Finite Group Databasebeta

← #31968124

Subgroups — #31968124

Order 18176.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598895C211yesmaximal, characteristic
2#22597437C211yesmaximal, characteristic, complemented
3#22597416C211yesmaximal, characteristic
4#22587239C212yesmaximal, complemented
5#22587233C212yesmaximal, complemented
6#22587149C212yesmaximal, complemented
7#22587143C212yesmaximal, complemented
8#22587101C211yesmaximal, characteristic, complemented
9#22586814C211yesmaximal, characteristic, complemented
10#22586624C211yesmaximal, characteristic, complemented
11C71×#1496C211yesmaximal, characteristic
12#17986556C2211yescharacteristic
13#17986395C411yescharacteristic
14#17986371C411yescharacteristic
15#17986351C411yescharacteristic
16#17986345C411yescharacteristic
17#17986226C412yes—
18#17986226C412yes—
19#17986127—22no—
20#17986126C2212yes—
21#17986123C2212yes—
22C4×#11765853C2212yes—
23C4×#11765853C2212yes—
24#17986077C2212yes—
25#17986071—22no—
26D4×C71⋊C8C2212yes—
27#17986057—22no—
28#17986056C2211yescharacteristic
29#17986052C2211yescharacteristic
30#17985966—22no—
31C2×C4×C71⋊C8C2211yescharacteristic
32#17985751C2212yes—
33#17985749C2212yes—
34#17985731C2212yes—
35#17985731C2212yes—
36#17985721C2211yescharacteristic
37#17985721C2211yescharacteristic
38C8×C71⋊C8C2211yescharacteristic
39C8×C71⋊C8C2211yescharacteristic
40#532×C71C2211yescharacteristic
41#432×C71C2211yescharacteristic
42#431×C71C2211yescharacteristic
43D4⋊C8×C71C2212yes—
44D4⋊C8×C71C2212yes—
45C2×#11765853C2×C412yes—
46C2×#11765853C2×C412yes—
47C2×#11765849—22no—
48C22×C71⋊C8—22no—
49C22×C71⋊C8C2×C411yescharacteristic
50C22×C71⋊C8C2×C411yescharacteristic
51#14556128C2×C411yescharacteristic
52#14556128C2×C411yescharacteristic
53#14556120—21no—
54#14556107—21no—
55#14556107C2×C412yes—
56C4×Dic142—21no—
57C4×Dic142C2312yes—
58#14556087—21no—
59#14556059C2×C412yes—
60#14556059C2×C412yes—
61C4⋊(C71⋊C8)D412yes—
62C4⋊(C71⋊C8)C2312yes—
63Dic71⋊C8—22no—
64Dic71⋊C8—22no—
65#14556051C2×C412yes—
66#14556051C2×C412yes—
67#14556049C2×C412yes—
68#14556049C2×C412yes—
69C22⋊(C71⋊C8)—22no—
70C22⋊(C71⋊C8)—22no—
71C22⋊(C71⋊C8)—21no—
72C22⋊(C71⋊C8)D412yes—
73#14556039—21no—
74#14556038—22no—
75C4×C71⋊C8C2×C411yescharacteristic
76C4×C71⋊C8C2×C411yescharacteristic
77C4×C71⋊C8C2311yescharacteristic
78C4×C71⋊C8C2311yescharacteristic
79C4×C71⋊C8C2311yescharacteristic
80C4×C71⋊C8C2311yescharacteristic
81C8×Dic71—22no—
82C2×#41×C71—21no—
83C4⋊Q8×C71C2×C411yescharacteristic
84#127×C71C2×C411yescharacteristic
85#124×C71—21no—
86#121×C71—21no—
87#121×C71C2×C412yes—
88C4×D4×C71—21no—
89C4×D4×C71C2312yes—
90C2×C42×C71C2311yescharacteristic
91C4⋊C8×C71—21no—
92C4⋊C8×C71C2312yes—
93C22⋊C8×C71—21no—
94C22⋊C8×C71—21no—
95#97×C71—21no—
96C4×C8×C71C2311yescharacteristic
97C4×C8×C71C2311yescharacteristic
98C2×Dic142—21no—
99C2×Dic142C22×C412yes—
100#11765853—24no—
101#11765853—24no—
102#11765849—82no—
103C2×C71⋊C8—22no—
104C2×C71⋊C8—22no—
105C2×C71⋊C8—22no—
106C2×C71⋊C8—82no—
107C2×C71⋊C8C22×C411yescharacteristic
108C2×C71⋊C8C22×C411yescharacteristic
109C2×C71⋊C8C22×C411yescharacteristic
110C2×C71⋊C8C22×C411yescharacteristic
111C2×C71⋊C8C2×D412yes—
112C2×C71⋊C8C2×D412yes—
113C4⋊Dic71—21no—
114C4⋊Dic71—21no—
115C4⋊Dic71—21no—
116C4⋊Dic71C22×C412yes—
117Dic71⋊C4—22no—
118Dic71⋊C4—22no—
119C22⋊Dic71—22no—
120C4×Dic71—22no—
121#41×C71—82no—
122C2×Q8×C71—21no—
123C2×D4×C71—22no—
124C2×D4×C71—21no—
125C2×D4×C71—21no—
126C2×D4×C71C22×C412yes—
127C22×C4×C71—22no—
128C22×C4×C71C22×C411yescharacteristic
129C22×C4×C71C2×D411yescharacteristic
130C22×C4×C71#4111yescharacteristic
131C2×C8×C71—22no—
132C2×C8×C71—21no—
133C2×C8×C71—21no—
134C4⋊C4×C71—21no—
135C4⋊C4×C71C22×C412yes—
136C22⋊C4×C71—22no—
137C22⋊C4×C71—22no—
138C42×C71C22×C411yescharacteristic
139C42×C71C2×D411yescharacteristic
140C42×C71#4111yescharacteristic
141C42×C71C2411yescharacteristic
142C23×C71—22no—
143C23×C71C4×D411yescharacteristic
144Q8×C71—81no—
145D4×C71—22no—
146D4×C71—22no—
147D4×C71—81no—
148D4×C71—81no—
149D4×C71—82no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—22no—
153C2×C4×C71—22no—
154C2×C4×C71—41no—
155C2×C4×C71—42no—
156C2×C4×C71—42no—
157C2×C4×C71C4×D411yescharacteristic
158C2×C4×C71C4×D411yescharacteristic
159C2×C4×C71C4×D411yescharacteristic
160C2×C4×C71C23×C411yescharacteristic
161C2×C4×C71C22×D411yescharacteristic
162C2×C4×C71C2×#4111yescharacteristic
163C2×Dic71—22no—
164C2×Dic71—22no—
165Dic142—22no—
166Dic142—22no—
167Dic142—81no—
168C8×C71—41no—
169C8×C71—41no—
170C8×C71—42no—
171C71⋊C8—21no—
172C71⋊C8—21no—
173C71⋊C8—22no—
174C71⋊C8—22no—
175C71⋊C8—42no—
176C71⋊C8#13112yes—
177C71⋊C8#13112yes—
178C22×C71—21no—
179C22×C71—21no—
180C22×C71—21no—
181C22×C71—42no—
182C22×C71—42no—
183C22×C71—42no—
184C22×C71C2×C4×D411yescharacteristic
185C4×C71—21no—
186C4×C71—21no—
187C4×C71—21no—
188C4×C71—21no—
189C4×C71—42no—
190C4×C71C2×#13111yescharacteristic
191C4×C71C2×#13111yescharacteristic
192C4×C71#57712yes—
193Dic71—42no—
194Dic71—42no—
195#17971—711nomaximal, Sylow
196C2×C71—41no—
197C2×C71—82no—
198C2×C71#293111yescharacteristic
199C2×C71#294911yescharacteristic
200C2×C71#295711yescharacteristic
201#3167—711no—
202#2985—711no—
203#2981—711no—
204#1592—712no—
205#1590—712no—
206SD16⋊C8—712no—
207C8×SD16—712no—
208#1568—711no—
209#1497—711no—
210#1496D7111yescharacteristic
211#1450—711no—
212C71#1797111yescharacteristic, complemented, Sylow
213#533—711no—
214#532C2×D7111yescharacteristic
215#498—711no—
216#492—711no—
217#487—711no—
218#486—711no—
219#465—712no—
220#465—712no—
221#446—712no—
222#446—1422no—
223C8×Q8—712no—
224C4×SD16—712no—
225C4×SD16—712no—
226#436—712no—
227#435—1422no—
228C8×D4—712no—
229#433—1422no—
230#432—711no—
231#432C2×D7111yescharacteristic
232#431—711no—
233#431C2×D7111yescharacteristic
234#408—1422no—
235C2×C4×C8—711no—
236#335—712no—
237#334—712no—
238Q8⋊C8—712no—
239Q8⋊C8—712no—
240D4⋊C8C2×D7112yes—
241D4⋊C8C2×D7112yes—
242#322—711no—
243#322—711no—
244C82—711no—
245C82—711no—
246C2×#41—21no—
247C2×SD16—712no—
248C2×SD16—712no—
249C2×M16—1422no—
250C22×C8—711no—
251C22×C8—711no—
252C22×C8—1422no—
253C4⋊Q8—711no—
254C4⋊Q8—711no—
255C4⋊Q8C4×D7111yescharacteristic
256#127C4×D7111yescharacteristic
257#125—1421no—
258#124—21no—
259C22⋊Q8—712no—
260C22⋊Q8—1421no—
261#121—21no—
262#121C4×D7112yes—
263C4×Q8—712no—
264C4×Q8—1421no—
265C4×D4—21no—
266C4×D4C22×D7112yes—
267#117—1421no—
268C2×C42C22×D7111yescharacteristic
269#106—712no—
270#106—712no—
271C4⋊C8—21no—
272C4⋊C8—712no—
273C4⋊C8—712no—
274C4⋊C8—1422no—
275C4⋊C8—1422no—
276C4⋊C8C22×D7112yes—
277#103—712no—
278#103—712no—
279#102—712no—
280#102—712no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—712no—
284C22⋊C8—1422no—
285C22⋊C8—1422no—
286C22⋊C8—1421no—
287#97—21no—
288#97—1422no—
289#97—1421no—
290C4×C8—711no—
291C4×C8—711no—
292C4×C8—711no—
293C4×C8—711no—
294C4×C8—711no—
295C4×C8—711no—
296C4×C8—1422no—
297C4×C8C22×D7111yescharacteristic
298C4×C8C22×D7111yescharacteristic
299#41—82no—
300C2×Q8—21no—
301C2×Q8—712no—
302C2×Q8—1421no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×C4×D7112yes—
307C22×C4—22no—
308C22×C4C2×C4×D7111yescharacteristic
309C22×C4D4×D7111yescharacteristic
310C22×C4#1176586411yescharacteristic
311SD16—1424no—
312SD16—1424no—
313M16—5682no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—5682no—
318C2×C8—711no—
319C2×C8—711no—
320C2×C8—711no—
321C2×C8—711no—
322C2×C8—712no—
323C2×C8—712no—
324C2×C8—1422no—
325C2×C8—1422no—
326C2×C8—1422no—
327C4⋊C4—21no—
328C4⋊C4—712no—
329C4⋊C4—1422no—
330C4⋊C4—1422no—
331C4⋊C4—1421no—
332C4⋊C4—1421no—
333C4⋊C4—1421no—
334C4⋊C4C2×C4×D7112yes—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—1422no—
338C42—1422no—
339C42C2×C4×D7111yescharacteristic
340C42D4×D7111yescharacteristic
341C42#1176586411yescharacteristic
342C42C23×D7111yescharacteristic
343C23—22no—
344C23(C22⋊D71)⋊C411yescharacteristic
345Q8—81no—
346Q8—5681no—
347Q8—1422no—
348Q8—1422no—
349D4—22no—
350D4—22no—
351D4—81no—
352D4—81no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—1422no—
362C2×C4—1422no—
363C2×C4(C22⋊D71)⋊C411yescharacteristic
364C2×C4(C22⋊D71)⋊C411yescharacteristic
365C2×C4(C22⋊D71)⋊C411yescharacteristic
366C2×C4C22×C4×D7111yescharacteristic
367C2×C4C2×D4×D7111yescharacteristic
368C2×C4C2×#1176586411yescharacteristic
369C8—41no—
370C8—41no—
371C8—42no—
372C8—2842no—
373C8—712no—
374C8—712no—
375C8—1421no—
376C8—1421no—
377C8—1422no—
378C8—1422no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22C2×(C22⋊D71)⋊C411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—2842no—
392C4—2842no—
393C4#131×D7111yescharacteristic
394C4#131×D7111yescharacteristic
395C4#1798680912yes—
396C2—41no—
397C2—82no—
398C2#2259704811yescharacteristic
399C2#2259720211yescharacteristic
400C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1496C2
socleC22×C71C2×C4×D4
prime radical#532×C71C22
cosocleC42×C71C24
hypercenter#1496D71
hypocenterC71#17971
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22597437C2semidirect
4. #22587239C2semidirect
5. #22587233C2semidirect
6. #22587149C2semidirect
7. #22587143C2semidirect
8. #22587101C2semidirect
9. #22586814C2semidirect
10. #22586624C2semidirect
212. C71#17971semidirect

Maximal subgroups

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

subgroupindex
#225988952
#225974372
#225974162
#225872392
#225872332
#225871492
#225871432
#225871012
#225868142
#225866242
C71×#14962
#1797171