Finite Group Databasebeta

← #22599385

Subgroups — #22599385

Order 9088.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986603C211yesmaximal, characteristic, complemented
2#17986567C211yesmaximal, characteristic, complemented
3#17986375C211yesmaximal, characteristic, complemented
4#17986364C211yesmaximal, characteristic, complemented
5#17986353C211yesmaximal, characteristic, complemented
6#17986216C212yesmaximal, complemented
7#17986192C212yesmaximal, complemented
8#17986143C212yesmaximal, complemented
9#17986131C212yesmaximal, complemented
10#17986061C211yesmaximal, characteristic, complemented
11#496×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765850C2212yes—
15C2×#11765850C2212yescomplemented
16#14556125C2211yescharacteristic
17#14556124C2211yescharacteristic
18#14556120C2211yescharacteristic, complemented
19#14556117C2211yescharacteristic
20#14556109—22no—
21#14556101—22no—
22#14556100—22no—
23#14556099C2212yes—
24#14556098—22no—
25D4×Dic71C2212yescomplemented
26#14556085C2211yescharacteristic
27Dic71⋊C8C2212yescomplemented
28#14556052C2212yescomplemented
29#14556050C2212yes—
30#14556050C2212yescomplemented
31#14556049C2212yes—
32(C2×D71)⋊C8C2212yes—
33C4×C71⋊C8C2211yescharacteristic, complemented
34C2×SD16×C71C2212yescomplemented
35C2×D8×C71C2212yes—
36#127×C71C2211yescharacteristic, complemented
37#124×C71C2211yescharacteristic
38#97×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40#11765866—42no—
41C2×D284—21no—
42C2×D284—22no—
43C2×D284C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—22no—
46C2×C22⋊D71—21no—
47C2×C22⋊D71—22no—
48D4×D71—42no—
49C22×Dic71—22no—
50C2×C4×D71—21no—
51C2×C4×D71C2311yescharacteristic
52#11765850—42no—
53#11765850—42no—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71C2312yes—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58C22⋊Dic71—22no—
59C22⋊Dic71—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4D412yes—
62C4×Dic71C2311yescharacteristic
63C2×Q8×C71C2311yescharacteristic
64C2×D4×C71—22no—
65C2×D4×C71C2311yescharacteristic
66C2×D4×C71C2312yes—
67SD16×C71—42no—
68D8×C71—42no—
69C2×C8×C71C2312yes—
70C22⋊C4×C71—22no—
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C23×C71—22no—
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—42no—
77C22×D71—82no—
78C22×D71C2×D411yescharacteristic
79Q8×C71—22no—
80D4×C71—22no—
81D4×C71—22no—
82D4×C71—22no—
83D4×C71—82no—
84C2×C4×C71—21no—
85C2×C4×C71C2×D412yes—
86C2×C4×C71C2411yescharacteristic
87C22⋊D71—42no—
88C22⋊D71—82no—
89C22⋊D71—82no—
90C2×Dic71—22no—
91C2×Dic71—82no—
92C2×Dic71C2×D412yes—
93C2×Dic71C2×D411yescharacteristic
94D284—22no—
95D284—22no—
96D284—42no—
97D284—82no—
98C4×D71—22no—
99C4×D71—42no—
100C8×C71—22no—
101C71⋊C8—22no—
102C22×C71—41no—
103C22×C71—42no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71C22×D411yescharacteristic
108C2×D71—21no—
109C2×D71—22no—
110C2×D71—41no—
111C2×D71—41no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—84no—
115C4×C71—22no—
116C4×C71—41no—
117C4×C71H(5,𝔽2)12yes—
118Dic71—21no—
119Dic71—22no—
120Dic71—42no—
121C2×C71—81no—
122C2×C71—82no—
123C2×C71#53511yescharacteristic
124C2×C71#57612yes—
125D71—41no—
126D71—81no—
127D71—81no—
128#3220—711nomaximal, Sylow
129C71#322011yescharacteristic, complemented, Sylow
130#540—711no—
131#535—711no—
132#496D7111yescharacteristic, complemented
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×SD16C2×D7112yescomplemented
144C2×D8—712no—
145C2×D8—712no—
146C2×D8C2×D7112yes—
147#127—711no—
148#127—711no—
149#127C2×D7111yescharacteristic, complemented
150#125—711no—
151#124—711no—
152#124C2×D7111yescharacteristic
153#123—1422no—
154#121—712no—
155#121—1422no—
156#121—1422no—
157#120—1422no—
158C4×D4—712no—
159#117—711no—
160C4⋊C8—712no—
161#103—712no—
162#102—712no—
163#102—712no—
164#102—712no—
165C22⋊C8—712no—
166#97C2×D7111yescharacteristic, complemented
167C4×C8—711no—
168C24—1421no—
169#41—2842no—
170C2×Q8C22×D7111yescharacteristic
171C2×D4—22no—
172C2×D4—2842no—
173C2×D4—711no—
174C2×D4—711no—
175C2×D4—1421no—
176C2×D4—1421no—
177C2×D4—1422no—
178C2×D4—1422no—
179C2×D4—1422no—
180C2×D4C22×D7111yescharacteristic
181C2×D4C22×D7112yes—
182C22×C4—711no—
183C22×C4—1421no—
184C22×C4—1422no—
185SD16—42no—
186D8—42no—
187D8—2842no—
188D8—2842no—
189C2×C8—712no—
190C2×C8C22×D7112yes—
191C4⋊C4—712no—
192C4⋊C4—712no—
193C4⋊C4—1422no—
194C22⋊C4—22no—
195C22⋊C4—712no—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C22⋊C4—1422no—
199C42—711no—
200C42C22×D7111yescharacteristic
201C23—21no—
202C23—22no—
203C23—2842no—
204C23—5682no—
205C23—711no—
206C23—1421no—
207C23—1421no—
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—82no—
213D4—2842no—
214D4—2842no—
215D4—5682no—
216D4—5682no—
217D4—5682no—
218D4—1422no—
219D4—1422no—
220C2×C4—21no—
221C2×C4—2842no—
222C2×C4—5682no—
223C2×C4—712no—
224C2×C4—711no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4D4×D7112yes—
228C2×C4C23×D7111yescharacteristic
229C8—22no—
230C8—1422no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2841no—
237C22—2841no—
238C22—2842no—
239C22—2842no—
240C22—5684no—
241C22—1421no—
242C22—1422no—
243C22C2×D4×D7111yescharacteristic
244C4—22no—
245C4—41no—
246C4—2842no—
247C4—1421no—
248C4—1422no—
249C4#1455617312yes—
250C2—81no—
251C2—82no—
252C2—2841no—
253C2—5681no—
254C2—5681no—
255C2#1798656211yescharacteristic
256C2#1798680312yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#496×C71C2
socleC22×C71C22×D4
hypercenter#496D71
hypocenterC71#3220
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #17986603C2semidirect
2. #17986567C2semidirect
3. #17986375C2semidirect
4. #17986364C2semidirect
5. #17986353C2semidirect
6. #17986216C2semidirect
7. #17986192C2semidirect
8. #17986143C2semidirect
9. #17986131C2semidirect
10. #17986061C2semidirect
11. #496×C71C2semidirect
15. C2×#11765850C22semidirect
18. #14556120C22semidirect
25. D4×Dic71C22semidirect
27. Dic71⋊C8C22semidirect
28. #14556052C22semidirect
30. #14556050C22semidirect
33. C4×C71⋊C8C22semidirect
34. C2×SD16×C71C22semidirect
36. #127×C71C22semidirect
38. #97×C71C22semidirect
129. C71#3220semidirect
132. #496D71semidirect
143. C2×SD16C2×D71semidirect
149. #127C2×D71semidirect
166. #97C2×D71semidirect