Finite Group Databasebeta

← #22598731

Subgroups — #22598731

Order 9088.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#17986545C211yesmaximal, characteristic
2C2×#14556127C211yesmaximal, characteristic, complemented
3#17986360C212yesmaximal, complemented
4#17986357C212yesmaximal, complemented
5#17986348C212yesmaximal, complemented
6#17986344C212yesmaximal, complemented
7#17986001C211yesmaximal, characteristic, complemented
8#17986000C211yesmaximal, characteristic, complemented
9#17985998C211yesmaximal, characteristic, complemented
10M16×Dic71C211yesmaximal, characteristic, complemented
11#417×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×Dic142—21no—
14C2×#11765849C2211yescharacteristic
15#14556127—21no—
16#14556127—21no—
17#14556127C2212yes—
18#14556127C2212yes—
19#14556126C2211yescharacteristic
20#14556124C2211yescharacteristic
21#14556117C2211yescharacteristic
22#14556115C2211yescharacteristic
23#14556100—21no—
24#14556100—21no—
25C4×C22⋊D71—21no—
26C4×C22⋊D71—21no—
27C2×C4⋊Dic71C2211yescharacteristic
28C2×Dic71⋊C4—21no—
29C2×C4×Dic71C2211yescharacteristic
30#14556053C2212yes—
31#14556052C2212yes—
32#14556051C2212yes—
33#14556050C2212yes—
34#14556049C2212yes—
35#14556048C2212yes—
36#14556038C2212yes—
37C8×Dic71C2212yes—
38C2×#41×C71C2211yescharacteristic
39C2×M16×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#103×C71C2212yes—
42#102×C71C2212yes—
43#11765865—22no—
44#11765865—22no—
45C2×Dic142—22no—
46C2×Dic142—22no—
47C2×Dic142—44no—
48C2×Dic142C2311yescharacteristic
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—21no—
51C2×C22⋊D71—21no—
52C2×C22⋊D71—21no—
53C22×Dic71—21no—
54C22×Dic71D411yescharacteristic
55C22×Dic71D411yescharacteristic
56C2×C4×D71—21no—
57#11765849—22no—
58C2×C71⋊C8C2312yes—
59C4⋊Dic71—21no—
60C4⋊Dic71C2312yes—
61Dic71⋊C4—22no—
62Dic71⋊C4—22no—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65C22⋊Dic71—21no—
66C22⋊Dic71—21no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—21no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73#41×C71—22no—
74#41×C71—22no—
75C2×Q8×C71C2311yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71C2311yescharacteristic
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71C2311yescharacteristic
81M16×C71—22no—
82C2×C8×C71C2312yes—
83C4⋊C4×C71—21no—
84C4⋊C4×C71C2312yes—
85C23×C71—21no—
86C23×C71C2×D411yescharacteristic
87C22×D71—21no—
88Q8×C71—21no—
89Q8×C71—21no—
90D4×C71—21no—
91D4×C71—21no—
92D4×C71—42no—
93C2×C4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71#4112yes—
99C2×C4×C71#4112yes—
100C2×C4×C71C2411yescharacteristic
101C22⋊D71—42no—
102C22⋊D71—42no—
103C22⋊D71—42no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—42no—
109C2×Dic71C2×D411yescharacteristic
110C2×Dic71C2×D411yescharacteristic
111C2×Dic71C2×D411yescharacteristic
112C2×Dic71C2×D411yescharacteristic
113D284—21no—
114D284—21no—
115C4×D71—42no—
116Dic142—21no—
117Dic142—21no—
118Dic142—44no—
119Dic142—44no—
120C8×C71—22no—
121C71⋊C8—22no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—41no—
125C22×C71—41no—
126C22×C71—41no—
127C22×C71#12412yes—
128C22×C71C22×D411yescharacteristic
129C2×D71—41no—
130C2×D71—41no—
131C2×D71—41no—
132C4×C71—41no—
133C4×C71—42no—
134C4×C71#12412yes—
135C4×C71C2×#4111yescharacteristic
136C4×C71C2×#4111yescharacteristic
137Dic71—21no—
138Dic71—21no—
139Dic71—41no—
140Dic71—42no—
141Dic71AGL(1,ℤ/8)12yes—
142Dic71#13712yes—
143C2×C71—22no—
144C2×C71—81no—
145C2×C71C2×#12411yescharacteristic
146C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C71C2×#13711yescharacteristic
148D71—81no—
149#3141—711nomaximal, Sylow
150C71#314111yescharacteristic, complemented, Sylow
151#532—711no—
152C2×C4⋊Q8—711no—
153#489—712no—
154#489—712no—
155#487—712no—
156#486—712no—
157#417—711no—
158#417—711no—
159#417—711no—
160#417D7111yescharacteristic, complemented
161C4×M16—711no—
162C2×#41—711no—
163C2×#41C2×D7111yescharacteristic
164C22×Q8—1421no—
165C2×M16—711no—
166C2×M16C2×D7111yescharacteristic
167C4⋊Q8—712no—
168C4⋊Q8—712no—
169C4⋊Q8—711no—
170C4⋊Q8—1421no—
171C4⋊Q8—1421no—
172#127—711no—
173#124—711no—
174#124—711no—
175#121—1421no—
176#121—1421no—
177C4×D4—1421no—
178C4×D4—1421no—
179C2×C4⋊C4—711no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4C2×D7111yescharacteristic
182C2×C42—711no—
183#103—712no—
184#103—712no—
185#103—712no—
186#103C2×D7112yes—
187#102—712no—
188#102—712no—
189#102—712no—
190#102C2×D7112yes—
191#97—712no—
192C4×C8—712no—
193#41—22no—
194#41—22no—
195#41—1422no—
196#41—1422no—
197C2×Q8—2844no—
198C2×Q8—711no—
199C2×Q8—1422no—
200C2×Q8—1422no—
201C2×Q8C22×D7111yescharacteristic
202C2×D4—21no—
203C2×D4—711no—
204C2×D4—1421no—
205C2×D4—1421no—
206C2×D4—1421no—
207C2×D4C22×D7111yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—711no—
211C22×C4—711no—
212C22×C4—1421no—
213C22×C4—1421no—
214C22×C4C22×D7111yescharacteristic
215M16—22no—
216M16—1422no—
217C2×C8—712no—
218C2×C8C22×D7112yes—
219C4⋊C4—21no—
220C4⋊C4—712no—
221C4⋊C4—1422no—
222C4⋊C4—1422no—
223C4⋊C4—1421no—
224C4⋊C4—1421no—
225C4⋊C4—1421no—
226C4⋊C4C22×D7112yes—
227C22⋊C4—1421no—
228C22⋊C4—1421no—
229C22⋊C4—1421no—
230C22⋊C4—1421no—
231C42—711no—
232C42—711no—
233C42—711no—
234C42—711no—
235C23—21no—
236C23—1421no—
237C23D4×D7111yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—2844no—
241Q8—2844no—
242Q8—1421no—
243Q8—1421no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—2842no—
248D4—2842no—
249D4—2842no—
250D4—1421no—
251D4—1421no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—711no—
259C2×C4—711no—
260C2×C4—711no—
261C2×C4—711no—
262C2×C4—1421no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4D4×D7111yescharacteristic
267C2×C4#1176586412yes—
268C2×C4#1176586512yes—
269C2×C4C23×D7111yescharacteristic
270C8—22no—
271C8—1422no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—2841no—
278C22—2841no—
279C22—2841no—
280C22#1455611412yes—
281C22C2×D4×D7111yescharacteristic
282C4—41no—
283C4—42no—
284C4—2841no—
285C4—2842no—
286C4—712no—
287C4—712no—
288C4—1421no—
289C4—1421no—
290C4#1455611412yes—
291C4C2×#1176586411yescharacteristic
292C4C2×#1176586511yescharacteristic
293C2—22no—
294C2—81no—
295C2—5681no—
296C2C2×#1455611411yescharacteristic
297C2AGL(1,ℤ/8)×D7111yescharacteristic
298C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#417×C71C2
socleC22×C71C22×D4
hypercenter#417D71
hypocenterC71#3141
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#14556127C2semidirect
3. #17986360C2semidirect
4. #17986357C2semidirect
5. #17986348C2semidirect
6. #17986344C2semidirect
7. #17986001C2semidirect
8. #17986000C2semidirect
9. #17985998C2semidirect
10. M16×Dic71C2semidirect
11. #417×C71C2semidirect
150. C71#3141semidirect
160. #417D71semidirect