Finite Group Databasebeta

← #22601126

Subgroups — #22601126

Order 9088.

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

normal subgroups only

This group has 340 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#17986724C211yesmaximal, characteristic, complemented
2#17986710C211yesmaximal, characteristic
3#17986366C212yesmaximal, complemented
4#17986304C212yesmaximal, complemented
5#17986291C211yesmaximal, characteristic, complemented
6#17986203C212yesmaximal
7#17986201C211yesmaximal, characteristic
8#17986129C211yesmaximal, characteristic, complemented
9#17986102C212yesmaximal
10C4×#11765854C211yesmaximal, characteristic
11#487×C71C211yesmaximal, characteristic, complemented
12C2×#11765854C2211yescharacteristic
13C2×#11765852C2212yes—
14#14556126C2212yes—
15#14556119—21no—
16#14556119C2212yes—
17#14556118—22no—
18#14556108—21no—
19#14556104—22no—
20Q8×Dic71C2212yes—
21C4×Dic142C2211yescharacteristic
22C4×Dic142C2211yescharacteristic
23#14556093—21no—
24C4×D284C2211yescharacteristic
25#14556090C2212yes—
26C4⋊C4×D71—21no—
27#14556061C2212yes—
28#14556060C2211yescharacteristic
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556052C2212yes—
33#14556051C2212yes—
34#14556050C2211yescharacteristic
35#14556038C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142D411yescharacteristic
42C2×Dic142C2311yescharacteristic
43C2×D284C2311yescharacteristic
44C2×C4×D71—21no—
45C2×C4×D71—22no—
46#11765854—22no—
47#11765852—24no—
48C2×C71⋊C8C2312yes—
49C4⋊Dic71—22no—
50C4⋊Dic71—22no—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—21no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4D412yes—
59(C2×D71)⋊C4—21no—
60(C2×D71)⋊C4—22no—
61C4×Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×Q8×C71C2312yes—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71—21no—
71C4⋊C4×C71C2312yes—
72C42×C71C2311yescharacteristic
73C22×D71—21no—
74Q8×C71—22no—
75Q8×C71—22no—
76Q8×C71—81no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71C2×D412yes—
86C2×Dic71C2×D412yes—
87D284—21no—
88D284#4112yes—
89C4×D71—42no—
90C4×D71—82no—
91Dic142—22no—
92Dic142—21no—
93Dic142—21no—
94C8×C71—21no—
95C8×C71—21no—
96C71⋊C8—22no—
97C22×C71C22×D411yescharacteristic
98C2×D71—22no—
99C2×D71—41no—
100C2×D71—41no—
101C4×C71—21no—
102C4×C71—42no—
103C4×C71—42no—
104C4×C71#13712yes—
105C4×C71C2×#4111yescharacteristic
106C4×C712−1+411yescharacteristic
107Dic71—22no—
108Dic71—22no—
109Dic71—41no—
110Dic71—41no—
111Dic71—41no—
112C2×C71#54811yescharacteristic
113C2×C71C2×#13711yescharacteristic
114C2×C71#57711yescharacteristic
115D71—42no—
116#3372—711nomaximal, Sylow
117C71#337211yescharacteristic, complemented, Sylow
118#557—711no—
119#554—711no—
120#491—712no—
121#487D7111yescharacteristic, complemented
122#479—712no—
123#477—711no—
124#461—711no—
125#461—712no—
126#446—711no—
127#440—712no—
128C4×SD16—711no—
129C2×SD16—712no—
130C2×SD16—711no—
131C4⋊Q8—712no—
132C4⋊Q8C2×D7111yescharacteristic
133C4⋊Q8C2×D7111yescharacteristic
134#125—712no—
135#125—1422no—
136#125—1421no—
137C22⋊Q8—1421no—
138C22⋊Q8—1422no—
139C4×Q8—711no—
140C4×Q8—712no—
141C4×Q8—711no—
142C4×Q8—1421no—
143C4×D4—712no—
144C4×D4—711no—
145C2×C4⋊C4—1421no—
146#107—712no—
147#106—711no—
148C4⋊C8—712no—
149C4⋊C8—711no—
150#103—712no—
151#103—711no—
152#103—712no—
153#103C2×D7112yes—
154#103C2×D7112yes—
155#102—711no—
156#97—712no—
157C4×C8C2×D7111yescharacteristic
158C2×Q8—21no—
159C2×Q8—711no—
160C2×Q8—711no—
161C2×Q8C22×D7112yes—
162C2×D4—711no—
163C22×C4—1422no—
164C22×C4—1421no—
165SD16—1422no—
166SD16—1424no—
167C2×C8—712no—
168C2×C8C22×D7111yescharacteristic
169C2×C8C22×D7111yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—712no—
174C4⋊C4—711no—
175C4⋊C4—711no—
176C4⋊C4—711no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C4⋊C4—1421no—
183C4⋊C4C22×D7112yes—
184C22⋊C4—1422no—
185C22⋊C4—1421no—
186C42—712no—
187C42—1422no—
188C42—1421no—
189C42C22×D7111yescharacteristic
190C23—1421no—
191Q8—22no—
192Q8—22no—
193Q8—81no—
194Q8—1421no—
195Q8—1421no—
196Q8—1422no—
197D4—712no—
198D4—1421no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—2842no—
202C2×C4—5682no—
203C2×C4—712no—
204C2×C4—712no—
205C2×C4—1421no—
206C2×C4—1421no—
207C2×C4—1421no—
208C2×C4D4×D7111yescharacteristic
209C2×C4D4×D7111yescharacteristic
210C2×C4C23×D7111yescharacteristic
211C8—21no—
212C8—21no—
213C8—1422no—
214C22—2841no—
215C22—2841no—
216C22—1422no—
217C22C2×D4×D7111yescharacteristic
218C4—21no—
219C4—42no—
220C4—42no—
221C4—2841no—
222C4—2841no—
223C4—2841no—
224C4—1422no—
225C4—1422no—
226C4#1455616012yes—
227C4#41×D7111yescharacteristic
228C4#1455617511yescharacteristic
229C2—2842no—
230C2#1798665711yescharacteristic
231C2C2×#1455616011yescharacteristic
232C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3372
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986724C2semidirect
3. #17986366C2semidirect
4. #17986304C2semidirect
5. #17986291C2semidirect
8. #17986129C2semidirect
11. #487×C71C2semidirect
117. C71#3372semidirect
121. #487D71semidirect