Finite Group Databasebeta

← #1921×D71 #31999600

Subgroups — #1921×D71 #31999600

Order 18176.

1086 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D71C211yesmaximal, characteristic
2#389×D71C211yesmaximal, characteristic
3C4×C4⋊C4×D71C211yesmaximal, characteristic
4#22588935C211yesmaximal, characteristic, complemented
5#22588934C211yesmaximal, characteristic, complemented
6#22588933C211yesmaximal, characteristic, complemented
7#22588932C211yesmaximal, characteristic, complemented
8#22588931C211yesmaximal, characteristic, complemented
9#22588930C211yesmaximal, characteristic, complemented
10#22588929C211yesmaximal, characteristic, complemented
11#340×D71C211yesmaximal, characteristic
12#340×D71C211yesmaximal, characteristic
13#340×D71C211yesmaximal, characteristic
14#336×D71C211yesmaximal, characteristic
15C71×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D71C2211yescharacteristic
17C22×C8×D71C2211yescharacteristic
18C2×C4⋊C4×D71—21no—
19C2×C4⋊C4×D71C2211yescharacteristic
20C2×C4⋊C4×D71C2211yescharacteristic
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C4⋊C4×D71C2211yescharacteristic
23C2×C42×D71—21no—
24C2×C42×D71C2211yescharacteristic
25C2×#14556062C2211yescharacteristic
26C2×#14556061C2211yescharacteristic
27C2×#14556061C2211yescharacteristic
28#107×D71C418yescomplemented
29#17985898C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985896—21no—
32#17985895C2211yescharacteristic
33#17985894—21no—
34C4⋊C4×Dic71C2211yescharacteristic
35C4⋊C4×Dic71C2211yescharacteristic
36C4×Dic71⋊C4—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38#17985850—21no—
39#95×D71—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×D71—21no—
63C22×C4×D71—21no—
64C22×C4×D71D411yescharacteristic
65C22×C4×D71D411yescharacteristic
66C22×C4×D71C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C2×C8×D71—22no—
70C2×C8×D71—22no—
71C2×C8×D71—22no—
72C2×C8×D71—22no—
73C2×C8×D71—22no—
74C2×C8×D71C2×C414yes—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71C2311yescharacteristic
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
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×D71—22no—
89C4⋊C4×D71—22no—
90C4⋊C4×D71—22no—
91C4⋊C4×D71—44no—
92C4⋊C4×D71C2×C414yes—
93C4⋊C4×D71C2×C414yes—
94C4⋊C4×D71Q814yes—
95C2×C4×Dic71—21no—
96C2×C4×Dic71—21no—
97C2×C4×Dic71—21no—
98C2×C4×Dic71D411yescharacteristic
99C2×C4×Dic71D411yescharacteristic
100C2×C4×Dic71C2311yescharacteristic
101C42×D71—22no—
102C42×D71—22no—
103C42×D71—44no—
104#14556062C2×C414yescomplemented
105#14556061C2×C414yescomplemented
106#14556061C2×C414yescomplemented
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C71⋊#95—21no—
111C71⋊#95—21no—
112C71⋊#95—21no—
113C22×C8×C71C2311yescharacteristic
114C22×C8×C71C2311yescharacteristic
115C2×C4⋊C4×C71—21no—
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C4⋊C4×C71C2311yescharacteristic
120C2×C42×C71—21no—
121C2×C42×C71C2311yescharacteristic
122#107×C71C2×C414yescomplemented
123#95×C71—21no—
124#95×C71—21no—
125C23×D71C2×D411yescharacteristic
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C22×Dic71C2×D411yescharacteristic
131C22×Dic71C2×D411yescharacteristic
132C22×Dic71C2×D411yescharacteristic
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—24no—
136C2×C4×D71—24no—
137C2×C4×D71—24no—
138C2×C4×D71—22no—
139C2×C4×D71—22no—
140C2×C4×D71—22no—
141C2×C4×D71—42no—
142C2×C4×D71—42no—
143C2×C4×D71—42no—
144C2×C4×D71—42no—
145C2×C4×D71—42no—
146C2×C4×D71—42no—
147C2×C4×D71—42no—
148C2×C4×D71—42no—
149C2×C4×D71—42no—
150C2×C4×D71Q1614yes—
151C2×C4×D71C22×C412yes—
152C2×C4×D71C2×Q812yes—
153C2×C4×D71#4112yes—
154C2×C4×D71#4112yes—
155C2×C4×D71#4112yes—
156C2×C4×D71#4112yes—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8—21no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8C22×C412yes—
163C8×D71—28no—
164C8×D71—44no—
165C4⋊Dic71—21no—
166C4⋊Dic71—21no—
167C4⋊Dic71—21no—
168C4⋊Dic71—21no—
169C4⋊Dic71—21no—
170C4⋊Dic71—42no—
171C4⋊Dic71C22×C412yes—
172C4⋊Dic71C22×C412yes—
173C4⋊Dic71C2×Q812yes—
174Dic71⋊C4—24no—
175Dic71⋊C4—24no—
176Dic71⋊C4—24no—
177Dic71⋊C4—42no—
178Dic71⋊C4—42no—
179Dic71⋊C4—42no—
180C4×Dic71—21no—
181C4×Dic71—21no—
182C4×Dic71—22no—
183C4×Dic71—22no—
184C4×Dic71—42no—
185C4×Dic71—42no—
186C4×Dic71—42no—
187C22×C4×C71—21no—
188C22×C4×C71—21no—
189C22×C4×C71—21no—
190C22×C4×C71—21no—
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2411yescharacteristic
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71—21no—
198C2×C8×C71—21no—
199C2×C8×C71C22×C412yes—
200C4⋊C4×C71—21no—
201C4⋊C4×C71—21no—
202C4⋊C4×C71—21no—
203C4⋊C4×C71—21no—
204C4⋊C4×C71—21no—
205C4⋊C4×C71—42no—
206C4⋊C4×C71C22×C412yes—
207C4⋊C4×C71C22×C412yes—
208C4⋊C4×C71C2×Q812yes—
209C42×C71—21no—
210C42×C71—21no—
211C42×C71—42no—
212C23×C71C22×D411yescharacteristic
213C22×D71C4×D412yes—
214C22×D71C22⋊Q812yes—
215C22×D71#12312yes—
216C22×D71C2×Q1612yes—
217C22×D71#13512yes—
218C22×D71AGL(1,ℤ/8)12yes—
219C22×D71AGL(1,ℤ/8)12yes—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71C2×Q1612yes—
238C2×C4×C71C23×C411yescharacteristic
239C2×C4×C71C22×Q811yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—21no—
250C2×Dic71—21no—
251C2×Dic71—21no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71C4×D412yes—
262C2×Dic71C22⋊Q812yes—
263C2×Dic71#12312yes—
264C2×Dic71C2×Q1612yes—
265C4×D71—28no—
266C4×D71—48no—
267C4×D71—48no—
268C4×D71—48no—
269C4×D71—44no—
270C4×D71—84no—
271C4×D71#11714yes—
272C4×D71C4×Q814yes—
273C4×D71#12514yes—
274C4×D71#12614yes—
275C8×C71—22no—
276C8×C71—41no—
277C71⋊C8—22no—
278C71⋊C8—41no—
279C22×C71C2×C4×D411yescharacteristic
280C22×C71C2×C22⋊Q811yescharacteristic
281C22×C71C2×#12311yescharacteristic
282C22×C71C22×Q1611yescharacteristic
283C22×C71C2×#13511yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
286C2×D71#38714yes—
287C2×D71C4×Q1614yes—
288C2×D71#44214yes—
289C2×D71#47714yes—
290C2×D71#47814yes—
291C2×D71#48214yes—
292C2×D71#48414yes—
293C4×C71—22no—
294C4×C71—42no—
295C4×C71—41no—
296C4×C71—42no—
297C4×C71—42no—
298C4×C71—81no—
299C4×C71C2×#11711yescharacteristic
300C4×C71C2×C4×Q811yescharacteristic
301C4×C71C2×#12511yescharacteristic
302C4×C71C2×#12611yescharacteristic
303Dic71—22no—
304Dic71—42no—
305Dic71—41no—
306Dic71—42no—
307Dic71—42no—
308Dic71—81no—
309Dic71#38714yes—
310C2×#1921—711nomaximal, Sylow
311C2×C71C2×#38711yescharacteristic
312C2×C71C2×C4×Q1611yescharacteristic
313C2×C71C2×#44211yescharacteristic
314C2×C71C2×#47811yescharacteristic
315C2×C71C2×#47711yescharacteristic
316C2×C71C2×#48211yescharacteristic
317C2×C71C2×#48411yescharacteristic
318D71#192118yesdirect product factor
319C22×#107—711no—
320C2×#389—711no—
321C2×C4×C4⋊C4—711no—
322#1921—711no—
323#1921—711no—
324#1921—711no—
325#1921—711no—
326#1921—711no—
327#1921—711no—
328#1921—711no—
329#1921D7111yescharacteristic, direct product factor
330C2×#340—711no—
331C2×#340—711no—
332C2×#340—711no—
333C2×#336—711no—
334C71C2×#192111yescharacteristic, complemented, Sylow
335C23×C8—711no—
336C23×C8—711no—
337C22×C4⋊C4—711no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—1421no—
342C22×C42—711no—
343C22×C42—1421no—
344C2×#107—711no—
345C2×#107—711no—
346C2×#107—711no—
347C2×#107—718no—
348C2×#107C2×D7111yescharacteristic
349#389—711no—
350#389—711no—
351#389—711no—
352#389—1421no—
353#389—1421no—
354#389C2×D7111yescharacteristic
355C4×C4⋊C4—711no—
356C4×C4⋊C4—711no—
357C4×C4⋊C4—711no—
358C4×C4⋊C4—1421no—
359C4×C4⋊C4—1421no—
360C4×C4⋊C4C2×D7111yescharacteristic
361C2×#95—1421no—
362C2×#95—1421no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#340—711no—
371#340—711no—
372#340C2×D7111yescharacteristic
373#340C2×D7111yescharacteristic
374#340C2×D7111yescharacteristic
375#336—711no—
376#336—711no—
377#336—711no—
378#336C2×D7111yescharacteristic
379C23×C4—711no—
380C23×C4—711no—
381C23×C4—711no—
382C23×C4—1421no—
383C23×C4—1421no—
384C23×C4—1421no—
385C23×C4—1421no—
386C22×C8—711no—
387C22×C8—714no—
388C22×C8—711no—
389C22×C8—1422no—
390C22×C8—1422no—
391C22×C8—1422no—
392C22×C8—1422no—
393C22×C8—1422no—
394C22×C8C22×D7111yescharacteristic
395C22×C8C22×D7111yescharacteristic
396C2×C4⋊C4—21no—
397C2×C4⋊C4—2844no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—711no—
400C2×C4⋊C4—711no—
401C2×C4⋊C4—714no—
402C2×C4⋊C4—714no—
403C2×C4⋊C4—714no—
404C2×C4⋊C4—711no—
405C2×C4⋊C4—1422no—
406C2×C4⋊C4—1422no—
407C2×C4⋊C4—1422no—
408C2×C4⋊C4—1422no—
409C2×C4⋊C4—1422no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1421no—
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C4⋊C4C22×D7111yescharacteristic
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C42—21no—
422C2×C42—2844no—
423C2×C42—711no—
424C2×C42—711no—
425C2×C42—711no—
426C2×C42—1422no—
427C2×C42—1422no—
428C2×C42—1421no—
429C2×C42—1421no—
430C2×C42—1421no—
431C2×C42C22×D7111yescharacteristic
432#107—714no—
433#107—714no—
434#107—714no—
435#107C4×D7114yescomplemented
436#95—21no—
437#95—21no—
438#95—1421no—
439#95—1421no—
440#95—1421no—
441#95—1421no—
442#95—1421no—
443#95—1421no—
444C24—711no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—2842no—
450C22×C4—2842no—
451C22×C4—2842no—
452C22×C4—2842no—
453C22×C4—2842no—
454C22×C4—2842no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—711no—
459C22×C4—712no—
460C22×C4—712no—
461C22×C4—712no—
462C22×C4—712no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—711no—
466C22×C4—714no—
467C22×C4—711no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1424no—
471C22×C4—1424no—
472C22×C4—1424no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1421no—
477C22×C4—1421no—
478C22×C4—1421no—
479C22×C4—1421no—
480C22×C4D4×D7111yescharacteristic
481C22×C4D4×D7111yescharacteristic
482C22×C4C23×D7111yescharacteristic
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—2844no—
489C2×C8—712no—
490C2×C8—1428no—
491C2×C8—1421no—
492C2×C8—1421no—
493C2×C8—1421no—
494C2×C8—1421no—
495C2×C8—1421no—
496C2×C8C2×C4×D7112yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—2842no—
504C4⋊C4—2842no—
505C4⋊C4—2842no—
506C4⋊C4—2842no—
507C4⋊C4—712no—
508C4⋊C4—712no—
509C4⋊C4—712no—
510C4⋊C4—1424no—
511C4⋊C4—1424no—
512C4⋊C4—1424no—
513C4⋊C4—1421no—
514C4⋊C4—1421no—
515C4⋊C4—1421no—
516C4⋊C4—1421no—
517C4⋊C4—1421no—
518C4⋊C4C2×C4×D7112yes—
519C4⋊C4C2×C4×D7112yes—
520C4⋊C4Q8×D7112yes—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—2842no—
525C42—2842no—
526C42—2842no—
527C42—1422no—
528C42—1422no—
529C42—1421no—
530C42—1421no—
531C23—712no—
532C23—712no—
533C23—712no—
534C23—712no—
535C23—712no—
536C23—712no—
537C23—712no—
538C23C2×D4×D7111yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—2848no—
557C2×C4—2848no—
558C2×C4—2848no—
559C2×C4—2844no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—5684no—
570C2×C4—712no—
571C2×C4—712no—
572C2×C4—712no—
573C2×C4—714no—
574C2×C4—714no—
575C2×C4—714no—
576C2×C4—714no—
577C2×C4—712no—
578C2×C4—1428no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1422no—
582C2×C4—1421no—
583C2×C4—1421no—
584C2×C4—1421no—
585C2×C4—1421no—
586C2×C4—1421no—
587C2×C4Q16×D7112yes—
588C2×C4C22×C4×D7111yescharacteristic
589C2×C4C2×Q8×D7111yescharacteristic
590C2×C4#41×D7111yescharacteristic
591C2×C4#41×D7111yescharacteristic
592C2×C4#41×D7111yescharacteristic
593C2×C4#41×D7111yescharacteristic
594C8—22no—
595C8—41no—
596C8—2841no—
597C8—1422no—
598C22—714no—
599C22—714no—
600C22—714no—
601C22—714no—
602C22—714no—
603C22—714no—
604C22—714no—
605C22C4×D4×D7111yescharacteristic
606C22C22⋊Q8×D7111yescharacteristic
607C22#123×D7111yescharacteristic
608C22C2×Q16×D7111yescharacteristic
609C22#135×D7111yescharacteristic
610C22AGL(1,ℤ/8)×D7111yescharacteristic
611C22AGL(1,ℤ/8)×D7111yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—2842no—
619C4—2842no—
620C4—2841no—
621C4—2842no—
622C4—5681no—
623C4—714no—
624C4—1422no—
625C4#117×D7111yescharacteristic
626C4C4×Q8×D7111yescharacteristic
627C4#125×D7111yescharacteristic
628C4#126×D7111yescharacteristic
629C2—718no—
630C2#387×D7111yescharacteristic
631C2C4×Q16×D7111yescharacteristic
632C2#442×D7111yescharacteristic
633C2#478×D7111yescharacteristic
634C2#477×D7111yescharacteristic
635C2#482×D7111yescharacteristic
636C2#484×D7111yescharacteristic
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
hypocenterC71C2×#1921
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. #22588935C2semidirect
5. #22588934C2semidirect
6. #22588933C2semidirect
7. #22588932C2semidirect
8. #22588931C2semidirect
9. #22588930C2semidirect
10. #22588929C2semidirect
15. C71×#1921C2semidirect
28. #107×D71C4semidirect
104. #14556062C2×C4semidirect
105. #14556061C2×C4semidirect
106. #14556061C2×C4semidirect
122. #107×C71C2×C4semidirect
318. D71#1921direct (N × H)
329. #1921D71direct (N × H)
334. C71C2×#1921semidirect
435. #107C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D712
#389×D712
C4×C4⋊C4×D712
#225889352
#225889342
#225889332
#225889322
#225889312
#225889302
#225889292
#340×D712
#336×D712
C71×#19212
C2×#192171