Finite Group Databasebeta

← #22598387

Subgroups — #22598387

Order 9088.

454 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#17986533C211yesmaximal, characteristic, complemented
2#17986334C211yesmaximal, characteristic, complemented
3#17986317C211yesmaximal, characteristic, complemented
4#17986299C212yesmaximal, complemented
5#17986214C212yesmaximal, complemented
6#17986150C211yesmaximal, characteristic, complemented
7#17986133C211yesmaximal, characteristic
8C4×#11765853C212yesmaximal, complemented
9C4×#11765850C212yesmaximal, complemented
10#17986052C211yesmaximal, characteristic, complemented
11C2×C4×D4×C71C211yesmaximal, characteristic, complemented
12C2×#11765853C2212yes—
13C2×#11765850C2212yes—
14#14556122—21no—
15#14556120C2211yescharacteristic, complemented
16#14556116C2211yescharacteristic, complemented
17#14556111—21no—
18#14556107C2211yescharacteristic
19#14556102C2211yescharacteristic
20C4×Dic142C2211yescharacteristic, complemented
21C4×D284C2211yescharacteristic, complemented
22C4×C22⋊D71—21no—
23#14556085—21no—
24#14556061C2212yescomplemented
25#14556059C2212yescomplemented
26C4⋊(C71⋊C8)C2212yescomplemented
27#14556051C2212yescomplemented
28#14556049C2212yes—
29#14556049C2212yes—
30#14556048C2212yescomplemented
31C22⋊(C71⋊C8)C2211yescharacteristic
32C22⋊(C71⋊C8)C2211yescharacteristic
33C4×C71⋊C8C2212yescomplemented
34C22×D4×C71C2211yescharacteristic
35C23×C4×C71—21no—
36C4×D4×C71—21no—
37C4×D4×C71—21no—
38C4×D4×C71C2212yes—
39C4×D4×C71C2212yes—
40C2×C4⋊C4×C71C2211yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C42×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71—21no—
47#11765853—24no—
48#11765850—24no—
49C2×C71⋊C8C2312yes—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53Dic71⋊C4—21no—
54Dic71⋊C4—21no—
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57C22⋊Dic71—21no—
58(C2×D71)⋊C4—21no—
59(C2×D71)⋊C4—21no—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—21no—
62C4×Dic71—21no—
63C24×C71—21no—
64C2×D4×C71—24no—
65C2×D4×C71—21no—
66C2×D4×C71C2312yes—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—44no—
71C22×C4×C71D411yescharacteristic
72C22×C4×C71D411yescharacteristic
73C22×C4×C71C2311yescharacteristic
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C22⋊C4×C71—22no—
77C22⋊C4×C71—22no—
78C42×C71D411yescharacteristic
79C42×C71D411yescharacteristic
80C42×C71C2311yescharacteristic
81C42×C71C2311yescharacteristic
82C23×C71—24no—
83C23×C71—22no—
84C23×C71—42no—
85C23×C71—42no—
86C23×C71C2×D411yescharacteristic
87C22×D71—21no—
88D4×C71—24no—
89D4×C71—22no—
90D4×C71#4114yes—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—44no—
96C2×C4×C71—42no—
97C2×C4×C71—44no—
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2411yescharacteristic
104C22⋊D71—42no—
105C2×Dic71—21no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108D284—22no—
109C4×D71—42no—
110Dic142—22no—
111C71⋊C8—22no—
112C71⋊C8—22no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71—24no—
116C22×C71—48no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71#13512yes—
120C22×C71C22×D411yescharacteristic
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—41no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—21no—
127C4×C71—21no—
128C4×C71—42no—
129C4×C71AGL(1,ℤ/8)12yes—
130C4×C71C2×#4112yes—
131Dic71—41no—
132Dic71—41no—
133Dic71—41no—
134C2×C71—22no—
135C2×C71—44no—
136C2×C71#52511yescharacteristic
137C2×C71C2×#13511yescharacteristic
138C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
139D71—81no—
140#3105—711nomaximal, Sylow
141C71#310511yescharacteristic, complemented, Sylow
142#529—711no—
143C2×C4×D4D7111yescharacteristic, complemented
144#484—711no—
145#481—711no—
146#478—712no—
147#463—712no—
148#450—711no—
149#447—711no—
150C4×SD16—712no—
151C4×D8—712no—
152#431—711no—
153C22×D4C2×D7111yescharacteristic
154C23×C4—21no—
155C2×SD16—712no—
156C2×D8—712no—
157#126—1421no—
158#125—711no—
159#124—711no—
160#123—1421no—
161C22⋊Q8—711no—
162#121—711no—
163C4×Q8—711no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—711no—
167C4×D4—1421no—
168C4×D4C2×D7112yes—
169C4×D4C2×D7112yes—
170#117—1421no—
171C2×C4⋊C4C2×D7111yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D7111yescharacteristic
174#107—712no—
175#106—712no—
176C4⋊C8—712no—
177#103—712no—
178#102—712no—
179#102—712no—
180#102—712no—
181C22⋊C8—711no—
182C22⋊C8—711no—
183C4×C8—712no—
184C24—21no—
185C2×Q8—711no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—711no—
189C2×D4—1421no—
190C2×D4C22×D7112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1421no—
196C22×C4C22⋊D7111yescharacteristic
197C22×C4C22⋊D7111yescharacteristic
198C22×C4C22×D7111yescharacteristic
199SD16—1424no—
200D8—1424no—
201C2×C8—712no—
202C2×C8—712no—
203C4⋊C4—21no—
204C4⋊C4—711no—
205C4⋊C4—711no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4—1421no—
210C4⋊C4C22×D7112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—1421no—
214C22⋊C4—1421no—
215C22⋊C4—1421no—
216C22⋊C4—1421no—
217C22⋊C4—1421no—
218C42—1421no—
219C42C22⋊D7111yescharacteristic
220C42C22⋊D7111yescharacteristic
221C42C22×D7111yescharacteristic
222C42C22×D7111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1421no—
228C23C2×C22⋊D7111yescharacteristic
229Q8—1422no—
230D4—24no—
231D4—22no—
232D4—2842no—
233D4—1422no—
234D4#1176586514yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—2842no—
243C2×C4—1421no—
244C2×C4—1421no—
245C2×C4—1421no—
246C2×C4C2×C22⋊D7111yescharacteristic
247C2×C4C2×C22⋊D7111yescharacteristic
248C2×C4C2×C22⋊D7111yescharacteristic
249C2×C4C2×C22⋊D7111yescharacteristic
250C2×C4C2×C22⋊D7111yescharacteristic
251C2×C4C23×D7111yescharacteristic
252C8—1422no—
253C8—1422no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—2841no—
261C22—2841no—
262C22—2841no—
263C22#1455614812yes—
264C22C22×C22⋊D7111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—2841no—
271C4—2841no—
272C4—2841no—
273C4#1455615412yes—
274C4C2×#1176586512yes—
275C2—22no—
276C2—44no—
277C2—5681no—
278C2#1798651011yescharacteristic
279C2C2×#1455614811yescharacteristic
280C2C2×#1455615411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C4×D4×C71C2
socleC22×C71C22×D4
hypercenterC2×C4×D4D71
hypocenterC71#3105
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #17986533C2semidirect
2. #17986334C2semidirect
3. #17986317C2semidirect
4. #17986299C2semidirect
5. #17986214C2semidirect
6. #17986150C2semidirect
8. C4×#11765853C2semidirect
9. C4×#11765850C2semidirect
10. #17986052C2semidirect
11. C2×C4×D4×C71C2semidirect
15. #14556120C22semidirect
16. #14556116C22semidirect
20. C4×Dic142C22semidirect
21. C4×D284C22semidirect
24. #14556061C22semidirect
25. #14556059C22semidirect
26. C4⋊(C71⋊C8)C22semidirect
27. #14556051C22semidirect
30. #14556048C22semidirect
33. C4×C71⋊C8C22semidirect
141. C71#3105semidirect
143. C2×C4×D4D71semidirect