Finite Group Databasebeta

← #22591747

Subgroups — #22591747

Order 9088.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14556111C211yesmaximal, characteristic
3C2×#14556087C211yesmaximal, characteristic, complemented
4#17985932C212yesmaximal, complemented
5#17985921C212yesmaximal, complemented
6#17985894C212yesmaximal, complemented
7#17985889C212yesmaximal, complemented
8#17985878C211yesmaximal, characteristic
9#17985872C211yesmaximal, characteristic, complemented
10#17985862C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13#14556111—24no—
14#14556111—24no—
15#14556087—24no—
16C2×C4⋊Dic71C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×C22⋊Dic71C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2211yescharacteristic
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2211yescharacteristic
28C2×C4×Dic71C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C71⋊#95C2212yes—
33C23×C4×C71C2211yescharacteristic
34C23×C4×C71C2211yescharacteristic
35C23×C4×C71C2211yescharacteristic
36C2×C4⋊C4×C71—24no—
37C2×C4⋊C4×C71C2212yes—
38C2×C4⋊C4×C71C2212yes—
39C23×D71C2311yescharacteristic
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2312yes—
48C4⋊Dic71—24no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—24no—
52Dic71⋊C4—24no—
53C22⋊Dic71—22no—
54C22⋊Dic71—22no—
55C22⋊Dic71D414yes—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60C4×Dic71—24no—
61C24×C71C2311yescharacteristic
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—24no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71D414yes—
70C22×C4×C71C2312yes—
71C22×C4×C71C2311yescharacteristic
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2312yes—
74C4⋊C4×C71—24no—
75C4⋊C4×C71—24no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71C2×D412yes—
79C23×C71C2×D412yes—
80C23×C71#4112yes—
81C23×C71C2411yescharacteristic
82C22×D71—21no—
83C22×D71—21no—
84C22×D71—22no—
85C22×D71—21no—
86C22×D71—22no—
87C2×C4×C71—28no—
88C2×C4×C71—28no—
89C2×C4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—44no—
99C2×C4×C71C2×D412yes—
100C2×C4×C71C2×D412yes—
101C2×C4×C71#4114yes—
102C22⋊D71—44no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—21no—
109C2×Dic71—21no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—21no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71C2×D414yes—
116C22×C71—22no—
117C22×C71—22no—
118C22×C71—24no—
119C22×C71—24no—
120C22×C71#12114yes—
121C22×C71C22×D411yescharacteristic
122C22×C71C22×D411yescharacteristic
123C22×C71C2×#4111yescharacteristic
124C22×C71C2×#4112yes—
125C22×C712−1+412yes—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71—41no—
129C2×D71—42no—
130C2×D71—42no—
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—24no—
134C4×C71—42no—
135Dic71—24no—
136Dic71—41no—
137Dic71—41no—
138Dic71—41no—
139Dic71—42no—
140C2×C71—24no—
141C2×C71C2×#12111yescharacteristic
142C2×C71#52511yescharacteristic
143C2×C71#53611yescharacteristic
144C2×C71#54712yes—
145C2×C71#55212yes—
146D71—81no—
147#2413—711nomaximal, Sylow
148C71#241311yescharacteristic, complemented, Sylow
149C2×#123—711no—
150C2×#123—711no—
151C2×#117—711no—
152C22×C4⋊C4D7111yescharacteristic, complemented
153#398—712no—
154#396—712no—
155#389—712no—
156#388—712no—
157#386—711no—
158#385—711no—
159#381—711no—
160C22×D4—711no—
161C23×C4C2×D7111yescharacteristic
162C23×C4C2×D7111yescharacteristic
163C23×C4C2×D7111yescharacteristic
164#123—1424no—
165#123—1424no—
166#117—1424no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—711no—
170C2×C4⋊C4—712no—
171C2×C4⋊C4—711no—
172C2×C4⋊C4—712no—
173C2×C4⋊C4C2×D7112yes—
174C2×C4⋊C4C2×D7112yes—
175C2×C22⋊C4—711no—
176C2×C22⋊C4—711no—
177C2×C22⋊C4—711no—
178C2×C22⋊C4—711no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—711no—
181C2×C22⋊C4—712no—
182C2×C42—712no—
183#95—712no—
184#95—712no—
185#95—712no—
186#95—712no—
187C24—711no—
188C24C22×D7111yescharacteristic
189C2×D4—1422no—
190C2×D4—1422no—
191C2×D4—1422no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—712no—
200C22×C4—711no—
201C22×C4—712no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4C22⋊D7114yes—
205C22×C4C22×D7112yes—
206C22×C4C22×D7111yescharacteristic
207C22×C4C22×D7111yescharacteristic
208C22×C4C22×D7112yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—1424no—
212C4⋊C4—1424no—
213C4⋊C4—1422no—
214C4⋊C4—1422no—
215C4⋊C4—1424no—
216C22⋊C4—714no—
217C22⋊C4—1424no—
218C22⋊C4—1422no—
219C22⋊C4—1422no—
220C22⋊C4—1422no—
221C22⋊C4—1422no—
222C22⋊C4—1424no—
223C42—1424no—
224C23—22no—
225C23—22no—
226C23—1421no—
227C23—1421no—
228C23—1421no—
229C23—1422no—
230C23—1422no—
231C23D4×D7112yes—
232C23C2×C22⋊D7112yes—
233C23#1176586412yes—
234C23C23×D7111yescharacteristic
235D4—2844no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—714no—
249C2×C4—1422no—
250C2×C4—1421no—
251C2×C4—1421no—
252C2×C4—1421no—
253C2×C4—1421no—
254C2×C4—1421no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4C2×C22⋊D7112yes—
262C2×C4C2×C22⋊D7112yes—
263C2×C4#1176586514yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—22no—
268C22—2841no—
269C22—2842no—
270C22—2842no—
271C22—2841no—
272C22—2841no—
273C22#1455610014yes—
274C22C2×D4×D7111yescharacteristic
275C22C22×C22⋊D7111yescharacteristic
276C22C2×#1176586411yescharacteristic
277C22C2×#1176586512yes—
278C22#1455617612yes—
279C4—22no—
280C4—22no—
281C4—24no—
282C4—42no—
283C4—2841no—
284C4—2842no—
285C4—2841no—
286C4—2841no—
287C4—1424no—
288C2—24no—
289C2—5681no—
290C2C2×#1455610011yescharacteristic
291C2#1798651011yescharacteristic
292C2#1798657311yescharacteristic
293C2#1798665012yes—
294C2#1798668412yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC22×C4⋊C4×C71C2
hypercenterC22×C4⋊C4D71
hypocenterC71#2413
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#14556087C2semidirect
4. #17985932C2semidirect
5. #17985921C2semidirect
6. #17985894C2semidirect
7. #17985889C2semidirect
9. #17985872C2semidirect
10. #17985862C2semidirect
11. C22×C4⋊C4×C71C2semidirect
148. C71#2413semidirect
152. C22×C4⋊C4D71semidirect