Finite Group Databasebeta

← #31967291

Subgroups — #31967291

Order 18176.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597423C211yesmaximal, characteristic, complemented
2#22597381C211yesmaximal, characteristic
3C2×C4×#11765853C211yesmaximal, characteristic
4#22587239C214yesmaximal, complemented
5#22587233C214yesmaximal, complemented
6#22587086C211yesmaximal, characteristic, complemented
7#22586736C211yesmaximal, characteristic, complemented
8#22586624C211yesmaximal, characteristic, complemented
9C71×#1478C211yesmaximal, characteristic
10C22×#11765853C411yescharacteristic
11C2×C4×Dic142C2211yescharacteristic
12#17986129—22no—
13#17986126C2214yescomplemented
14C4×#11765853C412yes—
15C4×#11765853C412yes—
16C4×#11765853C2212yes—
17C4×#11765853C2212yes—
18#17986077—22no—
19#17986077C2214yes—
20#17986053—22no—
21#17986052C2211yescharacteristic
22#17986052C2211yescharacteristic
23C2×#14556059C411yescharacteristic
24C2×#14556051C411yescharacteristic
25C2×#14556049C411yescharacteristic
26#17985963—22no—
27C2×C4×C71⋊C8C2211yescharacteristic
28#17985751C2212yescomplemented
29#17985749C2212yescomplemented
30#17985731C2214yescomplemented
31#17985721C2212yescomplemented
32C8×C71⋊C8C2212yescomplemented
33C2×C4×D4×C71C2211yescharacteristic
34#431×C71C2211yescharacteristic
35#431×C71C2211yescharacteristic
36D4⋊C8×C71C2214yes—
37C22×Dic142C2×C411yescharacteristic
38C2×#11765853—24no—
39C2×#11765853C2×C412yes—
40C2×#11765853C2×C412yes—
41C2×#11765849—24no—
42C22×C71⋊C8C2×C411yescharacteristic
43C22×C71⋊C8C2×C411yescharacteristic
44C4×Dic142—21no—
45C4×Dic142—21no—
46C4×Dic142C2×C412yes—
47C4×Dic142C2312yes—
48C2×C4⋊Dic71C2×C411yescharacteristic
49C2×Dic71⋊C4—21no—
50C2×C4×Dic71—21no—
51#14556059C2×C412yes—
52#14556059C2×C412yes—
53C4⋊(C71⋊C8)D412yes—
54C4⋊(C71⋊C8)C2312yes—
55Dic71⋊C8—24no—
56#14556051C2×C412yes—
57#14556051C2×C412yes—
58#14556049C2×C412yes—
59#14556049C2×C412yes—
60C22⋊(C71⋊C8)—24no—
61C22⋊(C71⋊C8)—21no—
62C22⋊(C71⋊C8)D412yes—
63#14556038—24no—
64C4×C71⋊C8—21no—
65C4×C71⋊C8C2×C411yescharacteristic
66C4×C71⋊C8C2×C411yescharacteristic
67C4×C71⋊C8C2311yescharacteristic
68C4×C71⋊C8C2311yescharacteristic
69C4×C71⋊C8C2312yes—
70C22×D4×C71C2×C411yescharacteristic
71C23×C4×C71—21no—
72C4×D4×C71—21no—
73C4×D4×C71—21no—
74C4×D4×C71C2×C412yes—
75C4×D4×C71C2312yes—
76C2×C4⋊C4×C71C2×C411yescharacteristic
77C2×C22⋊C4×C71—21no—
78C2×C42×C71C2311yescharacteristic
79C4⋊C8×C71—21no—
80C4⋊C8×C71C2312yes—
81C22⋊C8×C71—21no—
82C22⋊C8×C71—21no—
83C4×C8×C71—21no—
84C4×C8×C71C2312yes—
85C2×Dic142—22no—
86C2×Dic142—22no—
87C2×Dic142—21no—
88C2×Dic142C22×C412yes—
89C22×Dic71—21no—
90C22×Dic71—21no—
91#11765853—24no—
92#11765853—24no—
93#11765849—84no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8—22no—
96C2×C71⋊C8M1614yescomplemented
97C2×C71⋊C8C22×C411yescharacteristic
98C2×C71⋊C8C22×C411yescharacteristic
99C2×C71⋊C8C22×C411yescharacteristic
100C2×C71⋊C8C22×C411yescharacteristic
101C2×C71⋊C8C2×D414yes—
102C4⋊Dic71—21no—
103C4⋊Dic71C22×C412yes—
104Dic71⋊C4—22no—
105Dic71⋊C4—22no—
106C4×Dic71—22no—
107C4×Dic71—22no—
108C24×C71—21no—
109C2×D4×C71—21no—
110C2×D4×C71—22no—
111C2×D4×C71—22no—
112C2×D4×C71C22×C412yes—
113C22×C4×C71—21no—
114C22×C4×C71—22no—
115C22×C4×C71—22no—
116C22×C4×C71—82no—
117C22×C4×C71C22×C411yescharacteristic
118C22×C4×C71C2×D411yescharacteristic
119C22×C4×C71#4111yescharacteristic
120C2×C8×C71—22no—
121C2×C8×C71—22no—
122C4⋊C4×C71—21no—
123C4⋊C4×C71C22×C412yes—
124C22⋊C4×C71—22no—
125C22⋊C4×C71—22no—
126C42×C71C22×C411yescharacteristic
127C42×C71C2×D411yescharacteristic
128C42×C71#4111yescharacteristic
129C42×C71C2411yescharacteristic
130C23×C71—22no—
131C23×C71—42no—
132C23×C71—81no—
133C23×C71—81no—
134C23×C71C4×D411yescharacteristic
135D4×C71—22no—
136D4×C71—22no—
137D4×C71—42no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—41no—
141C2×C4×C71—44no—
142C2×C4×C71—82no—
143C2×C4×C71—81no—
144C2×C4×C71C4×D411yescharacteristic
145C2×C4×C71C4×D411yescharacteristic
146C2×C4×C71C4×D411yescharacteristic
147C2×C4×C71C2×M1612yes—
148C2×C4×C71#13112yes—
149C2×C4×C71C23×C411yescharacteristic
150C2×C4×C71C22×D411yescharacteristic
151C2×C4×C71C2×#4111yescharacteristic
152C2×Dic71—22no—
153C2×Dic71—22no—
154C2×Dic71—42no—
155C2×Dic71—81no—
156Dic142—22no—
157Dic142—22no—
158Dic142—42no—
159C8×C71—42no—
160C8×C71—42no—
161C71⋊C8—21no—
162C71⋊C8—21no—
163C71⋊C8—24no—
164C71⋊C8—42no—
165C71⋊C8C2×M1612yes—
166C71⋊C8C2×M1612yes—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—42no—
170C22×C71—42no—
171C22×C71—42no—
172C22×C71—84no—
173C22×C71#43612yes—
174C22×C71C2×C4×D411yescharacteristic
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—42no—
179C4×C71#43612yes—
180C4×C71C22×M1611yescharacteristic
181C4×C71C2×#13111yescharacteristic
182C4×C71#57712yes—
183Dic71—42no—
184Dic71—42no—
185#17896—711nomaximal, Sylow
186C2×C71—22no—
187C2×C71—82no—
188C2×C71C2×#43611yescharacteristic
189C2×C71#294911yescharacteristic
190C2×C71#295711yescharacteristic
191#2983—711no—
192#2975—711no—
193C2×C4×SD16—711no—
194#1592—714no—
195#1590—714no—
196#1564—711no—
197#1479—711no—
198#1478D7111yescharacteristic
199#1450—711no—
200C71#1789611yescharacteristic, complemented, Sylow
201C22×SD16—711no—
202C2×C4×Q8—711no—
203C2×C4×D4C2×D7111yescharacteristic
204#446—714no—
205#446—1422no—
206C4×SD16—712no—
207C4×SD16—712no—
208C4×SD16—712no—
209C4×SD16—712no—
210#436—714no—
211#436—1422no—
212#432—1422no—
213#431—711no—
214#431—711no—
215#431C2×D7111yescharacteristic
216#431C2×D7111yescharacteristic
217C2×#106—711no—
218C2×#103—711no—
219C2×#102—711no—
220#407—1422no—
221C2×C4×C8—711no—
222#335—712no—
223#334—712no—
224Q8⋊C8—714no—
225D4⋊C8C2×D7114yes—
226#322—712no—
227C82—712no—
228C22×Q8—711no—
229C22×D4C4×D7111yescharacteristic
230C23×C4—21no—
231C2×SD16—712no—
232C2×SD16—712no—
233C2×SD16—1424no—
234C2×M16—1424no—
235C22×C8—711no—
236C22×C8—711no—
237C4×Q8—712no—
238C4×Q8—712no—
239C4×Q8—1421no—
240C4×Q8—1421no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C4×D7112yes—
244C4×D4C22×D7112yes—
245C2×C4⋊C4—711no—
246C2×C4⋊C4—1421no—
247C2×C4⋊C4C4×D7111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42—1421no—
250C2×C42C22×D7111yescharacteristic
251#106—712no—
252#106—712no—
253C4⋊C8—21no—
254C4⋊C8—712no—
255C4⋊C8—712no—
256C4⋊C8—1424no—
257C4⋊C8C22×D7112yes—
258#103—712no—
259#103—712no—
260#102—712no—
261#102—712no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—712no—
265C22⋊C8—1424no—
266C22⋊C8—1421no—
267#97—1424no—
268C4×C8—21no—
269C4×C8—711no—
270C4×C8—711no—
271C4×C8—711no—
272C4×C8—712no—
273C4×C8—711no—
274C4×C8—1421no—
275C4×C8C22×D7112yes—
276C24—21no—
277C2×Q8—712no—
278C2×Q8—1422no—
279C2×Q8—1422no—
280C2×Q8—1421no—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4C2×C4×D7112yes—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—82no—
289C22×C4—1421no—
290C22×C4—1421no—
291C22×C4C2×C4×D7111yescharacteristic
292C22×C4D4×D7111yescharacteristic
293C22×C4#1176586411yescharacteristic
294SD16—1424no—
295SD16—1424no—
296M16—5684no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—711no—
300C2×C8—711no—
301C2×C8—714no—
302C2×C8—711no—
303C2×C8—711no—
304C2×C8—714no—
305C2×C8—1422no—
306C2×C8—1422no—
307C4⋊C4—21no—
308C4⋊C4—712no—
309C4⋊C4—1422no—
310C4⋊C4—1422no—
311C4⋊C4—1421no—
312C4⋊C4C2×C4×D7112yes—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C42—1422no—
316C42—1422no—
317C42C2×C4×D7111yescharacteristic
318C42D4×D7111yescharacteristic
319C42#1176586411yescharacteristic
320C42C23×D7111yescharacteristic
321C23—22no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23(C22⋊D71)⋊C411yescharacteristic
326Q8—2842no—
327Q8—1422no—
328Q8—1422no—
329D4—22no—
330D4—22no—
331D4—42no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—2842no—
339C2×C4—5681no—
340C2×C4—1422no—
341C2×C4—1422no—
342C2×C4(C22⋊D71)⋊C411yescharacteristic
343C2×C4(C22⋊D71)⋊C411yescharacteristic
344C2×C4(C22⋊D71)⋊C411yescharacteristic
345C2×C4M16×D7112yes—
346C2×C4#1455613412yes—
347C2×C4C22×C4×D7111yescharacteristic
348C2×C4C2×D4×D7111yescharacteristic
349C2×C4C2×#1176586411yescharacteristic
350C8—42no—
351C8—42no—
352C8—2842no—
353C8—712no—
354C8—712no—
355C8—1421no—
356C8—1421no—
357C8—1424no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22#1798607412yes—
365C22C2×(C22⋊D71)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—2842no—
371C4—2842no—
372C4#1798607412yes—
373C4C2×M16×D7111yescharacteristic
374C4C2×#1455613411yescharacteristic
375C4#1798680912yes—
376C2—22no—
377C2—82no—
378C2C2×#1798607411yescharacteristic
379C2#2259720211yescharacteristic
380C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1478C2
socleC22×C71C2×C4×D4
prime radicalC22×D4×C71C2×C4
cosocleC42×C71C24
hypercenter#1478D71
hypocenterC71#17896
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
1. #22597423C2semidirect
4. #22587239C2semidirect
5. #22587233C2semidirect
6. #22587086C2semidirect
7. #22586736C2semidirect
8. #22586624C2semidirect
13. #17986126C22semidirect
28. #17985751C22semidirect
29. #17985749C22semidirect
30. #17985731C22semidirect
31. #17985721C22semidirect
32. C8×C71⋊C8C22semidirect
96. C2×C71⋊C8M16semidirect
200. C71#17896semidirect

Maximal subgroups

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

subgroupindex
#225974232
#225973812
C2×C4×#117658532
#225872392
#225872332
#225870862
#225867362
#225866242
C71×#14782
#1789671