Finite Group Databasebeta

← #22600952

Subgroups — #22600952

Order 9088.

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

normal subgroups only

This group has 342 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#17986723C211yesmaximal, characteristic
2#17986709C211yesmaximal, characteristic
3#17986395C211yesmaximal, characteristic, complemented
4#17986339C212yesmaximal, complemented
5#17986331C212yesmaximal, complemented
6#17986295C211yesmaximal, characteristic, complemented
7#17986256C212yesmaximal, complemented
8C4×#11765855C211yesmaximal, characteristic, complemented
9#17986076C211yesmaximal, characteristic
10#17986038C212yesmaximal, complemented
11#475×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2211yescharacteristic
13C2×#11765848C2212yes—
14#14556128C2211yescharacteristic
15#14556123—22no—
16#14556120—21no—
17#14556117—21no—
18#14556108C2212yes—
19#14556106—22no—
20#14556104C2212yes—
21Q8×Dic71—21no—
22C4×Dic142C2211yescharacteristic
23C4×D284—21no—
24C4×D284C2211yescharacteristic
25#14556086C2212yes—
26C4⋊C4×D71C2212yes—
27#14556062C2212yes—
28#14556061C2211yescharacteristic
29#14556059C2212yes—
30#14556059C2212yes—
31#14556053C2212yes—
32#14556052C2211yescharacteristic
33#14556051C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C4⋊Q8×C71C2211yescharacteristic
37C4×Q8×C71C2211yescharacteristic
38#106×C71C2212yes—
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142—21no—
42C2×Dic142C2311yescharacteristic
43C2×D284D411yescharacteristic
44C2×C4×D71—22no—
45C2×C4×D71C2312yes—
46#11765855—22no—
47#11765848—24no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2311yescharacteristic
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2312yes—
55Dic71⋊C4—21no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4D412yes—
62C4×Dic71—21no—
63C4×Dic71—22no—
64C2×Q8×C71—21no—
65C2×Q8×C71C2311yescharacteristic
66C2×C8×C71C2312yes—
67C4⋊C4×C71—21no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71C2312yes—
70C4⋊C4×C71C2311yescharacteristic
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73C22×D71C2×D412yes—
74Q8×C71—21no—
75Q8×C71—21no—
76Q8×C71—42no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71—22no—
86C2×Dic71C2×D412yes—
87D284—22no—
88C4×D71—22no—
89C4×D71—22no—
90C4×D71—82no—
91Dic142—21no—
92Dic142—21no—
93Dic142—81no—
94C8×C71—22no—
95C71⋊C8—21no—
96C71⋊C8#4112yes—
97C22×C71C22×D411yescharacteristic
98C2×D71—22no—
99C2×D71—22no—
100C2×D71—22no—
101C4×C71—21no—
102C4×C71—41no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71#13712yes—
106C4×C71C2×#4111yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—22no—
109Dic71—41no—
110Dic71—42no—
111Dic71—41no—
112C2×C71#54711yescharacteristic
113C2×C71C2×#13711yescharacteristic
114C2×C71#57711yescharacteristic
115D71—42no—
116#3356—711nomaximal, Sylow
117C71#335611yescharacteristic, complemented, Sylow
118#556—711no—
119#554—711no—
120#498—711no—
121#485—712no—
122#483—712no—
123#477—711no—
124#475D7111yescharacteristic, complemented
125#470—712no—
126C4×Q16—711no—
127#436—711no—
128#428—712no—
129C2×Q16—711no—
130C2×M16—712no—
131C4⋊Q8—711no—
132C4⋊Q8C2×D7111yescharacteristic
133#126—1422no—
134#125—1421no—
135#124—1421no—
136C22⋊Q8—712no—
137C22⋊Q8—712no—
138C22⋊Q8—1422no—
139C4×Q8—711no—
140C4×Q8—1421no—
141C4×Q8C2×D7111yescharacteristic
142C4×D4—711no—
143C4×D4—1421no—
144#117—712no—
145C2×C4⋊C4—712no—
146#107—712no—
147#107—711no—
148#106—712no—
149#106—712no—
150#106C2×D7112yes—
151C4⋊C8C2×D7111yescharacteristic
152#103—711no—
153#103—712no—
154#103—711no—
155#103C2×D7112yes—
156C22⋊C8—712no—
157C4×C8—711no—
158C2×Q8—21no—
159C2×Q8—711no—
160C2×Q8—1421no—
161C2×Q8C22×D7111yescharacteristic
162C2×D4—711no—
163C22×C4—712no—
164C22×C4—1422no—
165Q16—1422no—
166M16—1424no—
167C2×C8—711no—
168C2×C8—711no—
169C2×C8C22×D7112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—711no—
173C4⋊C4—712no—
174C4⋊C4—711no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1421no—
179C4⋊C4—1421no—
180C4⋊C4C22×D7111yescharacteristic
181C4⋊C4C22×D7112yes—
182C22⋊C4—712no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C42—21no—
187C42—1422no—
188C42—1421no—
189C42C22×D7111yescharacteristic
190C23—712no—
191Q8—21no—
192Q8—21no—
193Q8—42no—
194Q8—5681no—
195Q8—1421no—
196Q8—1421no—
197D4—1422no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—5682no—
202C2×C4—712no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4D4×D7111yescharacteristic
209C2×C4D4×D7111yescharacteristic
210C2×C4C23×D7111yescharacteristic
211C8—22no—
212C8—712no—
213C8—1421no—
214C22—1422no—
215C22—1422no—
216C22—1422no—
217C22C2×D4×D7111yescharacteristic
218C4—21no—
219C4—41no—
220C4—42no—
221C4—41no—
222C4—2841no—
223C4—2842no—
224C4—2841no—
225C4—1422no—
226C4#1455615712yes—
227C4#41×D7111yescharacteristic
228C4#1455617611yescharacteristic
229C2—2842no—
230C2#1798665111yescharacteristic
231C2C2×#1455615711yescharacteristic
232C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#475×C71C2
socleC22×C71C22×D4
hypercenter#475D71
hypocenterC71#3356
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #17986395C2semidirect
4. #17986339C2semidirect
5. #17986331C2semidirect
6. #17986295C2semidirect
7. #17986256C2semidirect
8. C4×#11765855C2semidirect
10. #17986038C2semidirect
11. #475×C71C2semidirect
117. C71#3356semidirect
124. #475D71semidirect