Finite Group Databasebeta

← #31967374

Subgroups — #31967374

Order 18176.

698 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#22598344C211yesmaximal, characteristic
2#22597275C211yesmaximal, characteristic, complemented
3C2×#17986064C211yesmaximal, characteristic
4#22587200C214yesmaximal, complemented
5#22587155C214yesmaximal, complemented
6#22587114C211yesmaximal, characteristic, complemented
7#22586766C211yesmaximal, characteristic
8#22586633C211yesmaximal, characteristic, complemented
9C71×#1481C211yesmaximal, characteristic, complemented
10C22×C8×D71—21no—
11C2×C4×D284C2211yescharacteristic
12#17986327C412yes—
13#17986286C412yes—
14#17986201C412yes—
15#17986149C412yes—
16#17986130—21no—
17#17986130C2212yescomplemented
18Q8×C71⋊C8—21no—
19Q8×C71⋊C8C2212yescomplemented
20C4×#11765852C2214yes—
21#17986064—21no—
22#17986064—21no—
23#17986064C2212yes—
24#17986064C2212yes—
25#17986052C2211yescharacteristic
26#17986026—21no—
27#17986023C2211yescharacteristic
28C2×(C2×D71)⋊C8—21no—
29#17985955—21no—
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985751C2214yescomplemented
32#17985728C2214yes—
33#17985721C2212yescomplemented
34#17985720C2212yescomplemented
35#533×C71C2211yescharacteristic
36#431×C71C2211yescharacteristic
37C2×C4⋊C8×C71C2211yescharacteristic
38Q8⋊C8×C71C2214yescomplemented
39C22×D284C2×C411yescharacteristic
40C22×C4×D71—21no—
41C2×#11765852C2×C414yes—
42C2×#11765849—21no—
43C22×C71⋊C8D411yescharacteristic
44C22×C71⋊C8D411yescharacteristic
45C2×C8×D71—22no—
46C2×C8×D71—22no—
47C2×C8×D71—44no—
48C4×D284—21no—
49C4×D284—21no—
50C4×D284C2×C412yes—
51C4×D284C2312yes—
52C2×C4⋊Dic71C2×C411yescharacteristic
53C2×(C2×D71)⋊C4—21no—
54#14556059C2×C414yes—
55C4⋊(C71⋊C8)—22no—
56C4⋊(C71⋊C8)—22no—
57C4⋊(C71⋊C8)—21no—
58C4⋊(C71⋊C8)C2×C412yes—
59C4⋊(C71⋊C8)C2312yes—
60#14556052C2×C414yes—
61#14556048C2×C414yes—
62C22⋊(C71⋊C8)C2×C412yes—
63(C2×D71)⋊C8—22no—
64(C2×D71)⋊C8—22no—
65#14556039—22no—
66C4×C71⋊C8—22no—
67C4×C71⋊C8D411yescharacteristic
68C4×C71⋊C8D411yescharacteristic
69C4×C71⋊C8C2312yes—
70C4×C71⋊C8C2311yescharacteristic
71C4×C71⋊C8C2311yescharacteristic
72C22×C8×C71—21no—
73C4⋊Q8×C71C2×C411yescharacteristic
74C4⋊Q8×C71C2×C411yescharacteristic
75#125×C71—21no—
76C22⋊Q8×C71—21no—
77C22⋊Q8×C71C2×C412yes—
78C4×Q8×C71—21no—
79C4×Q8×C71C2312yes—
80#117×C71—21no—
81C2×C42×C71C2311yescharacteristic
82C4⋊C8×C71—21no—
83C4⋊C8×C71—21no—
84C4⋊C8×C71C2312yes—
85C22⋊C8×C71—21no—
86C4×C8×C71C2312yes—
87C23×D71—21no—
88C2×D284—21no—
89C2×D284—24no—
90C2×D284C22×C412yes—
91C22×Dic71—21no—
92C2×C4×D71—22no—
93C2×C4×D71—22no—
94C2×C4×D71—42no—
95C2×C4×D71—42no—
96#11765852—28no—
97#11765849—42no—
98C2×C71⋊C8—22no—
99C2×C71⋊C8—22no—
100C2×C71⋊C8—22no—
101C2×C71⋊C8C22×C414yes—
102C2×C71⋊C8C2×D411yescharacteristic
103C2×C71⋊C8C2×D411yescharacteristic
104C2×C71⋊C8C2×D411yescharacteristic
105C2×C71⋊C8C2×D411yescharacteristic
106C8×D71—44no—
107C8×D71—44no—
108C4⋊Dic71—21no—
109C4⋊Dic71C22×C412yes—
110(C2×D71)⋊C4—22no—
111(C2×D71)⋊C4—22no—
112C2×Q8×C71—21no—
113C2×Q8×C71C22×C412yes—
114C22×C4×C71C22×C411yescharacteristic
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71#4111yescharacteristic
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71—42no—
120C4⋊C4×C71—21no—
121C4⋊C4×C71—21no—
122C4⋊C4×C71—21no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—22no—
125C4⋊C4×C71C22×C412yes—
126C22⋊C4×C71—22no—
127C42×C71—22no—
128C42×C71C22×C411yescharacteristic
129C42×C71C2×D411yescharacteristic
130C42×C71#4111yescharacteristic
131C42×C71C2411yescharacteristic
132C23×C71C4×D411yescharacteristic
133C22×D71—24no—
134C22×D71—22no—
135C22×D71—42no—
136C22×D71—42no—
137Q8×C71—24no—
138Q8×C71—42no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—41no—
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71C4×D411yescharacteristic
145C2×C4×C71C4×D411yescharacteristic
146C2×C4×C71AGL(1,ℤ/8)12yes—
147C2×C4×C71#13712yes—
148C2×C4×C71C23×C411yescharacteristic
149C2×C4×C71C22×D411yescharacteristic
150C2×C4×C71C2×#4111yescharacteristic
151C2×Dic71—22no—
152C2×Dic71—42no—
153D284—24no—
154D284—22no—
155D284#13114yes—
156C4×D71—42no—
157C4×D71—42no—
158C4×D71—44no—
159C8×C71—42no—
160C8×C71—42no—
161C71⋊C8—24no—
162C71⋊C8—21no—
163C71⋊C8—21no—
164C71⋊C8—21no—
165C71⋊C8—21no—
166C71⋊C8—42no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71#44012yes—
170C22×C71C2×C4×D411yescharacteristic
171C2×D71—24no—
172C2×D71—48no—
173C2×D71—42no—
174C2×D71—42no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—42no—
179C4×C71—42no—
180C4×C71#44012yes—
181C4×C71C2×#13112yes—
182C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
183C4×C71C2×#13711yescharacteristic
184Dic71—42no—
185#17905—711nomaximal, Sylow
186C2×C71—22no—
187C2×C71#292911yescharacteristic
188C2×C71C2×#44011yescharacteristic
189C2×C71#295511yescharacteristic
190D71—44no—
191#3101—711no—
192#2960—711no—
193C2×C8×D4—711no—
194#1585—714no—
195SD16⋊C8—714no—
196#1571—711no—
197#1486—711no—
198#1481D7111yescharacteristic, complemented
199#1452—711no—
200C71#1790511yescharacteristic, complemented, Sylow
201C23×C8—1421no—
202#533C2×D7111yescharacteristic
203C2×C4×D4—711no—
204#483—712no—
205#476—712no—
206#461—712no—
207#450—712no—
208#446—712no—
209#446—1421no—
210C8×Q8—712no—
211C8×Q8—1421no—
212C4×SD16—714no—
213C8×D4—712no—
214C8×D4—712no—
215C8×D4—1421no—
216C8×D4—1421no—
217#431—711no—
218#431C2×D7111yescharacteristic
219#424—1421no—
220#423—711no—
221C2×C4⋊C8C2×D7111yescharacteristic
222C2×C22⋊C8—1421no—
223#405—1421no—
224C2×C4×C8—711no—
225#335—714no—
226Q8⋊C8C2×D7114yescomplemented
227D4⋊C8—714no—
228#322—712no—
229#322—712no—
230C22×D4—711no—
231C23×C4—1421no—
232C2×SD16—714no—
233C2×M16—1421no—
234C22×C8—21no—
235C22×C8—2844no—
236C22×C8—711no—
237C22×C8—711no—
238C22×C8—1422no—
239C22×C8—1422no—
240C4⋊Q8C4×D7111yescharacteristic
241C4⋊Q8C4×D7111yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8C4×D7112yes—
245C4×Q8—21no—
246C4×Q8C22×D7112yes—
247C4×D4—712no—
248C4×D4—712no—
249C4×D4—1421no—
250C4×D4—1421no—
251#117—21no—
252C2×C4⋊C4—711no—
253C2×C22⋊C4—1421no—
254C2×C42C22×D7111yescharacteristic
255#106—714no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—712no—
259C4⋊C8—712no—
260C4⋊C8—1422no—
261C4⋊C8—1422no—
262C4⋊C8—1421no—
263C4⋊C8C22×D7112yes—
264#103—714no—
265#102—714no—
266C22⋊C8—21no—
267C22⋊C8—712no—
268C22⋊C8—1422no—
269C22⋊C8—1422no—
270#97—1422no—
271C4×C8—711no—
272C4×C8—711no—
273C4×C8—712no—
274C4×C8—711no—
275C4×C8—711no—
276C4×C8—1422no—
277C4×C8C22×D7112yes—
278C24—1421no—
279C2×Q8—21no—
280C2×Q8C2×C4×D7112yes—
281C2×D4—712no—
282C2×D4—1424no—
283C2×D4—1421no—
284C22×C4—2842no—
285C22×C4—2842no—
286C22×C4—1421no—
287C22×C4—1422no—
288C22×C4—1422no—
289C22×C4C2×C4×D7111yescharacteristic
290C22×C4D4×D7111yescharacteristic
291C22×C4#1176586411yescharacteristic
292SD16—1428no—
293M16—2842no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—2844no—
298C2×C8—2844no—
299C2×C8—714no—
300C2×C8—711no—
301C2×C8—711no—
302C2×C8—711no—
303C2×C8—711no—
304C2×C8—1422no—
305C2×C8—1422no—
306C2×C8—1422no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—712no—
313C4⋊C4—1421no—
314C4⋊C4C2×C4×D7112yes—
315C22⋊C4—22no—
316C22⋊C4—1422no—
317C22⋊C4—1422no—
318C42—22no—
319C42C2×C4×D7111yescharacteristic
320C42D4×D7111yescharacteristic
321C42#1176586411yescharacteristic
322C42C23×D7111yescharacteristic
323C23—2842no—
324C23—2842no—
325C23—1424no—
326C23—1422no—
327C23(C22⋊D71)⋊C411yescharacteristic
328Q8—24no—
329Q8—42no—
330D4—714no—
331D4—1424no—
332D4—1422no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—2842no—
338C2×C4—2844no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4—1422no—
342C2×C4(C22⋊D71)⋊C411yescharacteristic
343C2×C4(C22⋊D71)⋊C411yescharacteristic
344C2×C4(C22⋊D71)⋊C411yescharacteristic
345C2×C4#1455615512yes—
346C2×C4#1455616012yes—
347C2×C4C22×C4×D7111yescharacteristic
348C2×C4C2×D4×D7111yescharacteristic
349C2×C4C2×#1176586411yescharacteristic
350C8—42no—
351C8—42no—
352C8—2842no—
353C8—1424no—
354C8—1421no—
355C8—1421no—
356C8—1421no—
357C8—1421no—
358C22—21no—
359C22—21no—
360C22—2848no—
361C22—2842no—
362C22—2842no—
363C22—1424no—
364C22#1798610212yes—
365C22C2×(C22⋊D71)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—2842no—
372C4#1798610212yes—
373C4#131×D7112yes—
374C4C2×#1455615511yescharacteristic
375C4C2×#1455616011yescharacteristic
376C2—22no—
377C2—2844no—
378C2#2259703711yescharacteristic
379C2C2×#1798610211yescharacteristic
380C2#2259723911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1481C2
socleC22×C71C2×C4×D4
prime radicalC22×D284C2×C4
cosocleC42×C71C24
hypercenter#1481D71
hypocenterC71#17905
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
2. #22597275C2semidirect
4. #22587200C2semidirect
5. #22587155C2semidirect
6. #22587114C2semidirect
8. #22586633C2semidirect
9. C71×#1481C2semidirect
17. #17986130C22semidirect
19. Q8×C71⋊C8C22semidirect
31. #17985751C22semidirect
33. #17985721C22semidirect
34. #17985720C22semidirect
38. Q8⋊C8×C71C22semidirect
198. #1481D71semidirect
200. C71#17905semidirect
226. Q8⋊C8C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225983442
#225972752
C2×#179860642
#225872002
#225871552
#225871142
#225867662
#225866332
C71×#14812
#1790571