Finite Group Databasebeta

← #22600622

Subgroups — #22600622

Order 9088.

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

normal subgroups only

This group has 374 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#17986664C211yesmaximal, characteristic
3#17986339C212yesmaximal, complemented
4#17986281C211yesmaximal, characteristic, complemented
5#17986256C212yesmaximal, complemented
6#17986206C211yesmaximal, characteristic, complemented
7#17986156C212yesmaximal, complemented
8C4×Dic284C211yesmaximal, characteristic, complemented
9#17986078C211yesmaximal, characteristic
10#17986006C212yesmaximal, complemented
11#487×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2212yes—
13C2×Dic284C2211yescharacteristic
14C2×#11765855C2212yes—
15C2×#11765848C2212yes—
16#14556127—21no—
17#14556123—22no—
18#14556117—21no—
19#14556108C2212yes—
20#14556106—22no—
21#14556104C2212yes—
22Q8×Dic71—21no—
23C4×Dic142C2211yescharacteristic
24C4×Dic142C2211yescharacteristic
25C4×D284C2211yescharacteristic
26#14556090—21no—
27#14556086C2212yes—
28#14556062C2211yescharacteristic
29#14556059C2212yes—
30C4⋊(C71⋊C8)C2211yescharacteristic
31#14556053C2211yescharacteristic
32#14556053C2211yescharacteristic
33#14556052C2212yes—
34#14556051C2212yes—
35(C2×D71)⋊C8C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142—22no—
42C2×Dic142C2311yescharacteristic
43C2×Dic142C2311yescharacteristic
44Q8×D71—24no—
45Q8×D71—42no—
46C2×D284D411yescharacteristic
47C2×C4×D71—22no—
48C2×C4×D71C2312yes—
49Dic284—22no—
50#11765855—42no—
51#11765848—42no—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71—22no—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56C4⋊Dic71C2311yescharacteristic
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59Dic71⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4D412yes—
64C4×Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71—22no—
67C2×Q8×C71—21no—
68C2×Q8×C71C2312yes—
69C2×C8×C71C2311yescharacteristic
70C2×C8×C71C2311yescharacteristic
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C42×C71C2311yescharacteristic
76C22×D71C2×D412yes—
77Q8×C71—22no—
78Q8×C71—22no—
79Q8×C71—42no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71C2×D412yes—
90D284—22no—
91C4×D71—22no—
92C4×D71—22no—
93C4×D71—44no—
94Dic142—21no—
95Dic142—21no—
96Dic142—21no—
97Dic142—44no—
98Dic142#4112yes—
99C8×C71—21no—
100C8×C71—21no—
101C71⋊C8—22no—
102C22×C71C22×D411yescharacteristic
103C2×D71—22no—
104C2×D71—22no—
105C2×D71—22no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71—42no—
109C4×C71#13712yes—
110C4×C71C2×#4111yescharacteristic
111C4×C71H(5,𝔽2)11yescharacteristic
112Dic71—22no—
113Dic71—22no—
114Dic71—41no—
115Dic71—41no—
116Dic71—41no—
117C2×C71#54611yescharacteristic
118C2×C71C2×#13711yescharacteristic
119C2×C71#57811yescharacteristic
120D71—42no—
121#3328—711nomaximal, Sylow
122C71#332811yescharacteristic, complemented, Sylow
123#556—711no—
124D4×Q8—711no—
125#487D7111yescharacteristic, complemented
126#485—712no—
127#475—711no—
128#470—712no—
129#462—711no—
130#451—712no—
131C4×Q16—711no—
132#436—711no—
133#419—712no—
134C22×Q8—712no—
135C2×Q16—712no—
136C2×Q16—711no—
137C2×M16—712no—
138C4⋊Q8—1421no—
139C4⋊Q8C2×D7111yescharacteristic
140C4⋊Q8C2×D7111yescharacteristic
141#126—1422no—
142#124—1421no—
143C22⋊Q8—712no—
144C22⋊Q8—712no—
145C22⋊Q8—1422no—
146C4×Q8—711no—
147C4×Q8—711no—
148C4×Q8—1421no—
149C4×D4—711no—
150C4×D4—1421no—
151#117—712no—
152#107—711no—
153#106—712no—
154C4⋊C8—711no—
155#103—712no—
156#103—711no—
157#103—711no—
158#103—712no—
159#103C2×D7112yes—
160#103C2×D7112yes—
161C22⋊C8—712no—
162C4×C8C2×D7111yescharacteristic
163C2×Q8—21no—
164C2×Q8—2842no—
165C2×Q8—711no—
166C2×Q8—711no—
167C2×Q8—1422no—
168C2×Q8—1424no—
169C2×Q8C22×D7112yes—
170C2×D4—711no—
171C22×C4—712no—
172C22×C4—1422no—
173Q16—2842no—
174Q16—1422no—
175M16—2842no—
176C2×C8—712no—
177C2×C8C22×D7111yescharacteristic
178C2×C8C22×D7111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—711no—
183C4⋊C4—711no—
184C4⋊C4—711no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1421no—
188C4⋊C4—1421no—
189C4⋊C4C22×D7112yes—
190C22⋊C4—712no—
191C22⋊C4—1422no—
192C22⋊C4—1422no—
193C22⋊C4—1422no—
194C42—1422no—
195C42—1421no—
196C42—1421no—
197C42C22×D7111yescharacteristic
198C23—712no—
199Q8—22no—
200Q8—22no—
201Q8—42no—
202Q8—2844no—
203Q8—712no—
204Q8—1421no—
205Q8—1421no—
206Q8—1421no—
207D4—1422no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2844no—
211C2×C4—712no—
212C2×C4—1421no—
213C2×C4—1421no—
214C2×C4—1421no—
215C2×C4—1421no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4D4×D7111yescharacteristic
219C2×C4D4×D7111yescharacteristic
220C2×C4C23×D7111yescharacteristic
221C8—21no—
222C8—21no—
223C8—1422no—
224C22—1422no—
225C22—1422no—
226C22—1422no—
227C22C2×D4×D7111yescharacteristic
228C4—21no—
229C4—42no—
230C4—42no—
231C4—2841no—
232C4—2841no—
233C4—2841no—
234C4—1422no—
235C4—1422no—
236C4#1455615712yes—
237C4#41×D7111yescharacteristic
238C4#1455617411yescharacteristic
239C2—2842no—
240C2#1798664211yescharacteristic
241C2C2×#1455615711yescharacteristic
242C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3328
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
3. #17986339C2semidirect
4. #17986281C2semidirect
5. #17986256C2semidirect
6. #17986206C2semidirect
7. #17986156C2semidirect
8. C4×Dic284C2semidirect
10. #17986006C2semidirect
11. #487×C71C2semidirect
122. C71#3328semidirect
125. #487D71semidirect