Finite Group Databasebeta

← C71×#19363 #31840235

Subgroups — C71×#19363 #31840235

Order 18176.

596 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3239C211yesmaximal, characteristic
2C71×#3218C211yesmaximal, characteristic
3C2×#487×C71C211yesmaximal, characteristic, complemented
4C71×#2966C211yesmaximal, characteristic
5C71×#1700C212yesmaximal, complemented
6C71×#1699C212yesmaximal, complemented
7C71×#1680C212yesmaximal, complemented
8C71×#1679C212yesmaximal, complemented
9C71×#1568C211yesmaximal, characteristic, complemented
10C71×#1556C211yesmaximal, characteristic, complemented
11C71×#1556C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C71C2211yescharacteristic
13C2×C4⋊Q8×C71C2211yescharacteristic
14#529×C71C2211yescharacteristic
15#501×C71—21no—
16#497×C71—21no—
17#491×C71—21no—
18#489×C71—21no—
19#487×C71—21no—
20#487×C71—21no—
21#487×C71C2212yes—
22#487×C71C2212yes—
23#484×C71—21no—
24#481×C71—21no—
25#475×C71C2212yes—
26#474×C71C2212yes—
27#462×C71C2212yes—
28#461×C71C2212yes—
29#451×C71—21no—
30#448×C71—21no—
31#446×C71C2211yescharacteristic
32C8×Q8×C71C2211yescharacteristic
33#436×C71C2211yescharacteristic
34#435×C71—21no—
35C8×D4×C71—21no—
36C8×D4×C71C2211yescharacteristic
37#433×C71—21no—
38#432×C71C2211yescharacteristic
39#432×C71C2211yescharacteristic
40#432×C71C2211yescharacteristic
41C2×#103×C71—21no—
42C2×#103×C71—21no—
43#405×C71—21no—
44C2×C4×C8×C71C2211yescharacteristic
45#335×C71C2212yes—
46#334×C71C2212yes—
47#332×C71C2212yes—
48#332×C71C2212yes—
49#332×C71C2212yes—
50#332×C71C2212yes—
51C22×Q8×C71—21no—
52C22×Q8×C71—21no—
53C2×Q16×C71—22no—
54C2×SD16×C71—22no—
55C2×M16×C71—21no—
56C22×C8×C71—21no—
57C22×C8×C71D411yescharacteristic
58C22×C8×C71D411yescharacteristic
59C4⋊Q8×C71—21no—
60C4⋊Q8×C71—21no—
61C4⋊Q8×C71—21no—
62C4⋊Q8×C71—21no—
63C4⋊Q8×C71—21no—
64C4⋊Q8×C71—21no—
65C4⋊Q8×C71C2312yes—
66C4⋊Q8×C71C2312yes—
67#126×C71—21no—
68#125×C71D411yescharacteristic
69#124×C71D411yescharacteristic
70#123×C71—21no—
71C22⋊Q8×C71D411yescharacteristic
72#121×C71D411yescharacteristic
73C4×Q8×C71C2311yescharacteristic
74C4×D4×C71—21no—
75C4×D4×C71C2311yescharacteristic
76#117×C71—21no—
77C2×C4⋊C4×C71—21no—
78C2×C4⋊C4×C71—21no—
79C2×C4⋊C4×C71—21no—
80C2×C4⋊C4×C71—21no—
81C2×C42×C71C2311yescharacteristic
82#107×C71—22no—
83#106×C71—22no—
84C4⋊C8×C71—21no—
85C4⋊C8×C71—21no—
86C4⋊C8×C71C2311yescharacteristic
87C4⋊C8×C71C2311yescharacteristic
88C4⋊C8×C71C2312yes—
89C4⋊C8×C71C2312yes—
90#103×C71—22no—
91#103×C71—22no—
92#103×C71—22no—
93#103×C71—22no—
94#103×C71—22no—
95#103×C71—22no—
96#103×C71—22no—
97#102×C71—22no—
98C22⋊C8×C71—21no—
99C22⋊C8×C71—21no—
100C22⋊C8×C71—21no—
101C22⋊C8×C71—21no—
102C22⋊C8×C71—21no—
103#97×C71—21no—
104#97×C71—21no—
105#97×C71—21no—
106#97×C71—21no—
107C4×C8×C71—21no—
108C4×C8×C71D411yescharacteristic
109C4×C8×C71D411yescharacteristic
110C4×C8×C71C2311yescharacteristic
111C4×C8×C71C2311yescharacteristic
112C2×Q8×C71—22no—
113C2×Q8×C71—22no—
114C2×Q8×C71—42no—
115C2×Q8×C71—42no—
116C2×Q8×C71—41no—
117C2×Q8×C71—41no—
118C2×Q8×C71—82no—
119C2×Q8×C71C2×D411yescharacteristic
120C2×D4×C71—21no—
121C2×D4×C71C2×D411yescharacteristic
122C22×C4×C71—21no—
123C22×C4×C71—41no—
124C22×C4×C71—41no—
125C22×C4×C71C2×D411yescharacteristic
126C22×C4×C71C2×D411yescharacteristic
127C22×C4×C71#4111yescharacteristic
128Q16×C71—82no—
129SD16×C71—82no—
130M16×C71—42no—
131C2×C8×C71—21no—
132C2×C8×C71—21no—
133C2×C8×C71—22no—
134C2×C8×C71—22no—
135C2×C8×C71—41no—
136C2×C8×C71—41no—
137C2×C8×C71—42no—
138C2×C8×C71C2×D411yescharacteristic
139C2×C8×C71C2×D411yescharacteristic
140C2×C8×C71C2×D411yescharacteristic
141C2×C8×C71C2×D411yescharacteristic
142C4⋊C4×C71—21no—
143C4⋊C4×C71—21no—
144C4⋊C4×C71—21no—
145C4⋊C4×C71—21no—
146C4⋊C4×C71—22no—
147C4⋊C4×C71—22no—
148C4⋊C4×C71—22no—
149C4⋊C4×C71—22no—
150C4⋊C4×C71—41no—
151C4⋊C4×C71—41no—
152C4⋊C4×C71—41no—
153C4⋊C4×C71—41no—
154C4⋊C4×C71C2×D411yescharacteristic
155C4⋊C4×C71C2×D411yescharacteristic
156C22⋊C4×C71—21no—
157C22⋊C4×C71—21no—
158C22⋊C4×C71—21no—
159C22⋊C4×C71—21no—
160C22⋊C4×C71—21no—
161C42×C71—21no—
162C42×C71C2×D411yescharacteristic
163C42×C71C2×D411yescharacteristic
164C42×C71#4111yescharacteristic
165C42×C71C2411yescharacteristic
166C23×C71—21no—
167C23×C71#12111yescharacteristic
168Q8×C71—22no—
169Q8×C71—42no—
170Q8×C71—42no—
171Q8×C71—42no—
172Q8×C71—42no—
173Q8×C71—82no—
174Q8×C71—82no—
175D4×C71—22no—
176D4×C71—42no—
177C2×C4×C71—22no—
178C2×C4×C71—21no—
179C2×C4×C71—21no—
180C2×C4×C71—22no—
181C2×C4×C71—21no—
182C2×C4×C71—21no—
183C2×C4×C71—21no—
184C2×C4×C71—41no—
185C2×C4×C71—41no—
186C2×C4×C71—42no—
187C2×C4×C71—42no—
188C2×C4×C71—82no—
189C2×C4×C71—82no—
190C2×C4×C71#12111yescharacteristic
191C2×C4×C71#12111yescharacteristic
192C2×C4×C71#12111yescharacteristic
193C2×C4×C71C22×D411yescharacteristic
194C2×C4×C71C22×D411yescharacteristic
195C2×C4×C71C2×#4111yescharacteristic
196C8×C71—21no—
197C8×C71—21no—
198C8×C71—21no—
199C8×C71—21no—
200C8×C71—41no—
201C8×C71—41no—
202C8×C71—42no—
203C8×C71—42no—
204C22×C71—21no—
205C22×C71—21no—
206C22×C71—41no—
207C22×C71—41no—
208C22×C71—41no—
209C22×C71#47312yes—
210C22×C71C2×#12111yescharacteristic
211C4×C71—21no—
212C4×C71—21no—
213C4×C71—21no—
214C4×C71—21no—
215C4×C71—41no—
216C4×C71—41no—
217C4×C71—41no—
218C4×C71—82no—
219C4×C71—82no—
220C4×C71#57511yescharacteristic
221C4×C71#57511yescharacteristic
222C4×C71#57712yes—
223#19363C7111yesmaximal, characteristic, direct product factor, Sylow
224C2×C71—22no—
225C2×C71—81no—
226C2×C71#304011yescharacteristic
227C2×C71#305711yescharacteristic
228C2×C71C2×#47311yescharacteristic
229#3239C2×C7111yescharacteristic
230#3218C2×C7111yescharacteristic
231C2×#487C2×C7111yescharacteristic, complemented
232#2966C2×C7111yescharacteristic
233#1700C2×C7112yescomplemented
234#1699C2×C7112yescomplemented
235#1680C2×C7112yescomplemented
236#1679C2×C7112yescomplemented
237#1568C2×C7111yescharacteristic, complemented
238#1556C2×C7111yescharacteristic, complemented
239#1556C2×C7111yescharacteristic, complemented
240C71#1936311yescharacteristic, direct product factor, Sylow
241C2×C4⋊Q8C22×C7111yescharacteristic
242C2×C4⋊Q8C22×C7111yescharacteristic
243#529C22×C7111yescharacteristic
244#501—21no—
245#497—21no—
246#491—21no—
247#489—21no—
248#487—21no—
249#487—21no—
250#487C22×C7112yes—
251#487C22×C7112yes—
252#484—21no—
253#481—21no—
254#475C22×C7112yes—
255#474C22×C7112yes—
256#462C22×C7112yes—
257#461C22×C7112yes—
258#451—21no—
259#448—21no—
260#446C22×C7111yescharacteristic
261C8×Q8C22×C7111yescharacteristic
262#436C22×C7111yescharacteristic
263#435—21no—
264C8×D4—21no—
265C8×D4C22×C7111yescharacteristic
266#433—21no—
267#432C22×C7111yescharacteristic
268#432C22×C7111yescharacteristic
269#432C22×C7111yescharacteristic
270C2×#103—21no—
271C2×#103—21no—
272#405—21no—
273C2×C4×C8C22×C7111yescharacteristic
274#335C22×C7112yes—
275#334C22×C7112yes—
276#332C22×C7112yes—
277#332C22×C7112yes—
278#332C22×C7112yes—
279#332C22×C7112yes—
280C22×Q8—21no—
281C22×Q8—21no—
282C2×Q16—22no—
283C2×SD16—22no—
284C2×M16—21no—
285C22×C8—21no—
286C22×C8D4×C7111yescharacteristic
287C22×C8D4×C7111yescharacteristic
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C23×C7112yes—
295C4⋊Q8C23×C7112yes—
296#126—21no—
297#125D4×C7111yescharacteristic
298#124D4×C7111yescharacteristic
299#123—21no—
300C22⋊Q8D4×C7111yescharacteristic
301#121D4×C7111yescharacteristic
302C4×Q8C23×C7111yescharacteristic
303C4×D4—21no—
304C4×D4C23×C7111yescharacteristic
305#117—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C42C23×C7111yescharacteristic
311#107—22no—
312#106—22no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8C23×C7111yescharacteristic
316C4⋊C8C23×C7111yescharacteristic
317C4⋊C8C23×C7112yes—
318C4⋊C8C23×C7112yes—
319#103—22no—
320#103—22no—
321#103—22no—
322#103—22no—
323#103—22no—
324#103—22no—
325#103—22no—
326#102—22no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332#97—21no—
333#97—21no—
334#97—21no—
335#97—21no—
336C4×C8—21no—
337C4×C8D4×C7111yescharacteristic
338C4×C8D4×C7111yescharacteristic
339C4×C8C23×C7111yescharacteristic
340C4×C8C23×C7111yescharacteristic
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—82no—
348C2×Q8C2×D4×C7111yescharacteristic
349C2×D4—21no—
350C2×D4C2×D4×C7111yescharacteristic
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4C2×D4×C7111yescharacteristic
355C22×C4C2×D4×C7111yescharacteristic
356C22×C4#41×C7111yescharacteristic
357Q16—82no—
358SD16—82no—
359M16—42no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8C2×D4×C7111yescharacteristic
368C2×C8C2×D4×C7111yescharacteristic
369C2×C8C2×D4×C7111yescharacteristic
370C2×C8C2×D4×C7111yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4C2×D4×C7111yescharacteristic
384C4⋊C4C2×D4×C7111yescharacteristic
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—21no—
390C42—21no—
391C42C2×D4×C7111yescharacteristic
392C42C2×D4×C7111yescharacteristic
393C42#41×C7111yescharacteristic
394C42C24×C7111yescharacteristic
395C23—21no—
396C23#121×C7111yescharacteristic
397Q8—22no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—82no—
403Q8—82no—
404D4—22no—
405D4—42no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#121×C7111yescharacteristic
420C2×C4#121×C7111yescharacteristic
421C2×C4#121×C7111yescharacteristic
422C2×C4C22×D4×C7111yescharacteristic
423C2×C4C22×D4×C7111yescharacteristic
424C2×C4C2×#41×C7111yescharacteristic
425C8—21no—
426C8—21no—
427C8—21no—
428C8—21no—
429C8—41no—
430C8—41no—
431C8—42no—
432C8—42no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22#473×C7112yes—
439C22C2×#121×C7111yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4#575×C7111yescharacteristic
450C4#575×C7111yescharacteristic
451C4#577×C7112yes—
452C2—22no—
453C2—81no—
454C2C71×#304011yescharacteristic
455C2C71×#305711yescharacteristic
456C2C2×#473×C7111yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radical#121×C71D4
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
3. C2×#487×C71C2semidirect
5. C71×#1700C2semidirect
6. C71×#1699C2semidirect
7. C71×#1680C2semidirect
8. C71×#1679C2semidirect
9. C71×#1568C2semidirect
10. C71×#1556C2semidirect
11. C71×#1556C2semidirect
223. #19363C71direct (N × H)
231. C2×#487C2×C71semidirect
233. #1700C2×C71semidirect
234. #1699C2×C71semidirect
235. #1680C2×C71semidirect
236. #1679C2×C71semidirect
237. #1568C2×C71semidirect
238. #1556C2×C71semidirect
239. #1556C2×C71semidirect
240. C71#19363direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32392
C71×#32182
C2×#487×C712
C71×#29662
C71×#17002
C71×#16992
C71×#16802
C71×#16792
C71×#15682
C71×#15562
#1936371