Finite Group Databasebeta

← C71×#61932 #31882804

Subgroups — C71×#61932 #31882804

Order 18176.

1866 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3548C211yesmaximal, characteristic, complemented
2C71×#3508C211yesmaximal, characteristic, complemented
3C71×#3399C214yesmaximal, complemented
4C71×#3391C214yesmaximal, complemented
5C71×#3366C214yesmaximal, complemented
6C71×#3361C214yesmaximal, complemented
7C71×#3220C214yesmaximal, complemented
8C71×#3205C212yesmaximal, complemented
9C71×#3122C214yesmaximal, complemented
10C71×#3106C212yesmaximal, complemented
11C71×#2986C211yesmaximal, characteristic, complemented
12#562×C71—22no—
13#559×C71C2214yes—
14#554×C71C2214yescomplemented
15#550×C71C2214yes—
16#550×C71C2214yescomplemented
17#547×C71—22no—
18#546×C71—24no—
19#540×C71—22no—
20#540×C71C2214yes—
21#537×C71—24no—
22#535×C71C2212yes—
23#533×C71C2211yescharacteristic
24#532×C71—22no—
25#532×C71C2212yes—
26C2×#127×C71C2211yescharacteristic
27#518×C71C2212yes—
28C4×#41×C71C2211yescharacteristic
29#496×C71C2214yescomplemented
30#493×C71C2214yes—
31#493×C71C2214yescomplemented
32#490×C71C2214yescomplemented
33#488×C71C2214yescomplemented
34#486×C71C2214yes—
35#482×C71C2214yes—
36#478×C71C2218yescomplemented
37#477×C71C2212yescomplemented
38#476×C71C2212yescomplemented
39#463×C71C2218yescomplemented
40#460×C71C2212yescomplemented
41#459×C71C2214yes—
42#459×C71C2214yes—
43#459×C71C2212yescomplemented
44#459×C71C2218yescomplemented
45#459×C71C2218yescomplemented
46#449×C71C2214yes—
47#447×C71C2218yes—
48#446×C71C2214yescomplemented
49C8×Q8×C71C2214yescomplemented
50#442×C71C2218yescomplemented
51C4×SD16×C71C2214yescomplemented
52C4×D8×C71C2218yescomplemented
53C4×D8×C71C2214yescomplemented
54#433×C71C2214yes—
55#431×C71C2212yes—
56C2×#41×C71—24no—
57C2×#41×C71—24no—
58C2×#41×C71—22no—
59C2×#41×C71C2311yescharacteristic
60C22×D4×C71—22no—
61C22×D4×C71C2312yes—
62C2×SD16×C71C2314yes—
63C2×D8×C71C2318yes—
64C2×D8×C71C2314yes—
65C2×D8×C71C2318yes—
66C4⋊Q8×C71D412yes—
67C4⋊Q8×C71C2312yes—
68#127×C71—21no—
69#127×C71—21no—
70#127×C71C2314yes—
71#127×C71C2314yes—
72#127×C71C2314yes—
73#127×C71C2312yes—
74#125×C71D412yes—
75#125×C71C2314yes—
76#124×C71—24no—
77#124×C71—22no—
78#124×C71C2314yes—
79#123×C71—28no—
80#123×C71—28no—
81C22⋊Q8×C71—24no—
82C22⋊Q8×C71—28no—
83C22⋊Q8×C71D414yes—
84#121×C71—22no—
85#121×C71—24no—
86#121×C71—24no—
87#121×C71—24no—
88#121×C71—28no—
89#121×C71—28no—
90#121×C71—28no—
91#121×C71C2314yes—
92#120×C71—24no—
93C4×Q8×C71C2312yes—
94C4×Q8×C71C2314yes—
95C4×D4×C71—22no—
96C4×D4×C71—24no—
97C4×D4×C71—24no—
98C4×D4×C71—22no—
99C4×D4×C71C2312yes—
100C4×D4×C71C2314yes—
101C4×D4×C71C2318yes—
102C4×D4×C71C2318yes—
103#117×C71—21no—
104#117×C71C2311yescharacteristic
105#117×C71C2312yes—
106C2×C4⋊C4×C71—24no—
107C2×C22⋊C4×C71—24no—
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110#107×C71C2314yes—
111#106×C71C2314yes—
112C4⋊C8×C71C2312yes—
113C4⋊C8×C71C2312yes—
114C4⋊C8×C71C2318yes—
115#103×C71C2314yes—
116#102×C71C2314yes—
117#102×C71C2314yes—
118#102×C71C2318yes—
119#102×C71C2314yes—
120#102×C71C2318yes—
121C22⋊C8×C71C2314yes—
122#97×C71C2314yes—
123C4×C8×C71C2314yes—
124C4×C8×C71C2314yes—
125C24×C71—22no—
126#41×C71—42no—
127#41×C71—84no—
128#41×C71—88no—
129#41×C71—88no—
130C2×Q8×C71—22no—
131C2×Q8×C71C2×D414yes—
132C2×Q8×C71C2411yescharacteristic
133C2×D4×C71—21no—
134C2×D4×C71—24no—
135C2×D4×C71—22no—
136C2×D4×C71—22no—
137C2×D4×C71—24no—
138C2×D4×C71—28no—
139C2×D4×C71—28no—
140C2×D4×C71—22no—
141C2×D4×C71—24no—
142C2×D4×C71—44no—
143C2×D4×C71—84no—
144C2×D4×C71C2414yes—
145C2×D4×C71C2411yescharacteristic
146C2×D4×C71C2414yes—
147C22×C4×C71—21no—
148C22×C4×C71—22no—
149C22×C4×C71—24no—
150C22×C4×C71—28no—
151C22×C4×C71—28no—
152C22×C4×C71C2×D412yes—
153C22×C4×C71C2411yescharacteristic
154SD16×C71—44no—
155D8×C71—44no—
156D8×C71—48no—
157D8×C71—48no—
158C2×C8×C71C2414yes—
159C2×C8×C71C2414yes—
160C4⋊C4×C71—22no—
161C4⋊C4×C71—28no—
162C4⋊C4×C71—22no—
163C4⋊C4×C71C2×D414yes—
164C4⋊C4×C71C2×D414yes—
165C4⋊C4×C71C2×D418yes—
166C4⋊C4×C71C2414yes—
167C4⋊C4×C71C2412yes—
168C22⋊C4×C71—21no—
169C22⋊C4×C71—28no—
170C22⋊C4×C71—22no—
171C22⋊C4×C71—28no—
172C22⋊C4×C71—24no—
173C22⋊C4×C71—28no—
174C22⋊C4×C71—24no—
175C22⋊C4×C71C2×D414yes—
176C42×C71—21no—
177C42×C71—22no—
178C42×C71C2×D411yescharacteristic
179C42×C71C2×D411yescharacteristic
180C42×C71C2412yes—
181C42×C71C2414yes—
182C23×C71—24no—
183C23×C71—24no—
184C23×C71—21no—
185C23×C71—44no—
186C23×C71—84no—
187C23×C71C22×D411yescharacteristic
188Q8×C71—28no—
189Q8×C71—22no—
190Q8×C71—82no—
191D4×C71—24no—
192D4×C71—24no—
193D4×C71—28no—
194D4×C71—22no—
195D4×C71—44no—
196D4×C71—42no—
197D4×C71—48no—
198D4×C71—44no—
199D4×C71—84no—
200D4×C71—82no—
201D4×C71—88no—
202D4×C71—88no—
203D4×C71—84no—
204C2×C4×C71—24no—
205C2×C4×C71—22no—
206C2×C4×C71—21no—
207C2×C4×C71—22no—
208C2×C4×C71—24no—
209C2×C4×C71—42no—
210C2×C4×C71—44no—
211C2×C4×C71—48no—
212C2×C4×C71—88no—
213C2×C4×C71—88no—
214C2×C4×C71C22×D411yescharacteristic
215C2×C4×C71C22×D412yes—
216C2×C4×C71C22×D418yes—
217C2×C4×C71C22×D412yes—
218C2×C4×C71C2511yescharacteristic
219C8×C71—24no—
220C8×C71—24no—
221C22×C71—21no—
222C22×C71—22no—
223C22×C71—41no—
224C22×C71—44no—
225C22×C71—44no—
226C22×C71—42no—
227C22×C71—44no—
228C22×C71—44no—
229C22×C71—48no—
230C22×C71—164no—
231C22×C71C23×D411yescharacteristic
232C4×C71—21no—
233C4×C71—22no—
234C4×C71—28no—
235C4×C71—41no—
236C4×C71—42no—
237C4×C71—44no—
238C4×C71#57614yes—
239C4×C71#58512yes—
240#61932C7111yesmaximal, characteristic, direct product factor, Sylow
241C2×C71—41no—
242C2×C71—81no—
243C2×C71—84no—
244C2×C71—84no—
245C2×C71#348711yescharacteristic
246C2×C71C2×#57612yes—
247#3548C2×C7111yescharacteristic, complemented
248#3508C2×C7111yescharacteristic, complemented
249#3399C2×C7114yescomplemented
250#3391C2×C7114yescomplemented
251#3366C2×C7114yescomplemented
252#3361C2×C7114yescomplemented
253#3220C2×C7114yescomplemented
254#3205C2×C7112yescomplemented
255#3122C2×C7114yescomplemented
256#3106C2×C7112yescomplemented
257#2986C2×C7111yescharacteristic, complemented
258C71#6193211yescharacteristic, direct product factor, Sylow
259#562—22no—
260#559C22×C7114yes—
261#554C22×C7114yescomplemented
262#550C22×C7114yes—
263#550C22×C7114yescomplemented
264#547—22no—
265#546—24no—
266#540—22no—
267#540C22×C7114yes—
268#537—24no—
269#535C22×C7112yes—
270#533C22×C7111yescharacteristic
271#532—22no—
272#532C22×C7112yes—
273C2×#127C22×C7111yescharacteristic
274#518C22×C7112yes—
275C4×#41C22×C7111yescharacteristic
276#496C22×C7114yescomplemented
277#493C22×C7114yes—
278#493C22×C7114yescomplemented
279#490C22×C7114yescomplemented
280#488C22×C7114yescomplemented
281#486C22×C7114yes—
282#482C22×C7114yes—
283#478C22×C7118yescomplemented
284#477C22×C7112yescomplemented
285#476C22×C7112yescomplemented
286#463C22×C7118yescomplemented
287#460C22×C7112yescomplemented
288#459C22×C7114yes—
289#459C22×C7114yes—
290#459C22×C7112yescomplemented
291#459C22×C7118yescomplemented
292#459C22×C7118yescomplemented
293#449C22×C7114yes—
294#447C22×C7118yes—
295#446C22×C7114yescomplemented
296C8×Q8C22×C7114yescomplemented
297#442C22×C7118yescomplemented
298C4×SD16C22×C7114yescomplemented
299C4×D8C22×C7118yescomplemented
300C4×D8C22×C7114yescomplemented
301#433C22×C7114yes—
302#431C22×C7112yes—
303C2×#41—24no—
304C2×#41—24no—
305C2×#41—22no—
306C2×#41C23×C7111yescharacteristic
307C22×D4—22no—
308C22×D4C23×C7112yes—
309C2×SD16C23×C7114yes—
310C2×D8C23×C7118yes—
311C2×D8C23×C7114yes—
312C2×D8C23×C7118yes—
313C4⋊Q8D4×C7112yes—
314C4⋊Q8C23×C7112yes—
315#127—21no—
316#127—21no—
317#127C23×C7114yes—
318#127C23×C7112yes—
319#127C23×C7114yes—
320#127C23×C7114yes—
321#125D4×C7112yes—
322#125C23×C7114yes—
323#124—24no—
324#124—22no—
325#124C23×C7114yes—
326#123—28no—
327#123—28no—
328C22⋊Q8—24no—
329C22⋊Q8—28no—
330C22⋊Q8D4×C7114yes—
331#121—22no—
332#121—24no—
333#121—24no—
334#121—24no—
335#121—28no—
336#121—28no—
337#121—28no—
338#121C23×C7114yes—
339#120—24no—
340C4×Q8C23×C7112yes—
341C4×Q8C23×C7114yes—
342C4×D4—22no—
343C4×D4—24no—
344C4×D4—24no—
345C4×D4—22no—
346C4×D4C23×C7112yes—
347C4×D4C23×C7118yes—
348C4×D4C23×C7114yes—
349C4×D4C23×C7118yes—
350#117—21no—
351#117C23×C7111yescharacteristic
352#117C23×C7112yes—
353C2×C4⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C42—21no—
356C2×C42C23×C7111yescharacteristic
357#107C23×C7114yes—
358#106C23×C7114yes—
359C4⋊C8C23×C7112yes—
360C4⋊C8C23×C7112yes—
361C4⋊C8C23×C7118yes—
362#103C23×C7114yes—
363#102C23×C7114yes—
364#102C23×C7114yes—
365#102C23×C7118yes—
366#102C23×C7114yes—
367#102C23×C7118yes—
368C22⋊C8C23×C7114yes—
369#97C23×C7114yes—
370C4×C8C23×C7114yes—
371C4×C8C23×C7114yes—
372C24—22no—
373#41—42no—
374#41—84no—
375#41—88no—
376#41—88no—
377C2×Q8—22no—
378C2×Q8C2×D4×C7114yes—
379C2×Q8C24×C7111yescharacteristic
380C2×D4—21no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—44no—
390C2×D4—84no—
391C2×D4C24×C7114yes—
392C2×D4C24×C7111yescharacteristic
393C2×D4C24×C7114yes—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4C2×D4×C7112yes—
400C22×C4C24×C7111yescharacteristic
401SD16—44no—
402D8—44no—
403D8—48no—
404D8—48no—
405C2×C8C24×C7114yes—
406C2×C8C24×C7114yes—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4C2×D4×C7114yes—
411C4⋊C4C2×D4×C7114yes—
412C4⋊C4C2×D4×C7118yes—
413C4⋊C4C24×C7114yes—
414C4⋊C4C24×C7112yes—
415C22⋊C4—21no—
416C22⋊C4—28no—
417C22⋊C4—22no—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4C2×D4×C7114yes—
423C42—21no—
424C42—22no—
425C42C2×D4×C7111yescharacteristic
426C42C2×D4×C7111yescharacteristic
427C42C24×C7112yes—
428C42C24×C7114yes—
429C23—24no—
430C23—24no—
431C23—21no—
432C23—44no—
433C23—84no—
434C23C22×D4×C7111yescharacteristic
435Q8—28no—
436Q8—22no—
437Q8—82no—
438D4—24no—
439D4—24no—
440D4—28no—
441D4—22no—
442D4—44no—
443D4—42no—
444D4—48no—
445D4—44no—
446D4—84no—
447D4—82no—
448D4—88no—
449D4—88no—
450D4—84no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4C22×D4×C7111yescharacteristic
462C2×C4C22×D4×C7112yes—
463C2×C4C22×D4×C7118yes—
464C2×C4C22×D4×C7112yes—
465C2×C4C25×C7111yescharacteristic
466C8—24no—
467C8—24no—
468C22—22no—
469C22—21no—
470C22—42no—
471C22—44no—
472C22—44no—
473C22—48no—
474C22—41no—
475C22—44no—
476C22—44no—
477C22—164no—
478C22C23×D4×C7111yescharacteristic
479C4—22no—
480C4—28no—
481C4—21no—
482C4—42no—
483C4—44no—
484C4—41no—
485C4#576×C7114yes—
486C4#585×C7112yes—
487C2—41no—
488C2—84no—
489C2—81no—
490C2—84no—
491C2C71×#348711yescharacteristic
492C2C2×#576×C7112yes—
subgroupG/H
center, socleC22×C71C23×D4
commutator, Frattini, cosocleC2×C4C25×C71
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ #533×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3548C2semidirect
2. C71×#3508C2semidirect
3. C71×#3399C2semidirect
4. C71×#3391C2semidirect
5. C71×#3366C2semidirect
6. C71×#3361C2semidirect
7. C71×#3220C2semidirect
8. C71×#3205C2semidirect
9. C71×#3122C2semidirect
10. C71×#3106C2semidirect
11. C71×#2986C2semidirect
14. #554×C71C22semidirect
16. #550×C71C22semidirect
29. #496×C71C22semidirect
31. #493×C71C22semidirect
32. #490×C71C22semidirect
33. #488×C71C22semidirect
36. #478×C71C22semidirect
37. #477×C71C22semidirect
38. #476×C71C22semidirect
39. #463×C71C22semidirect
40. #460×C71C22semidirect
43. #459×C71C22semidirect
44. #459×C71C22semidirect
45. #459×C71C22semidirect
48. #446×C71C22semidirect
49. C8×Q8×C71C22semidirect
50. #442×C71C22semidirect
51. C4×SD16×C71C22semidirect
52. C4×D8×C71C22semidirect
53. C4×D8×C71C22semidirect
240. #61932C71direct (N × H)
247. #3548C2×C71semidirect
248. #3508C2×C71semidirect
249. #3399C2×C71semidirect
250. #3391C2×C71semidirect
251. #3366C2×C71semidirect
252. #3361C2×C71semidirect
253. #3220C2×C71semidirect
254. #3205C2×C71semidirect
255. #3122C2×C71semidirect
256. #3106C2×C71semidirect
257. #2986C2×C71semidirect
258. C71#61932direct (N × H)
261. #554C22×C71semidirect
263. #550C22×C71semidirect
276. #496C22×C71semidirect
278. #493C22×C71semidirect
279. #490C22×C71semidirect
280. #488C22×C71semidirect
283. #478C22×C71semidirect
284. #477C22×C71semidirect
285. #476C22×C71semidirect
286. #463C22×C71semidirect
287. #460C22×C71semidirect
290. #459C22×C71semidirect
291. #459C22×C71semidirect
292. #459C22×C71semidirect
295. #446C22×C71semidirect
296. C8×Q8C22×C71semidirect
297. #442C22×C71semidirect
298. C4×SD16C22×C71semidirect
299. C4×D8C22×C71semidirect
300. C4×D8C22×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#35482
C71×#35082
C71×#33992
C71×#33912
C71×#33662
C71×#33612
C71×#32202
C71×#32052
C71×#31222
C71×#31062
C71×#29862
#6193271