Finite Group Databasebeta

← #22593911

Subgroups — #22593911

Order 9088.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#14556085C211yesmaximal, characteristic
2#17985930C211yesmaximal, characteristic, complemented
3#17985925C211yesmaximal, characteristic
4#17985918C211yesmaximal, characteristic, complemented
5#17985910C211yesmaximal, characteristic
6#17985890C212yesmaximal
7#17985883C212yesmaximal, complemented
8#17985880C212yesmaximal, complemented
9#17985870C212yesmaximal
10#17985867C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C71C211yesmaximal, characteristic, complemented
12C22×C4×D71C2211yescharacteristic
13#14556085—24no—
14C2×Dic71⋊C4C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2211yescharacteristic
17C2×Dic71⋊C4C2211yescharacteristic
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2211yescharacteristic
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×C4×Dic71C2211yescharacteristic
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C22×Q8×C71C2211yescharacteristic
32C4×Q8×C71—24no—
33C2×C4⋊C4×C71C2212yes—
34C2×C4⋊C4×C71C2211yescharacteristic
35C2×C42×C71C2212yes—
36C2×C42×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C22×Dic71C2312yes—
39C22×Dic71C2312yes—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C2×C4×D71—22no—
44C2×C4×D71—22no—
45C2×C4×D71—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—24no—
49Dic71⋊C4—24no—
50(C2×D71)⋊C4—24no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—24no—
54C4×Dic71—22no—
55C2×Q8×C71—22no—
56C2×Q8×C71—24no—
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C4⋊C4×C71—24no—
63C4⋊C4×C71—22no—
64C42×C71—24no—
65C42×C71D414yes—
66C23×C71C2411yescharacteristic
67C22×D71—21no—
68C22×D71—22no—
69C22×D71—21no—
70C22×D71—22no—
71C22×D71#4112yes—
72Q8×C71—28no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2×D412yes—
84C2×C4×C71C2×D412yes—
85C2×C4×C71#4114yes—
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71#4112yes—
99C4×D71—28no—
100C22×C71C22×D411yescharacteristic
101C22×C71C2×#4111yescharacteristic
102C22×C71C2×#4111yescharacteristic
103C22×C71C2×#4112yes—
104C22×C712−1+412yes—
105C2×D71—22no—
106C2×D71—24no—
107C2×D71—41no—
108C2×D71—41no—
109C2×D71—42no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—24no—
113C4×C71—22no—
114C4×C71—42no—
115Dic71—22no—
116Dic71—41no—
117Dic71—41no—
118Dic71—42no—
119Dic71—42no—
120C2×C71#52511yescharacteristic
121C2×C71#52511yescharacteristic
122C2×C71#53611yescharacteristic
123C2×C71#55212yes—
124C2×C71#55612yes—
125D71—42no—
126#2599—711nomaximal, Sylow
127C71#259911yescharacteristic, complemented, Sylow
128C2×C4×Q8D7111yescharacteristic, complemented
129C2×#117—711no—
130#398—711no—
131#397—711no—
132#395—711no—
133#393—711no—
134#388—712no—
135#387—712no—
136#387—712no—
137#385—712no—
138#383—711no—
139C22×Q8C2×D7111yescharacteristic
140C23×C4—711no—
141C4×Q8—24no—
142#117—1424no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7112yes—
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C4—711no—
150C2×C22⋊C4—711no—
151C2×C22⋊C4—712no—
152C2×C22⋊C4—712no—
153C2×C42—711no—
154C2×C42C2×D7112yes—
155C2×C42C2×D7111yescharacteristic
156#95—712no—
157#95—712no—
158#95—711no—
159#95—711no—
160#95—712no—
161#95—712no—
162#95—712no—
163#95—712no—
164C24—711no—
165C2×Q8—22no—
166C2×Q8—24no—
167C22×C4—711no—
168C22×C4—712no—
169C22×C4—712no—
170C22×C4—711no—
171C22×C4—711no—
172C22×C4—1422no—
173C22×C4—1422no—
174C22×C4—1422no—
175C22×C4C22×D7112yes—
176C22×C4C22×D7111yescharacteristic
177C22×C4C22×D7111yescharacteristic
178C22×C4C22×D7111yescharacteristic
179C22×C4C22×D7112yes—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—1424no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C4⋊C4—1424no—
186C22⋊C4—1424no—
187C22⋊C4—1422no—
188C22⋊C4—1422no—
189C22⋊C4—1424no—
190C42—24no—
191C42—1422no—
192C42C22⋊D7114yes—
193C23—712no—
194C23—1421no—
195C23—1421no—
196C23—1422no—
197C23—1422no—
198C23C23×D7111yescharacteristic
199Q8—28no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—712no—
210C2×C4—1428no—
211C2×C4—1421no—
212C2×C4—1421no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4C2×C22⋊D7112yes—
224C2×C4C2×C22⋊D7112yes—
225C2×C4C2×C22⋊D7112yes—
226C2×C4#1176586514yes—
227C22—2841no—
228C22—2842no—
229C22—2841no—
230C22—1424no—
231C22—1422no—
232C22C22×C22⋊D7111yescharacteristic
233C22#41×D7111yescharacteristic
234C22#41×D7111yescharacteristic
235C22C2×#1176586512yes—
236C22#1455617612yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—24no—
241C4—42no—
242C4—2841no—
243C4—2842no—
244C4—2842no—
245C4—2841no—
246C4—1422no—
247C2—2842no—
248C2#1798650611yescharacteristic
249C2#1798651011yescharacteristic
250C2#1798657311yescharacteristic
251C2#1798669012yes—
252C2#1798672112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×C4×Q8×C71C2
hypercenterC2×C4×Q8D71
hypocenterC71#2599
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #17985930C2semidirect
4. #17985918C2semidirect
7. #17985883C2semidirect
8. #17985880C2semidirect
10. #17985867C2semidirect
11. C2×C4×Q8×C71C2semidirect
127. C71#2599semidirect
128. C2×C4×Q8D71semidirect