Finite Group Databasebeta

← #22594021

Subgroups — #22594021

Order 9088.

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

normal subgroups only

This group has 576 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×#14556116C211yesmaximal, characteristic, complemented
2C2×#14556085C211yesmaximal, characteristic, complemented
3#17985920C211yesmaximal, characteristic, complemented
4#17985917C211yesmaximal, characteristic, complemented
5#17985912C211yesmaximal, characteristic, complemented
6#17985909C211yesmaximal, characteristic
7#17985901C212yesmaximal, complemented
8#17985891C212yesmaximal, complemented
9#17985890C212yesmaximal, complemented
10#17985877C212yesmaximal, complemented
11#376×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2212yes—
14C22×D284C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16#14556116—24no—
17#14556085—24no—
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2211yescharacteristic
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×C4×Dic71C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C42×C71C2212yes—
33C2×C42×C71C2211yescharacteristic
34#95×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C23×D71C2312yes—
38C23×D71C2311yescharacteristic
39C2×Dic142—22no—
40C2×Dic142—42no—
41C2×D284—24no—
42C2×D284—22no—
43C2×D284—24no—
44C2×D284—42no—
45C2×D284—42no—
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C22×Dic71C2311yescharacteristic
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C4⋊Dic71—42no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—42no—
63(C2×D71)⋊C4—42no—
64C4×Dic71—22no—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2312yes—
70C42×C71—24no—
71C42×C71D414yescomplemented
72C23×C71C2411yescharacteristic
73C22×D71—21no—
74C22×D71—21no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71#4112yes—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D412yes—
96C2×C4×C71C2×D412yes—
97C2×C4×C71#4114yes—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71#4112yes—
108D284—48no—
109D284—44no—
110C4×D71—28no—
111Dic142—44no—
112C22×C71C22×D411yescharacteristic
113C22×C71C2×#4111yescharacteristic
114C22×C71C2×#4111yescharacteristic
115C22×C71C2×#4112yes—
116C22×C71H(5,𝔽2)12yes—
117C2×D71—22no—
118C2×D71—24no—
119C2×D71—41no—
120C2×D71—41no—
121C2×D71—42no—
122C2×D71—42no—
123C2×D71—42no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—42no—
128C2×D71—42no—
129C4×C71—22no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—24no—
133C4×C71—42no—
134Dic71—22no—
135Dic71—41no—
136Dic71—42no—
137Dic71—41no—
138C2×C71#52511yescharacteristic
139C2×C71#53211yescharacteristic
140C2×C71#53511yescharacteristic
141C2×C71#55112yes—
142C2×C71#55512yes—
143D71—42no—
144D71—82no—
145#2610—711nomaximal, Sylow
146C71#261011yescharacteristic, complemented, Sylow
147C2×#124—711no—
148C2×#117—711no—
149#395—711no—
150#394—711no—
151#393—711no—
152#392—711no—
153#390—712no—
154#388—712no—
155#388—712no—
156#386—712no—
157#376D7111yescharacteristic, complemented
158C22×Q8—711no—
159C22×D4—711no—
160C22×D4—712no—
161C23×C4—711no—
162#124—1424no—
163#117—1424no—
164C2×C4⋊C4—712no—
165C2×C4⋊C4—711no—
166C2×C4⋊C4—711no—
167C2×C22⋊C4—712no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—711no—
170C2×C22⋊C4—711no—
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
175C2×C42—711no—
176C2×C42C2×D7112yes—
177C2×C42C2×D7111yescharacteristic
178#95—712no—
179#95—712no—
180#95C2×D7112yes—
181#95C2×D7111yescharacteristic
182#95C2×D7111yescharacteristic
183C24—711no—
184C24—712no—
185C2×Q8—2842no—
186C2×Q8—1422no—
187C2×D4—2842no—
188C2×D4—2842no—
189C2×D4—1424no—
190C2×D4—1422no—
191C2×D4—1424no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—712no—
195C22×C4—711no—
196C22×C4—1422no—
197C22×C4—1422no—
198C22×C4—1422no—
199C22×C4C22×D7112yes—
200C22×C4C22×D7111yescharacteristic
201C22×C4C22×D7111yescharacteristic
202C22×C4C22×D7111yescharacteristic
203C22×C4C22×D7112yes—
204C4⋊C4—2842no—
205C4⋊C4—1422no—
206C4⋊C4—1422no—
207C22⋊C4—2842no—
208C22⋊C4—2842no—
209C22⋊C4—1424no—
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C22⋊C4—1424no—
213C22⋊C4—1424no—
214C22⋊C4—1424no—
215C42—24no—
216C42—1422no—
217C42D28414yescomplemented
218C23—712no—
219C23—1422no—
220C23—1422no—
221C23—1422no—
222C23—1422no—
223C23—1422no—
224C23—1422no—
225C23—1422no—
226C23—1421no—
227C23—1421no—
228C23—1422no—
229C23—1422no—
230C23C23×D7111yescharacteristic
231Q8—2844no—
232D4—2844no—
233D4—2848no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—712no—
244C2×C4—1428no—
245C2×C4—1421no—
246C2×C4—1421no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4—1422no—
252C2×C4—1421no—
253C2×C4—1421no—
254C2×C4C2×D28412yes—
255C2×C4C2×D28412yes—
256C2×C4C2×D28412yes—
257C2×C4#1176586514yes—
258C22—2841no—
259C22—2842no—
260C22—2841no—
261C22—2842no—
262C22—2842no—
263C22—2842no—
264C22—2842no—
265C22—2842no—
266C22—2842no—
267C22—2842no—
268C22—1424no—
269C22—1422no—
270C22C22×D28411yescharacteristic
271C22#41×D7111yescharacteristic
272C22#41×D7111yescharacteristic
273C22C2×#1176586512yes—
274C22#1455617412yes—
275C4—22no—
276C4—24no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—2842no—
281C4—2841no—
282C4—2841no—
283C4—1422no—
284C2—2842no—
285C2—5682no—
286C2#1798651111yescharacteristic
287C2#1798654911yescharacteristic
288C2#1798656811yescharacteristic
289C2#1798667312yes—
290C2#1798671212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#376×C71C2
hypercenter#376D71
hypocenterC71#2610
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
1. C2×#14556116C2semidirect
2. C2×#14556085C2semidirect
3. #17985920C2semidirect
4. #17985917C2semidirect
5. #17985912C2semidirect
7. #17985901C2semidirect
8. #17985891C2semidirect
9. #17985890C2semidirect
10. #17985877C2semidirect
11. #376×C71C2semidirect
71. C42×C71D4semidirect
146. C71#2610semidirect
157. #376D71semidirect
217. C42D284semidirect