Finite Group Databasebeta

← #1803×D71 #31996907

Subgroups — #1803×D71 #31996907

Order 18176.

1598 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D71C211yesmaximal, characteristic
2C2×C4⋊C8×D71C211yesmaximal, characteristic
3C2×#103×D71C212yesmaximal
4#384×D71C211yesmaximal, characteristic
5#22588346C211yesmaximal, characteristic, complemented
6#22588345C211yesmaximal, characteristic, complemented
7#22588344C212yesmaximal, complemented
8#22588343C212yesmaximal, complemented
9#22588342C211yesmaximal, characteristic, complemented
10#340×D71C212yesmaximal
11C71×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D71C2211yescharacteristic
13C22×C8×D71C2212yes—
14C2×Q8×Dic71C2211yescharacteristic
15C2×C4×Dic142—21no—
16C2×C4×Dic142C2211yescharacteristic
17C2×#14556093—21no—
18C2×#14556093C2211yescharacteristic
19C4×Q8×D71—24no—
20C4×Q8×D71C418yescomplemented
21C2×C4⋊C4×D71—21no—
22C2×C4⋊C4×D71—21no—
23C2×C4⋊C4×D71—21no—
24C2×C4⋊C4×D71C2212yes—
25C2×C4⋊C4×D71C2211yescharacteristic
26C2×C42×D71—21no—
27C2×C42×D71C2211yescharacteristic
28C2×C4⋊(C71⋊C8)C2211yescharacteristic
29C2×Dic71⋊C8C2212yes—
30C4⋊C8×D71—24no—
31C2×#14556053C2212yes—
32C2×#14556052C2212yes—
33C2×#14556051C2212yes—
34#103×D71—28no—
35#17985869C2211yescharacteristic
36#17985868—21no—
37#17985868—21no—
38#17985868C2212yes—
39#17985762C2212yes—
40#17985761C2212yes—
41#17985761C2212yes—
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4⋊C8×C71C2211yescharacteristic
44C2×#103×C71C2212yes—
45#384×C71C2211yescharacteristic
46#340×C71C2212yes—
47C22×Dic142—21no—
48C22×Dic142C2311yescharacteristic
49C2×Q8×D71—22no—
50C2×Q8×D71—22no—
51C2×Q8×D71—22no—
52C2×Q8×D71C2×C414yes—
53C2×Q8×D71D414yes—
54C2×Q8×D71Q814yes—
55C22×C4×D71—21no—
56C22×C4×D71—21no—
57C22×C4×D71—22no—
58C22×C4×D71D411yescharacteristic
59C22×C4×D71D411yescharacteristic
60C22×C4×D71C2311yescharacteristic
61C22×C71⋊C8C2312yes—
62C2×C8×D71—24no—
63C2×C8×D71—24no—
64C2×C8×D71—24no—
65Q8×Dic71—22no—
66Q8×Dic71C2×C414yescomplemented
67C4×Dic142—28no—
68C4×Dic142—22no—
69C4×Dic142C2×C414yescomplemented
70#14556093—28no—
71#14556093—22no—
72#14556093C2×C414yescomplemented
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71C2312yes—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×Dic71⋊C4—22no—
81C2×Dic71⋊C4—22no—
82C2×Dic71⋊C4—22no—
83C2×Dic71⋊C4—21no—
84C4⋊C4×D71—28no—
85C4⋊C4×D71—22no—
86C4⋊C4×D71—24no—
87C4⋊C4×D71—24no—
88C4⋊C4×D71—44no—
89C4⋊C4×D71—44no—
90C4⋊C4×D71C2×C414yes—
91C2×C4×Dic71—21no—
92C2×C4×Dic71—21no—
93C2×C4×Dic71—21no—
94C2×C4×Dic71D411yescharacteristic
95C2×C4×Dic71D411yescharacteristic
96C2×C4×Dic71C2311yescharacteristic
97C42×D71—28no—
98C42×D71C2×C414yes—
99C42×D71D414yes—
100C4⋊(C71⋊C8)—22no—
101Dic71⋊C8—24no—
102#14556053—24no—
103#14556052—24no—
104#14556051—24no—
105C22×Q8×C71C2311yescharacteristic
106C22×C8×C71C2312yes—
107C4×Q8×C71—22no—
108C4×Q8×C71C2×C414yescomplemented
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71C2312yes—
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C42×C71—21no—
115C2×C42×C71C2311yescharacteristic
116C4⋊C8×C71—22no—
117#103×C71—24no—
118C23×D71C2×D411yescharacteristic
119C2×Dic142—21no—
120C2×Dic142—21no—
121C2×Dic142—21no—
122C2×Dic142—24no—
123C2×Dic142—24no—
124C2×Dic142—24no—
125C2×Dic142C22×C412yes—
126C2×Dic142C2×D412yes—
127C2×Dic142C2×Q812yes—
128Q8×D71—28no—
129Q8×D71—28no—
130Q8×D71—28no—
131Q8×D71C4⋊C4116yescomplemented
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—22no—
135C22×Dic71C2×D411yescharacteristic
136C22×Dic71C2×D411yescharacteristic
137C22×Dic71C2×D411yescharacteristic
138C2×C4×D71—24no—
139C2×C4×D71—24no—
140C2×C4×D71—24no—
141C2×C4×D71—22no—
142C2×C4×D71—22no—
143C2×C4×D71—24no—
144C2×C4×D71—42no—
145C2×C4×D71—42no—
146C2×C4×D71—44no—
147C2×C4×D71—44no—
148C2×C4×D71—44no—
149C2×C4×D71C22⋊C414yes—
150C2×C4×D71C22⋊C414yes—
151C2×C4×D71SD1614yes—
152C2×C4×D71Q1614yes—
153C2×C4×D71C22×C412yes—
154C2×C4×D71C2×D412yes—
155C2×C4×D71C2×D412yes—
156C2×C4×D71C2×Q812yes—
157C2×C4×D71#4112yes—
158C2×C4×D71#4112yes—
159C2×C71⋊C8—22no—
160C2×C71⋊C8—22no—
161C2×C71⋊C8—22no—
162C8×D71—48no—
163C4⋊Dic71—21no—
164C4⋊Dic71—24no—
165C4⋊Dic71—22no—
166C4⋊Dic71—22no—
167C4⋊Dic71—42no—
168C4⋊Dic71—42no—
169C4⋊Dic71C22×C412yes—
170Dic71⋊C4—24no—
171Dic71⋊C4—24no—
172Dic71⋊C4—24no—
173Dic71⋊C4—44no—
174Dic71⋊C4—44no—
175Dic71⋊C4—44no—
176C4×Dic71—24no—
177C4×Dic71—24no—
178C4×Dic71—24no—
179C4×Dic71C22⋊C414yes—
180C4×Dic71C22⋊C414yes—
181C4×Dic71C22×C412yes—
182C4×Dic71C2×D412yes—
183C2×Q8×C71—21no—
184C2×Q8×C71—21no—
185C2×Q8×C71—21no—
186C2×Q8×C71C22×C412yes—
187C2×Q8×C71C2×D412yes—
188C2×Q8×C71C2×Q812yes—
189C22×C4×C71—21no—
190C22×C4×C71—21no—
191C22×C4×C71—22no—
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2411yescharacteristic
195C2×C8×C71—22no—
196C2×C8×C71—22no—
197C2×C8×C71—22no—
198C4⋊C4×C71—24no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—22no—
201C4⋊C4×C71—22no—
202C4⋊C4×C71—42no—
203C4⋊C4×C71—42no—
204C4⋊C4×C71C22×C412yes—
205C42×C71—24no—
206C42×C71C22×C412yes—
207C42×C71C2×D412yes—
208C23×C71C22×D411yescharacteristic
209C22×D71C2×C22⋊C412yes—
210C22×D71#12112yes—
211C22×D71C22⋊Q812yes—
212C22×D71C2×SD1612yes—
213C22×D71C2×Q1612yes—
214C22×D71AGL(1,ℤ/8)12yes—
215C22×D71#13712yes—
216Q8×C71—22no—
217Q8×C71—22no—
218Q8×C71—22no—
219Q8×C71C2×C4⋊C414yescomplemented
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—22no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71C2×C22⋊C412yes—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×SD1612yes—
234C2×C4×C71C2×Q1612yes—
235C2×C4×C71C23×C411yescharacteristic
236C2×C4×C71C22×D411yescharacteristic
237C2×C4×C71C22×D411yescharacteristic
238C2×C4×C71C22×Q811yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—22no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71C2×C22⋊C412yes—
253C2×Dic71C2×C22⋊C412yes—
254C2×Dic71C2×C22⋊C412yes—
255C2×Dic71#12112yes—
256C2×Dic71C22⋊Q812yes—
257C2×Dic71C2×SD1612yes—
258C2×Dic71C2×Q1612yes—
259C4×D71—216no—
260C4×D71—28no—
261C4×D71—48no—
262C4×D71—88no—
263C4×D71#103116yescomplemented
264C4×D71C2×C4⋊C414yes—
265C4×D71#11714yes—
266C4×D71#12114yes—
267C4×D71C22⋊Q814yes—
268Dic142—22no—
269Dic142—22no—
270Dic142—22no—
271Dic142—216no—
272Dic142C2×C4⋊C414yescomplemented
273C8×C71—42no—
274C71⋊C8—42no—
275C22×C71C22×C22⋊C411yescharacteristic
276C22×C71C2×#12111yescharacteristic
277C22×C71C2×C22⋊Q811yescharacteristic
278C22×C71C22×SD1611yescharacteristic
279C22×C71C22×Q1611yescharacteristic
280C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C71C2×#13711yescharacteristic
282C2×D71#38014yes—
283C2×D71C2×#10314yes—
284C2×D71#41714yes—
285C2×D71#46014yes—
286C2×D71#46214yes—
287C2×D71#47514yes—
288C2×D71#47714yes—
289C4×C71—22no—
290C4×C71—24no—
291C4×C71—42no—
292C4×C71—82no—
293C4×C71C2×#10314yescomplemented
294C4×C71C22×C4⋊C411yescharacteristic
295C4×C71C2×#11711yescharacteristic
296C4×C71C2×#12111yescharacteristic
297C4×C71C2×C22⋊Q811yescharacteristic
298Dic71—22no—
299Dic71—24no—
300Dic71—42no—
301Dic71—82no—
302Dic71#38014yes—
303Dic71C2×#10314yescomplemented
304C2×#1803—711nomaximal, Sylow
305C2×C71C2×#38011yescharacteristic
306C2×C71C22×#10311yescharacteristic
307C2×C71C2×#41711yescharacteristic
308C2×C71C2×#46011yescharacteristic
309C2×C71C2×#46211yescharacteristic
310C2×C71C2×#47511yescharacteristic
311C2×C71C2×#47711yescharacteristic
312D71#180318yesdirect product factor
313C22×C4×Q8—711no—
314C22×C4⋊C8—711no—
315C22×#103—712no—
316C2×#384—711no—
317#1803—711no—
318#1803—712no—
319#1803—711no—
320#1803—712no—
321#1803—711no—
322#1803D7111yescharacteristic, direct product factor
323C2×#340—712no—
324C71C2×#180311yescharacteristic, complemented, Sylow
325C23×Q8—711no—
326C23×C8—712no—
327C2×C4×Q8—711no—
328C2×C4×Q8—718no—
329C2×C4×Q8—711no—
330C2×C4×Q8—711no—
331C2×C4×Q8—1421no—
332C2×C4×Q8—1421no—
333C2×C4×Q8—1424no—
334C2×C4×Q8C2×D7111yescharacteristic
335C22×C4⋊C4—711no—
336C22×C4⋊C4—712no—
337C22×C4⋊C4—1421no—
338C22×C4⋊C4—1421no—
339C22×C4⋊C4—1421no—
340C22×C42—711no—
341C22×C42—1421no—
342C2×C4⋊C8—712no—
343C2×C4⋊C8—711no—
344C2×C4⋊C8—1424no—
345C2×C4⋊C8C2×D7111yescharacteristic
346C2×#103—712no—
347C2×#103—712no—
348C2×#103—712no—
349C2×#103—1428no—
350C2×#103C2×D7112yes—
351#384—712no—
352#384—711no—
353#384—1421no—
354#384—1421no—
355#384C2×D7111yescharacteristic
356#340—712no—
357#340—712no—
358#340—712no—
359#340C2×D7112yes—
360C22×Q8—714no—
361C22×Q8—714no—
362C22×Q8—714no—
363C22×Q8—711no—
364C22×Q8—1421no—
365C22×Q8—1422no—
366C22×Q8—1422no—
367C22×Q8—1422no—
368C22×Q8C22×D7111yescharacteristic
369C23×C4—711no—
370C23×C4—711no—
371C23×C4—711no—
372C23×C4—1422no—
373C23×C4—1421no—
374C23×C4—1421no—
375C22×C8—712no—
376C22×C8—1424no—
377C22×C8—1424no—
378C22×C8—1424no—
379C22×C8C22×D7112yes—
380C4×Q8—22no—
381C4×Q8—714no—
382C4×Q8—714no—
383C4×Q8—714no—
384C4×Q8—1428no—
385C4×Q8—1422no—
386C4×Q8—1428no—
387C4×Q8—1422no—
388C4×Q8—1422no—
389C4×Q8C4×D7114yescomplemented
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—2844no—
394C2×C4⋊C4—2844no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—714no—
397C2×C4⋊C4—712no—
398C2×C4⋊C4—1424no—
399C2×C4⋊C4—1424no—
400C2×C4⋊C4—1422no—
401C2×C4⋊C4—1428no—
402C2×C4⋊C4—1422no—
403C2×C4⋊C4—1422no—
404C2×C4⋊C4—1422no—
405C2×C4⋊C4—1421no—
406C2×C4⋊C4—1421no—
407C2×C4⋊C4—1421no—
408C2×C4⋊C4—1421no—
409C2×C4⋊C4—1421no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4C22×D7112yes—
412C2×C4⋊C4C22×D7111yescharacteristic
413C2×C42—21no—
414C2×C42—711no—
415C2×C42—711no—
416C2×C42—711no—
417C2×C42—714no—
418C2×C42—714no—
419C2×C42—1428no—
420C2×C42—1421no—
421C2×C42—1421no—
422C2×C42—1421no—
423C2×C42C22×D7111yescharacteristic
424C4⋊C8—22no—
425C4⋊C8—1424no—
426C4⋊C8—1422no—
427#103—24no—
428#103—1424no—
429#103—1424no—
430#103—1424no—
431C24—711no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—7116no—
436C2×Q8—712no—
437C2×Q8—712no—
438C2×Q8—712no—
439C2×Q8—1424no—
440C2×Q8—1424no—
441C2×Q8—1424no—
442C2×Q8—1428no—
443C2×Q8—1428no—
444C2×Q8—1428no—
445C2×Q8—1421no—
446C2×Q8—1421no—
447C2×Q8—1421no—
448C2×Q8C2×C4×D7112yes—
449C2×Q8D4×D7112yes—
450C2×Q8Q8×D7112yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—2842no—
455C22×C4—2842no—
456C22×C4—2844no—
457C22×C4—2844no—
458C22×C4—2844no—
459C22×C4—711no—
460C22×C4—711no—
461C22×C4—712no—
462C22×C4—712no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—712no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—714no—
469C22×C4—714no—
470C22×C4—711no—
471C22×C4—714no—
472C22×C4—1424no—
473C22×C4—1424no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1424no—
477C22×C4—1424no—
478C22×C4—1422no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4D4×D7111yescharacteristic
482C22×C4D4×D7111yescharacteristic
483C22×C4C23×D7111yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—2848no—
488C2×C8—1422no—
489C2×C8—1422no—
490C2×C8—1422no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—2842no—
498C4⋊C4—2844no—
499C4⋊C4—2844no—
500C4⋊C4—2844no—
501C4⋊C4—2842no—
502C4⋊C4—712no—
503C4⋊C4—1424no—
504C4⋊C4—1424no—
505C4⋊C4—1424no—
506C4⋊C4—1422no—
507C4⋊C4—1422no—
508C4⋊C4—1421no—
509C4⋊C4—1424no—
510C4⋊C4C2×C4×D7112yes—
511C42—24no—
512C42—712no—
513C42—712no—
514C42—714no—
515C42—714no—
516C42—1424no—
517C42—1424no—
518C42—1424no—
519C42C2×C4×D7112yes—
520C42D4×D7112yes—
521C23—712no—
522C23—712no—
523C23—712no—
524C23—712no—
525C23—712no—
526C23—712no—
527C23—712no—
528C23C2×D4×D7111yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—714no—
533Q8—14216no—
534Q8—1422no—
535Q8—1422no—
536Q8—1422no—
537Q8C4⋊C4×D7114yescomplemented
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—2848no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—5688no—
556C2×C4—712no—
557C2×C4—712no—
558C2×C4—712no—
559C2×C4—712no—
560C2×C4—712no—
561C2×C4—712no—
562C2×C4—714no—
563C2×C4—714no—
564C2×C4—714no—
565C2×C4—714no—
566C2×C4—7116no—
567C2×C4—712no—
568C2×C4—1428no—
569C2×C4—14216no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1422no—
574C2×C4—1421no—
575C2×C4—1421no—
576C2×C4C22⋊C4×D7112yes—
577C2×C4C22⋊C4×D7112yes—
578C2×C4SD16×D7112yes—
579C2×C4Q16×D7112yes—
580C2×C4C22×C4×D7111yescharacteristic
581C2×C4C2×D4×D7111yescharacteristic
582C2×C4C2×D4×D7111yescharacteristic
583C2×C4C2×Q8×D7111yescharacteristic
584C2×C4#41×D7111yescharacteristic
585C2×C4#41×D7111yescharacteristic
586C8—42no—
587C8—2842no—
588C22—714no—
589C22—714no—
590C22—714no—
591C22—714no—
592C22—714no—
593C22—714no—
594C22—714no—
595C22C2×C22⋊C4×D7111yescharacteristic
596C22#121×D7111yescharacteristic
597C22C22⋊Q8×D7111yescharacteristic
598C22C2×SD16×D7111yescharacteristic
599C22C2×Q16×D7111yescharacteristic
600C22AGL(1,ℤ/8)×D7111yescharacteristic
601C22#137×D7111yescharacteristic
602C4—22no—
603C4—24no—
604C4—42no—
605C4—82no—
606C4—2842no—
607C4—5682no—
608C4—714no—
609C4—714no—
610C4—1422no—
611C4—1424no—
612C4#103×D7114yescomplemented
613C4C2×C4⋊C4×D7111yescharacteristic
614C4#117×D7111yescharacteristic
615C4#121×D7111yescharacteristic
616C4C22⋊Q8×D7111yescharacteristic
617C2—718no—
618C2#380×D7111yescharacteristic
619C2C2×#103×D7111yescharacteristic
620C2#417×D7111yescharacteristic
621C2#460×D7111yescharacteristic
622C2#462×D7111yescharacteristic
623C2#475×D7111yescharacteristic
624C2#477×D7111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1803C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1803D71
hypocenterC71C2×#1803
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #22588346C2semidirect
6. #22588345C2semidirect
7. #22588344C2semidirect
8. #22588343C2semidirect
9. #22588342C2semidirect
11. C71×#1803C2semidirect
20. C4×Q8×D71C4semidirect
66. Q8×Dic71C2×C4semidirect
69. C4×Dic142C2×C4semidirect
72. #14556093C2×C4semidirect
108. C4×Q8×C71C2×C4semidirect
131. Q8×D71C4⋊C4semidirect
219. Q8×C71C2×C4⋊C4semidirect
263. C4×D71#103semidirect
272. Dic142C2×C4⋊C4semidirect
293. C4×C71C2×#103semidirect
303. Dic71C2×#103semidirect
312. D71#1803direct (N × H)
322. #1803D71direct (N × H)
324. C71C2×#1803semidirect
389. C4×Q8C4×D71semidirect
537. Q8C4⋊C4×D71semidirect
612. C4#103×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D712
C2×C4⋊C8×D712
C2×#103×D712
#384×D712
#225883462
#225883452
#225883442
#225883432
#225883422
#340×D712
C71×#18032
C2×#180371