Finite Group Databasebeta

← C71×#19500 #31840372

Subgroups — C71×#19500 #31840372

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3229C211yesmaximal, characteristic, complemented
2C71×#3107C211yesmaximal, characteristic
3C71×#3079C211yesmaximal, characteristic
4C71×#2991C211yesmaximal, characteristic
5C71×#1703C212yesmaximal, complemented
6C71×#1691C212yesmaximal, complemented
7C71×#1684C212yesmaximal, complemented
8C71×#1672C212yesmaximal, complemented
9C71×#1574C211yesmaximal, characteristic, complemented
10C71×#1561C211yesmaximal, characteristic, complemented
11C71×#1548C211yesmaximal, characteristic, complemented
12#533×C71C2211yescharacteristic
13C2×C4⋊Q8×C71C2211yescharacteristic
14C4×#41×C71C2211yescharacteristic
15#498×C71C2212yes—
16#487×C71C2212yes—
17#485×C71—21no—
18#483×C71—22no—
19#477×C71C2212yes—
20#476×C71C2212yes—
21#475×C71—21no—
22#475×C71—21no—
23#475×C71—21no—
24#475×C71C2211yescharacteristic
25#474×C71—21no—
26#474×C71C2211yescharacteristic
27#462×C71C2211yescharacteristic
28#461×C71C2211yescharacteristic
29#452×C71—21no—
30C8×Q8×C71C2212yes—
31C4×Q16×C71—21no—
32C4×SD16×C71—21no—
33#436×C71—22no—
34#436×C71C2212yes—
35#431×C71—21no—
36#431×C71C2211yescharacteristic
37#431×C71C2211yescharacteristic
38#428×C71—21no—
39#423×C71—21no—
40#423×C71C2211yescharacteristic
41#423×C71C2211yescharacteristic
42#417×C71—21no—
43#335×C71C2212yes—
44#334×C71C2212yes—
45#333×C71C2212yes—
46#333×C71C2212yes—
47#332×C71C2212yes—
48#332×C71C2212yes—
49C2×#41×C71D411yescharacteristic
50C22×Q8×C71—21no—
51C2×Q16×C71—21no—
52C2×SD16×C71—21no—
53C2×M16×C71—21no—
54C2×M16×C71—21no—
55C2×M16×C71—22no—
56C4⋊Q8×C71—21no—
57C4⋊Q8×C71—21no—
58C4⋊Q8×C71—21no—
59C4⋊Q8×C71C2311yescharacteristic
60C4⋊Q8×C71C2311yescharacteristic
61C4⋊Q8×C71C2312yes—
62#125×C71—21no—
63C22⋊Q8×C71—21no—
64C22⋊Q8×C71—21no—
65C4×Q8×C71—21no—
66C4×Q8×C71—21no—
67C4×Q8×C71D411yescharacteristic
68C4×Q8×C71C2311yescharacteristic
69C4×D4×C71—21no—
70C4×D4×C71—21no—
71C4×D4×C71D411yescharacteristic
72C4×D4×C71C2311yescharacteristic
73#117×C71—21no—
74#117×C71—21no—
75#117×C71D411yescharacteristic
76C2×C4⋊C4×C71—21no—
77C2×C4⋊C4×C71—21no—
78C2×C42×C71—21no—
79C2×C42×C71C2311yescharacteristic
80#107×C71—22no—
81#107×C71—21no—
82#106×C71—22no—
83#106×C71—22no—
84#106×C71—22no—
85#106×C71—21no—
86C4⋊C8×C71—22no—
87C4⋊C8×C71—21no—
88C4⋊C8×C71—21no—
89C4⋊C8×C71D411yescharacteristic
90C4⋊C8×C71D411yescharacteristic
91C4⋊C8×C71C2312yes—
92C4⋊C8×C71C2312yes—
93C4⋊C8×C71C2311yescharacteristic
94C4⋊C8×C71C2311yescharacteristic
95#103×C71—22no—
96#103×C71—21no—
97#103×C71—22no—
98#103×C71—22no—
99#103×C71—21no—
100#103×C71—21no—
101#102×C71—22no—
102#102×C71—21no—
103C22⋊C8×C71—21no—
104C22⋊C8×C71—22no—
105C22⋊C8×C71D412yes—
106C4×C8×C71—22no—
107C4×C8×C71—21no—
108C4×C8×C71C2312yes—
109#41×C71—22no—
110#41×C71—41no—
111C2×Q8×C71—21no—
112C2×Q8×C71—21no—
113C2×Q8×C71—22no—
114C2×Q8×C71—41no—
115C2×Q8×C71—82no—
116C2×Q8×C71C2×D411yescharacteristic
117C2×D4×C71—21no—
118C2×D4×C71C2×D411yescharacteristic
119C22×C4×C71—21no—
120C22×C4×C71—21no—
121C22×C4×C71—21no—
122C22×C4×C71—41no—
123C22×C4×C71C2×D411yescharacteristic
124C22×C4×C71C2×D411yescharacteristic
125C22×C4×C71#4111yescharacteristic
126Q16×C71—42no—
127SD16×C71—42no—
128M16×C71—42no—
129M16×C71—82no—
130M16×C71—81no—
131C2×C8×C71—21no—
132C2×C8×C71—21no—
133C2×C8×C71—22no—
134C2×C8×C71—22no—
135C2×C8×C71C2×D412yes—
136C2×C8×C71C2×D412yes—
137C4⋊C4×C71—21no—
138C4⋊C4×C71—21no—
139C4⋊C4×C71—22no—
140C4⋊C4×C71—21no—
141C4⋊C4×C71—22no—
142C4⋊C4×C71—21no—
143C4⋊C4×C71—21no—
144C4⋊C4×C71—21no—
145C4⋊C4×C71—41no—
146C4⋊C4×C71—41no—
147C4⋊C4×C71—41no—
148C4⋊C4×C71—41no—
149C4⋊C4×C71C2×D411yescharacteristic
150C4⋊C4×C71C2×D411yescharacteristic
151C22⋊C4×C71—21no—
152C22⋊C4×C71—21no—
153C22⋊C4×C71—21no—
154C22⋊C4×C71—41no—
155C42×C71—21no—
156C42×C71—21no—
157C42×C71—21no—
158C42×C71—41no—
159C42×C71C2×D411yescharacteristic
160C42×C71C2×D411yescharacteristic
161C42×C71#4111yescharacteristic
162C42×C71C2411yescharacteristic
163C23×C71—21no—
164C23×C71#12111yescharacteristic
165Q8×C71—21no—
166Q8×C71—41no—
167Q8×C71—41no—
168Q8×C71—41no—
169Q8×C71—41no—
170Q8×C71—82no—
171Q8×C71—82no—
172Q8×C71#13712yes—
173D4×C71—21no—
174D4×C71—21no—
175D4×C71—42no—
176C2×C4×C71—21no—
177C2×C4×C71—21no—
178C2×C4×C71—21no—
179C2×C4×C71—21no—
180C2×C4×C71—21no—
181C2×C4×C71—22no—
182C2×C4×C71—21no—
183C2×C4×C71—21no—
184C2×C4×C71—42no—
185C2×C4×C71—41no—
186C2×C4×C71—41no—
187C2×C4×C71—42no—
188C2×C4×C71—41no—
189C2×C4×C71—41no—
190C2×C4×C71—42no—
191C2×C4×C71—82no—
192C2×C4×C71#12111yescharacteristic
193C2×C4×C71#12111yescharacteristic
194C2×C4×C71#12111yescharacteristic
195C2×C4×C71C22×D411yescharacteristic
196C2×C4×C71C22×D411yescharacteristic
197C2×C4×C71C2×#4111yescharacteristic
198C8×C71—22no—
199C8×C71—22no—
200C8×C71—41no—
201C8×C71—42no—
202C8×C71—41no—
203C8×C71—42no—
204C22×C71—21no—
205C22×C71—21no—
206C22×C71—21no—
207C22×C71—41no—
208C22×C71—41no—
209C22×C71—41no—
210C22×C71C2×#12111yescharacteristic
211C4×C71—21no—
212C4×C71—21no—
213C4×C71—22no—
214C4×C71—21no—
215C4×C71—41no—
216C4×C71—41no—
217C4×C71—81no—
218C4×C71—81no—
219C4×C71—82no—
220C4×C71#47112yes—
221C4×C71C2×#13711yescharacteristic
222C4×C71#57511yescharacteristic
223C4×C71#57711yescharacteristic
224C4×C71#57811yescharacteristic
225#19500C7111yesmaximal, characteristic, direct product factor, Sylow
226C2×C71—41no—
227C2×C71—81no—
228C2×C71#304611yescharacteristic
229C2×C71#305611yescharacteristic
230C2×C71C2×#47111yescharacteristic
231#3229C2×C7111yescharacteristic, complemented
232#3107C2×C7111yescharacteristic
233#3079C2×C7111yescharacteristic
234#2991C2×C7111yescharacteristic
235#1703C2×C7112yescomplemented
236#1691C2×C7112yescomplemented
237#1684C2×C7112yescomplemented
238#1672C2×C7112yescomplemented
239#1574C2×C7111yescharacteristic, complemented
240#1561C2×C7111yescharacteristic, complemented
241#1548C2×C7111yescharacteristic, complemented
242C71#1950011yescharacteristic, direct product factor, Sylow
243#533C22×C7111yescharacteristic
244C2×C4⋊Q8C22×C7111yescharacteristic
245C4×#41C22×C7111yescharacteristic
246#498C22×C7112yes—
247#487C22×C7112yes—
248#485—21no—
249#483—22no—
250#477C22×C7112yes—
251#476C22×C7112yes—
252#475—21no—
253#475—21no—
254#475—21no—
255#475C22×C7111yescharacteristic
256#474—21no—
257#474C22×C7111yescharacteristic
258#462C22×C7111yescharacteristic
259#461C22×C7111yescharacteristic
260#452—21no—
261C8×Q8C22×C7112yes—
262C4×Q16—21no—
263C4×SD16—21no—
264#436—22no—
265#436C22×C7112yes—
266#431—21no—
267#431C22×C7111yescharacteristic
268#431C22×C7111yescharacteristic
269#428—21no—
270#423—21no—
271#423C22×C7111yescharacteristic
272#423C22×C7111yescharacteristic
273#417—21no—
274#335C22×C7112yes—
275#334C22×C7112yes—
276#333C22×C7112yes—
277#333C22×C7112yes—
278#332C22×C7112yes—
279#332C22×C7112yes—
280C2×#41D4×C7111yescharacteristic
281C22×Q8—21no—
282C2×Q16—21no—
283C2×SD16—21no—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—22no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8C23×C7112yes—
291C4⋊Q8C23×C7111yescharacteristic
292C4⋊Q8C23×C7111yescharacteristic
293#125—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8D4×C7111yescharacteristic
299C4×Q8C23×C7111yescharacteristic
300C4×D4—21no—
301C4×D4—21no—
302C4×D4D4×C7111yescharacteristic
303C4×D4C23×C7111yescharacteristic
304#117—21no—
305#117—21no—
306#117D4×C7111yescharacteristic
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C42—21no—
310C2×C42C23×C7111yescharacteristic
311#107—22no—
312#107—21no—
313#106—22no—
314#106—22no—
315#106—22no—
316#106—21no—
317C4⋊C8—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8D4×C7111yescharacteristic
321C4⋊C8D4×C7111yescharacteristic
322C4⋊C8C23×C7112yes—
323C4⋊C8C23×C7112yes—
324C4⋊C8C23×C7111yescharacteristic
325C4⋊C8C23×C7111yescharacteristic
326#103—22no—
327#103—21no—
328#103—22no—
329#103—22no—
330#103—21no—
331#103—21no—
332#102—22no—
333#102—21no—
334C22⋊C8—21no—
335C22⋊C8—22no—
336C22⋊C8D4×C7112yes—
337C4×C8—22no—
338C4×C8—21no—
339C4×C8C23×C7112yes—
340#41—22no—
341#41—41no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—82no—
347C2×Q8C2×D4×C7111yescharacteristic
348C2×D4—21no—
349C2×D4C2×D4×C7111yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4C2×D4×C7111yescharacteristic
355C22×C4C2×D4×C7111yescharacteristic
356C22×C4#41×C7111yescharacteristic
357Q16—42no—
358SD16—42no—
359M16—42no—
360M16—82no—
361M16—81no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8C2×D4×C7112yes—
367C2×C8C2×D4×C7112yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4C2×D4×C7111yescharacteristic
381C4⋊C4C2×D4×C7111yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—41no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—41no—
390C42C2×D4×C7111yescharacteristic
391C42C2×D4×C7111yescharacteristic
392C42#41×C7111yescharacteristic
393C42C24×C7111yescharacteristic
394C23—21no—
395C23#121×C7111yescharacteristic
396Q8—21no—
397Q8—41no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—82no—
402Q8—82no—
403Q8#137×C7112yes—
404D4—21no—
405D4—21no—
406D4—42no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4#121×C7111yescharacteristic
424C2×C4#121×C7111yescharacteristic
425C2×C4#121×C7111yescharacteristic
426C2×C4C22×D4×C7111yescharacteristic
427C2×C4C22×D4×C7111yescharacteristic
428C2×C4C2×#41×C7111yescharacteristic
429C8—22no—
430C8—22no—
431C8—41no—
432C8—42no—
433C8—41no—
434C8—42no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22C2×#121×C7111yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—21no—
446C4—41no—
447C4—41no—
448C4—81no—
449C4—81no—
450C4—82no—
451C4#471×C7112yes—
452C4C2×#137×C7111yescharacteristic
453C4#575×C7111yescharacteristic
454C4#577×C7111yescharacteristic
455C4#578×C7111yescharacteristic
456C2—41no—
457C2—81no—
458C2C71×#304611yescharacteristic
459C2C71×#305611yescharacteristic
460C2C2×#471×C7111yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radicalC2×#41×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
1. C71×#3229C2semidirect
5. C71×#1703C2semidirect
6. C71×#1691C2semidirect
7. C71×#1684C2semidirect
8. C71×#1672C2semidirect
9. C71×#1574C2semidirect
10. C71×#1561C2semidirect
11. C71×#1548C2semidirect
225. #19500C71direct (N × H)
231. #3229C2×C71semidirect
235. #1703C2×C71semidirect
236. #1691C2×C71semidirect
237. #1684C2×C71semidirect
238. #1672C2×C71semidirect
239. #1574C2×C71semidirect
240. #1561C2×C71semidirect
241. #1548C2×C71semidirect
242. C71#19500direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32292
C71×#31072
C71×#30792
C71×#29912
C71×#17032
C71×#16912
C71×#16842
C71×#16722
C71×#15742
C71×#15612
C71×#15482
#1950071