Finite Group Databasebeta

← #31999628

Subgroups — #31999628

Order 18176.

990 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986040C211yesmaximal, characteristic
2#22590821C211yesmaximal, characteristic
3C4×C4⋊C4×D71C211yesmaximal, characteristic
4#22588929C211yesmaximal, characteristic, complemented
5#22588924C211yesmaximal, characteristic, complemented
6#22588923C211yesmaximal, characteristic, complemented
7#22588919C211yesmaximal, characteristic, complemented
8#22588918C211yesmaximal, characteristic, complemented
9#22588914C211yesmaximal, characteristic, complemented
10#22588913C211yesmaximal, characteristic, complemented
11#22588058C211yesmaximal, characteristic
12#22588058C211yesmaximal, characteristic
13#22588058C211yesmaximal, characteristic
14#22588032C211yesmaximal, characteristic
15C71×#1921C211yesmaximal, characteristic, complemented
16C22×#11765848C2211yescharacteristic
17C22×#11765848C2211yescharacteristic
18C2×#14556086C2211yescharacteristic
19C2×#14556086C2211yescharacteristic
20C2×C4⋊C4×D71C2211yescharacteristic
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C42×D71—21no—
23C2×C42×D71C2211yescharacteristic
24#17986040C418yescomplemented
25C2×#14556061C2211yescharacteristic
26C2×#14556060C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28#17985898C2211yescharacteristic
29#17985897C2211yescharacteristic
30#17985895C2211yescharacteristic
31#17985873—21no—
32#17985870—21no—
33C4⋊C4×Dic71—21no—
34C4⋊C4×Dic71C2211yescharacteristic
35C4⋊C4×Dic71C2211yescharacteristic
36C4×Dic71⋊C4—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38#17985850—21no—
39#17985850—21no—
40#95×D71—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×D71—21no—
61C22×C4×D71—21no—
62C22×C4×D71D411yescharacteristic
63C22×C4×D71D411yescharacteristic
64C22×C4×D71C2311yescharacteristic
65C2×#11765848—22no—
66C2×#11765848—22no—
67C2×#11765848—22no—
68C2×#11765848—22no—
69C2×#11765848—22no—
70C2×#11765848C2×C414yes—
71C22×C71⋊C8C2311yescharacteristic
72C22×C71⋊C8C2311yescharacteristic
73#14556086—22no—
74#14556086—22no—
75#14556086—22no—
76#14556086C2×C414yes—
77C2×C4⋊Dic71—21no—
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×C4⋊Dic71C2311yescharacteristic
82C2×Dic71⋊C4—21no—
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C4⋊C4×D71—22no—
87C4⋊C4×D71—22no—
88C4⋊C4×D71C2×C414yes—
89C4⋊C4×D71Q814yes—
90C2×(C2×D71)⋊C4—21no—
91C2×(C2×D71)⋊C4—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—
102C42×D71—44no—
103#14556061C2×C414yescomplemented
104#14556060C2×C414yescomplemented
105#14556059C2×C414yescomplemented
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C71⋊#95—21no—
111C71⋊#95—21no—
112C22×C8×C71C2311yescharacteristic
113C22×C8×C71C2311yescharacteristic
114C2×C4⋊C4×C71—21no—
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C42×C71—21no—
120C2×C42×C71C2311yescharacteristic
121#107×C71C2×C414yescomplemented
122#95×C71—21no—
123#95×C71—21no—
124C23×D71C2×D411yescharacteristic
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C22×Dic71C2×D411yescharacteristic
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—24no—
136C2×C4×D71—24no—
137C2×C4×D71—22no—
138C2×C4×D71—41no—
139C2×C4×D71—41no—
140C2×C4×D71—42no—
141C2×C4×D71—42no—
142C2×C4×D71—42no—
143C2×C4×D71—42no—
144C2×C4×D71C22×C412yes—
145C2×C4×D71C2×Q812yes—
146C2×C4×D71#4112yes—
147C2×C4×D71#4112yes—
148C2×C4×D71#4112yes—
149C2×C4×D71#4112yes—
150#11765848—28no—
151#11765848—44no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8C22×C412yes—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163C4⋊Dic71—42no—
164C4⋊Dic71C22×C412yes—
165C4⋊Dic71C22×C412yes—
166C4⋊Dic71C2×Q812yes—
167Dic71⋊C4—24no—
168Dic71⋊C4—24no—
169Dic71⋊C4—42no—
170Dic71⋊C4—42no—
171(C2×D71)⋊C4—24no—
172(C2×D71)⋊C4—42no—
173C4×Dic71—21no—
174C4×Dic71—21no—
175C4×Dic71—24no—
176C4×Dic71—22no—
177C4×Dic71—22no—
178C4×Dic71—42no—
179C4×Dic71—42no—
180C4×Dic71—42no—
181C4×Dic71—42no—
182C22×C4×C71—21no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—21no—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2411yescharacteristic
189C2×C8×C71—21no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—21no—
193C2×C8×C71—21no—
194C2×C8×C71C22×C412yes—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—42no—
201C4⋊C4×C71C22×C412yes—
202C4⋊C4×C71C22×C412yes—
203C4⋊C4×C71C2×Q812yes—
204C42×C71—21no—
205C42×C71—21no—
206C42×C71—42no—
207C23×C71C22×D411yescharacteristic
208C22×D71—21no—
209C22×D71—21no—
210C22×D71—21no—
211C22×D71—21no—
212C22×D71C4×D412yes—
213C22×D71C22⋊Q812yes—
214C22×D71#12312yes—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
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#13712yes—
233C2×C4×C71C23×C411yescharacteristic
234C2×C4×C71C22×Q811yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×C4×C71C2×#4111yescharacteristic
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—21no—
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×Dic71C4×D412yes—
257C2×Dic71C22⋊Q812yes—
258C2×Dic71#12312yes—
259C2×Dic71AGL(1,ℤ/8)12yes—
260C4×D71—28no—
261C4×D71—48no—
262C4×D71—44no—
263C4×D71—48no—
264C4×D71#11714yes—
265C4×D71C4×Q814yes—
266C4×D71#12514yes—
267C4×D71#12614yes—
268C8×C71—22no—
269C8×C71—41no—
270C71⋊C8—22no—
271C71⋊C8—41no—
272C22×C71C2×C4×D411yescharacteristic
273C22×C71C2×C22⋊Q811yescharacteristic
274C22×C71C2×#12311yescharacteristic
275C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C71C2×#13711yescharacteristic
277C22×C71#57511yescharacteristic
278C22×C71#57511yescharacteristic
279C2×D71—22no—
280C2×D71—22no—
281C2×D71—22no—
282C2×D71—22no—
283C2×D71—22no—
284C2×D71—22no—
285C2×D71#38714yes—
286C4×C71—22no—
287C4×C71—42no—
288C4×C71—41no—
289C4×C71—42no—
290C4×C71—42no—
291C4×C71—81no—
292C4×C71C2×#11711yescharacteristic
293C4×C71C2×C4×Q811yescharacteristic
294C4×C71C2×#12511yescharacteristic
295C4×C71C2×#12611yescharacteristic
296Dic71—22no—
297Dic71—42no—
298Dic71—41no—
299Dic71—42no—
300Dic71—42no—
301Dic71—81no—
302Dic71#38714yes—
303#21033—711nomaximal, Sylow
304C2×C71C2×#38711yescharacteristic
305C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C71C4×#13711yescharacteristic
307C2×C71#307711yescharacteristic
308C2×C71#307811yescharacteristic
309C2×C71#309211yescharacteristic
310C2×C71#309311yescharacteristic
311D71—24no—
312C2×#428—711no—
313#2296—711no—
314C2×C4×C4⋊C4—711no—
315#1921—711no—
316#1921D7111yescharacteristic, complemented
317#1920—711no—
318#1920—711no—
319#1919—711no—
320#1919—711no—
321#1918—711no—
322#1918—711no—
323#1736—711no—
324#1736—711no—
325#1736—711no—
326#1728—711no—
327C71#2103311yescharacteristic, complemented, Sylow
328C22×M16—711no—
329C22×M16—711no—
330C2×#117—711no—
331C2×#117—711no—
332C22×C4⋊C4—711no—
333C22×C4⋊C4—711no—
334C22×C42—711no—
335C22×C42—1421no—
336#428—718no—
337C2×#107—711no—
338C2×#107C2×D7111yescharacteristic
339C2×#106—711no—
340C2×#106—711no—
341#389—711no—
342#389—711no—
343#389—711no—
344#389C2×D7111yescharacteristic
345#385—1421no—
346#385—1421no—
347C4×C4⋊C4—711no—
348C4×C4⋊C4—711no—
349C4×C4⋊C4—711no—
350C4×C4⋊C4—1421no—
351C4×C4⋊C4—1421no—
352C4×C4⋊C4—1421no—
353C4×C4⋊C4—1421no—
354C4×C4⋊C4C2×D7111yescharacteristic
355C2×#95—1421no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340C2×D7111yescharacteristic
366#340C2×D7111yescharacteristic
367#340C2×D7111yescharacteristic
368#336—711no—
369#336—711no—
370#336—711no—
371#336C2×D7111yescharacteristic
372C23×C4—711no—
373C23×C4—711no—
374C23×C4—711no—
375C23×C4—1421no—
376C23×C4—1421no—
377C2×M16—714no—
378C2×M16—1422no—
379C2×M16—1422no—
380C2×M16—1422no—
381C2×M16—1422no—
382C2×M16—1422no—
383C22×C8—711no—
384C22×C8—711no—
385C22×C8C22×D7111yescharacteristic
386C22×C8C22×D7111yescharacteristic
387#117—714no—
388#117—1422no—
389#117—1422no—
390#117—1422no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—711no—
393C2×C4⋊C4—711no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—714no—
396C2×C4⋊C4—714no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—1422no—
399C2×C4⋊C4—1422no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4—1421no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4—1421no—
404C2×C4⋊C4—1421no—
405C2×C4⋊C4C22×D7111yescharacteristic
406C2×C4⋊C4C22×D7111yescharacteristic
407C2×C4⋊C4C22×D7111yescharacteristic
408C2×C4⋊C4C22×D7111yescharacteristic
409C2×C22⋊C4—1421no—
410C2×C22⋊C4—1421no—
411C2×C42—21no—
412C2×C42—2844no—
413C2×C42—711no—
414C2×C42—711no—
415C2×C42—711no—
416C2×C42—1422no—
417C2×C42—1422no—
418C2×C42—1421no—
419C2×C42—1421no—
420C2×C42—1421no—
421C2×C42—1421no—
422C2×C42—1421no—
423C2×C42C22×D7111yescharacteristic
424#107—714no—
425#107C4×D7114yescomplemented
426#106—714no—
427#106—714no—
428#95—21no—
429#95—21no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436C24—711no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2841no—
442C22×C4—2841no—
443C22×C4—2842no—
444C22×C4—2842no—
445C22×C4—2842no—
446C22×C4—2842no—
447C22×C4—711no—
448C22×C4—712no—
449C22×C4—712no—
450C22×C4—712no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—712no—
454C22×C4—711no—
455C22×C4—711no—
456C22×C4—1422no—
457C22×C4—1424no—
458C22×C4—1424no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1422no—
466C22×C4D4×D7111yescharacteristic
467C22×C4D4×D7111yescharacteristic
468C22×C4C23×D7111yescharacteristic
469M16—2844no—
470M16—1428no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—712no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8C2×C4×D7112yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—2842no—
490C4⋊C4—2842no—
491C4⋊C4—2842no—
492C4⋊C4—712no—
493C4⋊C4—712no—
494C4⋊C4—712no—
495C4⋊C4—1424no—
496C4⋊C4—1424no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4—1421no—
502C4⋊C4C2×C4×D7112yes—
503C4⋊C4C2×C4×D7112yes—
504C4⋊C4Q8×D7112yes—
505C22⋊C4—2842no—
506C22⋊C4—1424no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—2842no—
511C42—2842no—
512C42—2842no—
513C42—2842no—
514C42—1422no—
515C42—1422no—
516C42—1424no—
517C42—1421no—
518C42—1421no—
519C23—712no—
520C23—712no—
521C23—712no—
522C23—1421no—
523C23—1421no—
524C23—1421no—
525C23—1421no—
526C23C2×D4×D7111yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—2848no—
545C2×C4—2844no—
546C2×C4—2848no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—712no—
557C2×C4—712no—
558C2×C4—712no—
559C2×C4—714no—
560C2×C4—714no—
561C2×C4—714no—
562C2×C4—714no—
563C2×C4—712no—
564C2×C4—1428no—
565C2×C4—1422no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—1421no—
569C2×C4—1421no—
570C2×C4—1421no—
571C2×C4—1421no—
572C2×C4—1421no—
573C2×C4#1455616012yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4C2×Q8×D7111yescharacteristic
576C2×C4#41×D7111yescharacteristic
577C2×C4#41×D7111yescharacteristic
578C2×C4#41×D7111yescharacteristic
579C2×C4#41×D7111yescharacteristic
580C8—22no—
581C8—41no—
582C8—2841no—
583C8—1422no—
584C22—714no—
585C22—1422no—
586C22—1422no—
587C22—1422no—
588C22—1422no—
589C22—1422no—
590C22—1422no—
591C22C4×D4×D7111yescharacteristic
592C22C22⋊Q8×D7111yescharacteristic
593C22#123×D7111yescharacteristic
594C22AGL(1,ℤ/8)×D7111yescharacteristic
595C22C2×#1455616011yescharacteristic
596C22#1798679511yescharacteristic
597C22#1798679611yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—2842no—
605C4—2842no—
606C4—2841no—
607C4—2842no—
608C4—5681no—
609C4—714no—
610C4—1422no—
611C4#117×D7111yescharacteristic
612C4C4×Q8×D7111yescharacteristic
613C4#125×D7111yescharacteristic
614C4#126×D7111yescharacteristic
615C2—1424no—
616C2#387×D7111yescharacteristic
617C2#2259715111yescharacteristic
618C2C4×#1455616011yescharacteristic
619C2#2259813511yescharacteristic
620C2#2259814611yescharacteristic
621C2#2259826711yescharacteristic
622C2#2259826811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1921C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1921D71
hypocenterC71#21033
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #22588929C2semidirect
5. #22588924C2semidirect
6. #22588923C2semidirect
7. #22588919C2semidirect
8. #22588918C2semidirect
9. #22588914C2semidirect
10. #22588913C2semidirect
15. C71×#1921C2semidirect
24. #17986040C4semidirect
103. #14556061C2×C4semidirect
104. #14556060C2×C4semidirect
105. #14556059C2×C4semidirect
121. #107×C71C2×C4semidirect
316. #1921D71semidirect
327. C71#21033semidirect
425. #107C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860402
#225908212
C4×C4⋊C4×D712
#225889292
#225889242
#225889232
#225889192
#225889182
#225889142
#225889132
#225880582
#225880322
C71×#19212
#2103371