Finite Group Databasebeta

← #22599390

Subgroups — #22599390

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#17986604C211yesmaximal, characteristic, complemented
2#17986389C211yesmaximal, characteristic, complemented
3#17986375C211yesmaximal, characteristic, complemented
4#17986363C211yesmaximal, characteristic, complemented
5#17986353C211yesmaximal, characteristic, complemented
6#17986211C212yesmaximal, complemented
7#17986191C212yesmaximal, complemented
8#17986144C212yesmaximal, complemented
9#17986133C212yesmaximal, complemented
10#17986060C211yesmaximal, characteristic, complemented
11#535×C71C211yesmaximal, characteristic, complemented
12C2×#11765852C2212yescomplemented
13C2×#11765850C2212yes—
14C2×#11765850C2212yes—
15C2×#11765850C2212yescomplemented
16#14556125C2211yescharacteristic, complemented
17#14556120C2211yescharacteristic, complemented
18#14556113—22no—
19#14556102C2212yes—
20#14556101—22no—
21C4×D284C2212yescomplemented
22C4⋊(C71⋊C8)C2212yescomplemented
23#14556052C2212yescomplemented
24#14556049C2212yes—
25#14556049C2212yes—
26#14556049C2212yescomplemented
27C22⋊(C71⋊C8)C2212yes—
28#14556039C2211yescharacteristic, complemented
29C4×C71⋊C8C2211yescharacteristic, complemented
30C2×#41×C71C2211yescharacteristic
31C22×D4×C71C2211yescharacteristic
32#127×C71C2211yescharacteristic
33#127×C71C2211yescharacteristic
34#124×C71C2211yescharacteristic
35#124×C71C2211yescharacteristic
36#121×C71—22no—
37#120×C71—22no—
38#117×C71C2211yescharacteristic
39C2×D284—22no—
40C2×D284C2312yes—
41C2×C22⋊D71—22no—
42C2×C4×D71—22no—
43#11765852—42no—
44#11765850—42no—
45#11765850—42no—
46#11765850—42no—
47C2×C71⋊C8C2312yes—
48C2×C71⋊C8C2312yes—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C24×C71—21no—
54#41×C71—42no—
55C2×Q8×C71C2311yescharacteristic
56C2×D4×C71—21no—
57C2×D4×C71—21no—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—42no—
61C2×D4×C71C2311yescharacteristic
62C2×D4×C71C2311yescharacteristic
63C2×D4×C71C2311yescharacteristic
64C22×C4×C71—21no—
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71D412yes—
67C22⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C22⋊C4×C71D412yes—
70C42×C71C2311yescharacteristic
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—21no—
75C23×C71—42no—
76C23×C71—82no—
77C23×C71C2×D411yescharacteristic
78C22×D71—22no—
79Q8×C71—22no—
80D4×C71—22no—
81D4×C71—22no—
82D4×C71—22no—
83D4×C71—42no—
84D4×C71—42no—
85D4×C71—82no—
86D4×C71—82no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—42no—
90C2×C4×C71C2×D412yes—
91C2×C4×C71C2×D412yes—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2411yescharacteristic
94C22⋊D71—82no—
95C2×Dic71—22no—
96D284—22no—
97D284—22no—
98D284—82no—
99C4×D71—82no—
100C71⋊C8—22no—
101C71⋊C8—22no—
102C22×C71—21no—
103C22×C71—22no—
104C22×C71—41no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71—42no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71—84no—
111C22×C71C22×D411yescharacteristic
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—42no—
115C4×C71—21no—
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—41no—
119C4×C71H(5,𝔽2)12yes—
120Dic71—42no—
121C2×C71—41no—
122C2×C71—81no—
123C2×C71—81no—
124C2×C71—81no—
125C2×C71#53511yescharacteristic
126C2×C71#57612yes—
127D71—82no—
128#3220—711nomaximal, Sylow
129C71#322011yescharacteristic, complemented, Sylow
130#540—711no—
131#535D7111yescharacteristic, complemented
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×#41C2×D7111yescharacteristic
142C22×D4C2×D7111yescharacteristic
143C2×SD16—712no—
144C2×D8—712no—
145C2×D8—712no—
146C2×D8—712no—
147#127—711no—
148#127C2×D7111yescharacteristic
149#127C2×D7111yescharacteristic
150#125—711no—
151#124C2×D7111yescharacteristic
152#124C2×D7111yescharacteristic
153#123—1422no—
154#121—22no—
155#121—712no—
156#121—1422no—
157#120—22no—
158C4×D4—712no—
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—21no—
169#41—42no—
170C2×Q8C22×D7111yescharacteristic
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—712no—
177C2×D4—1422no—
178C2×D4—1422no—
179C2×D4C22×D7111yescharacteristic
180C2×D4C22×D7111yescharacteristic
181C2×D4C22×D7111yescharacteristic
182C22×C4—21no—
183C22×C4—1422no—
184C22×C4C22×D7111yescharacteristic
185SD16—2842no—
186D8—2842no—
187D8—2842no—
188D8—2842no—
189C2×C8—712no—
190C2×C8—712no—
191C4⋊C4—712no—
192C4⋊C4—1422no—
193C4⋊C4C22⋊D7112yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C22⋊C4C22⋊D7112yes—
199C42C22×D7111yescharacteristic
200C42C22×D7111yescharacteristic
201C23—21no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23—82no—
206C23—1422no—
207C23C2×C22⋊D7111yescharacteristic
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—42no—
213D4—42no—
214D4—82no—
215D4—82no—
216D4—5682no—
217D4—5682no—
218D4—1422no—
219D4—1422no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—5682no—
224C2×C4—1422no—
225C2×C4C2×C22⋊D7112yes—
226C2×C4C2×C22⋊D7112yes—
227C2×C4C2×C22⋊D7111yescharacteristic
228C2×C4C23×D7111yescharacteristic
229C8—1422no—
230C8—1422no—
231C22—21no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—84no—
240C22—2842no—
241C22—2842no—
242C22—2842no—
243C22C22×C22⋊D7111yescharacteristic
244C4—21no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—2842no—
249C4#1455617312yes—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—81no—
254C2—5682no—
255C2#1798656611yescharacteristic
256C2#1798680212yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#535×C71C2
socleC22×C71C22×D4
hypercenter#535D71
hypocenterC71#3220
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #17986604C2semidirect
2. #17986389C2semidirect
3. #17986375C2semidirect
4. #17986363C2semidirect
5. #17986353C2semidirect
6. #17986211C2semidirect
7. #17986191C2semidirect
8. #17986144C2semidirect
9. #17986133C2semidirect
10. #17986060C2semidirect
11. #535×C71C2semidirect
12. C2×#11765852C22semidirect
15. C2×#11765850C22semidirect
16. #14556125C22semidirect
17. #14556120C22semidirect
21. C4×D284C22semidirect
22. C4⋊(C71⋊C8)C22semidirect
23. #14556052C22semidirect
26. #14556049C22semidirect
28. #14556039C22semidirect
29. C4×C71⋊C8C22semidirect
129. C71#3220semidirect
131. #535D71semidirect