Finite Group Databasebeta

← #22594083

Subgroups — #22594083

Order 9088.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14556113C212yesmaximal, complemented
2C2×#14556103C211yesmaximal, characteristic
3C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
4#17985930C211yesmaximal, characteristic, complemented
5#17985923C212yesmaximal, complemented
6#17985921C212yesmaximal, complemented
7#17985917C211yesmaximal, characteristic, complemented
8#17985916C211yesmaximal, characteristic, complemented
9#17985877C212yesmaximal, complemented
10#17985858C211yesmaximal, characteristic, complemented
11#398×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C22×D284C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16C22×C4×D71C2211yescharacteristic
17#14556113—28no—
18#14556103—24no—
19C2×C4⋊Dic71C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×Dic71⋊C4C2212yes—
24C4⋊C4×D71—24no—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2212yes—
30C2×(C2×D71)⋊C4C2212yes—
31C71⋊#95C2211yescharacteristic
32C71⋊#95C2212yes—
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2212yes—
35C2×C4⋊C4×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37#95×C71C2212yes—
38C23×D71C2312yes—
39C23×D71C2311yescharacteristic
40C2×D284—24no—
41C2×D284—24no—
42C2×D284—24no—
43C2×D284—22no—
44C2×D284—42no—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—42no—
53C2×C4×D71—42no—
54C2×C4×D71D414yes—
55C2×C4×D71D414yescomplemented
56C4⋊Dic71—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4—24no—
60Dic71⋊C4—42no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—24no—
63(C2×D71)⋊C4—24no—
64(C2×D71)⋊C4—24no—
65(C2×D71)⋊C4—42no—
66(C2×D71)⋊C4—42no—
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2312yes—
69C22×C4×C71C2312yes—
70C22×C4×C71C2311yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C4⋊C4×C71—22no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—24no—
75C23×C71C2411yescharacteristic
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—22no—
85C22×D71C2×D412yes—
86C22×D71C2×D412yes—
87C22×D71#4112yes—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71C2×D412yes—
100C2×C4×C71C2×D412yes—
101C2×Dic71—21no—
102C2×Dic71—21no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—21no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71C2×D412yes—
110C2×Dic71C2×D412yes—
111D284—44no—
112D284—48no—
113C4×D71—28no—
114C4×D71—44no—
115C4×D71—44no—
116C22×C71C22×D411yescharacteristic
117C22×C71C22×D411yescharacteristic
118C22×C71C2×#4111yescharacteristic
119C22×C71H(5,𝔽2)12yes—
120C22×C712−1+412yes—
121C2×D71—22no—
122C2×D71—22no—
123C2×D71—24no—
124C2×D71—24no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—42no—
128C2×D71—42no—
129C2×D71—42no—
130C2×D71—42no—
131C2×D71—42no—
132C2×D71#12114yes—
133C4×C71—22no—
134C4×C71—22no—
135C4×C71—42no—
136C4×C71—41no—
137C4×C71—42no—
138Dic71—22no—
139Dic71—22no—
140Dic71—41no—
141Dic71—42no—
142C2×C71C2×#12111yescharacteristic
143C2×C71#53411yescharacteristic
144C2×C71#53611yescharacteristic
145C2×C71#53712yes—
146C2×C71#53912yes—
147D71—24no—
148D71—82no—
149#2618—711nomaximal, Sylow
150C71#261811yescharacteristic, complemented, Sylow
151C2×#123—712no—
152C2×#121—711no—
153C22×C4⋊C4—711no—
154#398—711no—
155#398D7111yescharacteristic, complemented
156#396—712no—
157#396—712no—
158#394—711no—
159#394—711no—
160#386—712no—
161#380—711no—
162C22×D4—711no—
163C22×D4—712no—
164C23×C4—711no—
165C23×C4—711no—
166C23×C4—711no—
167#123—1428no—
168#121—1424no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—711no—
171C2×C4⋊C4—711no—
172C2×C4⋊C4—711no—
173C2×C4⋊C4—712no—
174C2×C4⋊C4—1424no—
175C2×C4⋊C4C2×D7111yescharacteristic
176C2×C4⋊C4C2×D7111yescharacteristic
177C2×C4⋊C4C2×D7112yes—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—712no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—712no—
183C2×C22⋊C4—712no—
184#95—711no—
185#95—712no—
186#95C2×D7111yescharacteristic
187#95C2×D7112yes—
188C24—711no—
189C24—712no—
190C2×D4—2842no—
191C2×D4—1424no—
192C2×D4—1424no—
193C2×D4—1424no—
194C2×D4—1422no—
195C22×C4—2842no—
196C22×C4—2842no—
197C22×C4—714no—
198C22×C4—711no—
199C22×C4—711no—
200C22×C4—714no—
201C22×C4—712no—
202C22×C4—711no—
203C22×C4—1422no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4C22×D7112yes—
207C22×C4C22×D7112yes—
208C22×C4C22×D7111yescharacteristic
209C22×C4C22×D7111yescharacteristic
210C22×C4C22×D7111yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—2842no—
215C4⋊C4—1422no—
216C4⋊C4—1424no—
217C4⋊C4—1422no—
218C4⋊C4—1422no—
219C22⋊C4—2842no—
220C22⋊C4—2842no—
221C22⋊C4—1424no—
222C22⋊C4—1424no—
223C22⋊C4—1424no—
224C22⋊C4—1424no—
225C23—712no—
226C23—712no—
227C23—712no—
228C23—1422no—
229C23—1422no—
230C23—1422no—
231C23—1422no—
232C23—1422no—
233C23—1422no—
234C23—1422no—
235C23—1422no—
236C23—1422no—
237C23C23×D7111yescharacteristic
238D4—2844no—
239D4—2848no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—2844no—
252C2×C4—2844no—
253C2×C4—712no—
254C2×C4—712no—
255C2×C4—1428no—
256C2×C4—1422no—
257C2×C4—1421no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1422no—
261C2×C4—1421no—
262C2×C4—1421no—
263C2×C4—1422no—
264C2×C4D4×D7112yes—
265C2×C4D4×D7112yes—
266C22—2842no—
267C22—2842no—
268C22—2842no—
269C22—2842no—
270C22—2842no—
271C22—2842no—
272C22—2842no—
273C22—714no—
274C22—1422no—
275C22—1422no—
276C22—1424no—
277C22—1424no—
278C22C2×D4×D7111yescharacteristic
279C22C2×D4×D7111yescharacteristic
280C22#41×D7111yescharacteristic
281C22#1455617412yes—
282C22#1455617612yes—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—42no—
287C4—42no—
288C4—2841no—
289C4—2842no—
290C4—1422no—
291C4—1422no—
292C2—5682no—
293C2—1424no—
294C2#121×D7111yescharacteristic
295C2#1798656011yescharacteristic
296C2#1798657411yescharacteristic
297C2#1798657912yes—
298C2#1798659712yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2618
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14556113C2semidirect
3. C2×C4⋊C4×D71C2semidirect
4. #17985930C2semidirect
5. #17985923C2semidirect
6. #17985921C2semidirect
7. #17985917C2semidirect
8. #17985916C2semidirect
9. #17985877C2semidirect
10. #17985858C2semidirect
11. #398×C71C2semidirect
55. C2×C4×D71D4semidirect
150. C71#2618semidirect
155. #398D71semidirect