Finite Group Databasebeta

← #22594497

Subgroups — #22594497

Order 9088.

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

normal subgroups only

This group has 498 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×#14556113C211yesmaximal, characteristic
2C2×#14556108C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic, complemented
4#17985928C212yesmaximal, complemented
5#17985919C211yesmaximal, characteristic, complemented
6#17985917C212yesmaximal
7#17985912C211yesmaximal, characteristic, complemented
8#17985896C212yesmaximal, complemented
9#17985891C212yesmaximal, complemented
10#17985846C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15#14556113—24no—
16#14556108—24no—
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2211yescharacteristic
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×D284—22no—
42C2×D284—42no—
43C22×Dic71C2312yes—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C2×C4×D71—42no—
48C2×C4×D71D414yes—
49C4⋊Dic71—24no—
50Dic71⋊C4—22no—
51Dic71⋊C4—24no—
52Dic71⋊C4—42no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—42no—
60C4×Dic71—42no—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2312yes—
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C23×C71C2411yescharacteristic
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—21no—
72C22×D71—21no—
73C22×D71—21no—
74C22×D71—22no—
75C22×D71—22no—
76C22×D71—21no—
77C22×D71—21no—
78C22×D71C2×D412yes—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71C2×D412yes—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71C2×D412yes—
101C2×Dic71#4114yes—
102D284—44no—
103C4×D71—44no—
104Dic142—44no—
105C22×C71C22×D411yescharacteristic
106C22×C71C2×#4112yes—
107C22×C71H(5,𝔽2)12yes—
108C22×C712−1+411yescharacteristic
109C22×C712−1+411yescharacteristic
110C2×D71—22no—
111C2×D71—22no—
112C2×D71—22no—
113C2×D71—42no—
114C2×D71—42no—
115C2×D71—41no—
116C2×D71—41no—
117C2×D71—41no—
118C2×D71—42no—
119C2×D71—42no—
120C4×C71—22no—
121C4×C71—42no—
122C4×C71—41no—
123C4×C71—41no—
124C4×C71—42no—
125Dic71—22no—
126Dic71—24no—
127Dic71—41no—
128Dic71—42no—
129C2×C71#53211yescharacteristic
130C2×C71#53411yescharacteristic
131C2×C71#53611yescharacteristic
132C2×C71#53912yes—
133C2×C71#54212yes—
134D71—42no—
135D71—81no—
136#2663—711nomaximal, Sylow
137C71#266311yescharacteristic, complemented, Sylow
138C2×#123—711no—
139C2×C22⋊Q8—711no—
140#400D7111yescharacteristic, complemented
141#399—711no—
142#397—712no—
143#395—711no—
144#394—712no—
145#393—711no—
146#389—712no—
147#388—712no—
148C4×C22⋊C4—711no—
149C22×Q8—711no—
150C22×D4—711no—
151C23×C4—711no—
152#123—1424no—
153C22⋊Q8—1424no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—711no—
157C2×C4⋊C4—712no—
158C2×C4⋊C4C2×D7111yescharacteristic
159C2×C4⋊C4C2×D7111yescharacteristic
160C2×C22⋊C4—712no—
161C2×C22⋊C4—712no—
162C2×C22⋊C4—711no—
163C2×C22⋊C4—711no—
164C2×C22⋊C4—711no—
165C2×C22⋊C4—712no—
166C2×C22⋊C4—712no—
167C2×C42—712no—
168#95—711no—
169#95—712no—
170#95—712no—
171#95C2×D7111yescharacteristic
172#95C2×D7112yes—
173#95C2×D7112yes—
174C24—711no—
175C24—711no—
176C2×Q8—2842no—
177C2×Q8—1422no—
178C2×D4—2842no—
179C2×D4—1422no—
180C22×C4—2842no—
181C22×C4—714no—
182C22×C4—712no—
183C22×C4—711no—
184C22×C4—711no—
185C22×C4—712no—
186C22×C4C22×D7111yescharacteristic
187C22×C4C22×D7112yes—
188C22×C4C22×D7112yes—
189C22×C4C22×D7111yescharacteristic
190C22×C4C22×D7111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2842no—
194C4⋊C4—1424no—
195C4⋊C4—1422no—
196C4⋊C4—1424no—
197C22⋊C4—2842no—
198C22⋊C4—1424no—
199C22⋊C4—1424no—
200C22⋊C4—1422no—
201C22⋊C4—1424no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C42—2842no—
205C23—712no—
206C23—1421no—
207C23—1422no—
208C23—1422no—
209C23—1421no—
210C23—1421no—
211C23—1422no—
212C23—1422no—
213C23—1421no—
214C23—1421no—
215C23C23×D7111yescharacteristic
216Q8—2844no—
217D4—2844no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—2844no—
231C2×C4—712no—
232C2×C4—714no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4D4×D7112yes—
242C22—2842no—
243C22—2842no—
244C22—2841no—
245C22—2841no—
246C22—2842no—
247C22—2842no—
248C22—2841no—
249C22—1422no—
250C22—1422no—
251C22—1422no—
252C22C2×D4×D7111yescharacteristic
253C22#41×D7112yes—
254C22#1455617412yes—
255C22#1455617511yescharacteristic
256C22#1455617611yescharacteristic
257C4—22no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—42no—
262C4—2842no—
263C4—2841no—
264C4—1422no—
265C4—1424no—
266C2—2842no—
267C2—5681no—
268C2#1798654811yescharacteristic
269C2#1798656011yescharacteristic
270C2#1798657211yescharacteristic
271C2#1798659712yes—
272C2#1798661512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2663
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#14556108C2semidirect
3. #17985934C2semidirect
4. #17985928C2semidirect
5. #17985919C2semidirect
7. #17985912C2semidirect
8. #17985896C2semidirect
9. #17985891C2semidirect
10. #17985846C2semidirect
11. #400×C71C2semidirect
137. C71#2663semidirect
140. #400D71semidirect