Finite Group Databasebeta

← #22598136

Subgroups — #22598136

Order 9088.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#17986552C211yesmaximal, characteristic, complemented
2C2×#14556090C211yesmaximal, characteristic
3#17986302C212yesmaximal, complemented
4#17986298C212yesmaximal, complemented
5#17986288C212yesmaximal, complemented
6#17986278C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic, complemented
8#17986022C211yesmaximal, characteristic, complemented
9#17986005C211yesmaximal, characteristic
10#17986003C211yesmaximal, characteristic
11#428×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C4×D71—21no—
14C2×#11765849C2211yescharacteristic
15#14556127C2211yescharacteristic, complemented
16#14556127C2211yescharacteristic, complemented
17#14556119C2211yescharacteristic, complemented
18#14556119C2211yescharacteristic, complemented
19#14556105—21no—
20#14556105—21no—
21C4×Dic142—21no—
22C4×Dic142—21no—
23#14556090—21no—
24#14556090—21no—
25#14556090C2212yes—
26#14556090C2212yes—
27#14556087C2211yescharacteristic
28C2×(C2×D71)⋊C4—21no—
29C2×C4×Dic71C2211yescharacteristic
30#14556061C2212yescomplemented
31#14556059C2212yescomplemented
32Dic71⋊C8C2212yescomplemented
33Dic71⋊C8C2212yescomplemented
34#14556050C2212yes—
35#14556050C2212yes—
36#14556048C2212yes—
37#14556048C2212yes—
38C2×M16×C71C2211yescharacteristic
39#117×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#107×C71C2212yescomplemented
42#106×C71C2212yescomplemented
43C23×D71—21no—
44C2×Dic142—21no—
45C2×Dic142—21no—
46C2×D284—21no—
47C2×D284—22no—
48C2×D284Q814yescomplemented
49C2×D284C2312yes—
50C22×Dic71D411yescharacteristic
51C22×Dic71D411yescharacteristic
52C2×C4×D71—22no—
53C2×C4×D71—22no—
54C2×C4×D71—44no—
55#11765849—22no—
56C2×C71⋊C8C2312yes—
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59C4⋊Dic71C2311yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68C4×Dic71—21no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73C22×C4×C71—21no—
74C22×C4×C71C2311yescharacteristic
75M16×C71—22no—
76C2×C8×C71C2312yes—
77C4⋊C4×C71—21no—
78C4⋊C4×C71C2311yescharacteristic
79C4⋊C4×C71C2312yes—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C42×C71—21no—
83C23×C71C2×D411yescharacteristic
84C22×D71—24no—
85C22×D71—22no—
86C22×D71—42no—
87C22×D71—42no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—42no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×Q812yes—
94C2×C4×C71#4112yes—
95C2×C4×C71C2411yescharacteristic
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—41no—
100C2×Dic71C2×D411yescharacteristic
101C2×Dic71C2×D411yescharacteristic
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D411yescharacteristic
104D284—24no—
105D284—22no—
106D284C2×Q814yes—
107C4×D71—44no—
108C4×D71—44no—
109Dic142—42no—
110Dic142—42no—
111C8×C71—22no—
112C71⋊C8—22no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71C22⋊Q812yes—
116C22×C71C22×D411yescharacteristic
117C2×D71—24no—
118C2×D71—48no—
119C2×D71—42no—
120C2×D71—42no—
121C4×C71—41no—
122C4×C71—42no—
123C4×C71—41no—
124C4×C71C22⋊Q812yes—
125C4×C71C22×Q811yescharacteristic
126C4×C71C2×#4111yescharacteristic
127Dic71—21no—
128Dic71—21no—
129Dic71—21no—
130Dic71—41no—
131Dic71—41no—
132Dic71AGL(1,ℤ/8)12yes—
133C2×C71—22no—
134C2×C71C2×C22⋊Q811yescharacteristic
135C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C71#57511yescharacteristic
137D71—44no—
138#3077—711nomaximal, Sylow
139C71#307711yescharacteristic, complemented, Sylow
140#533—711no—
141C2×C4×D4—711no—
142#478—712no—
143#478—712no—
144#476—712no—
145#474—712no—
146#428—711no—
147#428D7111yescharacteristic, complemented
148#423—711no—
149#418—711no—
150#418—711no—
151C22×D4—711no—
152C23×C4—1421no—
153C2×M16—711no—
154C2×M16C2×D7111yescharacteristic
155C4⋊Q8—711no—
156C4⋊Q8—711no—
157#125—711no—
158#125—711no—
159C22⋊Q8—1421no—
160C22⋊Q8—1421no—
161C4×Q8—1421no—
162C4×Q8—1421no—
163C4×D4—712no—
164C4×D4—712no—
165C4×D4—1421no—
166C4×D4—1421no—
167#117—711no—
168#117C2×D7111yescharacteristic
169C2×C4⋊C4C2×D7111yescharacteristic
170C2×C22⋊C4—1421no—
171C2×C42—711no—
172#107—712no—
173#107C2×D7112yescomplemented
174#106—712no—
175#106C2×D7112yescomplemented
176C4⋊C8—712no—
177C4⋊C8—712no—
178#102—712no—
179#102—712no—
180#102—712no—
181#102—712no—
182C24—1421no—
183C2×Q8—1421no—
184C2×Q8—1421no—
185C2×D4—714no—
186C2×D4—712no—
187C2×D4—1421no—
188C2×D4—1422no—
189C22×C4—21no—
190C22×C4—2844no—
191C22×C4—711no—
192C22×C4—711no—
193C22×C4—1422no—
194C22×C4—1422no—
195C22×C4C22×D7111yescharacteristic
196M16—22no—
197M16—1422no—
198C2×C8—712no—
199C2×C8C22×D7112yes—
200C4⋊C4—21no—
201C4⋊C4—711no—
202C4⋊C4—711no—
203C4⋊C4—1421no—
204C4⋊C4—1421no—
205C4⋊C4—1421no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4C22×D7111yescharacteristic
210C4⋊C4C22×D7111yescharacteristic
211C4⋊C4C22×D7112yes—
212C22⋊C4—21no—
213C22⋊C4—1422no—
214C22⋊C4—1422no—
215C22⋊C4—1421no—
216C42—21no—
217C42—711no—
218C42—711no—
219C42—711no—
220C42—711no—
221C42—1421no—
222C23—2842no—
223C23—2842no—
224C23—1422no—
225C23—1424no—
226C23D4×D7111yescharacteristic
227Q8—2842no—
228Q8—2842no—
229D4—714no—
230D4—1422no—
231D4—1424no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—2844no—
237C2×C4—2844no—
238C2×C4—2841no—
239C2×C4—711no—
240C2×C4—711no—
241C2×C4—711no—
242C2×C4—711no—
243C2×C4—1421no—
244C2×C4—1421no—
245C2×C4—1422no—
246C2×C4D4×D7111yescharacteristic
247C2×C4Q8×D7112yes—
248C2×C4#1176586512yes—
249C2×C4C23×D7111yescharacteristic
250C8—22no—
251C8—1422no—
252C22—21no—
253C22—21no—
254C22—2848no—
255C22—2842no—
256C22—2842no—
257C22—1424no—
258C22#1455610612yes—
259C22C2×D4×D7111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—2841no—
264C4—2841no—
265C4—712no—
266C4—1421no—
267C4—1421no—
268C4—1421no—
269C4#1455610612yes—
270C4C2×Q8×D7111yescharacteristic
271C4C2×#1176586511yescharacteristic
272C2—22no—
273C2—2844no—
274C2C2×#1455610611yescharacteristic
275C2AGL(1,ℤ/8)×D7111yescharacteristic
276C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3077
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #17986552C2semidirect
3. #17986302C2semidirect
4. #17986298C2semidirect
5. #17986288C2semidirect
6. #17986278C2semidirect
7. #17986039C2semidirect
8. #17986022C2semidirect
11. #428×C71C2semidirect
15. #14556127C22semidirect
16. #14556127C22semidirect
17. #14556119C22semidirect
18. #14556119C22semidirect
30. #14556061C22semidirect
31. #14556059C22semidirect
32. Dic71⋊C8C22semidirect
33. Dic71⋊C8C22semidirect
41. #107×C71C22semidirect
42. #106×C71C22semidirect
48. C2×D284Q8semidirect
139. C71#3077semidirect
147. #428D71semidirect
173. #107C2×D71semidirect
175. #106C2×D71semidirect