Finite Group Databasebeta

← #22598264

Subgroups — #22598264

Order 9088.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986512C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#17986333C212yesmaximal, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986319C212yesmaximal, complemented
6#17986313C211yesmaximal, characteristic, complemented
7#17986312C211yesmaximal, characteristic, complemented
8#17986038C212yesmaximal, complemented
9#17985999C212yesmaximal, complemented
10#17985961C211yesmaximal, characteristic, complemented
11#483×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C22×C4×D71—21no—
14C22×C4×D71C2211yescharacteristic
15C2×#11765848C2212yes—
16#14556113—21no—
17#14556104C2211yescharacteristic
18#14556103C2211yescharacteristic
19#14556102C2211yescharacteristic
20#14556098—21no—
21#14556090—21no—
22(C22⋊D71)⋊C4—21no—
23#14556086C2211yescharacteristic
24C2×C4⋊Dic71C2211yescharacteristic
25C2×Dic71⋊C4—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71—21no—
28C4⋊C4×D71C2212yes—
29C4⋊C4×D71C2212yes—
30#14556062C2212yescomplemented
31#14556061C2212yescomplemented
32#14556059C2212yescomplemented
33#14556052C2212yescomplemented
34#14556050C2212yescomplemented
35#14556048C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38C22⋊Q8×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#106×C71C2212yescomplemented
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44#11765866—24no—
45C2×D284—21no—
46C2×D284—21no—
47C2×D284C2311yescharacteristic
48C2×C22⋊D71—21no—
49C22×Dic71—21no—
50C22×Dic71D411yescharacteristic
51C2×C4×D71—24no—
52C2×C4×D71—22no—
53C2×C4×D71—22no—
54C2×C4×D71—21no—
55C2×C4×D71—44no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765848—24no—
61C2×C71⋊C8C2312yes—
62C4⋊Dic71—21no—
63C4⋊Dic71C2312yes—
64C4⋊Dic71C2311yescharacteristic
65Dic71⋊C4—22no—
66Dic71⋊C4—22no—
67Dic71⋊C4—21no—
68(C2×D71)⋊C4—21no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73C2×Q8×C71C2311yescharacteristic
74C22×C4×C71—21no—
75C22×C4×C71C2311yescharacteristic
76C2×C8×C71C2312yes—
77C4⋊C4×C71—21no—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2312yes—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C23×C71C2×D411yescharacteristic
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—21no—
87C22×D71C2×D411yescharacteristic
88C22×D71C2×D411yescharacteristic
89Q8×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—42no—
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—42no—
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—22no—
101C2×Dic71—42no—
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D411yescharacteristic
104D284—22no—
105D284—42no—
106D284—42no—
107C4×D71—22no—
108C4×D71—22no—
109C4×D71—22no—
110C4×D71—44no—
111C4×D71—44no—
112C4×D71—42no—
113C4×D71#4114yes—
114C8×C71—22no—
115C71⋊C8—22no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71AGL(1,ℤ/8)12yes—
119C22×C71C22×D411yescharacteristic
120C2×D71—28no—
121C2×D71—21no—
122C2×D71—21no—
123C2×D71—21no—
124C2×D71—21no—
125C2×D71—41no—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71#12312yes—
129C2×D71#13712yes—
130C4×C71—21no—
131C4×C71—41no—
132C4×C71—42no—
133C4×C71—41no—
134C4×C71C2×#4112yes—
135Dic71—21no—
136Dic71—42no—
137Dic71—41no—
138Dic71#12312yes—
139C2×C71—22no—
140C2×C71C2×#12311yescharacteristic
141C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C71C2×#13711yescharacteristic
143D71—22no—
144D71—22no—
145D71—81no—
146#3092—711nomaximal, Sylow
147C71#309211yescharacteristic, complemented, Sylow
148#525—711no—
149C22×C4⋊C4—711no—
150#484—712no—
151#483—711no—
152#483D7111yescharacteristic, complemented
153#481—712no—
154#480—711no—
155#480—711no—
156#428—712no—
157#417—712no—
158#407—711no—
159C2×#41—711no—
160C23×C4—711no—
161C23×C4—1421no—
162C2×M16—712no—
163#123—1421no—
164C22⋊Q8—711no—
165C22⋊Q8C2×D7111yescharacteristic
166#121—711no—
167#121—711no—
168#120—1421no—
169C4×D4—1421no—
170C4×D4—1421no—
171#117—711no—
172C2×C4⋊C4—711no—
173C2×C4⋊C4—712no—
174C2×C4⋊C4—712no—
175C2×C4⋊C4—1421no—
176C2×C4⋊C4—1421no—
177C2×C4⋊C4—1421no—
178C2×C4⋊C4C2×D7111yescharacteristic
179#107—712no—
180#107—712no—
181#106—712no—
182#106C2×D7112yescomplemented
183#103—712no—
184#103C2×D7112yescomplemented
185#102—712no—
186#102—712no—
187C22⋊C8—711no—
188C22⋊C8—712no—
189C22⋊C8C2×D7111yescharacteristic
190C24—711no—
191#41—1424no—
192C2×Q8C22×D7111yescharacteristic
193C2×D4—711no—
194C2×D4—1421no—
195C2×D4—1421no—
196C2×D4—1421no—
197C22×C4—21no—
198C22×C4—2844no—
199C22×C4—711no—
200C22×C4—711no—
201C22×C4—711no—
202C22×C4—711no—
203C22×C4—711no—
204C22×C4—1421no—
205C22×C4—1422no—
206C22×C4—1422no—
207C22×C4—1424no—
208C22×C4—1421no—
209C22×C4C22×D7111yescharacteristic
210M16—1424no—
211C2×C8—712no—
212C2×C8C22×D7112yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—712no—
216C4⋊C4—711no—
217C4⋊C4—1422no—
218C4⋊C4—1422no—
219C4⋊C4—1421no—
220C4⋊C4—1421no—
221C4⋊C4C22×D7112yes—
222C4⋊C4C22×D7111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1421no—
225C22⋊C4—1421no—
226C22⋊C4—1421no—
227C22⋊C4—1421no—
228C42—1421no—
229C23—711no—
230C23—711no—
231C23—1422no—
232C23—1422no—
233C23—1421no—
234C23—1422no—
235C23D4×D7111yescharacteristic
236Q8—22no—
237D4—2842no—
238D4—2842no—
239D4—2842no—
240D4—1422no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—2844no—
247C2×C4—2844no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—711no—
251C2×C4—714no—
252C2×C4—711no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1421no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4D4×D7111yescharacteristic
260C2×C4C23×D7111yescharacteristic
261C8—22no—
262C8—1422no—
263C22—21no—
264C22—21no—
265C22—2841no—
266C22—2841no—
267C22—2841no—
268C22—712no—
269C22—712no—
270C22—1428no—
271C22—1421no—
272C22—1421no—
273C22—1421no—
274C22—1421no—
275C22#1455615512yes—
276C22C2×D4×D7111yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—2842no—
282C4—2841no—
283C4—712no—
284C4—1421no—
285C4#41×D7112yes—
286C2—22no—
287C2—5681no—
288C2—1422no—
289C2—1422no—
290C2#123×D7111yescharacteristic
291C2C2×#1455615511yescharacteristic
292C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#483×C71C2
socleC22×C71C22×D4
hypercenter#483D71
hypocenterC71#3092
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D71C2semidirect
3. #17986333C2semidirect
4. #17986327C2semidirect
5. #17986319C2semidirect
6. #17986313C2semidirect
7. #17986312C2semidirect
8. #17986038C2semidirect
9. #17985999C2semidirect
10. #17985961C2semidirect
11. #483×C71C2semidirect
30. #14556062C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556052C22semidirect
34. #14556050C22semidirect
35. #14556048C22semidirect
40. #106×C71C22semidirect
41. #103×C71C22semidirect
147. C71#3092semidirect
152. #483D71semidirect
182. #106C2×D71semidirect
184. #103C2×D71semidirect