Finite Group Databasebeta

← #22599517

Subgroups — #22599517

Order 9088.

332 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 113 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#17986554C211yesmaximal, characteristic, complemented
2#17986531C211yesmaximal, characteristic
3#17986400C212yesmaximal, complemented
4#17986352C212yesmaximal, complemented
5#17986342C212yesmaximal, complemented
6#17986321C212yesmaximal
7#17986295C211yesmaximal, characteristic, complemented
8#17986289C211yesmaximal, characteristic, complemented
9#17986281C211yesmaximal, characteristic, complemented
10#17986051C211yesmaximal, characteristic, complemented
11#474×C71C211yesmaximal, characteristic, complemented
12#14556128C2211yescharacteristic
13#14556121—21no—
14#14556119C2212yes—
15#14556119C2211yescharacteristic
16#14556115C2211yescharacteristic
17#14556110—21no—
18#14556108—21no—
19#14556108C2211yescharacteristic
20#14556104—21no—
21#14556099C2211yescharacteristic
22Q8×Dic71—21no—
23C4×Dic142C2211yescharacteristic
24#14556093—21no—
25C4×C22⋊D71—21no—
26#14556087—21no—
27#14556086C2212yes—
28C42×D71C2211yescharacteristic
29#14556062C2212yes—
30#14556059C2212yes—
31#14556059C2212yes—
32C4⋊(C71⋊C8)C2211yescharacteristic
33#14556053C2212yes—
34#14556051C2212yes—
35#14556049C2212yes—
36(C2×D71)⋊C8C2212yes—
37C8×Dic71C2212yes—
38C4⋊Q8×C71C2211yescharacteristic
39C4×D4×C71C2211yescharacteristic
40#107×C71C2212yes—
41C4⋊C8×C71C2211yescharacteristic
42#102×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71D411yescharacteristic
47C2×C4×D71D411yescharacteristic
48C2×C4×D71C2311yescharacteristic
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—21no—
51C4⋊Dic71—21no—
52C4⋊Dic71—21no—
53C4⋊Dic71—22no—
54C4⋊Dic71C2312yes—
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58Dic71⋊C4—22no—
59C22⋊Dic71—21no—
60C22⋊Dic71—21no—
61C22⋊Dic71—21no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—21no—
64C4×Dic71—21no—
65C4×Dic71—22no—
66C4×Dic71D411yescharacteristic
67C4×Dic71D411yescharacteristic
68C4×Dic71C2311yescharacteristic
69C2×Q8×C71—21no—
70C2×D4×C71C2311yescharacteristic
71C22×C4×C71—21no—
72C2×C8×C71C2312yes—
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C4⋊C4×C71C2311yescharacteristic
76C22⋊C4×C71—21no—
77C42×C71C2311yescharacteristic
78C23×C71—21no—
79C22×D71C2×D411yescharacteristic
80Q8×C71—42no—
81D4×C71—21no—
82D4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—42no—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2411yescharacteristic
89C22⋊D71—42no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—21no—
93C2×Dic71C2×D411yescharacteristic
94C2×Dic71C2×D411yescharacteristic
95C2×Dic71C2×D411yescharacteristic
96C4×D71—22no—
97C4×D71—21no—
98C4×D71—21no—
99C4×D71—41no—
100Dic142—21no—
101Dic142—21no—
102Dic142—42no—
103C8×C71—22no—
104C71⋊C8—22no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71—41no—
108C22×C71C22×D411yescharacteristic
109C2×D71—21no—
110C2×D71—21no—
111C2×D71—21no—
112C4×C71—21no—
113C4×C71—41no—
114C4×C71—42no—
115C4×C71#13512yes—
116C4×C712−1+411yescharacteristic
117C4×C712−1+411yescharacteristic
118Dic71—21no—
119Dic71—21no—
120Dic71—21no—
121Dic71—41no—
122Dic71—42no—
123Dic71—41no—
124C2×C71—81no—
125C2×C71#53611yescharacteristic
126C2×C71C2×#13511yescharacteristic
127C2×C71#57511yescharacteristic
128D71—41no—
129#3233—711nomaximal, Sylow
130C71#323311yescharacteristic, complemented, Sylow
131#533—711no—
132#529—711no—
133#499—712no—
134#488—712no—
135#485—712no—
136#482—712no—
137#477—711no—
138#476—711no—
139#475—711no—
140#474D7111yescharacteristic, complemented
141#431—711no—
142C4⋊Q8—711no—
143C4⋊Q8C2×D7111yescharacteristic
144#126—1421no—
145#125—711no—
146#125—712no—
147#124—711no—
148#123—1421no—
149C22⋊Q8—711no—
150C22⋊Q8—1421no—
151C22⋊Q8—1421no—
152#121—711no—
153C4×Q8—711no—
154C4×Q8—1421no—
155C4×Q8—1421no—
156C4×D4—1421no—
157C4×D4C2×D7111yescharacteristic
158#117—712no—
159#117—1421no—
160C2×C42—711no—
161#107—712no—
162#107C2×D7112yes—
163#106—712no—
164#106—712no—
165C4⋊C8—711no—
166C4⋊C8C2×D7111yescharacteristic
167#103—712no—
168#103—712no—
169#102—712no—
170#102C2×D7112yes—
171C22⋊C8—712no—
172C4×C8—712no—
173C2×Q8—21no—
174C2×Q8—711no—
175C2×Q8—1421no—
176C2×D4—1421no—
177C2×D4C22×D7111yescharacteristic
178C22×C4—21no—
179C22×C4—711no—
180C22×C4—711no—
181C22×C4—711no—
182C2×C8—712no—
183C2×C8C22×D7112yes—
184C4⋊C4—21no—
185C4⋊C4—712no—
186C4⋊C4—711no—
187C4⋊C4—1421no—
188C4⋊C4—1422no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4—1422no—
192C4⋊C4—1421no—
193C4⋊C4—1421no—
194C4⋊C4C22×D7112yes—
195C4⋊C4C22×D7111yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—1422no—
198C22⋊C4—1421no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C42—711no—
203C42—711no—
204C42—711no—
205C42—1422no—
206C42—1421no—
207C42C22×D7111yescharacteristic
208C23—21no—
209C23—711no—
210Q8—42no—
211Q8—2842no—
212Q8—1421no—
213Q8—1421no—
214D4—21no—
215D4—21no—
216D4—2842no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—2841no—
221C2×C4—711no—
222C2×C4—711no—
223C2×C4—711no—
224C2×C4—1421no—
225C2×C4—1421no—
226C2×C4—1422no—
227C2×C4—1421no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4D4×D7111yescharacteristic
231C2×C4D4×D7111yescharacteristic
232C2×C4C23×D7111yescharacteristic
233C8—22no—
234C8—1422no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1421no—
239C22—1421no—
240C22—1421no—
241C22C2×D4×D7111yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—2841no—
246C4—2841no—
247C4—2842no—
248C4—1421no—
249C4—1421no—
250C4—1421no—
251C4#1455614712yes—
252C4#1455617511yescharacteristic
253C4#1455617611yescharacteristic
254C2—81no—
255C2—2841no—
256C2#1798657211yescharacteristic
257C2C2×#1455614711yescharacteristic
258C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#474×C71C2
socleC22×C71C22×D4
hypercenter#474D71
hypocenterC71#3233
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986400C2semidirect
4. #17986352C2semidirect
5. #17986342C2semidirect
7. #17986295C2semidirect
8. #17986289C2semidirect
9. #17986281C2semidirect
10. #17986051C2semidirect
11. #474×C71C2semidirect
130. C71#3233semidirect
140. #474D71semidirect