Finite Group Databasebeta

← C71×#19517 #31840389

Subgroups — C71×#19517 #31840389

Order 18176.

580 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3254C211yesmaximal, characteristic
2C71×#3246C211yesmaximal, characteristic
3C71×#3231C211yesmaximal, characteristic, complemented
4C71×#2981C211yesmaximal, characteristic
5C71×#1703C212yesmaximal, complemented
6C71×#1693C212yesmaximal, complemented
7C71×#1679C212yesmaximal, complemented
8C71×#1671C212yesmaximal, complemented
9C71×#1570C211yesmaximal, characteristic, complemented
10C71×#1559C211yesmaximal, characteristic, complemented
11C71×#1552C211yesmaximal, characteristic, complemented
12#533×C71C2211yescharacteristic
13#533×C71C2211yescharacteristic
14#532×C71C2211yescharacteristic
15#500×C71C2211yescharacteristic
16#498×C71C2211yescharacteristic
17#487×C71C2211yescharacteristic
18#487×C71C2211yescharacteristic
19#485×C71—22no—
20#479×C71—22no—
21#477×C71—21no—
22#476×C71C2212yes—
23#475×C71—21no—
24#474×C71C2212yes—
25#468×C71—22no—
26#465×C71—22no—
27#461×C71C2212yes—
28#461×C71C2212yes—
29#460×C71—21no—
30#460×C71—21no—
31#436×C71C2212yes—
32#435×C71—22no—
33C8×D4×C71C2212yes—
34#432×C71C2211yescharacteristic
35#431×C71C2211yescharacteristic
36#423×C71C2211yescharacteristic
37C2×C4⋊C8×C71C2211yescharacteristic
38#408×C71—22no—
39#334×C71C2212yes—
40#334×C71C2212yes—
41#333×C71C2212yes—
42#332×C71C2212yes—
43#332×C71C2212yes—
44#332×C71C2212yes—
45C2×#41×C71—21no—
46C2×SD16×C71—22no—
47C2×SD16×C71—22no—
48C2×M16×C71—21no—
49C2×M16×C71—22no—
50C22×C8×C71—21no—
51C22×C8×C71—22no—
52C4⋊Q8×C71D411yescharacteristic
53C4⋊Q8×C71C2311yescharacteristic
54C4⋊Q8×C71C2311yescharacteristic
55C4⋊Q8×C71C2311yescharacteristic
56C4⋊Q8×C71C2311yescharacteristic
57#127×C71D411yescharacteristic
58#125×C71—21no—
59#125×C71—21no—
60#124×C71—21no—
61C22⋊Q8×C71—21no—
62C22⋊Q8×C71—21no—
63C22⋊Q8×C71—21no—
64C22⋊Q8×C71—21no—
65#121×C71—21no—
66#121×C71D412yes—
67C4×Q8×C71—21no—
68C4×Q8×C71—21no—
69C4×Q8×C71—21no—
70C4×Q8×C71—21no—
71C4×D4×C71—21no—
72C4×D4×C71C2312yes—
73#117×C71—21no—
74#117×C71—21no—
75C2×C42×C71C2311yescharacteristic
76#107×C71—22no—
77#107×C71—22no—
78#106×C71—22no—
79#106×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#103×C71—22no—
90#102×C71—22no—
91#102×C71—22no—
92C22⋊C8×C71—21no—
93C22⋊C8×C71—22no—
94C22⋊C8×C71—22no—
95C22⋊C8×C71D412yes—
96#97×C71—21no—
97C4×C8×C71C2311yescharacteristic
98C4×C8×C71C2311yescharacteristic
99#41×C71—82no—
100C2×Q8×C71—21no—
101C2×Q8×C71—21no—
102C2×Q8×C71—21no—
103C2×Q8×C71—21no—
104C2×Q8×C71—21no—
105C2×D4×C71—22no—
106C2×D4×C71—21no—
107C2×D4×C71—21no—
108C2×D4×C71C2×D412yes—
109C22×C4×C71—22no—
110C22×C4×C71C2×D411yescharacteristic
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71#4111yescharacteristic
113SD16×C71—82no—
114SD16×C71—82no—
115M16×C71—82no—
116M16×C71—81no—
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71—22no—
120C2×C8×C71—82no—
121C2×C8×C71—81no—
122C2×C8×C71C2×D412yes—
123C2×C8×C71C2×D412yes—
124C4⋊C4×C71—21no—
125C4⋊C4×C71—21no—
126C4⋊C4×C71—21no—
127C4⋊C4×C71—21no—
128C4⋊C4×C71—21no—
129C4⋊C4×C71—21no—
130C4⋊C4×C71—21no—
131C4⋊C4×C71—21no—
132C4⋊C4×C71—21no—
133C4⋊C4×C71—41no—
134C4⋊C4×C71—41no—
135C4⋊C4×C71—41no—
136C4⋊C4×C71—41no—
137C4⋊C4×C71C2×D412yes—
138C22⋊C4×C71—22no—
139C22⋊C4×C71—22no—
140C22⋊C4×C71—41no—
141C22⋊C4×C71—41no—
142C42×C71—41no—
143C42×C71—41no—
144C42×C71C2×D411yescharacteristic
145C42×C71C2×D411yescharacteristic
146C42×C71#4111yescharacteristic
147C42×C71C2411yescharacteristic
148C23×C71—22no—
149C23×C71#12111yescharacteristic
150Q8×C71—41no—
151Q8×C71—41no—
152Q8×C71—42no—
153Q8×C71—41no—
154Q8×C71—41no—
155Q8×C71—81no—
156Q8×C71—81no—
157D4×C71—22no—
158D4×C71—22no—
159D4×C71—81no—
160D4×C71—81no—
161D4×C71—82no—
162C2×C4×C71—21no—
163C2×C4×C71—21no—
164C2×C4×C71—22no—
165C2×C4×C71—22no—
166C2×C4×C71—41no—
167C2×C4×C71—42no—
168C2×C4×C71—42no—
169C2×C4×C71—41no—
170C2×C4×C71—41no—
171C2×C4×C71—41no—
172C2×C4×C71—41no—
173C2×C4×C71#12111yescharacteristic
174C2×C4×C71#12111yescharacteristic
175C2×C4×C71#12111yescharacteristic
176C2×C4×C71C22×D411yescharacteristic
177C2×C4×C71C22×D411yescharacteristic
178C2×C4×C71C2×#4111yescharacteristic
179C8×C71—22no—
180C8×C71—22no—
181C8×C71—42no—
182C8×C71—42no—
183C8×C71—42no—
184C22×C71—21no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—42no—
188C22×C71—42no—
189C22×C71—42no—
190C22×C71C2×#12111yescharacteristic
191C4×C71—21no—
192C4×C71—21no—
193C4×C71—21no—
194C4×C71—21no—
195C4×C71—42no—
196C4×C71—81no—
197C4×C71—81no—
198C4×C71—81no—
199C4×C71—81no—
200C4×C71#57712yes—
201C4×C71#57711yescharacteristic
202C4×C71#57711yescharacteristic
203#19517C7111yesmaximal, characteristic, direct product factor, Sylow
204C2×C71—41no—
205C2×C71—82no—
206C2×C71#304211yescharacteristic
207C2×C71#306011yescharacteristic
208C2×C71#306911yescharacteristic
209#3254C2×C7111yescharacteristic
210#3246C2×C7111yescharacteristic
211#3231C2×C7111yescharacteristic, complemented
212#2981C2×C7111yescharacteristic
213#1703C2×C7112yescomplemented
214#1693C2×C7112yescomplemented
215#1679C2×C7112yescomplemented
216#1671C2×C7112yescomplemented
217#1570C2×C7111yescharacteristic, complemented
218#1559C2×C7111yescharacteristic, complemented
219#1552C2×C7111yescharacteristic, complemented
220C71#1951711yescharacteristic, direct product factor, Sylow
221#533C22×C7111yescharacteristic
222#533C22×C7111yescharacteristic
223#532C22×C7111yescharacteristic
224#500C22×C7111yescharacteristic
225#498C22×C7111yescharacteristic
226#487C22×C7111yescharacteristic
227#487C22×C7111yescharacteristic
228#485—22no—
229#479—22no—
230#477—21no—
231#476C22×C7112yes—
232#475—21no—
233#474C22×C7112yes—
234#468—22no—
235#465—22no—
236#461C22×C7112yes—
237#461C22×C7112yes—
238#460—21no—
239#460—21no—
240#436C22×C7112yes—
241#435—22no—
242C8×D4C22×C7112yes—
243#432C22×C7111yescharacteristic
244#431C22×C7111yescharacteristic
245#423C22×C7111yescharacteristic
246C2×C4⋊C8C22×C7111yescharacteristic
247#408—22no—
248#334C22×C7112yes—
249#334C22×C7112yes—
250#333C22×C7112yes—
251#332C22×C7112yes—
252#332C22×C7112yes—
253#332C22×C7112yes—
254C2×#41—21no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×M16—21no—
258C2×M16—22no—
259C22×C8—21no—
260C22×C8—22no—
261C4⋊Q8D4×C7111yescharacteristic
262C4⋊Q8C23×C7111yescharacteristic
263C4⋊Q8C23×C7111yescharacteristic
264C4⋊Q8C23×C7111yescharacteristic
265C4⋊Q8C23×C7111yescharacteristic
266#127D4×C7111yescharacteristic
267#125—21no—
268#125—21no—
269#124—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274#121—21no—
275#121D4×C7112yes—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×D4—21no—
281C4×D4C23×C7112yes—
282#117—21no—
283#117—21no—
284C2×C42C23×C7111yescharacteristic
285#107—22no—
286#107—22no—
287#106—22no—
288#106—22no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8D4×C7112yes—
292C4⋊C8C23×C7112yes—
293C4⋊C8C23×C7112yes—
294C4⋊C8C23×C7112yes—
295#103—22no—
296#103—22no—
297#103—22no—
298#103—22no—
299#102—22no—
300#102—22no—
301C22⋊C8—21no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8D4×C7112yes—
305#97—21no—
306C4×C8C23×C7111yescharacteristic
307C4×C8C23×C7111yescharacteristic
308#41—82no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C2×D4×C7112yes—
318C22×C4—22no—
319C22×C4C2×D4×C7111yescharacteristic
320C22×C4C2×D4×C7111yescharacteristic
321C22×C4#41×C7111yescharacteristic
322SD16—82no—
323SD16—82no—
324M16—82no—
325M16—81no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—82no—
330C2×C8—81no—
331C2×C8C2×D4×C7112yes—
332C2×C8C2×D4×C7112yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4C2×D4×C7112yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C42—41no—
352C42—41no—
353C42C2×D4×C7111yescharacteristic
354C42C2×D4×C7111yescharacteristic
355C42#41×C7111yescharacteristic
356C42C24×C7111yescharacteristic
357C23—22no—
358C23#121×C7111yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—42no—
362Q8—41no—
363Q8—41no—
364Q8—81no—
365Q8—81no—
366D4—22no—
367D4—22no—
368D4—81no—
369D4—81no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4#121×C7111yescharacteristic
383C2×C4#121×C7111yescharacteristic
384C2×C4#121×C7111yescharacteristic
385C2×C4C22×D4×C7111yescharacteristic
386C2×C4C22×D4×C7111yescharacteristic
387C2×C4C2×#41×C7111yescharacteristic
388C8—22no—
389C8—22no—
390C8—42no—
391C8—42no—
392C8—42no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22C2×#121×C7111yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4#577×C7112yes—
410C4#577×C7111yescharacteristic
411C4#577×C7111yescharacteristic
412C2—41no—
413C2—82no—
414C2C71×#304211yescharacteristic
415C2C71×#306011yescharacteristic
416C2C71×#306911yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radical#532×C71C22
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. C71×#3231C2semidirect
5. C71×#1703C2semidirect
6. C71×#1693C2semidirect
7. C71×#1679C2semidirect
8. C71×#1671C2semidirect
9. C71×#1570C2semidirect
10. C71×#1559C2semidirect
11. C71×#1552C2semidirect
203. #19517C71direct (N × H)
211. #3231C2×C71semidirect
213. #1703C2×C71semidirect
214. #1693C2×C71semidirect
215. #1679C2×C71semidirect
216. #1671C2×C71semidirect
217. #1570C2×C71semidirect
218. #1559C2×C71semidirect
219. #1552C2×C71semidirect
220. C71#19517direct (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×#32312
C71×#29812
C71×#17032
C71×#16932
C71×#16792
C71×#16712
C71×#15702
C71×#15592
C71×#15522
#1951771