Finite Group Databasebeta

← #22597565

Subgroups — #22597565

Order 9088.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D71C211yesmaximal, characteristic, complemented
2C2×#14556153C211yesmaximal, characteristic, complemented
3#17986506C211yesmaximal, characteristic, complemented
4#17986148C214yesmaximal, complemented
5#17986131C214yesmaximal, complemented
6#17986005C211yesmaximal, characteristic, complemented
7#17986004C211yesmaximal, characteristic, complemented
8#17985962C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
10C24×D71—21no—
11C22×D284C2211yescharacteristic
12C22×C22⋊D71—21no—
13C2×D4×D71—24no—
14C2×D4×D71—22no—
15C2×D4×D71—24no—
16C2×D4×D71C2214yes—
17C22×C4×D71C2211yescharacteristic
18#14556153—24no—
19C2×#11765852C2212yescomplemented
20C2×#11765850C2212yescomplemented
21C2×#11765849C2211yescharacteristic, complemented
22#14556109—21no—
23#14556104C2212yescomplemented
24#14556099C2212yescomplemented
25#14556096—21no—
26C4×C22⋊D71—22no—
27#14556087C2211yescharacteristic, complemented
28#14556050C2214yescomplemented
29#14556049C2214yescomplemented
30(C2×D71)⋊C8C2214yescomplemented
31C2×#41×C71C2211yescharacteristic, complemented
32C22×D4×C71C2211yescharacteristic
33AGL(1,ℤ/8)×C71—24no—
34C2×SD16×C71C2212yescomplemented
35C2×D8×C71C2212yescomplemented
36C2×M16×C71C2211yescharacteristic, complemented
37C23×D71—28no—
38C23×D71—21no—
39C23×D71—24no—
40C23×D71—48no—
41C23×D71D411yescharacteristic
42C2×D284—21no—
43C2×D284—22no—
44C2×D284D414yescomplemented
45C2×D284C2312yes—
46C2×C22⋊D71—28no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—24no—
49D4×D71—216no—
50D4×D71—28no—
51D4×D71—28no—
52C22×Dic71D411yescharacteristic
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55C2×C4×D71D412yes—
56C2×C4×D71C2312yes—
57#11765852—24no—
58#11765850—24no—
59#11765849—22no—
60C2×C71⋊C8C2312yescomplemented
61C4⋊Dic71C2312yescomplemented
62Dic71⋊C4—22no—
63C22⋊Dic71—21no—
64C22⋊Dic71—22no—
65(C2×D71)⋊C4—22no—
66C4×Dic71—21no—
67C24×C71—21no—
68#41×C71—22no—
69#41×C71—22no—
70C2×Q8×C71C2311yescharacteristic, complemented
71C2×D4×C71—21no—
72C2×D4×C71—21no—
73C2×D4×C71—22no—
74C2×D4×C71D414yescomplemented
75C2×D4×C71C2312yes—
76C2×D4×C71C2311yescharacteristic, complemented
77C22×C4×C71—21no—
78C22×C4×C71C2311yescharacteristic
79SD16×C71—24no—
80D8×C71—24no—
81M16×C71—22no—
82C2×C8×C71C2312yescomplemented
83C23×C71—24no—
84C23×C71—22no—
85C23×C71—21no—
86C23×C71—44no—
87C23×C71C2×D411yescharacteristic
88C22×D71—22no—
89C22×D71—24no—
90C22×D71—216no—
91C22×D71—24no—
92C22×D71—22no—
93C22×D71—432no—
94C22×D71—48no—
95C22×D71—44no—
96C22×D71—48no—
97C22×D71C2×D412yes—
98Q8×C71—21no—
99Q8×C71—21no—
100D4×C71—24no—
101D4×C71—22no—
102D4×C71—21no—
103D4×C71—21no—
104D4×C71—42no—
105D4×C71C2×D414yes—
106C2×C4×C71—21no—
107C2×C4×C71—42no—
108C2×C4×C71C2×D411yescharacteristic
109C2×C4×C71C2×D412yes—
110C2×C4×C71C2×D412yes—
111C2×C4×C71C2411yescharacteristic
112C22⋊D71—216no—
113C22⋊D71—44no—
114C2×Dic71—22no—
115C2×Dic71—22no—
116C2×Dic71C2×D412yes—
117D284—24no—
118D284—22no—
119D284C2×D414yes—
120C4×D71—24no—
121C4×D71—22no—
122C4×D71—22no—
123C8×C71—22no—
124C71⋊C8—22no—
125C22×C71—22no—
126C22×C71—24no—
127C22×C71—48no—
128C22×C71—44no—
129C22×C71—41no—
130C22×C71—42no—
131C22×C71#12012yes—
132C22×C71C22×D411yescharacteristic
133C2×D71—28no—
134C2×D71—22no—
135C2×D71—22no—
136C2×D71—24no—
137C2×D71—432no—
138C2×D71—48no—
139C2×D71—44no—
140C2×D71AGL(1,ℤ/8)14yescomplemented
141C4×C71—41no—
142C4×C71#12012yes—
143C4×C71C22×D411yescharacteristic
144C4×C71C22×D411yescharacteristic
145Dic71—22no—
146Dic71—42no—
147C2×C71—22no—
148C2×C71—44no—
149C2×C71—81no—
150C2×C71C2×#12011yescharacteristic
151C2×C71C2×AGL(1,ℤ/8)12yes—
152D71—24no—
153D71—44no—
154#3003—711nomaximal, Sylow
155C71#300311yescharacteristic, complemented, Sylow
156C23×D4—711no—
157C2×AGL(1,ℤ/8)—711no—
158C2×AGL(1,ℤ/8)D7111yescharacteristic, complemented
159#525—711no—
160#450—714no—
161#447—714no—
162#418—711no—
163#418—711no—
164#407—711no—
165C25—1421no—
166C2×#41C2×D7111yescharacteristic, complemented
167C22×D4—714no—
168C22×D4—711no—
169C22×D4—1422no—
170C22×D4—1421no—
171C22×D4—1424no—
172C22×D4—1424no—
173C22×D4C2×D7111yescharacteristic
174C23×C4—711no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1424no—
177C2×SD16—712no—
178C2×SD16C2×D7112yescomplemented
179C2×D8—712no—
180C2×D8C2×D7112yescomplemented
181C2×M16—711no—
182C2×M16C2×D7111yescharacteristic, complemented
183#123—1421no—
184C22⋊Q8—712no—
185#121—712no—
186#120—1421no—
187C4×D4—1422no—
188#117—711no—
189#102—714no—
190#102—714no—
191C22⋊C8—714no—
192C24—21no—
193C24—2848no—
194C24—711no—
195C24—1424no—
196C24—1428no—
197C24—1421no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D7111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—714no—
205C2×D4—712no—
206C2×D4—1424no—
207C2×D4—1428no—
208C2×D4—14216no—
209C2×D4—1422no—
210C2×D4—1428no—
211C2×D4—1428no—
212C2×D4—1421no—
213C2×D4—1422no—
214C2×D4C22⋊D7114yescomplemented
215C2×D4C22×D7112yes—
216C2×D4C22×D7111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—711no—
219C22×C4—712no—
220C22×C4—712no—
221C22×C4—1422no—
222C22×C4—1422no—
223C22×C4C22×D7111yescharacteristic
224SD16—24no—
225SD16—1424no—
226D8—24no—
227D8—1424no—
228M16—22no—
229M16—1422no—
230C2×C8—712no—
231C2×C8C22×D7112yescomplemented
232C4⋊C4—712no—
233C4⋊C4—1422no—
234C22⋊C4—1422no—
235C22⋊C4—1422no—
236C22⋊C4—1421no—
237C42—1421no—
238C23—22no—
239C23—24no—
240C23—21no—
241C23—44no—
242C23—2844no—
243C23—2848no—
244C23—28432no—
245C23—2848no—
246C23—712no—
247C23—14216no—
248C23—1424no—
249C23—1422no—
250C23—1424no—
251C23—1422no—
252C23D4×D7111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—24no—
259D4—42no—
260D4—2844no—
261D4—714no—
262D4—14216no—
263D4—1422no—
264D4—1424no—
265D4C2×C22⋊D7114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—712no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4—1424no—
274C2×C4D4×D7112yes—
275C2×C4D4×D7111yescharacteristic
276C2×C4C2×C22⋊D7112yes—
277C2×C4C23×D7111yescharacteristic
278C8—22no—
279C8—1422no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—2848no—
287C22—28432no—
288C22—2844no—
289C22—714no—
290C22—1428no—
291C22—1424no—
292C22—1422no—
293C22—1422no—
294C22#1455609612yes—
295C22C2×D4×D7111yescharacteristic
296C4—41no—
297C4—2842no—
298C4—1422no—
299C4#1455609612yes—
300C4C2×D4×D7111yescharacteristic
301C4C22×C22⋊D7111yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—2844no—
306C2—1424no—
307C2C2×#1455609611yescharacteristic
308C2AGL(1,ℤ/8)×D7112yes—
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#3003
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. C22×D4×D71C2semidirect
2. C2×#14556153C2semidirect
3. #17986506C2semidirect
4. #17986148C2semidirect
5. #17986131C2semidirect
6. #17986005C2semidirect
7. #17986004C2semidirect
8. #17985962C2semidirect
9. C2×AGL(1,ℤ/8)×C71C2semidirect
19. C2×#11765852C22semidirect
20. C2×#11765850C22semidirect
21. C2×#11765849C22semidirect
23. #14556104C22semidirect
24. #14556099C22semidirect
27. #14556087C22semidirect
28. #14556050C22semidirect
29. #14556049C22semidirect
30. (C2×D71)⋊C8C22semidirect
31. C2×#41×C71C22semidirect
34. C2×SD16×C71C22semidirect
35. C2×D8×C71C22semidirect
36. C2×M16×C71C22semidirect
44. C2×D284D4semidirect
60. C2×C71⋊C8C23semidirect
61. C4⋊Dic71C23semidirect
70. C2×Q8×C71C23semidirect
74. C2×D4×C71D4semidirect
76. C2×D4×C71C23semidirect
82. C2×C8×C71C23semidirect
140. C2×D71AGL(1,ℤ/8)semidirect
155. C71#3003semidirect
158. C2×AGL(1,ℤ/8)D71semidirect
166. C2×#41C2×D71semidirect
178. C2×SD16C2×D71semidirect
180. C2×D8C2×D71semidirect
182. C2×M16C2×D71semidirect
200. C2×Q8C22×D71semidirect
214. C2×D4C22⋊D71semidirect
216. C2×D4C22×D71semidirect
231. C2×C8C22×D71semidirect