Finite Group Databasebeta

← #22599521

Subgroups — #22599521

Order 9088.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986556C211yesmaximal, characteristic, complemented
2#17986533C211yesmaximal, characteristic
3#17986401C212yesmaximal, complemented
4#17986354C212yesmaximal, complemented
5#17986341C212yesmaximal, complemented
6#17986324C212yesmaximal
7#17986295C211yesmaximal, characteristic, complemented
8#17986290C211yesmaximal, characteristic, complemented
9#17986285C211yesmaximal, characteristic, complemented
10#17986280C211yesmaximal, characteristic, complemented
11#431×C71C211yesmaximal, characteristic, complemented
12#14556128C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556122—21no—
15#14556120C2212yes—
16#14556120C2211yescharacteristic
17#14556116C2211yescharacteristic
18#14556111—21no—
19#14556107—21no—
20#14556107—21no—
21#14556107C2211yescharacteristic
22#14556102C2211yescharacteristic
23C4×Dic142—21no—
24C4×Dic142—21no—
25C4×Dic142C2211yescharacteristic
26C4×D284C2211yescharacteristic
27C4×C22⋊D71—21no—
28#14556087C2212yes—
29#14556085—21no—
30#14556062C2212yes—
31#14556062C2212yes—
32#14556060C2212yes—
33#14556060C2212yes—
34#14556053C2212yes—
35#14556053C2212yes—
36#14556050C2212yes—
37#14556050C2212yes—
38C2×C42×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40C4⋊C8×C71C2211yescharacteristic
41C22⋊C8×C71C2212yes—
42C4×C8×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—21no—
48C2×C4×D71—21no—
49C4⋊Dic71—21no—
50C4⋊Dic71—21no—
51C4⋊Dic71C2312yes—
52C4⋊Dic71C2312yes—
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—21no—
61C22⋊Dic71—22no—
62C22⋊Dic71—21no—
63(C2×D71)⋊C4—21no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67C4×Dic71—21no—
68C4×Dic71—22no—
69C22×C4×C71D411yescharacteristic
70C22×C4×C71D411yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C2×C8×C71C2312yes—
73C2×C8×C71C2312yes—
74C42×C71D411yescharacteristic
75C42×C71D411yescharacteristic
76C42×C71C2311yescharacteristic
77C42×C71C2311yescharacteristic
78C23×C71C2×D411yescharacteristic
79C22×D71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—41no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—42no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96D284—21no—
97D284—21no—
98C4×D71—42no—
99Dic142—21no—
100Dic142—21no—
101Dic142—42no—
102Dic142—42no—
103C8×C71—22no—
104C8×C71—22no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71C22×D411yescharacteristic
109C2×D71—41no—
110C2×D71—41no—
111C2×D71—41no—
112C4×C71—21no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—21no—
116C4×C71#13512yes—
117C4×C712−1+411yescharacteristic
118C4×C712−1+411yescharacteristic
119Dic71—41no—
120Dic71—42no—
121Dic71—41no—
122Dic71—42no—
123Dic71—41no—
124C2×C71—41no—
125C2×C71#53611yescharacteristic
126C2×C71C2×#13511yescharacteristic
127C2×C71#57511yescharacteristic
128D71—81no—
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#474—711no—
141#431D7111yescharacteristic, complemented
142C4⋊Q8—711no—
143C4⋊Q8—711no—
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—711no—
157C4×D4—1421no—
158#117—712no—
159#117—1421no—
160C2×C42C2×D7111yescharacteristic
161#107—712no—
162#107—712no—
163#106—712no—
164#106—712no—
165C4⋊C8C2×D7111yescharacteristic
166C4⋊C8C2×D7111yescharacteristic
167#103—712no—
168#103—712no—
169#102—712no—
170#102—712no—
171C22⋊C8C2×D7112yes—
172C4×C8C2×D7112yes—
173C2×Q8—711no—
174C2×Q8—1421no—
175C2×Q8—1421no—
176C2×D4—711no—
177C2×D4—1421no—
178C22×C4—1421no—
179C22×C4D28411yescharacteristic
180C22×C4D28411yescharacteristic
181C22×C4C22×D7111yescharacteristic
182C2×C8C22×D7112yes—
183C2×C8C22×D7112yes—
184C4⋊C4—712no—
185C4⋊C4—712no—
186C4⋊C4—711no—
187C4⋊C4—711no—
188C4⋊C4—1421no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1421no—
195C4⋊C4—1421no—
196C22⋊C4—1422no—
197C22⋊C4—1421no—
198C22⋊C4—1421no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C42—1422no—
203C42—1421no—
204C42D28411yescharacteristic
205C42D28411yescharacteristic
206C42C22×D7111yescharacteristic
207C42C22×D7111yescharacteristic
208C23—1421no—
209C23C2×D28411yescharacteristic
210Q8—2842no—
211Q8—2842no—
212Q8—1421no—
213Q8—1421no—
214D4—2842no—
215D4—1421no—
216D4—1421no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—2842no—
222C2×C4—1421no—
223C2×C4—1421no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1421no—
227C2×C4C2×D28411yescharacteristic
228C2×C4C2×D28411yescharacteristic
229C2×C4C2×D28411yescharacteristic
230C2×C4C2×D28411yescharacteristic
231C2×C4C2×D28411yescharacteristic
232C2×C4C23×D7111yescharacteristic
233C8—22no—
234C8—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—2841no—
239C22—2841no—
240C22—2841no—
241C22C22×D28411yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—2841no—
247C4—2841no—
248C4—2842no—
249C4—2841no—
250C4—2842no—
251C4#1455615012yes—
252C4#1455617511yescharacteristic
253C4#1455617511yescharacteristic
254C2—41no—
255C2—5681no—
256C2#1798657511yescharacteristic
257C2C2×#1455615011yescharacteristic
258C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#431×C71C2
socleC22×C71C22×D4
hypercenter#431D71
hypocenterC71#3233
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #17986556C2semidirect
3. #17986401C2semidirect
4. #17986354C2semidirect
5. #17986341C2semidirect
7. #17986295C2semidirect
8. #17986290C2semidirect
9. #17986285C2semidirect
10. #17986280C2semidirect
11. #431×C71C2semidirect
130. C71#3233semidirect
141. #431D71semidirect