Finite Group Databasebeta

← #22594844

Subgroups — #22594844

Order 9088.

568 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14556110C212yesmaximal, complemented
2#17985941C212yesmaximal, complemented
3#17985921C212yesmaximal, complemented
4#17985908C211yesmaximal, characteristic
5#17985877C212yesmaximal, complemented
6#17985876C212yesmaximal, complemented
7#17985871C211yesmaximal, characteristic, complemented
8#17985858C212yesmaximal, complemented
9#397×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C23×Dic71C2211yescharacteristic
13C22×C4×D71C2212yes—
14#14556110—28no—
15C2×C4⋊Dic71C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×C22⋊Dic71C2211yescharacteristic
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C71⋊#95C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28C2×C22⋊C4×C71C2212yes—
29C2×C22⋊C4×C71C2211yescharacteristic
30#95×C71C2211yescharacteristic
31#95×C71C2212yes—
32C23×D71C2312yes—
33C2×D284—24no—
34C2×D284—41no—
35C2×C22⋊D71—24no—
36C2×C22⋊D71—24no—
37C2×C22⋊D71—42no—
38C22×Dic71—41no—
39C22×Dic71—42no—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C2×C4×D71—24no—
44C2×C4×D71—24no—
45C2×C4×D71—42no—
46C4⋊Dic71—24no—
47Dic71⋊C4—24no—
48Dic71⋊C4—42no—
49C22⋊Dic71D414yescomplemented
50(C2×D71)⋊C4—24no—
51(C2×D71)⋊C4—42no—
52(C2×D71)⋊C4—42no—
53(C2×D71)⋊C4—42no—
54C24×C71C2311yescharacteristic
55C22×C4×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C4⋊C4×C71—41no—
60C22⋊C4×C71—24no—
61C22⋊C4×C71—41no—
62C23×C71—22no—
63C23×C71—21no—
64C23×C71—22no—
65C23×C71—21no—
66C23×C71C2×D412yes—
67C23×C71C2411yescharacteristic
68C22×D71—22no—
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72C22×D71—22no—
73C22×D71—22no—
74C22×D71#4114yes—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
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×C71—22no—
84C2×C4×C71—22no—
85C22⋊D71—48no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—44no—
95C2×Dic71C2×D414yes—
96D284—44no—
97C4×D71—48no—
98C22×C71—22no—
99C22×C71—24no—
100C22×C71—42no—
101C22×C71—41no—
102C22×C71—41no—
103C22×C71C22×D411yescharacteristic
104C22×C71C2×#4112yes—
105C22×C71H(5,𝔽2)11yescharacteristic
106C22×C71H(5,𝔽2)12yes—
107C22×C712−1+411yescharacteristic
108C2×D71—24no—
109C2×D71—24no—
110C2×D71—24no—
111C2×D71—42no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—42no—
115C4×C71—42no—
116C4×C71—42no—
117C4×C71—41no—
118C4×C71—41no—
119Dic71—24no—
120Dic71—42no—
121Dic71—42no—
122C2×C71—42no—
123C2×C71#53711yescharacteristic
124C2×C71#53912yes—
125C2×C71#54612yes—
126C2×C71#55111yescharacteristic
127C2×C71#56011yescharacteristic
128D71—44no—
129#2693—711nomaximal, Sylow
130C71#269311yescharacteristic, complemented, Sylow
131C2×#123—712no—
132#400—712no—
133#397D7111yescharacteristic, complemented
134#396—712no—
135#392—711no—
136#386—712no—
137#386—712no—
138#385—711no—
139#380—712no—
140C22×D4—712no—
141C22×D4—711no—
142C23×C4—711no—
143C23×C4—712no—
144#123—1428no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—712no—
152C2×C22⋊C4—711no—
153C2×C22⋊C4—712no—
154C2×C22⋊C4C2×D7112yes—
155C2×C22⋊C4C2×D7111yescharacteristic
156#95—712no—
157#95—712no—
158#95—712no—
159#95—711no—
160#95C2×D7111yescharacteristic
161#95C2×D7112yes—
162C24—712no—
163C24C22×D7111yescharacteristic
164C2×D4—2841no—
165C2×D4—2842no—
166C2×D4—1424no—
167C2×D4—1424no—
168C2×D4—1424no—
169C22×C4—2842no—
170C22×C4—2841no—
171C22×C4—2842no—
172C22×C4—712no—
173C22×C4—712no—
174C22×C4—712no—
175C22×C4—1424no—
176C22×C4—1424no—
177C22×C4C22×D7111yescharacteristic
178C22×C4C22×D7112yes—
179C22×C4C22×D7112yes—
180C22×C4C22×D7111yescharacteristic
181C4⋊C4—41no—
182C4⋊C4—2842no—
183C4⋊C4—1424no—
184C4⋊C4—1424no—
185C22⋊C4—24no—
186C22⋊C4—41no—
187C22⋊C4—2842no—
188C22⋊C4—2842no—
189C22⋊C4—2842no—
190C22⋊C4—714no—
191C22⋊C4—1424no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—714no—
197C23—1422no—
198C23—1422no—
199C23—1422no—
200C23—1422no—
201C23—1422no—
202C23—1422no—
203C23D4×D7112yes—
204C23C23×D7111yescharacteristic
205D4—2844no—
206D4—2848no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—2848no—
218C2×C4—2844no—
219C2×C4—714no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1422no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—2842no—
234C22—2842no—
235C22—2842no—
236C22—2842no—
237C22—1424no—
238C22—1424no—
239C22—1424no—
240C22C2×D4×D7111yescharacteristic
241C22#41×D7112yes—
242C22#1455617311yescharacteristic
243C22#1455617412yes—
244C22#1455617511yescharacteristic
245C4—42no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—2842no—
250C4—2842no—
251C4—1424no—
252C2—42no—
253C2—2844no—
254C2#1798657611yescharacteristic
255C2#1798659012yes—
256C2#1798664112yes—
257C2#1798667911yescharacteristic
258C2#1798673911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2693
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14556110C2semidirect
2. #17985941C2semidirect
3. #17985921C2semidirect
5. #17985877C2semidirect
6. #17985876C2semidirect
7. #17985871C2semidirect
8. #17985858C2semidirect
9. #397×C71C2semidirect
49. C22⋊Dic71D4semidirect
130. C71#2693semidirect
133. #397D71semidirect