Finite Group Databasebeta

← C71×#19029 #31839901

Subgroups — C71×#19029 #31839901

Order 18176.

824 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3563C211yesmaximal, characteristic
2C71×#3238C211yesmaximal, characteristic
3C2×#501×C71C211yesmaximal, characteristic, complemented
4C71×#3139C211yesmaximal, characteristic, complemented
5C71×#1664C214yesmaximal, complemented
6C71×#1663C214yesmaximal, complemented
7C71×#1531C211yesmaximal, characteristic, complemented
8C71×#1528C211yesmaximal, characteristic, complemented
9C71×#1524C211yesmaximal, characteristic
10#564×C71C2212yes—
11#563×C71C2212yes—
12#557×C71—21no—
13#554×C71—21no—
14#552×C71—21no—
15#552×C71—22no—
16#552×C71—21no—
17#544×C71—21no—
18#544×C71—21no—
19#541×C71—21no—
20#541×C71—22no—
21#539×C71—21no—
22#533×C71C2211yescharacteristic
23C2×C4⋊Q8×C71C2211yescharacteristic
24#529×C71C2211yescharacteristic
25C2×#125×C71C2211yescharacteristic
26#501×C71—22no—
27#501×C71C2214yes—
28#501×C71C2212yes—
29#491×C71C2214yes—
30#489×C71C2212yes—
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
38C2×#107×C71—21no—
39C2×#106×C71—21no—
40#419×C71—22no—
41C4×M16×C71C2211yescharacteristic
42C2×#97×C71C2211yescharacteristic
43#330×C71C2212yes—
44#330×C71C2214yes—
45#330×C71C2214yes—
46#329×C71C2212yes—
47C22×Q8×C71—21no—
48C2×M16×C71—21no—
49C22×C8×C71—21no—
50C4⋊Q8×C71—22no—
51C4⋊Q8×C71—21no—
52C4⋊Q8×C71—22no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71—21no—
55C4⋊Q8×C71—21no—
56C4⋊Q8×C71D411yescharacteristic
57C4⋊Q8×C71D411yescharacteristic
58C4⋊Q8×C71C2312yes—
59#126×C71—22no—
60#126×C71—22no—
61#126×C71—22no—
62#126×C71—22no—
63#126×C71—21no—
64#125×C71—22no—
65#125×C71—22no—
66#125×C71—21no—
67#125×C71—22no—
68#125×C71—22no—
69#125×C71—22no—
70#125×C71—22no—
71#125×C71—22no—
72#125×C71—22no—
73#125×C71—21no—
74#125×C71—21no—
75#125×C71D412yes—
76#125×C71C2312yes—
77#125×C71C2312yes—
78#125×C71C2311yescharacteristic
79#124×C71C2311yescharacteristic
80#123×C71—22no—
81#123×C71—22no—
82#123×C71—21no—
83C22⋊Q8×C71—22no—
84C22⋊Q8×C71—21no—
85C22⋊Q8×C71—21no—
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×C71D411yescharacteristic
93C4×Q8×C71—21no—
94C4×Q8×C71—22no—
95C4×Q8×C71—21no—
96C4×Q8×C71—21no—
97C4×Q8×C71—21no—
98C4×Q8×C71D411yescharacteristic
99C4×D4×C71—22no—
100C4×D4×C71—21no—
101C4×D4×C71D411yescharacteristic
102#117×C71—22no—
103#117×C71—21no—
104#117×C71—22no—
105#117×C71—21no—
106#117×C71D412yes—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71D411yescharacteristic
113C2×C4⋊C4×C71D411yescharacteristic
114C2×C42×C71C2311yescharacteristic
115#107×C71—24no—
116#107×C71—22no—
117#106×C71—24no—
118#106×C71—22no—
119C4⋊C8×C71—21no—
120#103×C71—22no—
121#103×C71—24no—
122#103×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—
130C2×Q8×C71—22no—
131C2×Q8×C71—21no—
132C2×Q8×C71—21no—
133C2×Q8×C71—22no—
134C2×Q8×C71—44no—
135C2×Q8×C71—41no—
136C2×Q8×C71C2×D411yescharacteristic
137C2×D4×C71—21no—
138C2×D4×C71C2×D411yescharacteristic
139C22×C4×C71—21no—
140C22×C4×C71—22no—
141C22×C4×C71—21no—
142C22×C4×C71—21no—
143C22×C4×C71—41no—
144C22×C4×C71C2×D411yescharacteristic
145C22×C4×C71C2×D411yescharacteristic
146C22×C4×C71C2×D411yescharacteristic
147M16×C71—42no—
148C2×C8×C71—24no—
149C2×C8×C71—22no—
150C2×C8×C71—22no—
151C2×C8×C71—22no—
152C4⋊C4×C71—22no—
153C4⋊C4×C71—22no—
154C4⋊C4×C71—22no—
155C4⋊C4×C71—22no—
156C4⋊C4×C71—22no—
157C4⋊C4×C71—22no—
158C4⋊C4×C71—22no—
159C4⋊C4×C71—22no—
160C4⋊C4×C71—21no—
161C4⋊C4×C71—21no—
162C4⋊C4×C71—22no—
163C4⋊C4×C71—22no—
164C4⋊C4×C71—21no—
165C4⋊C4×C71—21no—
166C4⋊C4×C71—21no—
167C4⋊C4×C71—21no—
168C4⋊C4×C71—22no—
169C4⋊C4×C71—22no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—41no—
177C4⋊C4×C71—41no—
178C4⋊C4×C71C2×D412yes—
179C4⋊C4×C71C2×D412yes—
180C4⋊C4×C71C2×D412yes—
181C4⋊C4×C71C2×D411yescharacteristic
182C4⋊C4×C71C2×D412yes—
183C4⋊C4×C71C2×D411yescharacteristic
184C22⋊C4×C71—22no—
185C22⋊C4×C71—22no—
186C22⋊C4×C71—22no—
187C22⋊C4×C71—21no—
188C22⋊C4×C71—22no—
189C22⋊C4×C71—21no—
190C22⋊C4×C71—21no—
191C22⋊C4×C71—21no—
192C22⋊C4×C71—21no—
193C42×C71—22no—
194C42×C71—22no—
195C42×C71—22no—
196C42×C71—22no—
197C42×C71—21no—
198C42×C71C2×D411yescharacteristic
199C42×C71C2×D411yescharacteristic
200C42×C71C2×D411yescharacteristic
201C42×C71C2411yescharacteristic
202C23×C71—21no—
203C23×C71#12011yescharacteristic
204Q8×C71—41no—
205Q8×C71—44no—
206Q8×C71—42no—
207Q8×C71—81no—
208Q8×C71—82no—
209Q8×C71—82no—
210D4×C71—41no—
211D4×C71—42no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—22no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—81no—
226C2×C4×C71—82no—
227C2×C4×C71—82no—
228C2×C4×C71#12011yescharacteristic
229C2×C4×C71#12011yescharacteristic
230C2×C4×C71#12011yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C22×D411yescharacteristic
233C2×C4×C71C22×D411yescharacteristic
234C8×C71—24no—
235C8×C71—42no—
236C8×C71—42no—
237C22×C71—21no—
238C22×C71—21no—
239C22×C71—41no—
240C22×C71—41no—
241C22×C71—41no—
242C22×C71#45412yes—
243C22×C71C2×#12011yescharacteristic
244C4×C71—21no—
245C4×C71—21no—
246C4×C71—21no—
247C4×C71—21no—
248C4×C71—21no—
249C4×C71—42no—
250C4×C71—41no—
251C4×C71—41no—
252C4×C71—42no—
253C4×C71—42no—
254C4×C71—42no—
255C4×C71—41no—
256C4×C71—82no—
257C4×C71#45612yes—
258#19029C7111yesmaximal, characteristic, direct product factor, Sylow
259C2×C71—22no—
260C2×C71—81no—
261C2×C71C2×#45411yescharacteristic
262C2×C71C2×#45611yescharacteristic
263C2×C71#302211yescharacteristic
264#3563C2×C7111yescharacteristic
265#3238C2×C7111yescharacteristic
266C2×#501C2×C7111yescharacteristic, complemented
267#3139C2×C7111yescharacteristic, complemented
268#1664C2×C7114yescomplemented
269#1663C2×C7114yescomplemented
270#1531C2×C7111yescharacteristic, complemented
271#1528C2×C7111yescharacteristic, complemented
272#1524C2×C7111yescharacteristic
273C71#1902911yescharacteristic, direct product factor, Sylow
274#564C22×C7112yes—
275#563C22×C7112yes—
276#557—21no—
277#554—21no—
278#552—21no—
279#552—22no—
280#552—21no—
281#544—21no—
282#544—21no—
283#541—21no—
284#541—22no—
285#539—21no—
286#533C22×C7111yescharacteristic
287C2×C4⋊Q8C22×C7111yescharacteristic
288#529C22×C7111yescharacteristic
289C2×#125C22×C7111yescharacteristic
290#501—22no—
291#501C22×C7114yes—
292#501C22×C7112yes—
293#491C22×C7114yes—
294#489C22×C7112yes—
295#487—21no—
296#487—21no—
297#483—21no—
298#480—21no—
299#477—21no—
300#476—21no—
301#432C22×C7111yescharacteristic
302C2×#107—21no—
303C2×#106—21no—
304#419—22no—
305C4×M16C22×C7111yescharacteristic
306C2×#97C22×C7111yescharacteristic
307#330C22×C7112yes—
308#330C22×C7114yes—
309#330C22×C7114yes—
310#329C22×C7112yes—
311C22×Q8—21no—
312C2×M16—21no—
313C22×C8—21no—
314C4⋊Q8—22no—
315C4⋊Q8—21no—
316C4⋊Q8—22no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8D4×C7111yescharacteristic
321C4⋊Q8D4×C7111yescharacteristic
322C4⋊Q8C23×C7112yes—
323#126—22no—
324#126—22no—
325#126—22no—
326#126—22no—
327#126—21no—
328#125—22no—
329#125—22no—
330#125—21no—
331#125—22no—
332#125—22no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—21no—
338#125—21no—
339#125D4×C7112yes—
340#125C23×C7112yes—
341#125C23×C7111yescharacteristic
342#125C23×C7112yes—
343#124C23×C7111yescharacteristic
344#123—22no—
345#123—22no—
346#123—21no—
347C22⋊Q8—22no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8D4×C7111yescharacteristic
356#121D4×C7111yescharacteristic
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8D4×C7111yescharacteristic
363C4×D4—22no—
364C4×D4—21no—
365C4×D4D4×C7111yescharacteristic
366#117—22no—
367#117—21no—
368#117—22no—
369#117—21no—
370#117D4×C7112yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4D4×C7111yescharacteristic
377C2×C4⋊C4D4×C7111yescharacteristic
378C2×C42C23×C7111yescharacteristic
379#107—24no—
380#107—22no—
381#106—24no—
382#106—22no—
383C4⋊C8—21no—
384#103—22no—
385#103—24no—
386#103—24no—
387#102—22no—
388C22⋊C8—21no—
389#97—21no—
390#97C23×C7112yes—
391#97C23×C7112yes—
392#97C23×C7112yes—
393C4×C8—21no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—41no—
400C2×Q8C2×D4×C7111yescharacteristic
401C2×D4—21no—
402C2×D4C2×D4×C7111yescharacteristic
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—41no—
408C22×C4C2×D4×C7111yescharacteristic
409C22×C4C2×D4×C7111yescharacteristic
410C22×C4C2×D4×C7111yescharacteristic
411M16—42no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×D4×C7112yes—
443C4⋊C4C2×D4×C7112yes—
444C4⋊C4C2×D4×C7112yes—
445C4⋊C4C2×D4×C7111yescharacteristic
446C4⋊C4C2×D4×C7112yes—
447C4⋊C4C2×D4×C7111yescharacteristic
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42C2×D4×C7111yescharacteristic
463C42C2×D4×C7111yescharacteristic
464C42C2×D4×C7111yescharacteristic
465C42C24×C7111yescharacteristic
466C23—21no—
467C23#120×C7111yescharacteristic
468Q8—41no—
469Q8—44no—
470Q8—42no—
471Q8—81no—
472Q8—82no—
473Q8—82no—
474D4—41no—
475D4—42no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#120×C7111yescharacteristic
493C2×C4#120×C7111yescharacteristic
494C2×C4#120×C7111yescharacteristic
495C2×C4C22×D4×C7111yescharacteristic
496C2×C4C22×D4×C7111yescharacteristic
497C2×C4C22×D4×C7111yescharacteristic
498C8—24no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22#454×C7112yes—
507C22C2×#120×C7111yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—82no—
521C4#456×C7112yes—
522C2—22no—
523C2—81no—
524C2C2×#454×C7111yescharacteristic
525C2C2×#456×C7111yescharacteristic
526C2C71×#302211yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
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×#501×C71C2semidirect
4. C71×#3139C2semidirect
5. C71×#1664C2semidirect
6. C71×#1663C2semidirect
7. C71×#1531C2semidirect
8. C71×#1528C2semidirect
258. #19029C71direct (N × H)
266. C2×#501C2×C71semidirect
267. #3139C2×C71semidirect
268. #1664C2×C71semidirect
269. #1663C2×C71semidirect
270. #1531C2×C71semidirect
271. #1528C2×C71semidirect
273. C71#19029direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#35632
C71×#32382
C2×#501×C712
C71×#31392
C71×#16642
C71×#16632
C71×#15312
C71×#15282
C71×#15242
#1902971