Finite Group Databasebeta

← #22591670

Subgroups — #22591670

Order 9088.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#11765864C211yesmaximal, characteristic
2C2×#14556110C212yesmaximal, complemented
3C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#17985920C211yesmaximal, characteristic, complemented
5#17985919C211yesmaximal, characteristic, complemented
6#17985913C211yesmaximal, characteristic, complemented
7#17985875C212yesmaximal, complemented
8#17985874C212yesmaximal, complemented
9#17985871C212yesmaximal, complemented
10#17985861C211yesmaximal, characteristic, complemented
11#394×C71C211yesmaximal, characteristic, complemented
12C2×#11765864—28no—
13C2×#11765864—24no—
14C22×Dic142C2211yescharacteristic
15C22×C22⋊D71C2212yes—
16C23×Dic71C2212yes—
17C22×C4×D71C2211yescharacteristic
18C22×C4×D71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20#14556110—28no—
21C2×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊Dic71C2212yes—
23C2×Dic71⋊C4C2211yescharacteristic
24C2×Dic71⋊C4C2212yes—
25C2×Dic71⋊C4C2211yescharacteristic
26C4⋊C4×D71—24no—
27C2×C22⋊Dic71C2212yes—
28C2×C22⋊Dic71C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2211yescharacteristic
32C71⋊#95C2212yes—
33C71⋊#95C2212yes—
34C22×D4×C71C2211yescharacteristic
35C2×C4⋊C4×C71C2211yescharacteristic
36C2×C22⋊C4×C71C2212yes—
37C2×C22⋊C4×C71C2212yes—
38#95×C71C2211yescharacteristic
39C23×D71C2311yescharacteristic
40#11765864—232no—
41C2×Dic142—22no—
42C2×Dic142—24no—
43C2×C22⋊D71—24no—
44C2×C22⋊D71—24no—
45C2×C22⋊D71D418yes—
46C22×Dic71—24no—
47C22×Dic71—24no—
48C22×Dic71—24no—
49C22×Dic71C2311yescharacteristic
50C22×Dic71C2312yes—
51C22×Dic71C2311yescharacteristic
52C22×Dic71C2312yes—
53C22×Dic71C2311yescharacteristic
54C2×C4×D71—24no—
55C2×C4×D71—22no—
56C2×C4×D71—22no—
57C2×C4×D71—22no—
58C2×C4×D71—22no—
59C2×C4×D71—24no—
60C2×C4×D71D414yes—
61C4⋊Dic71—22no—
62C4⋊Dic71—24no—
63Dic71⋊C4—22no—
64Dic71⋊C4—22no—
65Dic71⋊C4—24no—
66C22⋊Dic71—24no—
67C22⋊Dic71—24no—
68(C2×D71)⋊C4—24no—
69C24×C71C2312yes—
70C2×D4×C71—24no—
71C2×D4×C71—22no—
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2312yes—
74C22×C4×C71C2311yescharacteristic
75C22×C4×C71C2311yescharacteristic
76C4⋊C4×C71—22no—
77C22⋊C4×C71—24no—
78C22⋊C4×C71—24no—
79C23×C71—22no—
80C23×C71—22no—
81C23×C71—22no—
82C23×C71—22no—
83C23×C71—22no—
84C23×C71—22no—
85C23×C71C2×D414yes—
86C23×C71C2411yescharacteristic
87C22×D71—21no—
88C22×D71—21no—
89C22×D71—22no—
90C22×D71C2×D414yes—
91C22×D71C2×D412yes—
92D4×C71—28no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—21no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71C2×D412yes—
102C2×C4×C71#4112yescomplemented
103C22⋊D71—216no—
104C2×Dic71—216no—
105C2×Dic71—21no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71—21no—
110C2×Dic71—22no—
111C2×Dic71—22no—
112C2×Dic71—21no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71C2×D412yes—
116C2×Dic71C2×D414yes—
117C2×Dic71#4112yescomplemented
118C4×D71—28no—
119C4×D71—28no—
120C4×D71—44no—
121Dic142—28no—
122C22×C71—24no—
123C22×C71—24no—
124C22×C71—24no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71C22×D411yescharacteristic
130C22×C71C22×D412yes—
131C22×C71C2×#4111yescharacteristic
132C22×C71C2×#4111yescharacteristic
133C22×C712−1+412yes—
134C2×D71—22no—
135C2×D71—22no—
136C2×D71—24no—
137C2×D71—24no—
138C2×D71#12014yes—
139C4×C71—22no—
140C4×C71—22no—
141C4×C71—41no—
142C4×C71—42no—
143Dic71—24no—
144Dic71—22no—
145Dic71—22no—
146Dic71—41no—
147Dic71—42no—
148C2×C71—44no—
149C2×C71C2×#12011yescharacteristic
150C2×C71#52511yescharacteristic
151C2×C71#53611yescharacteristic
152C2×C71#54712yes—
153C2×C71#54712yes—
154D71—24no—
155#2406—711nomaximal, Sylow
156C71#240611yescharacteristic, complemented, Sylow
157C22×#41—711no—
158C2×#123—712no—
159C22×C4⋊C4—711no—
160#395—711no—
161#395—711no—
162#394—711no—
163#394D7111yescharacteristic, complemented
164#386—712no—
165#385—712no—
166#385—712no—
167#381—711no—
168C2×#41—1424no—
169C2×#41—1428no—
170C22×Q8—711no—
171C22×D4—712no—
172C22×D4C2×D7111yescharacteristic
173C23×C4—711no—
174C23×C4—712no—
175C23×C4—711no—
176C23×C4—711no—
177#123—1428no—
178C2×C4⋊C4—711no—
179C2×C4⋊C4—711no—
180C2×C4⋊C4—711no—
181C2×C4⋊C4—712no—
182C2×C4⋊C4—712no—
183C2×C4⋊C4—1424no—
184C2×C4⋊C4C2×D7111yescharacteristic
185C2×C22⋊C4—712no—
186C2×C22⋊C4—712no—
187C2×C22⋊C4—712no—
188C2×C22⋊C4C2×D7112yes—
189C2×C22⋊C4C2×D7112yes—
190#95—712no—
191#95—711no—
192#95—712no—
193#95—712no—
194#95C2×D7111yescharacteristic
195C24—711no—
196C24C22×D7112yes—
197#41—14232no—
198C2×Q8—1424no—
199C2×Q8—1422no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—718no—
203C2×D4—1424no—
204C2×D4—1424no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4—712no—
208C22×C4—714no—
209C22×C4—712no—
210C22×C4—711no—
211C22×C4—1422no—
212C22×C4—1424no—
213C22×C4—1422no—
214C22×C4—1422no—
215C22×C4—1424no—
216C22×C4—1424no—
217C22×C4—1424no—
218C22×C4—1424no—
219C22×C4—1422no—
220C22×C4C22×D7112yes—
221C22×C4C22×D7111yescharacteristic
222C22×C4C22×D7111yescharacteristic
223C22×C4C22×D7111yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1422no—
226C4⋊C4—1424no—
227C4⋊C4—1422no—
228C4⋊C4—1424no—
229C4⋊C4—1422no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1424no—
233C22⋊C4—1424no—
234C22⋊C4—1424no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—714no—
242C23—712no—
243C23—1421no—
244C23—1421no—
245C23—1422no—
246C23D4×D7114yes—
247C23C23×D7111yescharacteristic
248Q8—1428no—
249D4—28no—
250D4—14216no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2844no—
260C2×C4—712no—
261C2×C4—714no—
262C2×C4—712no—
263C2×C4—1428no—
264C2×C4—1428no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4—1421no—
268C2×C4—1422no—
269C2×C4—1422no—
270C2×C4—14216no—
271C2×C4—1421no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4—1422no—
275C2×C4—1421no—
276C2×C4D4×D7112yes—
277C2×C4#1176586412yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—714no—
286C22—1422no—
287C22—1424no—
288C22—1424no—
289C22—1422no—
290C22C2×D4×D7111yescharacteristic
291C22C2×D4×D7112yes—
292C22#41×D7111yescharacteristic
293C22C2×#1176586411yescharacteristic
294C22#1455617512yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—2842no—
300C4—2841no—
301C4—1422no—
302C4—1424no—
303C4—1422no—
304C2—44no—
305C2—1424no—
306C2#120×D7111yescharacteristic
307C2#1798650911yescharacteristic
308C2#1798656911yescharacteristic
309C2#1798664712yes—
310C2#1798664912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#394×C71C2
hypercenter#394D71
hypocenterC71#2406
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14556110C2semidirect
3. C2×C4⋊C4×D71C2semidirect
4. #17985920C2semidirect
5. #17985919C2semidirect
6. #17985913C2semidirect
7. #17985875C2semidirect
8. #17985874C2semidirect
9. #17985871C2semidirect
10. #17985861C2semidirect
11. #394×C71C2semidirect
102. C2×C4×C71#41semidirect
117. C2×Dic71#41semidirect
156. C71#2406semidirect
163. #394D71semidirect
277. C2×C4#11765864semidirect