Finite Group Databasebeta

← #22594027

Subgroups — #22594027

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
2#17985919C211yesmaximal, characteristic, complemented
3#17985916C211yesmaximal, characteristic, complemented
4#17985911C211yesmaximal, characteristic, complemented
5#17985908C211yesmaximal, characteristic
6#17985899C212yesmaximal, complemented
7#17985890C212yesmaximal, complemented
8#17985889C212yesmaximal, complemented
9#17985878C212yesmaximal, complemented
10#17985845C211yesmaximal, characteristic, complemented
11C2×#117×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C22⋊D71C2212yes—
15#14556116—24no—
16C2×Dic71⋊C4C2212yes—
17C2×C22⋊Dic71C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×C4×Dic71C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C23×C4×C71C2211yescharacteristic
30#117×C71—24no—
31C2×C4⋊C4×C71C2211yescharacteristic
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C42×C71C2211yescharacteristic
36C2×C42×C71C2211yescharacteristic
37C23×D71C2312yes—
38C2×Dic142—22no—
39C2×Dic142—42no—
40C2×D284—22no—
41C2×D284—42no—
42C2×C22⋊D71—24no—
43C2×C22⋊D71—24no—
44C2×C22⋊D71—42no—
45C22×Dic71C2312yes—
46C22×Dic71C2312yes—
47C22×Dic71C2312yes—
48Dic71⋊C4—42no—
49C22⋊Dic71—42no—
50(C2×D71)⋊C4—24no—
51(C2×D71)⋊C4—24no—
52(C2×D71)⋊C4—24no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—42no—
55C4×Dic71—24no—
56C24×C71C2311yescharacteristic
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C22⋊C4×C71—22no—
70C42×C71—22no—
71C42×C71D414yescomplemented
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71#4112yes—
77C23×C71C2411yescharacteristic
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—22no—
85C2×C4×C71—28no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71C2×D412yes—
97C2×C4×C71C2×D412yes—
98C2×C4×C71C2×D412yes—
99C2×C4×C71#4112yes—
100C22⋊D71—48no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71#4114yes—
110D284—44no—
111Dic142—44no—
112C22×C71—22no—
113C22×C71—24no—
114C22×C71—41no—
115C22×C71—41no—
116C22×C71—42no—
117C22×C71C22×D411yescharacteristic
118C22×C71C2×#4112yes—
119C22×C71C2×#4111yescharacteristic
120C22×C71C2×#4111yescharacteristic
121C22×C71H(5,𝔽2)12yes—
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—22no—
133C4×C71—41no—
134C4×C71—41no—
135Dic71—24no—
136Dic71—42no—
137Dic71—42no—
138C2×C71—42no—
139C2×C71#52511yescharacteristic
140C2×C71#53211yescharacteristic
141C2×C71#53511yescharacteristic
142C2×C71#55112yes—
143C2×C71#55512yes—
144D71—82no—
145#2610—711nomaximal, Sylow
146C71#261011yescharacteristic, complemented, Sylow
147C2×#124—711no—
148C2×#117D7111yescharacteristic, complemented
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#376—711no—
158C22×Q8—711no—
159C22×D4—712no—
160C22×D4—711no—
161C23×C4C2×D7111yescharacteristic
162#124—1424no—
163#117—24no—
164C2×C4⋊C4—712no—
165C2×C4⋊C4C2×D7111yescharacteristic
166C2×C4⋊C4C2×D7111yescharacteristic
167C2×C22⋊C4—712no—
168C2×C22⋊C4—712no—
169C2×C22⋊C4—712no—
170C2×C22⋊C4—712no—
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4C2×D7111yescharacteristic
174C2×C22⋊C4C2×D7111yescharacteristic
175C2×C42—712no—
176C2×C42C2×D7111yescharacteristic
177C2×C42C2×D7111yescharacteristic
178#95—712no—
179#95—711no—
180#95—711no—
181#95—712no—
182#95—712no—
183C24—712no—
184C24C22×D7111yescharacteristic
185C2×Q8—2842no—
186C2×Q8—1422no—
187C2×D4—2842no—
188C2×D4—2842no—
189C2×D4—1424no—
190C2×D4—1424no—
191C2×D4—1422no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—712no—
196C22×C4—712no—
197C22×C4—712no—
198C22×C4C22×D7111yescharacteristic
199C22×C4C22×D7111yescharacteristic
200C22×C4C22×D7111yescharacteristic
201C22×C4C22×D7111yescharacteristic
202C22×C4C22×D7111yescharacteristic
203C22×C4C22×D7111yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—2842no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—2842no—
210C22⋊C4—2842no—
211C22⋊C4—1424no—
212C22⋊C4—1424no—
213C22⋊C4—1424no—
214C22⋊C4—1424no—
215C42—22no—
216C42—1424no—
217C42C22⋊D7114yescomplemented
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—1422no—
223C23—1422no—
224C23—1422no—
225C23—1422no—
226C23—1422no—
227C23—1422no—
228C23—1422no—
229C23#1176586512yes—
230C23C23×D7111yescharacteristic
231Q8—2844no—
232D4—2844no—
233D4—2848no—
234C2×C4—28no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—714no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4—1422no—
249C2×C4—1422no—
250C2×C4—1422no—
251C2×C4—1422no—
252C2×C4—1422no—
253C2×C4—1422no—
254C2×C4C2×C22⋊D7112yes—
255C2×C4C2×C22⋊D7112yes—
256C2×C4C2×C22⋊D7112yes—
257C2×C4#1176586512yes—
258C22—24no—
259C22—22no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—2842no—
264C22—2842no—
265C22—2842no—
266C22—2842no—
267C22—2842no—
268C22—2842no—
269C22—2842no—
270C22C22×C22⋊D7111yescharacteristic
271C22#41×D7112yes—
272C22C2×#1176586511yescharacteristic
273C22C2×#1176586511yescharacteristic
274C22#1455617412yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—2842no—
282C4—2842no—
283C4—1424no—
284C2—42no—
285C2—5682no—
286C2#1798651011yescharacteristic
287C2#1798654511yescharacteristic
288C2#1798656311yescharacteristic
289C2#1798667312yes—
290C2#1798671212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#117×C71C2
hypercenterC2×#117D71
hypocenterC71#2610
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14556116C2semidirect
2. #17985919C2semidirect
3. #17985916C2semidirect
4. #17985911C2semidirect
6. #17985899C2semidirect
7. #17985890C2semidirect
8. #17985889C2semidirect
9. #17985878C2semidirect
10. #17985845C2semidirect
11. C2×#117×C71C2semidirect
71. C42×C71D4semidirect
146. C71#2610semidirect
148. C2×#117D71semidirect
217. C42C22⋊D71semidirect