Finite Group Databasebeta

← #22597953

Subgroups — #22597953

Order 9088.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D71C211yesmaximal, characteristic, complemented
2#17986506C211yesmaximal, characteristic, complemented
3C2×#14556099C211yesmaximal, characteristic, complemented
4#17986249C212yesmaximal, complemented
5#17986242C212yesmaximal, complemented
6#17986232C212yesmaximal, complemented
7#17986225C212yesmaximal, complemented
8#17986041C211yesmaximal, characteristic, complemented
9#17986004C211yesmaximal, characteristic, complemented
10#17986000C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71—21no—
13C22×C4×D71C2211yescharacteristic
14C2×#11765849C2211yescharacteristic
15C2×#11765848C2212yescomplemented
16M16×D71—24no—
17C2×C8×D71C2212yescomplemented
18#14556109—21no—
19#14556104C2211yescharacteristic, complemented
20#14556104C2211yescharacteristic, complemented
21#14556099—21no—
22#14556099—21no—
23#14556099C2212yes—
24#14556099C2212yes—
25#14556099C2211yescharacteristic, complemented
26#14556099C2211yescharacteristic, complemented
27#14556096—21no—
28C4×C22⋊D71—21no—
29C4×C22⋊D71—21no—
30#14556087C2211yescharacteristic
31C2×C4⋊Dic71C2211yescharacteristic
32C2×C22⋊Dic71—21no—
33#14556062C2212yescomplemented
34#14556060C2212yescomplemented
35#14556052C2212yescomplemented
36#14556049C2212yescomplemented
37#14556049C2212yescomplemented
38#14556049C2212yescomplemented
39C2×#41×C71C2211yescharacteristic
40C22×D4×C71C2211yescharacteristic
41AGL(1,ℤ/8)×C71—24no—
42C2×SD16×C71C2212yescomplemented
43C2×D8×C71C2212yescomplemented
44C2×M16×C71C2211yescharacteristic
45C23×D71D411yescharacteristic
46C2×C22⋊D71—22no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—21no—
50C2×C22⋊D71—44no—
51C22×Dic71—21no—
52C22×Dic71D411yescharacteristic
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55C2×C4×D71D411yescharacteristic
56C2×C4×D71D411yescharacteristic
57C2×C4×D71C2311yescharacteristic
58C2×C4×D71C2311yescharacteristic
59#11765849—22no—
60#11765848—24no—
61C2×C71⋊C8C2312yes—
62C8×D71—24no—
63C4⋊Dic71—21no—
64C4⋊Dic71C2311yescharacteristic
65C4⋊Dic71C2311yescharacteristic
66C4⋊Dic71C2312yes—
67Dic71⋊C4—21no—
68Dic71⋊C4—21no—
69C22⋊Dic71—21no—
70C22⋊Dic71—22no—
71C22⋊Dic71—22no—
72C22⋊Dic71—21no—
73C22⋊Dic71—21no—
74(C2×D71)⋊C4—21no—
75(C2×D71)⋊C4—21no—
76C4×Dic71—21no—
77C24×C71—21no—
78#41×C71—22no—
79#41×C71—22no—
80C2×Q8×C71C2311yescharacteristic
81C2×D4×C71—21no—
82C2×D4×C71—22no—
83C2×D4×C71—21no—
84C2×D4×C71—22no—
85C2×D4×C71C2311yescharacteristic
86C2×D4×C71C2312yes—
87C22×C4×C71—21no—
88C22×C4×C71C2311yescharacteristic
89SD16×C71—24no—
90D8×C71—24no—
91M16×C71D414yescomplemented
92C2×C8×C71C2312yes—
93C23×C71—21no—
94C23×C71—22no—
95C23×C71—42no—
96C23×C71—42no—
97C23×C71—42no—
98C23×C71C2×D411yescharacteristic
99C22×D71—22no—
100C22×D71—22no—
101C22×D71—41no—
102C22×D71C2×D411yescharacteristic
103C22×D71C2×D411yescharacteristic
104Q8×C71—21no—
105Q8×C71—21no—
106D4×C71—21no—
107D4×C71—22no—
108D4×C71—22no—
109D4×C71—21no—
110D4×C71—42no—
111D4×C71—42no—
112C2×C4×C71—21no—
113C2×C4×C71—42no—
114C2×C4×C71C2×D411yescharacteristic
115C2×C4×C71C2×D412yes—
116C2×C4×C71#4112yes—
117C2×C4×C71C2411yescharacteristic
118C22⋊D71—42no—
119C22⋊D71—44no—
120C22⋊D71—42no—
121C22⋊D71—44no—
122C2×Dic71—22no—
123C2×Dic71—21no—
124C2×Dic71—21no—
125C2×Dic71—22no—
126C2×Dic71—42no—
127C2×Dic71C2×D411yescharacteristic
128C2×Dic71C2×D411yescharacteristic
129C4×D71—22no—
130C4×D71—22no—
131C4×D71—21no—
132C4×D71—21no—
133C4×D71—21no—
134C4×D71—21no—
135C8×C71C2×D414yescomplemented
136C71⋊C8—22no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—41no—
140C22×C71—42no—
141C22×C71—42no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—42no—
145C22×C71—84no—
146C22×C71#12112yes—
147C22×C71C22×D411yescharacteristic
148C2×D71—21no—
149C2×D71—21no—
150C2×D71—21no—
151C2×D71—21no—
152C2×D71—21no—
153C2×D71—44no—
154C2×D71AGL(1,ℤ/8)12yescomplemented
155C4×C71—41no—
156C4×C71#12112yes—
157C4×C71C22×D411yescharacteristic
158C4×C71C2×#4111yescharacteristic
159Dic71—21no—
160Dic71—21no—
161Dic71—42no—
162Dic71—41no—
163Dic71—41no—
164C2×C71—22no—
165C2×C71—81no—
166C2×C71—82no—
167C2×C71C2×#12111yescharacteristic
168C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C71#57511yescharacteristic
170D71—22no—
171D71—41no—
172#3055—711nomaximal, Sylow
173C71#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D7111yescharacteristic, complemented
175C22×M16—711no—
176#525—711no—
177C2×#121—711no—
178#469—712no—
179#468—712no—
180#466—712no—
181#465—712no—
182#428—711no—
183#418—711no—
184#417—711no—
185C2×#41C2×D7111yescharacteristic
186C22×D4—1421no—
187C22×D4C2×D7111yescharacteristic
188C23×C4—711no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D7112yescomplemented
191C2×D8C2×D7112yescomplemented
192C2×M16—711no—
193C2×M16—712no—
194C2×M16—1424no—
195C2×M16C2×D7111yescharacteristic
196C22×C8—712no—
197#123—1421no—
198C22⋊Q8—711no—
199C22⋊Q8—711no—
200#121—712no—
201#121—712no—
202#121—711no—
203#121—711no—
204#121—1421no—
205#121—1421no—
206#120—1421no—
207C4×D4—1421no—
208C4×D4—1421no—
209#117—711no—
210C2×C4⋊C4—711no—
211C2×C22⋊C4—1421no—
212#107—712no—
213#106—712no—
214#103—712no—
215#102—712no—
216#102—712no—
217#102—712no—
218C24—21no—
219C24—711no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D7111yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—2844no—
228C2×D4—1421no—
229C2×D4—1421no—
230C2×D4—1422no—
231C2×D4—1422no—
232C2×D4C22×D7111yescharacteristic
233C2×D4C22×D7112yes—
234C22×C4—21no—
235C22×C4—711no—
236C22×C4—711no—
237C22×C4—711no—
238C22×C4—711no—
239C22×C4—711no—
240C22×C4—1421no—
241C22×C4—1422no—
242C22×C4—1422no—
243C22×C4C22×D7111yescharacteristic
244SD16—24no—
245D8—24no—
246M16—1422no—
247M16—1424no—
248M16C22⋊D7114yescomplemented
249C2×C8—712no—
250C2×C8—1424no—
251C2×C8C22×D7112yes—
252C4⋊C4—711no—
253C4⋊C4—711no—
254C4⋊C4—712no—
255C4⋊C4—1421no—
256C4⋊C4—1421no—
257C4⋊C4—1421no—
258C22⋊C4—1421no—
259C22⋊C4—1421no—
260C22⋊C4—1422no—
261C22⋊C4—1422no—
262C22⋊C4—1421no—
263C22⋊C4—1421no—
264C22⋊C4—1421no—
265C42—1421no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—2841no—
272C23—711no—
273C23—711no—
274C23—1422no—
275C23—1422no—
276C23D4×D7111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—2844no—
286D4—2842no—
287D4—2844no—
288D4—2842no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—2842no—
292C2×C4—711no—
293C2×C4—711no—
294C2×C4—1421no—
295C2×C4—1421no—
296C2×C4—1422no—
297C2×C4—1422no—
298C2×C4—1421no—
299C2×C4—1421no—
300C2×C4—1421no—
301C2×C4—1421no—
302C2×C4—1422no—
303C2×C4—1422no—
304C2×C4D4×D7111yescharacteristic
305C2×C4C2×C22⋊D7112yes—
306C2×C4#1176586412yes—
307C2×C4C23×D7111yescharacteristic
308C8—1422no—
309C8C2×C22⋊D7114yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—2844no—
320C22—712no—
321C22—1421no—
322C22—1421no—
323C22—1421no—
324C22—1421no—
325C22—1421no—
326C22#1455609912yes—
327C22C2×D4×D7111yescharacteristic
328C4—41no—
329C4—2842no—
330C4—2841no—
331C4—2841no—
332C4—1421no—
333C4—1421no—
334C4#1455609912yes—
335C4C22×C22⋊D7111yescharacteristic
336C4C2×#1176586411yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—2841no—
341C2—1422no—
342C2C2×#1455609911yescharacteristic
343C2AGL(1,ℤ/8)×D7111yescharacteristic
344C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×AGL(1,ℤ/8)×C71C2
socleC22×C71C22×D4
hypercenterC2×AGL(1,ℤ/8)D71
hypocenterC71#3055
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D71C2semidirect
2. #17986506C2semidirect
3. C2×#14556099C2semidirect
4. #17986249C2semidirect
5. #17986242C2semidirect
6. #17986232C2semidirect
7. #17986225C2semidirect
8. #17986041C2semidirect
9. #17986004C2semidirect
10. #17986000C2semidirect
11. C2×AGL(1,ℤ/8)×C71C2semidirect
15. C2×#11765848C22semidirect
17. C2×C8×D71C22semidirect
19. #14556104C22semidirect
20. #14556104C22semidirect
25. #14556099C22semidirect
26. #14556099C22semidirect
33. #14556062C22semidirect
34. #14556060C22semidirect
35. #14556052C22semidirect
36. #14556049C22semidirect
37. #14556049C22semidirect
38. #14556049C22semidirect
42. C2×SD16×C71C22semidirect
43. C2×D8×C71C22semidirect
91. M16×C71D4semidirect
135. C8×C71C2×D4semidirect
154. C2×D71AGL(1,ℤ/8)semidirect
173. C71#3055semidirect
174. C2×AGL(1,ℤ/8)D71semidirect
190. C2×SD16C2×D71semidirect
191. C2×D8C2×D71semidirect
248. M16C22⋊D71semidirect
309. C8C2×C22⋊D71semidirect