
87C2×(C2×D71)⋊C4—21no—
88C2×(C2×D71)⋊C4—21no—
89C2×(C2×D71)⋊C4—21no—
90C2×C4×Dic71—21no—
91C2×C4×Dic71—21no—
92C2×C4×Dic71—21no—
93C2×C4×Dic71—21no—
94C2×C4×Dic71—21no—
95C2×C4×Dic71—21no—
96C2×C4×Dic71—21no—
97C2×C4×Dic71D411yescharacteristic
98C2×C4×Dic71D411yescharacteristic
99C2×C4×Dic71C2311yescharacteristic
100C42×D71—22no—
101C42×D71—22no—
102#14556062C2×C414yescomplemented
103#14556059C2×C414yescomplemented
104#14556059C2×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×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C42×C71—21no—
119C2×C42×C71C2311yescharacteristic
120#107×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—22no—
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—41no—
136C2×C4×D71—41no—
137C2×C4×D71C22×C412yes—
138C2×C4×D71C2×Q812yes—
139C2×C4×D71#4112yes—
140C2×C4×D71#4112yes—
141C2×C4×D71#4112yes—
142C2×C4×D71#4112yes—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8C22×C412yes—
149C8×D71—28no—
150C8×D71—44no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—42no—
157C4⋊Dic71C22×C412yes—
158C4⋊Dic71C22×C412yes—
159C4⋊Dic71C2×Q812yes—
160Dic71⋊C4—42no—
161Dic71⋊C4—42no—
162(C2×D71)⋊C4—24no—
163(C2×D71)⋊C4—24no—
164(C2×D71)⋊C4—24no—
165(C2×D71)⋊C4—42no—
166C4×Dic71—21no—
167C4×Dic71—21no—
168C4×Dic71—24no—
169C4×Dic71—24no—
170C4×Dic71—24no—
171C4×Dic71—22no—
172C4×Dic71—22no—
173C4×Dic71—42no—
174C4×Dic71—42no—
175C4×Dic71—42no—
176C4×Dic71—42no—
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×C71—21no—
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—42no—
196C4⋊C4×C71C22×C412yes—
197C4⋊C4×C71C22×C412yes—
198C4⋊C4×C71C2×Q812yes—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—42no—
202C23×C71C22×D411yescharacteristic
203C22×D71—21no—
204C22×D71—21no—
205C22×D71—21no—
206C22×D71—21no—
207C22×D71C4×D412yes—
208C22×D71C22⋊Q812yes—
209C22×D71#12312yes—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
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×C71—41no—
227C2×C4×C71#13512yes—
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—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—21no—
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—
251C2×Dic71C4×D412yes—
252C2×Dic71C22⋊Q812yes—
253C2×Dic71#12312yes—
254C2×Dic71#13512yes—
255C4×D71—44no—
256C4×D71—44no—
257C4×D71#11714yes—
258C4×D71C4×Q814yes—
259C4×D71#12514yes—
260C4×D71#12614yes—
261C8×C71—22no—
262C8×C71—41no—
263C71⋊C8—22no—
264C71⋊C8—41no—
265C22×C71C2×C4×D411yescharacteristic
266C22×C71C2×C22⋊Q811yescharacteristic
267C22×C71C2×#12311yescharacteristic
268C22×C71C2×#13511yescharacteristic
269C22×C71C2×#13511yescharacteristic
270C22×C71#57511yescharacteristic
271C22×C71#57511yescharacteristic
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71#38714yes—
279C4×C71—22no—
280C4×C71—42no—
281C4×C71—41no—
282C4×C71—42no—
283C4×C71—42no—
284C4×C71—81no—
285C4×C71C2×#11711yescharacteristic
286C4×C71C2×C4×Q811yescharacteristic
287C4×C71C2×#12511yescharacteristic
288C4×C71C2×#12611yescharacteristic
289Dic71—22no—
290Dic71—42no—
291Dic71—41no—
292Dic71—42no—
293Dic71—42no—
294Dic71—81no—
295Dic71#38714yes—
296#21032—711nomaximal, Sylow
297C2×C71C2×#38711yescharacteristic
298C2×C71C4×#13511yescharacteristic
299C2×C71#294311yescharacteristic
300C2×C71#307611yescharacteristic
301C2×C71#307611yescharacteristic
302C2×C71#309111yescharacteristic
303C2×C71#309111yescharacteristic
304D71—24no—
305C2×#427—711no—
306#2296—711no—
307C4×#117—711no—
308#1921—711no—
309#1921—711no—
310#1920—711no—
311#1920—711no—
312#1919—711no—
313#1919—711no—
314#1918—711no—
315#1918D7111yescharacteristic, complemented
316#1735—711no—
317#1735—711no—
318#1735—711no—
319C2×#336—711no—
320C71#2103211yescharacteristic, complemented, Sylow
321C23×C8—711no—
322C23×C8—711no—
323C2×#117—711no—
324C2×#117—711no—
325C2×#117—711no—
326C2×#117—711no—
327C22×C42—711no—
328#427—718no—
329C2×#107—711no—
330C2×#107C2×D7111yescharacteristic
331C2×#106—711no—
332C2×#106—711no—
333#389—711no—
334#389—711no—
335#389—711no—
336#389C2×D7111yescharacteristic
337#385—1421no—
338#385—1421no—
339C4×C4⋊C4—711no—
340C4×C4⋊C4—711no—
341C4×C4⋊C4—711no—
342C4×C4⋊C4—1421no—
343C4×C4⋊C4—1421no—
344C4×C4⋊C4C2×D7111yescharacteristic
345C4×C22⋊C4—1421no—
346C4×C22⋊C4—1421no—
347#376—1421no—
348C43—1421no—
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#340C2×D7111yescharacteristic
359#340C2×D7111yescharacteristic
360#340C2×D7111yescharacteristic
361#336—711no—
362#336—711no—
363#336—711no—
364#336C2×D7111yescharacteristic
365C23×C4—711no—
366C23×C4—711no—
367C23×C4—711no—
368C22×C8—711no—
369C22×C8—714no—
370C22×C8—711no—
371C22×C8—1422no—
372C22×C8—1422no—
373C22×C8—1422no—
374C22×C8—1422no—
375C22×C8—1422no—
376C22×C8C22×D7111yescharacteristic
377C22×C8C22×D7111yescharacteristic
378#117—714no—
379#117—714no—
380#117—714no—
381#117—1422no—
382#117—1422no—
383#117—1422no—
384#117—1422no—
385#117—1422no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C42—21no—
403C2×C42—711no—
404C2×C42—711no—
405C2×C42—711no—
406C2×C42—1422no—
407C2×C42—1422no—
408C2×C42—1421no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42—1421no—
413C2×C42—1421no—
414C2×C42—1421no—
415C2×C42C22×D7111yescharacteristic
416#107—714no—
417#107C4×D7114yescomplemented
418#106—714no—
419#106—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—2841no—
434C22×C4—2841no—
435C22×C4—711no—
436C22×C4—712no—
437C22×C4—712no—
438C22×C4—712no—
439C22×C4—712no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—711no—
443C22×C4—711no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1422no—
452C22×C4D4×D7111yescharacteristic
453C22×C4D4×D7111yescharacteristic
454C22×C4C23×D7111yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—2844no—
461C2×C8—712no—
462C2×C8—1428no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8C2×C4×D7112yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—2842no—
476C4⋊C4—2842no—
477C4⋊C4—2842no—
478C4⋊C4—712no—
479C4⋊C4—712no—
480C4⋊C4—712no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4C2×C4×D7112yes—
487C4⋊C4C2×C4×D7112yes—
488C4⋊C4Q8×D7112yes—
489C22⋊C4—2842no—
490C22⋊C4—1424no—
491C22⋊C4—1424no—
492C22⋊C4—1424no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—2842no—
497C42—2842no—
498C42—2842no—
499C42—2842no—
500C42—1422no—
501C42—1422no—
502C42—1424no—
503C42—1424no—
504C42—1424no—
505C42—1421no—
506C42—1421no—
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—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
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—41no—
532C2×C4—2844no—
533C2×C4—2844no—
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—712no—
544C2×C4—712no—
545C2×C4—712no—
546C2×C4—714no—
547C2×C4—714no—
548C2×C4—714no—
549C2×C4—714no—
550C2×C4—712no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4—1421no—
555C2×C4—1421no—
556C2×C4—1421no—
557C2×C4—1421no—
558C2×C4—1421no—
559C2×C4#1455614712yes—
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—41no—
568C8—2841no—
569C8—1422no—
570C22—714no—
571C22—1422no—
572C22—1422no—
573C22—1422no—
574C22—1422no—
575C22—1422no—
576C22—1422no—
577C22C4×D4×D7111yescharacteristic
578C22C22⋊Q8×D7111yescharacteristic
579C22#123×D7111yescharacteristic
580C22#135×D7111yescharacteristic
581C22C2×#1455614711yescharacteristic
582C22#1798679811yescharacteristic
583C22#1798679811yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—2842no—
591C4—2842no—
592C4—2841no—
593C4—2842no—
594C4—5681no—
595C4—714no—
596C4—1422no—
597C4#117×D7111yescharacteristic
598C4C4×Q8×D7111yescharacteristic
599C4#125×D7111yescharacteristic
600C4#126×D7111yescharacteristic
601C2—1424no—
602C2#387×D7111yescharacteristic
603C2C4×#1455614711yescharacteristic
604C2#2259713911yescharacteristic
605C2#2259812111yescharacteristic
606C2#2259812711yescharacteristic
607C2#2259824811yescharacteristic
608C2#2259825811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1918C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1918D71
hypocenterC71#21032
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
4. #22588933C2semidirect
5. #22588931C2semidirect
6. #22588925C2semidirect
7. #22588923C2semidirect
8. #22588921C2semidirect
9. #22588915C2semidirect
10. #22588913C2semidirect
15. C71×#1918C2semidirect
23. #17986036C4semidirect
102. #14556062C2×C4semidirect
103. #14556059C2×C4semidirect
104. #14556059C2×C4semidirect
120. #107×C71C2×C4semidirect
315. #1918D71semidirect
320. C71#21032semidirect
417. #107C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860362
#225908212
C4×#145560862
#225889332
#225889312
#225889252
#225889232
#225889212
#225889152
#225889132
#225880552
#336×D712
C71×#19182
#2103271