Finite Group Databasebeta

← C71×#19027 #31839899

Subgroups — C71×#19027 #31839899

Order 18176.

884 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3561C211yesmaximal, characteristic
2C71×#3238C211yesmaximal, characteristic, complemented
3C71×#3139C211yesmaximal, characteristic, complemented
4C2×#489×C71C211yesmaximal, characteristic, complemented
5C71×#1664C214yesmaximal, complemented
6C71×#1662C214yesmaximal, complemented
7C71×#1531C211yesmaximal, characteristic, complemented
8C71×#1527C211yesmaximal, characteristic, complemented
9C71×#1524C211yesmaximal, characteristic, complemented
10#563×C71C2212yes—
11#562×C71C2212yes—
12#556×C71—21no—
13#551×C71—22no—
14D4×Q8×C71—21no—
15#548×C71—21no—
16#546×C71—21no—
17#543×C71—21no—
18#542×C71—21no—
19#542×C71—22no—
20#538×C71—21no—
21#536×C71—21no—
22#533×C71C2211yescharacteristic
23C2×C4⋊Q8×C71C2211yescharacteristic
24#529×C71C2211yescharacteristic
25C2×#124×C71C2211yescharacteristic
26#501×C71C2212yescomplemented
27#491×C71C2214yescomplemented
28#489×C71—22no—
29#489×C71C2214yes—
30#489×C71C2212yescomplemented
31#487×C71—21no—
32#487×C71—21no—
33#483×C71—21no—
34#480×C71—21no—
35#477×C71—21no—
36#476×C71—21no—
37#432×C71C2211yescharacteristic
38#419×C71—22no—
39C2×#103×C71—21no—
40C2×#102×C71—21no—
41C4×M16×C71C2211yescharacteristic
42C2×#97×C71C2211yescharacteristic
43#330×C71C2212yescomplemented
44#330×C71C2214yescomplemented
45#329×C71C2214yes—
46#329×C71C2212yescomplemented
47C2×#41×C71—21no—
48C22×Q8×C71—21no—
49C22×Q8×C71—21no—
50C22×Q8×C71D411yescharacteristic
51C22×D4×C71D411yescharacteristic
52C2×M16×C71—21no—
53C22×C8×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71—22no—
56C4⋊Q8×C71—21no—
57C4⋊Q8×C71—21no—
58C4⋊Q8×C71—21no—
59C4⋊Q8×C71D411yescharacteristic
60C4⋊Q8×C71D411yescharacteristic
61C4⋊Q8×C71C2312yes—
62#126×C71—22no—
63#126×C71—22no—
64#126×C71—22no—
65#126×C71—21no—
66#125×C71C2312yes—
67#125×C71C2311yescharacteristic
68#124×C71—21no—
69#124×C71—22no—
70#124×C71—22no—
71#124×C71—21no—
72#124×C71—21no—
73#124×C71D412yes—
74#124×C71C2312yes—
75#124×C71C2311yescharacteristic
76#123×C71—22no—
77#123×C71—22no—
78#123×C71—21no—
79#123×C71—22no—
80#123×C71—22no—
81C22⋊Q8×C71—22no—
82C22⋊Q8×C71—22no—
83C22⋊Q8×C71—21no—
84C22⋊Q8×C71—22no—
85C22⋊Q8×C71—22no—
86C22⋊Q8×C71—22no—
87C22⋊Q8×C71—21no—
88C22⋊Q8×C71—21no—
89C22⋊Q8×C71—22no—
90C22⋊Q8×C71—22no—
91C22⋊Q8×C71D411yescharacteristic
92#121×C71—21no—
93#121×C71—22no—
94#121×C71—22no—
95#121×C71D411yescharacteristic
96#120×C71—22no—
97C4×Q8×C71—21no—
98C4×Q8×C71—21no—
99C4×Q8×C71—21no—
100C4×Q8×C71D411yescharacteristic
101C4×D4×C71—22no—
102C4×D4×C71—21no—
103C4×D4×C71—22no—
104C4×D4×C71—21no—
105C4×D4×C71D411yescharacteristic
106#117×C71—22no—
107#117×C71—21no—
108#117×C71D412yes—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C22⋊C4×C71—22no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C42×C71C2311yescharacteristic
115#107×C71—22no—
116#106×C71—22no—
117C4⋊C8×C71—21no—
118#103×C71—24no—
119#103×C71—22no—
120#103×C71—24no—
121#103×C71—24no—
122#102×C71—24no—
123#102×C71—22no—
124C22⋊C8×C71—21no—
125#97×C71—21no—
126#97×C71C2312yes—
127#97×C71C2312yes—
128#97×C71C2312yes—
129C4×C8×C71—21no—
130C24×C71—21no—
131#41×C71—82no—
132C2×Q8×C71—22no—
133C2×Q8×C71—21no—
134C2×Q8×C71—21no—
135C2×Q8×C71—21no—
136C2×Q8×C71—22no—
137C2×Q8×C71—42no—
138C2×Q8×C71—44no—
139C2×Q8×C71—41no—
140C2×Q8×C71—82no—
141C2×Q8×C71C2×D411yescharacteristic
142C2×Q8×C71C2×D412yes—
143C2×D4×C71—21no—
144C2×D4×C71—22no—
145C2×D4×C71—22no—
146C2×D4×C71—21no—
147C2×D4×C71—42no—
148C2×D4×C71C2×D411yescharacteristic
149C2×D4×C71C2×D412yes—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—41no—
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D411yescharacteristic
159M16×C71—42no—
160C2×C8×C71—22no—
161C2×C8×C71—22no—
162C2×C8×C71—22no—
163C2×C8×C71—42no—
164C4⋊C4×C71—22no—
165C4⋊C4×C71—22no—
166C4⋊C4×C71—22no—
167C4⋊C4×C71—22no—
168C4⋊C4×C71—21no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—22no—
178C4⋊C4×C71—41no—
179C4⋊C4×C71—41no—
180C4⋊C4×C71C2×D412yes—
181C4⋊C4×C71C2×D412yes—
182C4⋊C4×C71C2×D411yescharacteristic
183C4⋊C4×C71C2×D411yescharacteristic
184C22⋊C4×C71—22no—
185C22⋊C4×C71—22no—
186C22⋊C4×C71—22no—
187C22⋊C4×C71—22no—
188C22⋊C4×C71—22no—
189C22⋊C4×C71—21no—
190C22⋊C4×C71—22no—
191C22⋊C4×C71—22no—
192C22⋊C4×C71—22no—
193C22⋊C4×C71—21no—
194C22⋊C4×C71—21no—
195C22⋊C4×C71—22no—
196C22⋊C4×C71—22no—
197C22⋊C4×C71—21no—
198C22⋊C4×C71—21no—
199C42×C71—22no—
200C42×C71—22no—
201C42×C71—21no—
202C42×C71C2×D411yescharacteristic
203C42×C71C2×D411yescharacteristic
204C42×C71C2×D411yescharacteristic
205C42×C71C2411yescharacteristic
206C23×C71—22no—
207C23×C71—21no—
208C23×C71—42no—
209C23×C71—81no—
210C23×C71—81no—
211C23×C71#12011yescharacteristic
212Q8×C71—42no—
213Q8×C71—41no—
214Q8×C71—44no—
215Q8×C71—42no—
216Q8×C71—42no—
217Q8×C71—81no—
218Q8×C71—82no—
219Q8×C71—82no—
220D4×C71—42no—
221D4×C71—41no—
222D4×C71—42no—
223D4×C71—42no—
224D4×C71—82no—
225D4×C71—82no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—21no—
229C2×C4×C71—22no—
230C2×C4×C71—21no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—21no—
234C2×C4×C71—41no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—81no—
238C2×C4×C71—82no—
239C2×C4×C71—82no—
240C2×C4×C71—82no—
241C2×C4×C71—82no—
242C2×C4×C71#12011yescharacteristic
243C2×C4×C71#12011yescharacteristic
244C2×C4×C71#12011yescharacteristic
245C2×C4×C71C22×D411yescharacteristic
246C2×C4×C71C22×D411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C8×C71—42no—
249C8×C71—42no—
250C8×C71—42no—
251C22×C71—21no—
252C22×C71—21no—
253C22×C71—41no—
254C22×C71—42no—
255C22×C71—41no—
256C22×C71—42no—
257C22×C71—42no—
258C22×C71—41no—
259C22×C71—162no—
260C22×C71#45412yes—
261C22×C71C2×#12011yescharacteristic
262C4×C71—21no—
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—41no—
268C4×C71—42no—
269C4×C71—42no—
270C4×C71—41no—
271C4×C71—41no—
272C4×C71—42no—
273C4×C71—82no—
274C4×C71#45512yes—
275#19027C7111yesmaximal, characteristic, direct product factor, Sylow
276C2×C71—22no—
277C2×C71—81no—
278C2×C71—82no—
279C2×C71C2×#45511yescharacteristic
280C2×C71C2×#45411yescharacteristic
281C2×C71#302211yescharacteristic
282#3561C2×C7111yescharacteristic
283#3238C2×C7111yescharacteristic, complemented
284#3139C2×C7111yescharacteristic, complemented
285C2×#489C2×C7111yescharacteristic, complemented
286#1664C2×C7114yescomplemented
287#1662C2×C7114yescomplemented
288#1531C2×C7111yescharacteristic, complemented
289#1527C2×C7111yescharacteristic, complemented
290#1524C2×C7111yescharacteristic, complemented
291C71#1902711yescharacteristic, direct product factor, Sylow
292#563C22×C7112yes—
293#562C22×C7112yes—
294#556—21no—
295#551—22no—
296D4×Q8—21no—
297#548—21no—
298#546—21no—
299#543—21no—
300#542—21no—
301#542—22no—
302#538—21no—
303#536—21no—
304#533C22×C7111yescharacteristic
305C2×C4⋊Q8C22×C7111yescharacteristic
306#529C22×C7111yescharacteristic
307C2×#124C22×C7111yescharacteristic
308#501C22×C7112yescomplemented
309#491C22×C7114yescomplemented
310#489—22no—
311#489C22×C7114yes—
312#489C22×C7112yescomplemented
313#487—21no—
314#487—21no—
315#483—21no—
316#480—21no—
317#477—21no—
318#476—21no—
319#432C22×C7111yescharacteristic
320#419—22no—
321C2×#103—21no—
322C2×#102—21no—
323C4×M16C22×C7111yescharacteristic
324C2×#97C22×C7111yescharacteristic
325#330C22×C7114yescomplemented
326#330C22×C7112yescomplemented
327#329C22×C7114yes—
328#329C22×C7112yescomplemented
329C2×#41—21no—
330C22×Q8—21no—
331C22×Q8—21no—
332C22×Q8D4×C7111yescharacteristic
333C22×D4D4×C7111yescharacteristic
334C2×M16—21no—
335C22×C8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8D4×C7111yescharacteristic
342C4⋊Q8D4×C7111yescharacteristic
343C4⋊Q8C23×C7112yes—
344#126—22no—
345#126—22no—
346#126—22no—
347#126—21no—
348#125C23×C7111yescharacteristic
349#125C23×C7112yes—
350#124—21no—
351#124—22no—
352#124—22no—
353#124—21no—
354#124—21no—
355#124D4×C7112yes—
356#124C23×C7112yes—
357#124C23×C7111yescharacteristic
358#123—22no—
359#123—22no—
360#123—21no—
361#123—22no—
362#123—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8D4×C7111yescharacteristic
374#121—21no—
375#121—22no—
376#121—22no—
377#121D4×C7111yescharacteristic
378#120—22no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8D4×C7111yescharacteristic
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4D4×C7111yescharacteristic
388#117—22no—
389#117—21no—
390#117D4×C7112yes—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42C23×C7111yescharacteristic
397#107—22no—
398#106—22no—
399C4⋊C8—21no—
400#103—24no—
401#103—22no—
402#103—24no—
403#103—24no—
404#102—24no—
405#102—22no—
406C22⋊C8—21no—
407#97—21no—
408#97C23×C7112yes—
409#97C23×C7112yes—
410#97C23×C7112yes—
411C4×C8—21no—
412C24—21no—
413#41—82no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—41no—
422C2×Q8—82no—
423C2×Q8C2×D4×C7111yescharacteristic
424C2×Q8C2×D4×C7112yes—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—42no—
430C2×D4C2×D4×C7111yescharacteristic
431C2×D4C2×D4×C7112yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4C2×D4×C7111yescharacteristic
439C22×C4C2×D4×C7111yescharacteristic
440C22×C4C2×D4×C7111yescharacteristic
441M16—42no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—42no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4C2×D4×C7112yes—
463C4⋊C4C2×D4×C7112yes—
464C4⋊C4C2×D4×C7111yescharacteristic
465C4⋊C4C2×D4×C7111yescharacteristic
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42C2×D4×C7111yescharacteristic
485C42C2×D4×C7111yescharacteristic
486C42C2×D4×C7111yescharacteristic
487C42C24×C7111yescharacteristic
488C23—22no—
489C23—21no—
490C23—42no—
491C23—81no—
492C23—81no—
493C23#120×C7111yescharacteristic
494Q8—42no—
495Q8—41no—
496Q8—44no—
497Q8—42no—
498Q8—42no—
499Q8—81no—
500Q8—82no—
501Q8—82no—
502D4—42no—
503D4—41no—
504D4—42no—
505D4—42no—
506D4—82no—
507D4—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#120×C7111yescharacteristic
525C2×C4#120×C7111yescharacteristic
526C2×C4#120×C7111yescharacteristic
527C2×C4C22×D4×C7111yescharacteristic
528C2×C4C22×D4×C7111yescharacteristic
529C2×C4C22×D4×C7111yescharacteristic
530C8—42no—
531C8—42no—
532C8—42no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—162no—
542C22#454×C7112yes—
543C22C2×#120×C7111yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—82no—
556C4#455×C7112yes—
557C2—22no—
558C2—82no—
559C2—81no—
560C2C2×#455×C7111yescharacteristic
561C2C2×#454×C7111yescharacteristic
562C2C71×#302211yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
commutator, Frattini, cosocleC42C24×C71
prime radicalC71×#3561C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#3238C2semidirect
3. C71×#3139C2semidirect
4. C2×#489×C71C2semidirect
5. C71×#1664C2semidirect
6. C71×#1662C2semidirect
7. C71×#1531C2semidirect
8. C71×#1527C2semidirect
9. C71×#1524C2semidirect
26. #501×C71C22semidirect
27. #491×C71C22semidirect
30. #489×C71C22semidirect
43. #330×C71C22semidirect
44. #330×C71C22semidirect
46. #329×C71C22semidirect
275. #19027C71direct (N × H)
283. #3238C2×C71semidirect
284. #3139C2×C71semidirect
285. C2×#489C2×C71semidirect
286. #1664C2×C71semidirect
287. #1662C2×C71semidirect
288. #1531C2×C71semidirect
289. #1527C2×C71semidirect
290. #1524C2×C71semidirect
291. C71#19027direct (N × H)
308. #501C22×C71semidirect
309. #491C22×C71semidirect
312. #489C22×C71semidirect
325. #330C22×C71semidirect
326. #330C22×C71semidirect
328. #329C22×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#35612
C71×#32382
C71×#31392
C2×#489×C712
C71×#16642
C71×#16622
C71×#15312
C71×#15272
C71×#15242
#1902771