Finite Group Databasebeta

← #22601005

Subgroups — #22601005

Order 9088.

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

normal subgroups only

This group has 362 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#17986735C211yesmaximal, characteristic, complemented
2#17986710C211yesmaximal, characteristic, complemented
3#17986352C212yesmaximal, complemented
4#17986301C212yesmaximal, complemented
5#17986291C211yesmaximal, characteristic, complemented
6#17986213C212yesmaximal, complemented
7#17986191C211yesmaximal, characteristic
8C8×Dic142C211yesmaximal, characteristic, complemented
9C4×#11765854C211yesmaximal, characteristic, complemented
10#17986079C212yesmaximal, complemented
11#486×C71C211yesmaximal, characteristic, complemented
12C2×#11765854C2211yescharacteristic
13C2×#11765850C2212yes—
14#14556119C2212yes—
15#14556115—21no—
16#14556115C2212yes—
17#14556109—22no—
18#14556108—21no—
19#14556104—22no—
20C4×Dic142C2211yescharacteristic
21C4×Dic142C2211yescharacteristic
22C4×D284C2211yescharacteristic
23D4×Dic71—21no—
24D4×Dic71C2212yes—
25#14556090C2212yes—
26C4⋊C4×D71—21no—
27#14556061C2212yes—
28#14556060C2211yescharacteristic
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556051C2212yes—
33#14556050C2211yescharacteristic
34#14556049C2212yes—
35C8×Dic71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38#102×C71C2212yes—
39#102×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142D411yescharacteristic
42C2×Dic142C2311yescharacteristic
43C2×D284C2311yescharacteristic
44C22×Dic71—22no—
45C2×C4×D71—21no—
46C2×C4×D71—22no—
47#11765854—22no—
48#11765850—24no—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—22no—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—21no—
55Dic71⋊C4—22no—
56Dic71⋊C4D412yes—
57C22⋊Dic71—22no—
58C22⋊Dic71—22no—
59C22⋊Dic71—22no—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—21no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×D4×C71—21no—
66C2×D4×C71—21no—
67C2×D4×C71C2312yes—
68C2×C8×C71C2311yescharacteristic
69C2×C8×C71C2311yescharacteristic
70C4⋊C4×C71—21no—
71C4⋊C4×C71C2312yes—
72C42×C71C2311yescharacteristic
73C23×C71—22no—
74C22×D71—21no—
75Q8×C71—81no—
76D4×C71—22no—
77D4×C71—22no—
78D4×C71—42no—
79D4×C71—81no—
80C2×C4×C71—22no—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2411yescharacteristic
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—82no—
88C2×Dic71C2×D412yes—
89C2×Dic71C2×D412yes—
90D284—21no—
91D284#4112yes—
92C4×D71—42no—
93C4×D71—82no—
94Dic142—22no—
95Dic142—21no—
96Dic142—21no—
97C8×C71—21no—
98C8×C71—21no—
99C71⋊C8—22no—
100C22×C71—42no—
101C22×C71—42no—
102C22×C71—42no—
103C22×C71C22×D411yescharacteristic
104C2×D71—22no—
105C2×D71—41no—
106C2×D71—41no—
107C4×C71—21no—
108C4×C71—42no—
109C4×C71#13512yes—
110C4×C71C2×#4111yescharacteristic
111C4×C712−1+411yescharacteristic
112Dic71—22no—
113Dic71—22no—
114Dic71—41no—
115Dic71—41no—
116Dic71—41no—
117C2×C71—82no—
118C2×C71#54811yescharacteristic
119C2×C71C2×#13511yescharacteristic
120C2×C71#57611yescharacteristic
121D71—42no—
122#3361—711nomaximal, Sylow
123C71#336111yescharacteristic, complemented, Sylow
124#559—711no—
125#554—711no—
126#488—712no—
127#486D7111yescharacteristic, complemented
128#478—712no—
129#477—711no—
130#463—712no—
131#459—711no—
132C8×Q8—711no—
133C4×SD16—711no—
134C4×D8—712no—
135C2×SD16—711no—
136C2×D8—712no—
137C4⋊Q8C2×D7111yescharacteristic
138#127C2×D7111yescharacteristic
139#125—712no—
140#124—712no—
141#124—1421no—
142#123—1422no—
143C22⋊Q8—1421no—
144C22⋊Q8—1422no—
145C4×Q8—711no—
146C4×Q8—711no—
147C4×D4—712no—
148C4×D4—712no—
149C4×D4—711no—
150C4×D4—1421no—
151C2×C4⋊C4—1421no—
152#107—712no—
153#106—711no—
154C4⋊C8—712no—
155C4⋊C8—711no—
156#103—711no—
157#103—712no—
158#102—712no—
159#102—711no—
160#102C2×D7112yes—
161#102C2×D7112yes—
162C4×C8—712no—
163C4×C8C2×D7111yescharacteristic
164C2×Q8—21no—
165C2×Q8—711no—
166C2×Q8—711no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—711no—
170C2×D4C22×D7112yes—
171C22×C4—1422no—
172C22×C4—1421no—
173C22×C4—1422no—
174SD16—1422no—
175D8—1424no—
176C2×C8—712no—
177C2×C8C22×D7111yescharacteristic
178C2×C8C22×D7111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—712no—
181C4⋊C4—711no—
182C4⋊C4—711no—
183C4⋊C4—711no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C4⋊C4—1421no—
187C4⋊C4C22×D7112yes—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C22⋊C4—1422no—
191C22⋊C4—1422no—
192C22⋊C4—1421no—
193C42—712no—
194C42—1421no—
195C42C22×D7111yescharacteristic
196C23—22no—
197C23—1421no—
198Q8—81no—
199Q8—1421no—
200Q8—1421no—
201Q8—1422no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—81no—
206D4—712no—
207D4—1421no—
208C2×C4—22no—
209C2×C4—2842no—
210C2×C4—5682no—
211C2×C4—5682no—
212C2×C4—712no—
213C2×C4—712no—
214C2×C4—1421no—
215C2×C4—1421no—
216C2×C4—1421no—
217C2×C4D4×D7111yescharacteristic
218C2×C4D4×D7111yescharacteristic
219C2×C4C23×D7111yescharacteristic
220C8—21no—
221C8—21no—
222C8—1422no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—2841no—
227C22—2841no—
228C22—1422no—
229C22C2×D4×D7111yescharacteristic
230C4—21no—
231C4—42no—
232C4—2841no—
233C4—2841no—
234C4—2841no—
235C4—1422no—
236C4—1422no—
237C4#1455614912yes—
238C4#41×D7111yescharacteristic
239C4#1455617511yescharacteristic
240C2—82no—
241C2—2842no—
242C2#1798665711yescharacteristic
243C2C2×#1455614911yescharacteristic
244C2#1798680011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3361
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17986735C2semidirect
2. #17986710C2semidirect
3. #17986352C2semidirect
4. #17986301C2semidirect
5. #17986291C2semidirect
6. #17986213C2semidirect
8. C8×Dic142C2semidirect
9. C4×#11765854C2semidirect
10. #17986079C2semidirect
11. #486×C71C2semidirect
123. C71#3361semidirect
127. #486D71semidirect