Finite Group Databasebeta

← #22596376

Subgroups — #22596376

Order 9088.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556108C211yesmaximal, characteristic
2C2×#14556106C211yesmaximal, characteristic
3#17985941C212yesmaximal, complemented
4#17985937C211yesmaximal, characteristic, complemented
5#17985928C212yesmaximal
6#17985926C212yesmaximal
7#17985920C211yesmaximal, characteristic, complemented
8#17985920C212yesmaximal, complemented
9#17985919C211yesmaximal, characteristic, complemented
10#17985861C211yesmaximal, characteristic
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15#14556108—24no—
16#14556106—24no—
17C2×C4⋊Dic71C2212yes—
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×Dic142—42no—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2312yes—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2311yescharacteristic
47C2×C4×D71—42no—
48C2×C4×D71D414yes—
49C4⋊Dic71—24no—
50C4⋊Dic71—42no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—42no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—42no—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C23×C71C2411yescharacteristic
66C22×D71—22no—
67C22×D71—21no—
68C22×D71—22no—
69C22×D71—21no—
70C22×D71C2×D412yes—
71C2×C4×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—24no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71C2×D412yes—
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71C2×D412yes—
96C4×D71—44no—
97Dic142—44no—
98Dic142—44no—
99C22×C71C22×D411yescharacteristic
100C22×C71H(5,𝔽2)12yes—
101C22×C712−1+412yes—
102C22×C712−1+411yescharacteristic
103C22×C712−1+411yescharacteristic
104C2×D71—22no—
105C2×D71—22no—
106C2×D71—22no—
107C2×D71—42no—
108C2×D71—42no—
109C4×C71—22no—
110C4×C71—41no—
111C4×C71—42no—
112C4×C71—41no—
113C4×C71—42no—
114Dic71—22no—
115Dic71—41no—
116Dic71—42no—
117Dic71—41no—
118Dic71—42no—
119C2×C71#53411yescharacteristic
120C2×C71#53611yescharacteristic
121C2×C71#53611yescharacteristic
122C2×C71#56312yes—
123C2×C71#56312yes—
124D71—42no—
125#2817—711nomaximal, Sylow
126C71#281711yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—711no—
128C2×C22⋊Q8—711no—
129#400—711no—
130#400—712no—
131#400D7111yescharacteristic, complemented
132#397—712no—
133#397—712no—
134#395—711no—
135#395—712no—
136#395—711no—
137#381—711no—
138C22×Q8—711no—
139C22×Q8—711no—
140C23×C4—711no—
141C22⋊Q8—1424no—
142C22⋊Q8—1424no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4—712no—
145C2×C4⋊C4—711no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4—711no—
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C4⋊C4C2×D7111yescharacteristic
150C2×C22⋊C4—712no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4—711no—
153C2×C22⋊C4—712no—
154#95—712no—
155#95—711no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95—712no—
160#95C2×D7111yescharacteristic
161#95C2×D7112yes—
162#95C2×D7112yes—
163C24—711no—
164C2×Q8—2842no—
165C2×Q8—2842no—
166C2×Q8—1422no—
167C2×Q8—1422no—
168C22×C4—2842no—
169C22×C4—714no—
170C22×C4—711no—
171C22×C4—712no—
172C22×C4—711no—
173C22×C4—712no—
174C22×C4—711no—
175C22×C4C22×D7112yes—
176C22×C4C22×D7112yes—
177C22×C4C22×D7111yescharacteristic
178C22×C4C22×D7111yescharacteristic
179C22×C4C22×D7111yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—2842no—
183C4⋊C4—2842no—
184C4⋊C4—1422no—
185C4⋊C4—1424no—
186C4⋊C4—1422no—
187C22⋊C4—2842no—
188C22⋊C4—1422no—
189C22⋊C4—1424no—
190C22⋊C4—1422no—
191C23—712no—
192C23—1422no—
193C23—1421no—
194C23—1422no—
195C23—1421no—
196C23C23×D7111yescharacteristic
197Q8—2844no—
198Q8—2844no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—2844no—
211C2×C4—712no—
212C2×C4—1421no—
213C2×C4—1422no—
214C2×C4—1421no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1421no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4D4×D7112yes—
225C22—2842no—
226C22—2842no—
227C22—1422no—
228C22—1422no—
229C22—1422no—
230C22C2×D4×D7111yescharacteristic
231C22#1455617412yes—
232C22#1455617512yes—
233C22#1455617511yescharacteristic
234C22#1455617611yescharacteristic
235C4—22no—
236C4—41no—
237C4—42no—
238C4—41no—
239C4—42no—
240C4—2842no—
241C4—2841no—
242C4—2841no—
243C4—2842no—
244C4—1422no—
245C2—2842no—
246C2#1798656011yescharacteristic
247C2#1798656911yescharacteristic
248C2#1798657211yescharacteristic
249C2#1798675112yes—
250C2#1798675312yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2817
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17985941C2semidirect
4. #17985937C2semidirect
7. #17985920C2semidirect
8. #17985920C2semidirect
9. #17985919C2semidirect
11. #400×C71C2semidirect
126. C71#2817semidirect
131. #400D71semidirect