Finite Group Databasebeta

← #22599545

Subgroups — #22599545

Order 9088.

324 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 113 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#17986551C211yesmaximal, characteristic
2C2×#14556119C211yesmaximal, characteristic
3#17986407C212yesmaximal, complemented
4#17986366C212yesmaximal, complemented
5#17986334C211yesmaximal, characteristic
6#17986330C211yesmaximal, characteristic
7#17986327C211yesmaximal, characteristic
8#17986305C212yesmaximal, complemented
9#17986304C212yesmaximal, complemented
10#17986053C211yesmaximal, characteristic
11#484×C71C211yesmaximal, characteristic
12#14556127C2211yescharacteristic
13#14556126C2211yescharacteristic
14#14556119—21no—
15#14556119—21no—
16#14556119C2212yes—
17#14556119C2212yes—
18#14556118—21no—
19#14556107C2211yescharacteristic
20#14556105—21no—
21Q8×Dic71C2211yescharacteristic
22#14556093—21no—
23#14556093C2211yescharacteristic
24#14556084—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2211yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71C2211yescharacteristic
29#14556061C2212yes—
30#14556060C2212yes—
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33#14556053C2212yes—
34#14556052C2212yes—
35#14556051C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37#14556038C2212yes—
38C22⋊Q8×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#107×C71C2212yes—
41#103×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142C2311yescharacteristic
45C22×Dic71—21no—
46C22×Dic71D411yescharacteristic
47C22×Dic71D411yescharacteristic
48C2×C71⋊C8C2312yes—
49C4⋊Dic71—21no—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52C4⋊Dic71—21no—
53C4⋊Dic71C2312yes—
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—21no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62C22⋊Dic71—21no—
63C4×Dic71—21no—
64C4×Dic71—21no—
65C4×Dic71D411yescharacteristic
66C4×Dic71D411yescharacteristic
67C4×Dic71C2311yescharacteristic
68C4×Dic71C2311yescharacteristic
69C2×Q8×C71C2311yescharacteristic
70C22×C4×C71—21no—
71C22×C4×C71C2311yescharacteristic
72C2×C8×C71C2312yes—
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C4⋊C4×C71C2311yescharacteristic
77C22⋊C4×C71—21no—
78C23×C71C2×D411yescharacteristic
79Q8×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—42no—
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—22no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90C2×Dic71—22no—
91C2×Dic71—41no—
92C2×Dic71—42no—
93C2×Dic71C2×D411yescharacteristic
94C2×Dic71C2×D411yescharacteristic
95C2×Dic71C2×D411yescharacteristic
96C2×Dic71C2×D411yescharacteristic
97Dic142—22no—
98Dic142—42no—
99C8×C71—22no—
100C71⋊C8—22no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71#13712yes—
104C22×C71C22×D411yescharacteristic
105C4×C71—21no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—42no—
109C4×C712−1+412yes—
110Dic71—21no—
111Dic71—21no—
112Dic71—21no—
113Dic71—21no—
114Dic71—41no—
115Dic71—42no—
116Dic71—41no—
117C2×C71—22no—
118C2×C71#53611yescharacteristic
119C2×C71C2×#13711yescharacteristic
120C2×C71#57511yescharacteristic
121#3236—711nomaximal, Sylow
122C71#323611yescharacteristic, complemented, Sylow
123#533—711no—
124C2×#125—711no—
125#501—712no—
126#491—712no—
127#484—711no—
128#484D7111yescharacteristic
129#483—711no—
130#483—711no—
131#479—712no—
132#479—712no—
133#432—711no—
134C4⋊Q8—711no—
135C4⋊Q8—711no—
136#125—712no—
137#125—712no—
138#125—1421no—
139#125—1421no—
140#125—1421no—
141C22⋊Q8—711no—
142C22⋊Q8—1421no—
143C22⋊Q8C2×D7111yescharacteristic
144C4×Q8—711no—
145C4×Q8—711no—
146C4×Q8—1421no—
147#117—1421no—
148C2×C4⋊C4—711no—
149C2×C4⋊C4—1421no—
150C2×C4⋊C4—1421no—
151C2×C4⋊C4C2×D7111yescharacteristic
152C2×C42—711no—
153#107—712no—
154#107C2×D7112yes—
155#106—712no—
156#106—712no—
157C4⋊C8—712no—
158#103—712no—
159#103—712no—
160#103—712no—
161#103C2×D7112yes—
162C22⋊C8—711no—
163C22⋊C8C2×D7111yescharacteristic
164#97—712no—
165C2×Q8—711no—
166C2×Q8—1421no—
167C2×Q8C22×D7111yescharacteristic
168C22×C4—21no—
169C22×C4—711no—
170C22×C4—711no—
171C22×C4—1421no—
172C22×C4C22×D7111yescharacteristic
173C2×C8—712no—
174C2×C8C22×D7112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—712no—
178C4⋊C4—711no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1421no—
182C4⋊C4—1421no—
183C4⋊C4—1421no—
184C4⋊C4—1421no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1421no—
188C4⋊C4—1421no—
189C4⋊C4—1421no—
190C4⋊C4C22×D7112yes—
191C4⋊C4C22×D7111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1421no—
194C42—711no—
195C42—711no—
196C42—711no—
197C42—711no—
198C42—1421no—
199C42—1421no—
200C23D4×D7111yescharacteristic
201Q8—22no—
202Q8—2842no—
203Q8—1422no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2842no—
210C2×C4—2841no—
211C2×C4—711no—
212C2×C4—711no—
213C2×C4—711no—
214C2×C4—711no—
215C2×C4—1421no—
216C2×C4—1421no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4D4×D7111yescharacteristic
220C2×C4C23×D7111yescharacteristic
221C8—22no—
222C8—1422no—
223C22—21no—
224C22—21no—
225C22#1455616012yes—
226C22C2×D4×D7111yescharacteristic
227C4—21no—
228C4—41no—
229C4—42no—
230C4—41no—
231C4—2841no—
232C4—2842no—
233C4—2841no—
234C4—1421no—
235C4—1421no—
236C4—1421no—
237C4—1421no—
238C4#1455617512yes—
239C2—22no—
240C2#1798657111yescharacteristic
241C2C2×#1455616011yescharacteristic
242C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#484×C71C2
socleC22×C71C22×D4
hypercenter#484D71
hypocenterC71#3236
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #17986407C2semidirect
4. #17986366C2semidirect
8. #17986305C2semidirect
9. #17986304C2semidirect
122. C71#3236semidirect