Finite Group Databasebeta

← #22596355

Subgroups — #22596355

Order 9088.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556111C211yesmaximal, characteristic
2C2×#14556107C211yesmaximal, characteristic, complemented
3#17985931C211yesmaximal, characteristic, complemented
4#17985929C212yesmaximal, complemented
5#17985926C211yesmaximal, characteristic, complemented
6#17985924C212yesmaximal, complemented
7#17985921C212yesmaximal, complemented
8#17985920C211yesmaximal, characteristic, complemented
9#17985915C212yesmaximal
10#17985912C211yesmaximal, characteristic, complemented
11#380×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14#14556111—24no—
15#14556107—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2212yes—
23C2×C22⋊Dic71C2211yescharacteristic
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C23×C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2212yes—
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—42no—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—42no—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C4⋊Dic71—22no—
48C4⋊Dic71—42no—
49C4⋊Dic71—42no—
50Dic71⋊C4—24no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53C22⋊Dic71—24no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—42no—
58(C2×D71)⋊C4—42no—
59C24×C71C2311yescharacteristic
60C22×C4×C71—42no—
61C22×C4×C71D414yes—
62C22×C4×C71C2312yes—
63C22×C4×C71C2312yes—
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—42no—
67C22⋊C4×C71—42no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71—21no—
72C23×C71C2×D412yes—
73C23×C71C2411yescharacteristic
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—44no—
88C2×C4×C71C2×D412yes—
89C2×C4×C71C2×D412yes—
90C22⋊D71—44no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—22no—
104Dic142—44no—
105C22×C71—22no—
106C22×C71—22no—
107C22×C71—22no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71C22×D411yescharacteristic
111C22×C71H(5,𝔽2)12yes—
112C22×C71H(5,𝔽2)11yescharacteristic
113C22×C712−1+412yes—
114C22×C712−1+411yescharacteristic
115C2×D71—41no—
116C2×D71—41no—
117C2×D71—41no—
118C2×D71—42no—
119C2×D71—42no—
120C4×C71—22no—
121C4×C71—22no—
122C4×C71—42no—
123C4×C71—42no—
124Dic71—42no—
125Dic71—41no—
126Dic71—41no—
127Dic71—41no—
128Dic71—42no—
129C2×C71—42no—
130C2×C71#53411yescharacteristic
131C2×C71#53611yescharacteristic
132C2×C71#53711yescharacteristic
133C2×C71#56212yes—
134C2×C71#56312yes—
135D71—81no—
136#2814—711nomaximal, Sylow
137C71#281411yescharacteristic, complemented, Sylow
138C2×#123—711no—
139C2×C22⋊Q8—711no—
140#398—712no—
141#398—711no—
142#397—712no—
143#397—711no—
144#396—712no—
145#395—711no—
146#394—712no—
147#393—711no—
148#380D7111yescharacteristic, complemented
149C22×Q8—711no—
150C22×D4—711no—
151C23×C4C2×D7111yescharacteristic
152#123—1424no—
153C22⋊Q8—1424no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4—711no—
159C2×C4⋊C4—711no—
160C2×C4⋊C4C2×D7112yes—
161C2×C22⋊C4—712no—
162C2×C22⋊C4—712no—
163C2×C22⋊C4—712no—
164C2×C22⋊C4—711no—
165C2×C22⋊C4—711no—
166C2×C22⋊C4—711no—
167C2×C22⋊C4C2×D7112yes—
168#95—712no—
169#95—712no—
170#95—712no—
171#95—712no—
172#95C2×D7111yescharacteristic
173#95C2×D7111yescharacteristic
174C24—711no—
175C24C22×D7111yescharacteristic
176C2×Q8—2842no—
177C2×Q8—1422no—
178C2×D4—2842no—
179C2×D4—1422no—
180C22×C4—42no—
181C22×C4—711no—
182C22×C4—711no—
183C22×C4—712no—
184C22×C4—712no—
185C22×C4—711no—
186C22×C4D28414yes—
187C22×C4C22×D7112yes—
188C22×C4C22×D7112yes—
189C22×C4C22×D7111yescharacteristic
190C22×C4C22×D7111yescharacteristic
191C4⋊C4—42no—
192C4⋊C4—2842no—
193C4⋊C4—2842no—
194C4⋊C4—1424no—
195C4⋊C4—1422no—
196C4⋊C4—1422no—
197C4⋊C4—1422no—
198C22⋊C4—42no—
199C22⋊C4—2842no—
200C22⋊C4—2842no—
201C22⋊C4—1424no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C22⋊C4—1422no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—1421no—
210C23—1421no—
211C23—1422no—
212C23—1422no—
213C23—1421no—
214C23C2×D28412yes—
215C23C23×D7111yescharacteristic
216Q8—2844no—
217D4—2844no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—1421no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1421no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1421no—
239C2×C4—1422no—
240C2×C4C2×D28412yes—
241C2×C4C2×D28412yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—2841no—
248C22—2841no—
249C22—2841no—
250C22—2842no—
251C22—2842no—
252C22C22×D28411yescharacteristic
253C22#1455617312yes—
254C22#1455617311yescharacteristic
255C22#1455617512yes—
256C22#1455617611yescharacteristic
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—2841no—
262C4—2842no—
263C4—2841no—
264C4—2841no—
265C4—2842no—
266C2—42no—
267C2—5681no—
268C2#1798656111yescharacteristic
269C2#1798657511yescharacteristic
270C2#1798658011yescharacteristic
271C2#1798674412yes—
272C2#1798675012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#380×C71C2
hypercenter#380D71
hypocenterC71#2814
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14556107C2semidirect
3. #17985931C2semidirect
4. #17985929C2semidirect
5. #17985926C2semidirect
6. #17985924C2semidirect
7. #17985921C2semidirect
8. #17985920C2semidirect
10. #17985912C2semidirect
11. #380×C71C2semidirect
137. C71#2814semidirect
148. #380D71semidirect