Finite Group Databasebeta

← #22601397

Subgroups — #22601397

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#17986734C211yesmaximal, characteristic
2#17986724C211yesmaximal, characteristic, complemented
3#17986392C212yesmaximal, complemented
4#17986373C211yesmaximal, characteristic
5#17986223C212yesmaximal, complemented
6#17986203C212yesmaximal, complemented
7#17986200C211yesmaximal, characteristic
8#17986126C211yesmaximal, characteristic, complemented
9#17986102C212yesmaximal, complemented
10C4×#11765855C211yesmaximal, characteristic, complemented
11#462×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2211yescharacteristic
13C2×#11765854C2212yes—
14C2×#11765852C2212yes—
15C2×#11765852C2212yes—
16#14556126C2212yes—
17#14556125C2211yescharacteristic
18#14556119—21no—
19#14556118—22no—
20#14556117C2212yes—
21#14556116—21no—
22#14556113—22no—
23Q8×Dic71C2212yes—
24C4×Dic142C2211yescharacteristic
25C4×Dic142C2211yescharacteristic
26#14556093—21no—
27C4×D284—21no—
28#14556090C2212yes—
29#14556061C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556052C2211yescharacteristic
32#14556051C2211yescharacteristic
33#14556050C2212yes—
34#14556038C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×Q16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×Q8×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142D411yescharacteristic
42C2×Dic142C2311yescharacteristic
43C2×D284—21no—
44C2×D284—21no—
45C2×D284C2312yes—
46C2×C4×D71—22no—
47#11765855—22no—
48#11765854—42no—
49#11765852—24no—
50#11765852—42no—
51C2×C71⋊C8C2311yescharacteristic
52C2×C71⋊C8C2311yescharacteristic
53C4⋊Dic71—22no—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—21no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—21no—
64C4×Dic71—22no—
65C4×Dic71C2312yes—
66C2×Q8×C71C2312yes—
67C2×Q8×C71C2311yescharacteristic
68Q16×C71—42no—
69C2×C8×C71C2312yes—
70C4⋊C4×C71—21no—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2311yescharacteristic
74C42×C71—21no—
75C42×C71C2311yescharacteristic
76C22×D71—22no—
77Q8×C71—22no—
78Q8×C71—21no—
79Q8×C71—21no—
80Q8×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71C2×D412yes—
90C2×Dic71C2×D412yes—
91D284—22no—
92D284—22no—
93D284—42no—
94D284—81no—
95C4×D71—82no—
96Dic142—22no—
97Dic142—21no—
98Dic142—21no—
99C8×C71—22no—
100C71⋊C8—21no—
101C71⋊C8#4112yes—
102C22×C71C22×D411yescharacteristic
103C2×D71—42no—
104C2×D71—42no—
105C2×D71—42no—
106C4×C71—21no—
107C4×C71—41no—
108C4×C71—42no—
109C4×C71—41no—
110C4×C71#13712yes—
111C4×C71C2×#4111yescharacteristic
112C4×C71H(5,𝔽2)11yescharacteristic
113Dic71—22no—
114Dic71—22no—
115Dic71—41no—
116Dic71—41no—
117C2×C71#55011yescharacteristic
118C2×C71C2×#13711yescharacteristic
119C2×C71#57711yescharacteristic
120D71—82no—
121#3400—711nomaximal, Sylow
122C71#340011yescharacteristic, complemented, Sylow
123#559—711no—
124#557—711no—
125#497—712no—
126#492—711no—
127#464—712no—
128#462D7111yescharacteristic, complemented
129#461—712no—
130#460—711no—
131#446—711no—
132#440—712no—
133C4×Q16—711no—
134C2×Q16—711no—
135C2×Q16C2×D7112yes—
136C2×SD16—712no—
137C2×SD16—712no—
138C2×SD16—712no—
139C4⋊Q8—712no—
140C4⋊Q8C2×D7111yescharacteristic
141#127—711no—
142#125—1422no—
143#125—1421no—
144#124—712no—
145#124—1421no—
146#123—1422no—
147C4×Q8—711no—
148C4×Q8—712no—
149C4×Q8—711no—
150C4×Q8—1421no—
151C4×Q8C2×D7111yescharacteristic
152C4×D4—712no—
153C4×D4—1421no—
154#107—711no—
155C4⋊C8—712no—
156C4⋊C8C2×D7111yescharacteristic
157#103—711no—
158#103—711no—
159#103C2×D7112yes—
160#102—712no—
161#97—712no—
162C4×C8—711no—
163C2×Q8—711no—
164C2×Q8—711no—
165C2×Q8C22×D7111yescharacteristic
166C2×Q8C22×D7112yes—
167C2×D4—712no—
168C2×D4—1421no—
169C2×D4—1421no—
170C22×C4—1422no—
171Q16—42no—
172Q16—1422no—
173SD16—2842no—
174SD16—2842no—
175SD16—1424no—
176C2×C8—711no—
177C2×C8—711no—
178C2×C8C22×D7112yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—712no—
183C4⋊C4—711no—
184C4⋊C4—711no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1422no—
188C4⋊C4—1421no—
189C4⋊C4C22×D7111yescharacteristic
190C22⋊C4—1422no—
191C22⋊C4—1422no—
192C22⋊C4—1422no—
193C42—21no—
194C42—712no—
195C42—1422no—
196C42—1421no—
197C42C22×D7111yescharacteristic
198C23—1422no—
199Q8—22no—
200Q8—21no—
201Q8—22no—
202Q8—21no—
203Q8—1421no—
204Q8—1421no—
205Q8—1422no—
206D4—2842no—
207D4—5681no—
208D4—1422no—
209D4—1422no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—5682no—
214C2×C4—712no—
215C2×C4—712no—
216C2×C4—1421no—
217C2×C4—1421no—
218C2×C4D4×D7111yescharacteristic
219C2×C4D4×D7111yescharacteristic
220C2×C4C23×D7111yescharacteristic
221C8—22no—
222C8—712no—
223C8—1421no—
224C22—2842no—
225C22—2842no—
226C22—2842no—
227C22C2×D4×D7111yescharacteristic
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—2841no—
233C4—2841no—
234C4—1422no—
235C4—1422no—
236C4#1455616012yes—
237C4#41×D7111yescharacteristic
238C4#1455617311yescharacteristic
239C2—5682no—
240C2#1798666711yescharacteristic
241C2C2×#1455616011yescharacteristic
242C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#462×C71C2
socleC22×C71C22×D4
hypercenter#462D71
hypocenterC71#3400
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #17986724C2semidirect
3. #17986392C2semidirect
5. #17986223C2semidirect
6. #17986203C2semidirect
8. #17986126C2semidirect
9. #17986102C2semidirect
10. C4×#11765855C2semidirect
11. #462×C71C2semidirect
122. C71#3400semidirect
128. #462D71semidirect