Finite Group Databasebeta

← #22600648

Subgroups — #22600648

Order 9088.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986654C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic, complemented
3#17986349C211yesmaximal, characteristic, complemented
4#17986333C212yesmaximal, complemented
5#17986280C211yesmaximal, characteristic, complemented
6#17986242C212yesmaximal, complemented
7#17986143C212yesmaximal, complemented
8C4×#11765850C211yesmaximal, characteristic, complemented
9#17986076C211yesmaximal, characteristic, complemented
10#17985999C212yesmaximal, complemented
11#461×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yes—
13C2×#11765865—21no—
14C2×#11765850C2211yescharacteristic
15C2×#11765848C2212yes—
16#14556128C2211yescharacteristic
17#14556113—22no—
18#14556110—22no—
19#14556107—21no—
20#14556104C2212yes—
21#14556103—21no—
22#14556099C2212yes—
23C4×D284C2211yescharacteristic
24C4×D284C2211yescharacteristic
25C4⋊C4×D71—21no—
26C4⋊C4×D71C2212yes—
27#14556062C2212yes—
28#14556061C2211yescharacteristic
29#14556052C2212yes—
30#14556052C2212yes—
31#14556050C2212yes—
32#14556049C2211yescharacteristic
33#14556048C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×SD16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×D4×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41#11765866—24no—
42#11765866—42no—
43#11765865—44no—
44C2×Dic142—21no—
45C2×D284—22no—
46C2×D284D411yescharacteristic
47C2×D284C2311yescharacteristic
48C2×C22⋊D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—21no—
51C2×C4×D71—22no—
52C2×C4×D71C2312yes—
53#11765850—22no—
54#11765848—42no—
55C2×C71⋊C8C2311yescharacteristic
56C2×C71⋊C8C2311yescharacteristic
57C4⋊Dic71—21no—
58C4⋊Dic71D411yescharacteristic
59C4⋊Dic71C2311yescharacteristic
60C4⋊Dic71C2312yes—
61Dic71⋊C4—22no—
62Dic71⋊C4—22no—
63C22⋊Dic71—22no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4D412yes—
67C2×Q8×C71C2312yes—
68C2×D4×C71C2311yescharacteristic
69C22×C4×C71—21no—
70SD16×C71—42no—
71C2×C8×C71C2312yes—
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2311yescharacteristic
75C22⋊C4×C71—21no—
76C42×C71C2311yescharacteristic
77C23×C71—21no—
78C22×D71—21no—
79C22×D71C2×D412yes—
80Q8×C71—22no—
81Q8×C71—22no—
82D4×C71—21no—
83D4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C22⋊D71—44no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93C2×Dic71C2×D412yes—
94D284—22no—
95D284—21no—
96D284—44no—
97D284#4112yescomplemented
98C4×D71—22no—
99C4×D71—22no—
100C4×D71—44no—
101C4×D71—44no—
102C4×D71—42no—
103Dic142—42no—
104C8×C71—22no—
105C71⋊C8—21no—
106C71⋊C8—21no—
107C22×C71—41no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71C22×D411yescharacteristic
111C2×D71—22no—
112C2×D71—22no—
113C2×D71—22no—
114C2×D71—22no—
115C2×D71—41no—
116C2×D71—41no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71—41no—
120C4×C71AGL(1,ℤ/8)12yes—
121C4×C71C2×#4111yescharacteristic
122C4×C71H(5,𝔽2)11yescharacteristic
123Dic71—22no—
124Dic71—42no—
125Dic71—41no—
126C2×C71—81no—
127C2×C71#54611yescharacteristic
128C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C71#57811yescharacteristic
130D71—42no—
131D71—42no—
132#3330—711nomaximal, Sylow
133C71#333011yescharacteristic, complemented, Sylow
134#547—711no—
135#547—711no—
136#487—711no—
137#484—712no—
138#474—711no—
139#468—712no—
140#461D7111yescharacteristic, complemented
141#449—712no—
142C4×D8—711no—
143#436—711no—
144#417—712no—
145C2×#41—712no—
146C2×#41—1421no—
147C2×SD16C2×D7112yes—
148C2×D8—711no—
149C2×M16—712no—
150C4⋊Q8—711no—
151C4⋊Q8C2×D7111yescharacteristic
152#123—1422no—
153#123—1422no—
154C22⋊Q8—712no—
155C22⋊Q8—1421no—
156#121—712no—
157#121—1421no—
158C4×D4—711no—
159C4×D4—711no—
160C4×D4C2×D7111yescharacteristic
161C2×C4⋊C4—712no—
162C2×C4⋊C4—1421no—
163#107—712no—
164#107—711no—
165C4⋊C8C2×D7111yescharacteristic
166#103—712no—
167#103—712no—
168#103C2×D7112yes—
169#102—711no—
170#102—712no—
171#102—711no—
172C22⋊C8—712no—
173C4×C8—711no—
174#41—2844no—
175#41—2842no—
176#41—1424no—
177C2×Q8—1421no—
178C2×Q8C22×D7112yes—
179C2×D4—711no—
180C2×D4—711no—
181C2×D4—1422no—
182C2×D4—1422no—
183C2×D4C22×D7111yescharacteristic
184C22×C4—21no—
185C22×C4—712no—
186C22×C4—1422no—
187C22×C4—1422no—
188C22×C4—1421no—
189SD16—42no—
190D8—1422no—
191M16—2842no—
192C2×C8—711no—
193C2×C8—711no—
194C2×C8C22×D7112yes—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—711no—
198C4⋊C4—712no—
199C4⋊C4—711no—
200C4⋊C4—1422no—
201C4⋊C4—1422no—
202C4⋊C4—1421no—
203C4⋊C4C22×D7111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—712no—
206C22⋊C4—1422no—
207C22⋊C4—1422no—
208C22⋊C4—1421no—
209C42C22×D7111yescharacteristic
210C23—21no—
211C23—712no—
212C23—1421no—
213Q8—22no—
214Q8—22no—
215Q8—2842no—
216D4—21no—
217D4—21no—
218D4—2844no—
219D4—2844no—
220D4—712no—
221D4—1421no—
222D4—1422no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—2844no—
227C2×C4—2844no—
228C2×C4—2842no—
229C2×C4—712no—
230C2×C4—1421no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4D4×D7111yescharacteristic
235C2×C4D4×D7111yescharacteristic
236C2×C4C23×D7111yescharacteristic
237C8—22no—
238C8—1421no—
239C8—1421no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—2841no—
244C22—2841no—
245C22—1422no—
246C22—1422no—
247C22—1422no—
248C22—1422no—
249C22C2×D4×D7111yescharacteristic
250C4—21no—
251C4—42no—
252C4—41no—
253C4—2842no—
254C4—2841no—
255C4—1422no—
256C4#1455615512yes—
257C4#41×D7111yescharacteristic
258C4#1455617311yescharacteristic
259C2—81no—
260C2—2842no—
261C2—2842no—
262C2#1798664611yescharacteristic
263C2C2×#1455615511yescharacteristic
264C2#1798681511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#461×C71C2
socleC22×C71C22×D4
hypercenter#461D71
hypocenterC71#3330
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #17986654C2semidirect
2. #17986652C2semidirect
3. #17986349C2semidirect
4. #17986333C2semidirect
5. #17986280C2semidirect
6. #17986242C2semidirect
7. #17986143C2semidirect
8. C4×#11765850C2semidirect
9. #17986076C2semidirect
10. #17985999C2semidirect
11. #461×C71C2semidirect
97. D284#41semidirect
133. C71#3330semidirect
140. #461D71semidirect