Finite Group Databasebeta

← #22594038

Subgroups — #22594038

Order 9088.

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

normal subgroups only

This group has 476 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×#14556120C211yesmaximal, characteristic, complemented
2#17985938C211yesmaximal, characteristic, complemented
3#17985935C211yesmaximal, characteristic
4#17985929C211yesmaximal, characteristic, complemented
5#17985924C211yesmaximal, characteristic
6#17985894C212yesmaximal, complemented
7#17985893C212yesmaximal
8#17985883C212yesmaximal, complemented
9#17985872C212yesmaximal
10#17985845C211yesmaximal, characteristic, complemented
11C2×#117×C71C211yesmaximal, characteristic
12#14556120—24no—
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C4⋊Dic71C2211yescharacteristic
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×C22⋊Dic71C2212yes—
20C2×C22⋊Dic71C2212yes—
21C2×C4×Dic71C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2211yescharacteristic
24C71⋊#95C2211yescharacteristic
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C23×C4×C71C2211yescharacteristic
30#117×C71—24no—
31C2×C4⋊C4×C71C2211yescharacteristic
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C42×C71C2211yescharacteristic
36C2×C42×C71C2211yescharacteristic
37C22×Dic71C2312yes—
38C22×Dic71C2312yes—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C4⋊Dic71—22no—
42C4⋊Dic71—22no—
43Dic71⋊C4—24no—
44Dic71⋊C4—24no—
45Dic71⋊C4—24no—
46Dic71⋊C4—24no—
47C22⋊Dic71—24no—
48C22⋊Dic71—24no—
49C4×Dic71—24no—
50C24×C71C2311yescharacteristic
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71C2311yescharacteristic
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C22×C4×C71C2311yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C4⋊C4×C71—22no—
61C4⋊C4×C71—22no—
62C22⋊C4×C71—22no—
63C22⋊C4×C71—22no—
64C42×C71—22no—
65C42×C71D414yes—
66C23×C71—21no—
67C23×C71—21no—
68C23×C71—22no—
69C23×C71—22no—
70C23×C71#4112yes—
71C23×C71C2411yescharacteristic
72C2×C4×C71—28no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D412yes—
84C2×C4×C71C2×D412yes—
85C2×C4×C71C2×D412yes—
86C2×C4×C71#4112yes—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
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#4114yes—
99C22×C71—22no—
100C22×C71—24no—
101C22×C71—41no—
102C22×C71—41no—
103C22×C71—42no—
104C22×C71C22×D411yescharacteristic
105C22×C71C2×#4112yes—
106C22×C71C2×#4111yescharacteristic
107C22×C71C2×#4111yescharacteristic
108C22×C712−1+412yes—
109C4×C71—22no—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—22no—
113C4×C71—41no—
114C4×C71—41no—
115Dic71—24no—
116Dic71—42no—
117Dic71—42no—
118Dic71—42no—
119C2×C71—42no—
120C2×C71#52511yescharacteristic
121C2×C71#53211yescharacteristic
122C2×C71#53611yescharacteristic
123C2×C71#55212yes—
124C2×C71#55212yes—
125#2611—711nomaximal, Sylow
126C71#261111yescharacteristic, complemented, Sylow
127C2×#125—711no—
128C2×#117D7111yescharacteristic
129#400—711no—
130#399—711no—
131#398—711no—
132#397—711no—
133#389—712no—
134#388—712no—
135#387—712no—
136#385—712no—
137#376—711no—
138C23×C4C2×D7111yescharacteristic
139#125—1424no—
140#117—24no—
141C2×C4⋊C4—712no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7111yescharacteristic
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4C2×D7111yescharacteristic
152C2×C22⋊C4C2×D7111yescharacteristic
153C2×C42—712no—
154C2×C42C2×D7111yescharacteristic
155C2×C42C2×D7111yescharacteristic
156#95—712no—
157#95—711no—
158#95—711no—
159#95—712no—
160#95—712no—
161#95—712no—
162#95—712no—
163C24C22×D7111yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—712no—
168C22×C4—712no—
169C22×C4—712no—
170C22×C4—712no—
171C22×C4C22×D7111yescharacteristic
172C22×C4C22×D7111yescharacteristic
173C22×C4C22×D7111yescharacteristic
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—1424no—
180C4⋊C4—1424no—
181C4⋊C4—1424no—
182C4⋊C4—1422no—
183C4⋊C4—1422no—
184C4⋊C4—1424no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—1424no—
188C22⋊C4—1424no—
189C42—22no—
190C42—1424no—
191C42C22⋊D7114yes—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23#1176586512yes—
197C23C23×D7111yescharacteristic
198C2×C4—28no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—714no—
210C2×C4—1422no—
211C2×C4—1422no—
212C2×C4—1422no—
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×C4C2×C22⋊D7112yes—
222C2×C4C2×C22⋊D7112yes—
223C2×C4C2×C22⋊D7112yes—
224C2×C4#1176586512yes—
225C22—24no—
226C22—22no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22C22×C22⋊D7111yescharacteristic
231C22#41×D7112yes—
232C22C2×#1176586511yescharacteristic
233C22C2×#1176586511yescharacteristic
234C22#1455617512yes—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—2842no—
242C4—2842no—
243C4—2842no—
244C4—1424no—
245C2—42no—
246C2#1798651011yescharacteristic
247C2#1798654511yescharacteristic
248C2#1798657011yescharacteristic
249C2#1798668212yes—
250C2#1798668812yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#117×C71C2
hypercenterC2×#117D71
hypocenterC71#2611
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14556120C2semidirect
2. #17985938C2semidirect
4. #17985929C2semidirect
6. #17985894C2semidirect
8. #17985883C2semidirect
10. #17985845C2semidirect
126. C71#2611semidirect