Finite Group Databasebeta

← C71×#19507 #31840379

Subgroups — C71×#19507 #31840379

Order 18176.

588 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3235C211yesmaximal, characteristic
2C71×#3226C211yesmaximal, characteristic
3C71×#3209C211yesmaximal, characteristic, complemented
4C71×#2981C211yesmaximal, characteristic
5C71×#1703C212yesmaximal, complemented
6C71×#1689C212yesmaximal, complemented
7C71×#1683C212yesmaximal, complemented
8C71×#1669C212yesmaximal, complemented
9C71×#1572C211yesmaximal, characteristic, complemented
10C71×#1556C211yesmaximal, characteristic, complemented
11C71×#1554C211yesmaximal, characteristic, complemented
12#533×C71C2211yescharacteristic
13#532×C71C2211yescharacteristic
14C2×C4⋊Q8×C71C2211yescharacteristic
15#501×C71—21no—
16#500×C71—21no—
17#494×C71C2211yescharacteristic
18#494×C71C2211yescharacteristic
19#490×C71—21no—
20#487×C71C2211yescharacteristic
21#487×C71C2211yescharacteristic
22#486×C71—21no—
23#482×C71—22no—
24#480×C71—22no—
25#476×C71C2212yes—
26#474×C71C2212yes—
27#474×C71C2212yes—
28#474×C71C2212yes—
29#462×C71—22no—
30#451×C71—22no—
31#436×C71C2212yes—
32#435×C71—22no—
33C8×D4×C71C2212yes—
34#432×C71C2211yescharacteristic
35#432×C71C2211yescharacteristic
36#431×C71C2211yescharacteristic
37#431×C71C2211yescharacteristic
38#408×C71—22no—
39#335×C71C2212yes—
40#334×C71C2212yes—
41#333×C71C2212yes—
42#332×C71C2212yes—
43#332×C71C2212yes—
44#332×C71C2212yes—
45C2×#41×C71—21no—
46C22×Q8×C71—21no—
47C2×Q16×C71—22no—
48C2×Q16×C71—22no—
49C2×M16×C71—22no—
50C22×C8×C71—22no—
51C4⋊Q8×C71—21no—
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71D411yescharacteristic
55C4⋊Q8×C71C2311yescharacteristic
56C4⋊Q8×C71C2311yescharacteristic
57C4⋊Q8×C71C2311yescharacteristic
58C4⋊Q8×C71C2311yescharacteristic
59#127×C71D411yescharacteristic
60#125×C71—21no—
61#124×C71—21no—
62C22⋊Q8×C71—21no—
63C22⋊Q8×C71—21no—
64#121×C71—21no—
65#121×C71D412yes—
66C4×Q8×C71—21no—
67C4×Q8×C71—21no—
68C4×D4×C71—21no—
69C4×D4×C71C2312yes—
70#117×C71—21no—
71C2×C4⋊C4×C71—21no—
72C2×C4⋊C4×C71—21no—
73C2×C42×C71C2311yescharacteristic
74#107×C71—22no—
75#107×C71—22no—
76#107×C71—22no—
77#106×C71—22no—
78C4⋊C8×C71D412yes—
79C4⋊C8×C71C2312yes—
80C4⋊C8×C71C2312yes—
81C4⋊C8×C71C2312yes—
82#103×C71—22no—
83#103×C71—22no—
84#102×C71—22no—
85#102×C71—22no—
86#102×C71—22no—
87#102×C71—22no—
88C22⋊C8×C71—21no—
89C22⋊C8×C71—21no—
90C22⋊C8×C71—21no—
91C22⋊C8×C71—22no—
92C22⋊C8×C71—22no—
93C22⋊C8×C71D412yes—
94#97×C71—21no—
95#97×C71—21no—
96C4×C8×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×Q8×C71—41no—
106C2×Q8×C71—82no—
107C2×D4×C71—22no—
108C2×D4×C71—21no—
109C2×D4×C71—21no—
110C2×D4×C71C2×D412yes—
111C22×C4×C71—22no—
112C22×C4×C71—41no—
113C22×C4×C71C2×D411yescharacteristic
114C22×C4×C71C2×D411yescharacteristic
115C22×C4×C71#4111yescharacteristic
116Q16×C71—82no—
117Q16×C71—82no—
118M16×C71—82no—
119C2×C8×C71—22no—
120C2×C8×C71—22no—
121C2×C8×C71—22no—
122C2×C8×C71—82no—
123C2×C8×C71C2×D412yes—
124C2×C8×C71C2×D412yes—
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—21no—
134C4⋊C4×C71—41no—
135C4⋊C4×C71—41no—
136C4⋊C4×C71—41no—
137C4⋊C4×C71—41no—
138C4⋊C4×C71C2×D412yes—
139C22⋊C4×C71—22no—
140C22⋊C4×C71—22no—
141C22⋊C4×C71—41no—
142C42×C71—41no—
143C42×C71C2×D411yescharacteristic
144C42×C71C2×D411yescharacteristic
145C42×C71#4111yescharacteristic
146C42×C71C2411yescharacteristic
147C23×C71—22no—
148C23×C71#12111yescharacteristic
149Q8×C71—42no—
150Q8×C71—42no—
151Q8×C71—42no—
152Q8×C71—81no—
153Q8×C71—81no—
154Q8×C71—82no—
155D4×C71—24no—
156D4×C71—81no—
157D4×C71—81no—
158D4×C71—82no—
159C2×C4×C71—22no—
160C2×C4×C71—22no—
161C2×C4×C71—22no—
162C2×C4×C71—41no—
163C2×C4×C71—42no—
164C2×C4×C71—42no—
165C2×C4×C71—41no—
166C2×C4×C71—41no—
167C2×C4×C71—41no—
168C2×C4×C71—41no—
169C2×C4×C71—82no—
170C2×C4×C71#12111yescharacteristic
171C2×C4×C71#12111yescharacteristic
172C2×C4×C71#12111yescharacteristic
173C2×C4×C71C22×D411yescharacteristic
174C2×C4×C71C22×D411yescharacteristic
175C2×C4×C71C2×#4111yescharacteristic
176C8×C71—22no—
177C8×C71—22no—
178C8×C71—42no—
179C8×C71—42no—
180C8×C71—42no—
181C22×C71—21no—
182C22×C71—21no—
183C22×C71—21no—
184C22×C71—42no—
185C22×C71—42no—
186C22×C71—42no—
187C22×C71C2×#12111yescharacteristic
188C4×C71—21no—
189C4×C71—21no—
190C4×C71—21no—
191C4×C71—21no—
192C4×C71—42no—
193C4×C71—81no—
194C4×C71—81no—
195C4×C71—81no—
196C4×C71—81no—
197C4×C71#57612yes—
198C4×C71#57812yes—
199#19507C7111yesmaximal, characteristic, direct product factor, Sylow
200C2×C71—41no—
201C2×C71—82no—
202C2×C71#304111yescharacteristic
203C2×C71#305911yescharacteristic
204C2×C71#306911yescharacteristic
205#3235C2×C7111yescharacteristic
206#3226C2×C7111yescharacteristic
207#3209C2×C7111yescharacteristic, complemented
208#2981C2×C7111yescharacteristic
209#1703C2×C7112yescomplemented
210#1689C2×C7112yescomplemented
211#1683C2×C7112yescomplemented
212#1669C2×C7112yescomplemented
213#1572C2×C7111yescharacteristic, complemented
214#1556C2×C7111yescharacteristic, complemented
215#1554C2×C7111yescharacteristic, complemented
216C71#1950711yescharacteristic, direct product factor, Sylow
217#533C22×C7111yescharacteristic
218#532C22×C7111yescharacteristic
219C2×C4⋊Q8C22×C7111yescharacteristic
220#501—21no—
221#500—21no—
222#494C22×C7111yescharacteristic
223#494C22×C7111yescharacteristic
224#490—21no—
225#487C22×C7111yescharacteristic
226#487C22×C7111yescharacteristic
227#486—21no—
228#482—22no—
229#480—22no—
230#476C22×C7112yes—
231#474C22×C7112yes—
232#474C22×C7112yes—
233#474C22×C7112yes—
234#462—22no—
235#451—22no—
236#436C22×C7112yes—
237#435—22no—
238C8×D4C22×C7112yes—
239#432C22×C7111yescharacteristic
240#432C22×C7111yescharacteristic
241#431C22×C7111yescharacteristic
242#431C22×C7111yescharacteristic
243#408—22no—
244#335C22×C7112yes—
245#334C22×C7112yes—
246#333C22×C7112yes—
247#332C22×C7112yes—
248#332C22×C7112yes—
249#332C22×C7112yes—
250C2×#41—21no—
251C22×Q8—21no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×M16—22no—
255C22×C8—22no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8D4×C7111yescharacteristic
260C4⋊Q8C23×C7111yescharacteristic
261C4⋊Q8C23×C7111yescharacteristic
262C4⋊Q8C23×C7111yescharacteristic
263C4⋊Q8C23×C7111yescharacteristic
264#127D4×C7111yescharacteristic
265#125—21no—
266#124—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269#121—21no—
270#121D4×C7112yes—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×D4—21no—
274C4×D4C23×C7112yes—
275#117—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C42C23×C7111yescharacteristic
279#107—22no—
280#107—22no—
281#107—22no—
282#106—22no—
283C4⋊C8D4×C7112yes—
284C4⋊C8C23×C7112yes—
285C4⋊C8C23×C7112yes—
286C4⋊C8C23×C7112yes—
287#103—22no—
288#103—22no—
289#102—22no—
290#102—22no—
291#102—22no—
292#102—22no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8D4×C7112yes—
299#97—21no—
300#97—21no—
301C4×C8—21no—
302C4×C8C23×C7111yescharacteristic
303C4×C8C23×C7111yescharacteristic
304#41—82no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—41no—
311C2×Q8—82no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4C2×D4×C7112yes—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4C2×D4×C7111yescharacteristic
319C22×C4C2×D4×C7111yescharacteristic
320C22×C4#41×C7111yescharacteristic
321Q16—82no—
322Q16—82no—
323M16—82no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—82no—
328C2×C8C2×D4×C7112yes—
329C2×C8C2×D4×C7112yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4C2×D4×C7112yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C42—41no—
348C42C2×D4×C7111yescharacteristic
349C42C2×D4×C7111yescharacteristic
350C42#41×C7111yescharacteristic
351C42C24×C7111yescharacteristic
352C23—22no—
353C23#121×C7111yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—42no—
357Q8—81no—
358Q8—81no—
359Q8—82no—
360D4—24no—
361D4—81no—
362D4—81no—
363D4—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4#121×C7111yescharacteristic
376C2×C4#121×C7111yescharacteristic
377C2×C4#121×C7111yescharacteristic
378C2×C4C22×D4×C7111yescharacteristic
379C2×C4C22×D4×C7111yescharacteristic
380C2×C4C2×#41×C7111yescharacteristic
381C8—22no—
382C8—22no—
383C8—42no—
384C8—42no—
385C8—42no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22C2×#121×C7111yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#576×C7112yes—
403C4#578×C7112yes—
404C2—41no—
405C2—82no—
406C2C71×#304111yescharacteristic
407C2C71×#305911yescharacteristic
408C2C71×#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×#3209C2semidirect
5. C71×#1703C2semidirect
6. C71×#1689C2semidirect
7. C71×#1683C2semidirect
8. C71×#1669C2semidirect
9. C71×#1572C2semidirect
10. C71×#1556C2semidirect
11. C71×#1554C2semidirect
199. #19507C71direct (N × H)
207. #3209C2×C71semidirect
209. #1703C2×C71semidirect
210. #1689C2×C71semidirect
211. #1683C2×C71semidirect
212. #1669C2×C71semidirect
213. #1572C2×C71semidirect
214. #1556C2×C71semidirect
215. #1554C2×C71semidirect
216. C71#19507direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32352
C71×#32262
C71×#32092
C71×#29812
C71×#17032
C71×#16892
C71×#16832
C71×#16692
C71×#15722
C71×#15562
C71×#15542
#1950771