Finite Group Databasebeta

← #22600572

Subgroups — #22600572

Order 9088.

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

normal subgroups only

This group has 448 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#17986719C211yesmaximal, characteristic, complemented
2D4×D284C211yesmaximal, characteristic
3#17986346C211yesmaximal, characteristic, complemented
4#17986324C212yesmaximal, complemented
5#17986280C211yesmaximal, characteristic, complemented
6#17986245C212yesmaximal, complemented
7#17986200C211yesmaximal, characteristic, complemented
8#17986152C212yesmaximal, complemented
9C4×#11765854C211yesmaximal, characteristic, complemented
10#17986005C212yesmaximal, complemented
11#436×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C2×D4×D71—21no—
14C2×#11765854C2211yescharacteristic
15C2×#11765854C2212yescomplemented
16#14556128C2211yescharacteristic, complemented
17#14556125C2211yescharacteristic
18#14556121—22no—
19#14556115—21no—
20#14556107C2212yescomplemented
21#14556103—21no—
22#14556102C2212yes—
23#14556098—22no—
24C4×Dic142C2211yescharacteristic, complemented
25#14556093—21no—
26C4×D284C2211yescharacteristic
27#14556087C2212yescomplemented
28#14556062C2212yescomplemented
29#14556060C2211yescharacteristic, complemented
30#14556053C2211yescharacteristic, complemented
31#14556050C2211yescharacteristic
32#14556050C2212yes—
33#14556050C2212yescomplemented
34#14556050C2212yescomplemented
35#14556050C2212yescomplemented
36C2×M16×C71C2212yescomplemented
37C4×D4×C71C2211yescharacteristic
38C4⋊C8×C71C2211yescharacteristic, complemented
39C22⋊C8×C71C2212yescomplemented
40C4×C8×C71C2211yescharacteristic, complemented
41C23×D71—22no—
42C2×Dic142—21no—
43C2×Dic142C2311yescharacteristic
44C2×D284—24no—
45C2×D284—21no—
46C2×D284—22no—
47C2×D284—21no—
48C2×D284—42no—
49C2×D284C2312yes—
50C2×D284C2311yescharacteristic
51C2×C22⋊D71—22no—
52D4×D71—44no—
53C2×C4×D71—21no—
54#11765854—22no—
55#11765854—42no—
56C4⋊Dic71—21no—
57C4⋊Dic71C2311yescharacteristic
58C4⋊Dic71C2312yes—
59C4⋊Dic71C2311yescharacteristic
60Dic71⋊C4—22no—
61Dic71⋊C4—21no—
62C22⋊Dic71—22no—
63C22⋊Dic71—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—22no—
66C4×Dic71—22no—
67C4×Dic71—21no—
68C2×D4×C71D411yescharacteristic
69C22×C4×C71C2312yes—
70M16×C71—42no—
71C2×C8×C71C2311yescharacteristic
72C2×C8×C71C2311yescharacteristic
73C2×C8×C71C2312yes—
74C4⋊C4×C71D411yescharacteristic
75C22⋊C4×C71D412yes—
76C42×C71C2311yescharacteristic
77C23×C71C2×D412yes—
78C22×D71—21no—
79C22×D71—22no—
80C22×D71—44no—
81C22×D71—44no—
82C22×D71—82no—
83D4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D412yes—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—44no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95D284—22no—
96D284—21no—
97D284—22no—
98D284—44no—
99D284—42no—
100D284—81no—
101D284#4112yes—
102C4×D71—42no—
103Dic142—21no—
104Dic142—21no—
105Dic142—42no—
106C8×C71—21no—
107C8×C71—21no—
108C8×C71—22no—
109C22×C71—22no—
110C22×C71—22no—
111C22×C71—22no—
112C22×C71C22×D411yescharacteristic
113C2×D71—22no—
114C2×D71—41no—
115C2×D71—41no—
116C2×D71—42no—
117C2×D71—42no—
118C2×D71—42no—
119C2×D71—88no—
120C4×C71—22no—
121C4×C71—21no—
122C4×C71AGL(1,ℤ/8)12yes—
123C4×C71C2×#4111yescharacteristic
124C4×C71H(5,𝔽2)11yescharacteristic
125Dic71—41no—
126Dic71—42no—
127Dic71—41no—
128Dic71—41no—
129C2×C71—42no—
130C2×C71#54611yescharacteristic
131C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C71#57711yescharacteristic
133D71—42no—
134D71—82no—
135#3324—711nomaximal, Sylow
136C71#332411yescharacteristic, complemented, Sylow
137#556—711no—
138D42—711no—
139#486—711no—
140#482—712no—
141#474—711no—
142#468—712no—
143#460—711no—
144#450—712no—
145C4×SD16—711no—
146#436D7111yescharacteristic, complemented
147#418—712no—
148C22×D4—712no—
149C22×D4—1421no—
150C2×SD16—712no—
151C2×SD16—711no—
152C2×M16C2×D7112yescomplemented
153C4⋊Q8—711no—
154#127—711no—
155#126—1422no—
156#124—1421no—
157C22⋊Q8—712no—
158#121—712no—
159#121—1421no—
160#120—1422no—
161C4×Q8—711no—
162C4×Q8—1421no—
163C4×D4—711no—
164C4×D4C2×D7111yescharacteristic
165#117—712no—
166#107—712no—
167#106—711no—
168C4⋊C8C2×D7111yescharacteristic, complemented
169#103—711no—
170#102—711no—
171#102—712no—
172#102—712no—
173#102—712no—
174#102—712no—
175C22⋊C8C2×D7112yescomplemented
176C4×C8C2×D7111yescharacteristic, complemented
177C24—1422no—
178C2×Q8—711no—
179C2×Q8—1421no—
180C2×D4—2842no—
181C2×D4—2844no—
182C2×D4—711no—
183C2×D4—712no—
184C2×D4—1424no—
185C2×D4—1422no—
186C2×D4—1422no—
187C2×D4—1421no—
188C2×D4—1421no—
189C2×D4D28411yescharacteristic
190C22×C4—1421no—
191C22×C4C22×D7112yes—
192SD16—2842no—
193SD16—1422no—
194M16—42no—
195C2×C8C22×D7111yescharacteristic
196C2×C8C22×D7111yescharacteristic
197C2×C8C22×D7112yes—
198C4⋊C4—711no—
199C4⋊C4—712no—
200C4⋊C4—711no—
201C4⋊C4—1421no—
202C4⋊C4—1422no—
203C4⋊C4—1421no—
204C4⋊C4D28411yescharacteristic
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C22⋊C4—1421no—
208C22⋊C4—1422no—
209C22⋊C4D28412yes—
210C42—1422no—
211C42—1421no—
212C42C22×D7111yescharacteristic
213C23—2844no—
214C23—2844no—
215C23—5682no—
216C23—1422no—
217C23—1421no—
218C23C2×D28412yes—
219Q8—2842no—
220Q8—1421no—
221Q8—1421no—
222D4—22no—
223D4—2844no—
224D4—2844no—
225D4—2842no—
226D4—5681no—
227D4—712no—
228D4—1422no—
229D4—1422no—
230D4—1421no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—2842no—
234C2×C4—1421no—
235C2×C4—1422no—
236C2×C4—1421no—
237C2×C4—1421no—
238C2×C4C2×D28412yes—
239C2×C4C2×D28411yescharacteristic
240C2×C4C2×D28411yescharacteristic
241C2×C4C23×D7111yescharacteristic
242C8—21no—
243C8—21no—
244C8—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—2841no—
249C22—2841no—
250C22—2842no—
251C22—2842no—
252C22—2842no—
253C22—5688no—
254C22—1422no—
255C22C22×D28411yescharacteristic
256C4—22no—
257C4—21no—
258C4—2841no—
259C4—2841no—
260C4—2842no—
261C4—2841no—
262C4#1455615612yes—
263C4#41×D7111yescharacteristic
264C4#1455617311yescharacteristic
265C2—42no—
266C2—2842no—
267C2—5682no—
268C2#1798664411yescharacteristic
269C2C2×#1455615611yescharacteristic
270C2#1798681111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#436×C71C2
socleC22×C71C22×D4
hypercenter#436D71
hypocenterC71#3324
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #17986719C2semidirect
3. #17986346C2semidirect
4. #17986324C2semidirect
5. #17986280C2semidirect
6. #17986245C2semidirect
7. #17986200C2semidirect
8. #17986152C2semidirect
9. C4×#11765854C2semidirect
10. #17986005C2semidirect
11. #436×C71C2semidirect
15. C2×#11765854C22semidirect
16. #14556128C22semidirect
20. #14556107C22semidirect
24. C4×Dic142C22semidirect
27. #14556087C22semidirect
28. #14556062C22semidirect
29. #14556060C22semidirect
30. #14556053C22semidirect
33. #14556050C22semidirect
34. #14556050C22semidirect
35. #14556050C22semidirect
36. C2×M16×C71C22semidirect
38. C4⋊C8×C71C22semidirect
39. C22⋊C8×C71C22semidirect
40. C4×C8×C71C22semidirect
136. C71#3324semidirect
146. #436D71semidirect
152. C2×M16C2×D71semidirect
168. C4⋊C8C2×D71semidirect
175. C22⋊C8C2×D71semidirect
176. C4×C8C2×D71semidirect