Finite Group Databasebeta

← #22599384

Subgroups — #22599384

Order 9088.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986568C211yesmaximal, characteristic, complemented
2#17986386C211yesmaximal, characteristic, complemented
3#17986375C211yesmaximal, characteristic, complemented
4#17986365C211yesmaximal, characteristic, complemented
5#17986356C211yesmaximal, characteristic, complemented
6#17986214C212yesmaximal, complemented
7#17986194C212yesmaximal, complemented
8#17986142C212yesmaximal, complemented
9#17986132C212yesmaximal, complemented
10#17986060C211yesmaximal, characteristic, complemented
11#540×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C2×#11765853C2212yescomplemented
15C2×#11765850C2212yes—
16C2×#11765850C2212yes—
17C2×#11765850C2212yescomplemented
18#14556125C2211yescharacteristic
19#14556125C2211yescharacteristic
20#14556116C2211yescharacteristic
21#14556116C2211yescharacteristic
22#14556103—22no—
23#14556098—22no—
24C4⋊(C71⋊C8)C2212yescomplemented
25#14556051C2212yescomplemented
26#14556048C2212yes—
27#14556048C2212yes—
28#14556048C2212yescomplemented
29C22⋊(C71⋊C8)C2212yes—
30#14556039C2211yescharacteristic, complemented
31C4×C71⋊C8C2211yescharacteristic, complemented
32#127×C71C2211yescharacteristic, complemented
33#125×C71C2211yescharacteristic, complemented
34#123×C71—22no—
35#121×C71—22no—
36#121×C71C2212yes—
37C4×D4×C71C2212yescomplemented
38#117×C71C2211yescharacteristic
39C23×D71—21no—
40#11765865—42no—
41C2×Dic142C2311yescharacteristic
42C2×D284—21no—
43C2×D284—22no—
44C2×D284—22no—
45C2×D284—42no—
46C2×D284C2311yescharacteristic
47C2×D284C2311yescharacteristic
48C2×D284C2311yescharacteristic
49C2×C22⋊D71—21no—
50C2×C4×D71—21no—
51#11765853—42no—
52#11765850—42no—
53#11765850—42no—
54#11765850—42no—
55C2×C71⋊C8C2312yes—
56C2×C71⋊C8C2312yes—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59C2×D4×C71—22no—
60C2×D4×C71—22no—
61C2×D4×C71C2312yes—
62C22×C4×C71—22no—
63C22×C4×C71C2311yescharacteristic
64C4⋊C4×C71—22no—
65C4⋊C4×C71D412yes—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C22⋊C4×C71D412yes—
70C42×C71C2311yescharacteristic
71C42×C71C2311yescharacteristic
72C23×C71—22no—
73C23×C71C2×D411yescharacteristic
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—21no—
77C22×D71—42no—
78C22×D71—82no—
79D4×C71—22no—
80D4×C71—22no—
81D4×C71—82no—
82D4×C71—82no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—82no—
86C2×C4×C71C2×D412yes—
87C2×C4×C71C2×D412yes—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—42no—
91C2×Dic71—21no—
92D284—22no—
93D284—22no—
94D284—22no—
95D284—42no—
96D284—82no—
97D284—82no—
98C4×D71—42no—
99Dic142—22no—
100C71⋊C8—22no—
101C71⋊C8—22no—
102C22×C71—21no—
103C22×C71—22no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71C22×D411yescharacteristic
108C2×D71—41no—
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—42no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—84no—
115C4×C71—21no—
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—42no—
119C4×C71H(5,𝔽2)12yes—
120Dic71—41no—
121C2×C71—41no—
122C2×C71—82no—
123C2×C71#53511yescharacteristic
124C2×C71#57612yes—
125D71—81no—
126D71—81no—
127D71—81no—
128#3220—711nomaximal, Sylow
129C71#322011yescharacteristic, complemented, Sylow
130#540D7111yescharacteristic, complemented
131#535—711no—
132#496—711no—
133#493—711no—
134#490—711no—
135#488—711no—
136#463—712no—
137#459—712no—
138#449—712no—
139#447—712no—
140#433—711no—
141C2×#41—711no—
142C22×D4—711no—
143C2×SD16—712no—
144C2×D8—712no—
145C2×D8—712no—
146C2×D8—712no—
147#127—711no—
148#127—711no—
149#127C2×D7111yescharacteristic, complemented
150#125C2×D7111yescharacteristic, complemented
151#124—711no—
152#124—711no—
153#123—22no—
154#121—22no—
155#121—1422no—
156#121C2×D7112yes—
157#120—1422no—
158C4×D4C2×D7112yescomplemented
159#117C2×D7111yescharacteristic
160C4⋊C8—712no—
161#103—712no—
162#102—712no—
163#102—712no—
164#102—712no—
165C22⋊C8—712no—
166#97—711no—
167C4×C8—711no—
168C24—1421no—
169#41—2842no—
170C2×Q8—711no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—2842no—
174C2×D4—711no—
175C2×D4—711no—
176C2×D4—711no—
177C2×D4—1421no—
178C2×D4—1421no—
179C2×D4—1422no—
180C2×D4—1422no—
181C2×D4C22×D7112yes—
182C22×C4—22no—
183C22×C4—1421no—
184C22×C4C22×D7111yescharacteristic
185SD16—2842no—
186D8—2842no—
187D8—2842no—
188D8—2842no—
189C2×C8—712no—
190C2×C8—712no—
191C4⋊C4—22no—
192C4⋊C4C22⋊D7112yes—
193C4⋊C4C22×D7112yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C22⋊C4C22⋊D7112yes—
199C42C22×D7111yescharacteristic
200C42C22×D7111yescharacteristic
201C23—22no—
202C23—2842no—
203C23—5682no—
204C23—1421no—
205C23—1421no—
206C23—1421no—
207C23C2×C22⋊D7111yescharacteristic
208Q8—1422no—
209D4—22no—
210D4—22no—
211D4—82no—
212D4—82no—
213D4—2842no—
214D4—2842no—
215D4—5682no—
216D4—5682no—
217D4—1422no—
218D4—1422no—
219D4—1422no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—82no—
223C2×C4—2842no—
224C2×C4—1421no—
225C2×C4C2×C22⋊D7112yes—
226C2×C4C2×C22⋊D7112yes—
227C2×C4C2×C22⋊D7111yescharacteristic
228C2×C4C23×D7111yescharacteristic
229C8—1422no—
230C8—1422no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2841no—
237C22—2841no—
238C22—2841no—
239C22—2842no—
240C22—2842no—
241C22—2842no—
242C22—5684no—
243C22C22×C22⋊D7111yescharacteristic
244C4—21no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—2841no—
249C4#1455617412yes—
250C2—41no—
251C2—82no—
252C2—5681no—
253C2—5681no—
254C2—5681no—
255C2#1798656311yescharacteristic
256C2#1798680212yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#540×C71C2
socleC22×C71C22×D4
hypercenter#540D71
hypocenterC71#3220
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #17986568C2semidirect
2. #17986386C2semidirect
3. #17986375C2semidirect
4. #17986365C2semidirect
5. #17986356C2semidirect
6. #17986214C2semidirect
7. #17986194C2semidirect
8. #17986142C2semidirect
9. #17986132C2semidirect
10. #17986060C2semidirect
11. #540×C71C2semidirect
14. C2×#11765853C22semidirect
17. C2×#11765850C22semidirect
24. C4⋊(C71⋊C8)C22semidirect
25. #14556051C22semidirect
28. #14556048C22semidirect
30. #14556039C22semidirect
31. C4×C71⋊C8C22semidirect
32. #127×C71C22semidirect
33. #125×C71C22semidirect
37. C4×D4×C71C22semidirect
129. C71#3220semidirect
130. #540D71semidirect
149. #127C2×D71semidirect
150. #125C2×D71semidirect
158. C4×D4C2×D71semidirect