Finite Group Databasebeta

← C71×#61933 #31882805

Subgroups — C71×#61933 #31882805

Order 18176.

1754 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3556C211yesmaximal, characteristic, complemented
2C71×#3508C211yesmaximal, characteristic
3C71×#3400C214yesmaximal, complemented
4C71×#3392C214yesmaximal, complemented
5C71×#3368C214yesmaximal, complemented
6C71×#3365C214yesmaximal, complemented
7C71×#3222C214yesmaximal, complemented
8C71×#3208C212yesmaximal, complemented
9C71×#3123C214yesmaximal, complemented
10C71×#3107C212yesmaximal, complemented
11C71×#2986C211yesmaximal, characteristic, complemented
12#564×C71—22no—
13#559×C71C2214yes—
14#557×C71—22no—
15#557×C71C2214yescomplemented
16#557×C71C2214yescomplemented
17#550×C71C2214yes—
18#546×C71—24no—
19#541×C71—24no—
20#541×C71C2214yes—
21#540×C71—22no—
22#535×C71C2212yes—
23#533×C71C2212yes—
24#533×C71C2211yescharacteristic
25#532×C71—22no—
26C2×#127×C71C2211yescharacteristic
27#518×C71C2212yes—
28C4×#41×C71C2211yescharacteristic
29#497×C71C2214yescomplemented
30#492×C71C2214yes—
31#492×C71C2214yescomplemented
32#490×C71C2214yescomplemented
33#488×C71C2214yescomplemented
34#486×C71C2214yes—
35#485×C71C2214yes—
36#478×C71C2218yescomplemented
37#475×C71C2212yescomplemented
38#474×C71C2212yescomplemented
39#464×C71C2218yescomplemented
40#462×C71C2212yescomplemented
41#461×C71C2212yescomplemented
42#461×C71C2218yescomplemented
43#460×C71C2214yes—
44#460×C71C2214yes—
45#460×C71C2218yescomplemented
46#452×C71C2214yes—
47#450×C71C2218yes—
48#446×C71C2214yescomplemented
49C8×Q8×C71C2214yescomplemented
50#440×C71C2218yescomplemented
51C4×Q16×C71C2214yescomplemented
52C4×SD16×C71C2214yescomplemented
53C4×SD16×C71C2218yescomplemented
54#433×C71C2214yes—
55#431×C71C2212yes—
56C2×#41×C71—24no—
57C2×#41×C71C2311yescharacteristic
58C22×D4×C71—22no—
59C22×D4×C71C2312yes—
60C2×Q16×C71C2314yes—
61C2×SD16×C71C2318yes—
62C2×SD16×C71C2314yes—
63C2×SD16×C71C2318yes—
64C4⋊Q8×C71—24no—
65C4⋊Q8×C71D412yes—
66C4⋊Q8×C71C2314yes—
67C4⋊Q8×C71C2312yes—
68C4⋊Q8×C71C2312yes—
69#127×C71—21no—
70#127×C71—21no—
71#127×C71C2314yes—
72#127×C71C2314yes—
73#126×C71—28no—
74#125×C71—24no—
75#125×C71—28no—
76#125×C71—28no—
77#125×C71—28no—
78#125×C71—22no—
79#125×C71D412yes—
80#125×C71C2314yes—
81#124×C71—24no—
82#124×C71C2314yes—
83#123×C71—28no—
84C22⋊Q8×C71—22no—
85C22⋊Q8×C71D414yes—
86C22⋊Q8×C71C2314yes—
87#121×C71—24no—
88#121×C71—24no—
89#121×C71—28no—
90#120×C71—24no—
91C4×Q8×C71—24no—
92C4×Q8×C71—24no—
93C4×Q8×C71—22no—
94C4×Q8×C71C2314yes—
95C4×Q8×C71C2318yes—
96C4×Q8×C71C2312yes—
97C4×Q8×C71C2314yes—
98C4×D4×C71—22no—
99C4×D4×C71—24no—
100C4×D4×C71—24no—
101C4×D4×C71C2312yes—
102C4×D4×C71C2318yes—
103#117×C71—22no—
104#117×C71—21no—
105#117×C71C2311yescharacteristic
106#117×C71C2312yes—
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#103×C71C2314yes—
117#103×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—88no—
128C2×Q8×C71—22no—
129C2×Q8×C71C2×D414yes—
130C2×Q8×C71C2411yescharacteristic
131C2×Q8×C71C2414yes—
132C2×D4×C71—21no—
133C2×D4×C71—22no—
134C2×D4×C71—24no—
135C2×D4×C71—28no—
136C2×D4×C71—24no—
137C2×D4×C71—44no—
138C2×D4×C71—84no—
139C2×D4×C71C2414yes—
140C2×D4×C71C2411yescharacteristic
141C22×C4×C71—21no—
142C22×C4×C71—22no—
143C22×C4×C71—28no—
144C22×C4×C71C2×D412yes—
145C22×C4×C71C2411yescharacteristic
146Q16×C71—44no—
147SD16×C71—48no—
148SD16×C71—44no—
149SD16×C71—48no—
150C2×C8×C71C2414yes—
151C2×C8×C71C2414yes—
152C4⋊C4×C71—22no—
153C4⋊C4×C71—22no—
154C4⋊C4×C71—22no—
155C4⋊C4×C71—22no—
156C4⋊C4×C71—24no—
157C4⋊C4×C71—28no—
158C4⋊C4×C71—28no—
159C4⋊C4×C71—24no—
160C4⋊C4×C71—28no—
161C4⋊C4×C71C2×D414yes—
162C4⋊C4×C71C2×D414yes—
163C4⋊C4×C71C2×D418yes—
164C4⋊C4×C71C2414yes—
165C4⋊C4×C71C2412yes—
166C22⋊C4×C71—21no—
167C22⋊C4×C71—24no—
168C22⋊C4×C71—28no—
169C22⋊C4×C71—22no—
170C22⋊C4×C71—28no—
171C22⋊C4×C71C2×D414yes—
172C42×C71—21no—
173C42×C71—22no—
174C42×C71—24no—
175C42×C71—28no—
176C42×C71C2×D411yescharacteristic
177C42×C71C2×D411yescharacteristic
178C42×C71C2412yes—
179C42×C71C2414yes—
180C23×C71—24no—
181C23×C71—21no—
182C23×C71—44no—
183C23×C71—84no—
184C23×C71C22×D411yescharacteristic
185Q8×C71—28no—
186Q8×C71—24no—
187Q8×C71—24no—
188Q8×C71—22no—
189Q8×C71—44no—
190D4×C71—28no—
191D4×C71—22no—
192D4×C71—44no—
193D4×C71—42no—
194D4×C71—48no—
195D4×C71—84no—
196D4×C71—88no—
197C2×C4×C71—24no—
198C2×C4×C71—22no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—21no—
202C2×C4×C71—22no—
203C2×C4×C71—42no—
204C2×C4×C71—44no—
205C2×C4×C71—88no—
206C2×C4×C71C22×D411yescharacteristic
207C2×C4×C71C22×D418yes—
208C2×C4×C71C22×D412yes—
209C2×C4×C71C22×D412yes—
210C2×C4×C71C2511yescharacteristic
211C8×C71—24no—
212C8×C71—24no—
213C22×C71—21no—
214C22×C71—22no—
215C22×C71—41no—
216C22×C71—44no—
217C22×C71—42no—
218C22×C71—48no—
219C22×C71—164no—
220C22×C71C23×D411yescharacteristic
221C4×C71—21no—
222C4×C71—22no—
223C4×C71—28no—
224C4×C71—41no—
225C4×C71—44no—
226C4×C71—42no—
227C4×C71—44no—
228C4×C71#57714yes—
229C4×C71#58512yes—
230#61933C7111yesmaximal, characteristic, direct product factor, Sylow
231C2×C71—41no—
232C2×C71—81no—
233C2×C71—84no—
234C2×C71#348711yescharacteristic
235C2×C71C2×#57712yes—
236#3556C2×C7111yescharacteristic, complemented
237#3508C2×C7111yescharacteristic
238#3400C2×C7114yescomplemented
239#3392C2×C7114yescomplemented
240#3368C2×C7114yescomplemented
241#3365C2×C7114yescomplemented
242#3222C2×C7114yescomplemented
243#3208C2×C7112yescomplemented
244#3123C2×C7114yescomplemented
245#3107C2×C7112yescomplemented
246#2986C2×C7111yescharacteristic, complemented
247C71#6193311yescharacteristic, direct product factor, Sylow
248#564—22no—
249#559C22×C7114yes—
250#557—22no—
251#557C22×C7114yescomplemented
252#557C22×C7114yescomplemented
253#550C22×C7114yes—
254#546—24no—
255#541—24no—
256#541C22×C7114yes—
257#540—22no—
258#535C22×C7112yes—
259#533C22×C7112yes—
260#533C22×C7111yescharacteristic
261#532—22no—
262C2×#127C22×C7111yescharacteristic
263#518C22×C7112yes—
264C4×#41C22×C7111yescharacteristic
265#497C22×C7114yescomplemented
266#492C22×C7114yes—
267#492C22×C7114yescomplemented
268#490C22×C7114yescomplemented
269#488C22×C7114yescomplemented
270#486C22×C7114yes—
271#485C22×C7114yes—
272#478C22×C7118yescomplemented
273#475C22×C7112yescomplemented
274#474C22×C7112yescomplemented
275#464C22×C7118yescomplemented
276#462C22×C7112yescomplemented
277#461C22×C7112yescomplemented
278#461C22×C7118yescomplemented
279#460C22×C7114yes—
280#460C22×C7114yes—
281#460C22×C7118yescomplemented
282#452C22×C7114yes—
283#450C22×C7118yes—
284#446C22×C7114yescomplemented
285C8×Q8C22×C7114yescomplemented
286#440C22×C7118yescomplemented
287C4×Q16C22×C7114yescomplemented
288C4×SD16C22×C7114yescomplemented
289C4×SD16C22×C7118yescomplemented
290#433C22×C7114yes—
291#431C22×C7112yes—
292C2×#41—24no—
293C2×#41C23×C7111yescharacteristic
294C22×D4—22no—
295C22×D4C23×C7112yes—
296C2×Q16C23×C7114yes—
297C2×SD16C23×C7118yes—
298C2×SD16C23×C7114yes—
299C2×SD16C23×C7118yes—
300C4⋊Q8—24no—
301C4⋊Q8D4×C7112yes—
302C4⋊Q8C23×C7112yes—
303C4⋊Q8C23×C7114yes—
304C4⋊Q8C23×C7112yes—
305#127—21no—
306#127—21no—
307#127C23×C7114yes—
308#127C23×C7114yes—
309#126—28no—
310#125—24no—
311#125—28no—
312#125—28no—
313#125—28no—
314#125—22no—
315#125D4×C7112yes—
316#125C23×C7114yes—
317#124—24no—
318#124C23×C7114yes—
319#123—28no—
320C22⋊Q8—22no—
321C22⋊Q8D4×C7114yes—
322C22⋊Q8C23×C7114yes—
323#121—24no—
324#121—24no—
325#121—28no—
326#120—24no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—22no—
330C4×Q8C23×C7112yes—
331C4×Q8C23×C7114yes—
332C4×Q8C23×C7118yes—
333C4×Q8C23×C7114yes—
334C4×D4—22no—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4C23×C7112yes—
338C4×D4C23×C7118yes—
339#117—22no—
340#117—21no—
341#117C23×C7111yescharacteristic
342#117C23×C7112yes—
343C2×C22⋊C4—24no—
344C2×C42—21no—
345C2×C42C23×C7111yescharacteristic
346#107C23×C7114yes—
347#106C23×C7114yes—
348C4⋊C8C23×C7112yes—
349C4⋊C8C23×C7112yes—
350C4⋊C8C23×C7118yes—
351#103C23×C7114yes—
352#103C23×C7114yes—
353#103C23×C7114yes—
354#102C23×C7118yes—
355#102C23×C7114yes—
356#102C23×C7118yes—
357C22⋊C8C23×C7114yes—
358#97C23×C7114yes—
359C4×C8C23×C7114yes—
360C4×C8C23×C7114yes—
361C24—22no—
362#41—42no—
363#41—88no—
364C2×Q8—22no—
365C2×Q8C2×D4×C7114yes—
366C2×Q8C24×C7111yescharacteristic
367C2×Q8C24×C7114yes—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—44no—
374C2×D4—84no—
375C2×D4C24×C7114yes—
376C2×D4C24×C7111yescharacteristic
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4C2×D4×C7112yes—
381C22×C4C24×C7111yescharacteristic
382Q16—44no—
383SD16—48no—
384SD16—44no—
385SD16—48no—
386C2×C8C24×C7114yes—
387C2×C8C24×C7114yes—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4C2×D4×C7114yes—
398C4⋊C4C2×D4×C7114yes—
399C4⋊C4C2×D4×C7118yes—
400C4⋊C4C24×C7114yes—
401C4⋊C4C24×C7112yes—
402C22⋊C4—21no—
403C22⋊C4—24no—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—28no—
407C22⋊C4C2×D4×C7114yes—
408C42—21no—
409C42—22no—
410C42—24no—
411C42—28no—
412C42C2×D4×C7111yescharacteristic
413C42C2×D4×C7111yescharacteristic
414C42C24×C7112yes—
415C42C24×C7114yes—
416C23—24no—
417C23—21no—
418C23—44no—
419C23—84no—
420C23C22×D4×C7111yescharacteristic
421Q8—28no—
422Q8—24no—
423Q8—24no—
424Q8—22no—
425Q8—44no—
426D4—28no—
427D4—22no—
428D4—44no—
429D4—42no—
430D4—48no—
431D4—84no—
432D4—88no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—88no—
442C2×C4C22×D4×C7111yescharacteristic
443C2×C4C22×D4×C7118yes—
444C2×C4C22×D4×C7112yes—
445C2×C4C22×D4×C7112yes—
446C2×C4C25×C7111yescharacteristic
447C8—24no—
448C8—24no—
449C22—22no—
450C22—21no—
451C22—42no—
452C22—48no—
453C22—41no—
454C22—44no—
455C22—164no—
456C22C23×D4×C7111yescharacteristic
457C4—22no—
458C4—28no—
459C4—21no—
460C4—42no—
461C4—44no—
462C4—41no—
463C4—44no—
464C4#577×C7114yes—
465C4#585×C7112yes—
466C2—41no—
467C2—84no—
468C2—81no—
469C2C71×#348711yescharacteristic
470C2C2×#577×C7112yes—
subgroupG/H
center, socleC22×C71C23×D4
commutator, Frattini, cosocleC2×C4C25×C71
prime radicalC71×#3508C2
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×#3556C2semidirect
3. C71×#3400C2semidirect
4. C71×#3392C2semidirect
5. C71×#3368C2semidirect
6. C71×#3365C2semidirect
7. C71×#3222C2semidirect
8. C71×#3208C2semidirect
9. C71×#3123C2semidirect
10. C71×#3107C2semidirect
11. C71×#2986C2semidirect
15. #557×C71C22semidirect
16. #557×C71C22semidirect
29. #497×C71C22semidirect
31. #492×C71C22semidirect
32. #490×C71C22semidirect
33. #488×C71C22semidirect
36. #478×C71C22semidirect
37. #475×C71C22semidirect
38. #474×C71C22semidirect
39. #464×C71C22semidirect
40. #462×C71C22semidirect
41. #461×C71C22semidirect
42. #461×C71C22semidirect
45. #460×C71C22semidirect
48. #446×C71C22semidirect
49. C8×Q8×C71C22semidirect
50. #440×C71C22semidirect
51. C4×Q16×C71C22semidirect
52. C4×SD16×C71C22semidirect
53. C4×SD16×C71C22semidirect
230. #61933C71direct (N × H)
236. #3556C2×C71semidirect
238. #3400C2×C71semidirect
239. #3392C2×C71semidirect
240. #3368C2×C71semidirect
241. #3365C2×C71semidirect
242. #3222C2×C71semidirect
243. #3208C2×C71semidirect
244. #3123C2×C71semidirect
245. #3107C2×C71semidirect
246. #2986C2×C71semidirect
247. C71#61933direct (N × H)
251. #557C22×C71semidirect
252. #557C22×C71semidirect
265. #497C22×C71semidirect
267. #492C22×C71semidirect
268. #490C22×C71semidirect
269. #488C22×C71semidirect
272. #478C22×C71semidirect
273. #475C22×C71semidirect
274. #474C22×C71semidirect
275. #464C22×C71semidirect
276. #462C22×C71semidirect
277. #461C22×C71semidirect
278. #461C22×C71semidirect
281. #460C22×C71semidirect
284. #446C22×C71semidirect
285. C8×Q8C22×C71semidirect
286. #440C22×C71semidirect
287. C4×Q16C22×C71semidirect
288. C4×SD16C22×C71semidirect
289. C4×SD16C22×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#35562
C71×#35082
C71×#34002
C71×#33922
C71×#33682
C71×#33652
C71×#32222
C71×#32082
C71×#31232
C71×#31072
C71×#29862
#6193371