Finite Group Databasebeta

← C71×#19526 #31840398

Subgroups — C71×#19526 #31840398

Order 18176.

596 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3254C211yesmaximal, characteristic, complemented
2C71×#3246C211yesmaximal, characteristic, complemented
3C71×#3207C211yesmaximal, characteristic
4C71×#2985C211yesmaximal, characteristic, complemented
5C71×#1698C212yesmaximal, complemented
6C71×#1686C212yesmaximal, complemented
7C71×#1684C212yesmaximal, complemented
8C71×#1668C212yesmaximal, complemented
9C71×#1571C211yesmaximal, characteristic, complemented
10C71×#1557C211yesmaximal, characteristic, complemented
11C71×#1551C211yesmaximal, characteristic, complemented
12#533×C71C2211yescharacteristic
13#532×C71C2211yescharacteristic
14#532×C71C2211yescharacteristic
15#493×C71C2211yescharacteristic
16#492×C71C2211yescharacteristic
17#487×C71C2211yescharacteristic
18#486×C71C2211yescharacteristic
19#477×C71C2212yes—
20#476×C71—21no—
21#475×C71C2212yes—
22#474×C71—21no—
23#468×C71—22no—
24#465×C71—22no—
25#463×C71—22no—
26#461×C71—21no—
27#461×C71—21no—
28#460×C71C2212yes—
29#460×C71C2212yes—
30#449×C71—22no—
31#446×C71—22no—
32#446×C71C2212yes—
33C8×Q8×C71C2212yes—
34#433×C71—22no—
35#432×C71C2211yescharacteristic
36#431×C71C2211yescharacteristic
37#423×C71C2211yescharacteristic
38C2×C4⋊C8×C71C2211yescharacteristic
39#335×C71C2212yes—
40#335×C71C2212yes—
41#333×C71C2212yes—
42#332×C71C2212yes—
43#331×C71C2212yes—
44#331×C71C2212yes—
45C2×#41×C71—21no—
46C2×#41×C71—21no—
47C2×SD16×C71—22no—
48C2×SD16×C71—22no—
49C2×SD16×C71—22no—
50C2×D8×C71—22no—
51C2×M16×C71—21no—
52C22×C8×C71—21no—
53C4⋊Q8×C71D411yescharacteristic
54C4⋊Q8×C71D411yescharacteristic
55C4⋊Q8×C71C2311yescharacteristic
56C4⋊Q8×C71C2311yescharacteristic
57#127×C71C2311yescharacteristic
58#127×C71C2311yescharacteristic
59#125×C71—21no—
60#124×C71—21no—
61#124×C71—21no—
62C22⋊Q8×C71—21no—
63C22⋊Q8×C71D412yes—
64#121×C71—21no—
65#121×C71—21no—
66#121×C71—21no—
67#121×C71—21no—
68C4×Q8×C71—21no—
69C4×Q8×C71C2312yes—
70C4×D4×C71—21no—
71C4×D4×C71—21no—
72C4×D4×C71—21no—
73C4×D4×C71—21no—
74#117×C71—21no—
75C2×C42×C71C2311yescharacteristic
76#107×C71—22no—
77#106×C71—22no—
78C4⋊C8×C71—22no—
79C4⋊C8×C71—22no—
80C4⋊C8×C71—21no—
81C4⋊C8×C71—21no—
82C4⋊C8×C71D412yes—
83C4⋊C8×C71C2312yes—
84C4⋊C8×C71C2312yes—
85C4⋊C8×C71C2312yes—
86#103×C71—22no—
87#103×C71—22no—
88#103×C71—22no—
89#102×C71—22no—
90#102×C71—22no—
91#102×C71—22no—
92C22⋊C8×C71—21no—
93C22⋊C8×C71D412yes—
94#97×C71—22no—
95#97×C71—21no—
96C4×C8×C71—22no—
97C4×C8×C71C2311yescharacteristic
98C4×C8×C71C2311yescharacteristic
99#41×C71—82no—
100#41×C71—82no—
101C2×Q8×C71—21no—
102C2×Q8×C71—21no—
103C2×Q8×C71—21no—
104C2×Q8×C71C2×D412yes—
105C2×D4×C71—21no—
106C2×D4×C71—21no—
107C2×D4×C71—21no—
108C2×D4×C71—21no—
109C2×D4×C71—21no—
110C2×D4×C71—21no—
111C2×D4×C71—41no—
112C2×D4×C71—41no—
113C22×C4×C71—41no—
114C22×C4×C71—41no—
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71#4111yescharacteristic
118SD16×C71—82no—
119SD16×C71—82no—
120SD16×C71—82no—
121D8×C71—82no—
122M16×C71—81no—
123C2×C8×C71—22no—
124C2×C8×C71—22no—
125C2×C8×C71—22no—
126C2×C8×C71—81no—
127C2×C8×C71C2×D412yes—
128C2×C8×C71C2×D412yes—
129C4⋊C4×C71—21no—
130C4⋊C4×C71—21no—
131C4⋊C4×C71—21no—
132C4⋊C4×C71—22no—
133C4⋊C4×C71—22no—
134C4⋊C4×C71—21no—
135C4⋊C4×C71—21no—
136C4⋊C4×C71—21no—
137C4⋊C4×C71—21no—
138C4⋊C4×C71C2×D412yes—
139C22⋊C4×C71—22no—
140C22⋊C4×C71—41no—
141C22⋊C4×C71—41no—
142C22⋊C4×C71—41no—
143C22⋊C4×C71—41no—
144C42×C71—22no—
145C42×C71C2×D411yescharacteristic
146C42×C71C2×D411yescharacteristic
147C42×C71#4111yescharacteristic
148C42×C71C2411yescharacteristic
149C23×C71—41no—
150C23×C71—41no—
151C23×C71#12111yescharacteristic
152Q8×C71—22no—
153Q8×C71—22no—
154Q8×C71—42no—
155Q8×C71—81no—
156Q8×C71—81no—
157D4×C71—41no—
158D4×C71—41no—
159D4×C71—42no—
160D4×C71—42no—
161D4×C71—41no—
162D4×C71—41no—
163D4×C71—81no—
164D4×C71—81no—
165D4×C71—82no—
166D4×C71—82no—
167C2×C4×C71—21no—
168C2×C4×C71—22no—
169C2×C4×C71—21no—
170C2×C4×C71—22no—
171C2×C4×C71—22no—
172C2×C4×C71—41no—
173C2×C4×C71—41no—
174C2×C4×C71—41no—
175C2×C4×C71—82no—
176C2×C4×C71—82no—
177C2×C4×C71#12111yescharacteristic
178C2×C4×C71#12111yescharacteristic
179C2×C4×C71#12111yescharacteristic
180C2×C4×C71C22×D411yescharacteristic
181C2×C4×C71C22×D411yescharacteristic
182C2×C4×C71C2×#4111yescharacteristic
183C8×C71—22no—
184C8×C71—22no—
185C8×C71—42no—
186C8×C71—42no—
187C8×C71—42no—
188C22×C71—21no—
189C22×C71—21no—
190C22×C71—21no—
191C22×C71—81no—
192C22×C71—81no—
193C22×C71—81no—
194C22×C71—81no—
195C22×C71—81no—
196C22×C71—81no—
197C22×C71C2×#12111yescharacteristic
198C4×C71—21no—
199C4×C71—21no—
200C4×C71—21no—
201C4×C71—21no—
202C4×C71—42no—
203C4×C71—42no—
204C4×C71—81no—
205C4×C71—81no—
206C4×C71#57611yescharacteristic
207C4×C71#57712yes—
208C4×C71#57811yescharacteristic
209#19526C7111yesmaximal, characteristic, direct product factor, Sylow
210C2×C71—41no—
211C2×C71—161no—
212C2×C71—161no—
213C2×C71#304311yescharacteristic
214C2×C71#306111yescharacteristic
215C2×C71#306811yescharacteristic
216#3254C2×C7111yescharacteristic, complemented
217#3246C2×C7111yescharacteristic, complemented
218#3207C2×C7111yescharacteristic
219#2985C2×C7111yescharacteristic, complemented
220#1698C2×C7112yescomplemented
221#1686C2×C7112yescomplemented
222#1684C2×C7112yescomplemented
223#1668C2×C7112yescomplemented
224#1571C2×C7111yescharacteristic, complemented
225#1557C2×C7111yescharacteristic, complemented
226#1551C2×C7111yescharacteristic, complemented
227C71#1952611yescharacteristic, direct product factor, Sylow
228#533C22×C7111yescharacteristic
229#532C22×C7111yescharacteristic
230#532C22×C7111yescharacteristic
231#493C22×C7111yescharacteristic
232#492C22×C7111yescharacteristic
233#487C22×C7111yescharacteristic
234#486C22×C7111yescharacteristic
235#477C22×C7112yes—
236#476—21no—
237#475C22×C7112yes—
238#474—21no—
239#468—22no—
240#465—22no—
241#463—22no—
242#461—21no—
243#461—21no—
244#460C22×C7112yes—
245#460C22×C7112yes—
246#449—22no—
247#446—22no—
248#446C22×C7112yes—
249C8×Q8C22×C7112yes—
250#433—22no—
251#432C22×C7111yescharacteristic
252#431C22×C7111yescharacteristic
253#423C22×C7111yescharacteristic
254C2×C4⋊C8C22×C7111yescharacteristic
255#335C22×C7112yes—
256#335C22×C7112yes—
257#333C22×C7112yes—
258#332C22×C7112yes—
259#331C22×C7112yes—
260#331C22×C7112yes—
261C2×#41—21no—
262C2×#41—21no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×SD16—22no—
266C2×D8—22no—
267C2×M16—21no—
268C22×C8—21no—
269C4⋊Q8D4×C7111yescharacteristic
270C4⋊Q8D4×C7111yescharacteristic
271C4⋊Q8C23×C7111yescharacteristic
272C4⋊Q8C23×C7111yescharacteristic
273#127C23×C7111yescharacteristic
274#127C23×C7111yescharacteristic
275#125—21no—
276#124—21no—
277#124—21no—
278C22⋊Q8—21no—
279C22⋊Q8D4×C7112yes—
280#121—21no—
281#121—21no—
282#121—21no—
283#121—21no—
284C4×Q8—21no—
285C4×Q8C23×C7112yes—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290#117—21no—
291C2×C42C23×C7111yescharacteristic
292#107—22no—
293#106—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8D4×C7112yes—
299C4⋊C8C23×C7112yes—
300C4⋊C8C23×C7112yes—
301C4⋊C8C23×C7112yes—
302#103—22no—
303#103—22no—
304#103—22no—
305#102—22no—
306#102—22no—
307#102—22no—
308C22⋊C8—21no—
309C22⋊C8D4×C7112yes—
310#97—22no—
311#97—21no—
312C4×C8—22no—
313C4×C8C23×C7111yescharacteristic
314C4×C8C23×C7111yescharacteristic
315#41—82no—
316#41—82no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C2×D4×C7112yes—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C7111yescharacteristic
332C22×C4C2×D4×C7111yescharacteristic
333C22×C4#41×C7111yescharacteristic
334SD16—82no—
335SD16—82no—
336SD16—82no—
337D8—82no—
338M16—81no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—81no—
343C2×C8C2×D4×C7112yes—
344C2×C8C2×D4×C7112yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C2×D4×C7112yes—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C42—22no—
361C42C2×D4×C7111yescharacteristic
362C42C2×D4×C7111yescharacteristic
363C42#41×C7111yescharacteristic
364C42C24×C7111yescharacteristic
365C23—41no—
366C23—41no—
367C23#121×C7111yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—81no—
372Q8—81no—
373D4—41no—
374D4—41no—
375D4—42no—
376D4—42no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#121×C7111yescharacteristic
394C2×C4#121×C7111yescharacteristic
395C2×C4#121×C7111yescharacteristic
396C2×C4C22×D4×C7111yescharacteristic
397C2×C4C22×D4×C7111yescharacteristic
398C2×C4C2×#41×C7111yescharacteristic
399C8—22no—
400C8—22no—
401C8—42no—
402C8—42no—
403C8—42no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C2×#121×C7111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4#576×C7111yescharacteristic
423C4#577×C7112yes—
424C4#578×C7111yescharacteristic
425C2—41no—
426C2—161no—
427C2—161no—
428C2C71×#304311yescharacteristic
429C2C71×#306111yescharacteristic
430C2C71×#306811yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radicalC71×#3207C2
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×#3254C2semidirect
2. C71×#3246C2semidirect
4. C71×#2985C2semidirect
5. C71×#1698C2semidirect
6. C71×#1686C2semidirect
7. C71×#1684C2semidirect
8. C71×#1668C2semidirect
9. C71×#1571C2semidirect
10. C71×#1557C2semidirect
11. C71×#1551C2semidirect
209. #19526C71direct (N × H)
216. #3254C2×C71semidirect
217. #3246C2×C71semidirect
219. #2985C2×C71semidirect
220. #1698C2×C71semidirect
221. #1686C2×C71semidirect
222. #1684C2×C71semidirect
223. #1668C2×C71semidirect
224. #1571C2×C71semidirect
225. #1557C2×C71semidirect
226. #1551C2×C71semidirect
227. C71#19526direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32542
C71×#32462
C71×#32072
C71×#29852
C71×#16982
C71×#16862
C71×#16842
C71×#16682
C71×#15712
C71×#15572
C71×#15512
#1952671