Finite Group Databasebeta

← #22598737

Subgroups — #22598737

Order 9088.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986548C211yesmaximal, characteristic
2#125×D71C211yesmaximal, characteristic, complemented
3#17986366C212yesmaximal, complemented
4#17986362C212yesmaximal, complemented
5#17986356C211yesmaximal, characteristic, complemented
6#17986354C211yesmaximal, characteristic, complemented
7#17986353C211yesmaximal, characteristic, complemented
8#17985999C212yesmaximal, complemented
9#17985997C212yesmaximal, complemented
10C4×#11765848C211yesmaximal, characteristic, complemented
11#488×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×#11765864C2211yescharacteristic
14C2×#11765848C2212yes—
15#14556126C2211yescharacteristic
16#14556124C2211yescharacteristic
17#14556120C2211yescharacteristic
18#14556119C2212yes—
19#14556118—22no—
20#14556116C2211yescharacteristic
21#14556101—22no—
22(C22⋊D71)⋊C4—22no—
23C4⋊C4×D71—22no—
24C4⋊C4×D71C2212yes—
25C42×D71C2211yescharacteristic
26#14556053C2212yes—
27#14556052C2212yes—
28#14556051C2212yes—
29#14556050C2212yes—
30#14556049C2212yes—
31#14556048C2212yes—
32#14556038C2212yes—
33C4×C71⋊C8C2211yescharacteristic
34#125×C71C2211yescharacteristic
35#124×C71C2211yescharacteristic
36#103×C71C2212yes—
37#102×C71C2212yes—
38C4×C8×C71C2211yescharacteristic
39#11765866—24no—
40#11765864—24no—
41C2×Dic142C2311yescharacteristic
42C2×D284—21no—
43C2×D284C2311yescharacteristic
44C2×C22⋊D71—21no—
45C2×C22⋊D71—22no—
46C22×Dic71—21no—
47C2×C4×D71—22no—
48C2×C4×D71—21no—
49C2×C4×D71D412yes—
50C2×C4×D71C2311yescharacteristic
51#11765848—24no—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71—22no—
54C4⋊Dic71C2312yes—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59(C2×D71)⋊C4—22no—
60C4×Dic71D412yes—
61C4×Dic71C2311yescharacteristic
62C2×Q8×C71C2311yescharacteristic
63C2×D4×C71C2311yescharacteristic
64C2×C8×C71C2312yes—
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C42×C71C2311yescharacteristic
69C23×C71—21no—
70C22×D71—21no—
71C22×D71C2×D411yescharacteristic
72Q8×C71—22no—
73D4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—44no—
79C22⋊D71—42no—
80C2×Dic71—21no—
81C2×Dic71—22no—
82C2×Dic71—42no—
83C2×Dic71C2×D412yes—
84C2×Dic71C2×D411yescharacteristic
85D284—22no—
86D284—42no—
87C4×D71—42no—
88C4×D71—44no—
89C4×D71—42no—
90C4×D71#4114yes—
91Dic142—22no—
92C8×C71—22no—
93C71⋊C8—22no—
94C22×C71—41no—
95C22×C71—42no—
96C22×C71C22×D411yescharacteristic
97C2×D71—22no—
98C2×D71—41no—
99C2×D71—42no—
100C2×D71#12412yes—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—42no—
104C4×C71C2×#4112yes—
105Dic71—22no—
106Dic71—41no—
107Dic71—42no—
108Dic71#12412yes—
109C2×C71—81no—
110C2×C71C2×#12411yescharacteristic
111C2×C71#57512yes—
112D71—22no—
113D71—81no—
114#3142—711nomaximal, Sylow
115C71#314211yescharacteristic, complemented, Sylow
116#532—711no—
117C2×#125—711no—
118#491—712no—
119#490—712no—
120#488—711no—
121#488—711no—
122#488—711no—
123#488D7111yescharacteristic, complemented
124#417—712no—
125#417—712no—
126C4×M16—711no—
127C2×#41—711no—
128C2×#41—711no—
129C2×M16—712no—
130C4⋊Q8—711no—
131#127—711no—
132#125—712no—
133#125—711no—
134#125—1422no—
135#125C2×D7111yescharacteristic
136#124—711no—
137#124C2×D7111yescharacteristic
138#121—1422no—
139C4×D4—1422no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—1422no—
142C2×C42—711no—
143#103—712no—
144#103—712no—
145#103—712no—
146#103C2×D7112yes—
147#102—712no—
148#102—712no—
149#102—712no—
150#102C2×D7112yes—
151#97—712no—
152C4×C8—711no—
153C4×C8C2×D7111yescharacteristic
154#41—1424no—
155#41—1424no—
156C2×Q8—711no—
157C2×Q8C22×D7111yescharacteristic
158C2×D4—711no—
159C2×D4—1421no—
160C2×D4—1422no—
161C2×D4—1421no—
162C2×D4C22×D7111yescharacteristic
163C22×C4—712no—
164C22×C4—711no—
165C22×C4—1421no—
166C22×C4—1422no—
167C22×C4—1421no—
168M16—1424no—
169C2×C8—712no—
170C2×C8C22×D7112yes—
171C4⋊C4—22no—
172C4⋊C4—712no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4C22×D7112yes—
179C22⋊C4—22no—
180C22⋊C4—1422no—
181C42—712no—
182C42—711no—
183C42C22×D7111yescharacteristic
184C23—21no—
185C23—711no—
186C23—1421no—
187Q8—22no—
188Q8—1422no—
189D4—22no—
190D4—2842no—
191D4—2842no—
192D4—2844no—
193D4—1422no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—2842no—
197C2×C4—2844no—
198C2×C4—2842no—
199C2×C4—2842no—
200C2×C4—711no—
201C2×C4—714no—
202C2×C4—712no—
203C2×C4—1421no—
204C2×C4—1422no—
205C2×C4D4×D7112yes—
206C2×C4C23×D7111yescharacteristic
207C8—22no—
208C8—1422no—
209C22—41no—
210C22—42no—
211C22—2841no—
212C22—2842no—
213C22—712no—
214C22—1422no—
215C22C2×D4×D7111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—2841no—
220C4—2842no—
221C4—712no—
222C4—1422no—
223C4#41×D7112yes—
224C2—81no—
225C2—5681no—
226C2—1422no—
227C2#124×D7111yescharacteristic
228C2#1798679612yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#488×C71C2
socleC22×C71C22×D4
hypercenter#488D71
hypocenterC71#3142
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D71C2semidirect
3. #17986366C2semidirect
4. #17986362C2semidirect
5. #17986356C2semidirect
6. #17986354C2semidirect
7. #17986353C2semidirect
8. #17985999C2semidirect
9. #17985997C2semidirect
10. C4×#11765848C2semidirect
11. #488×C71C2semidirect
115. C71#3142semidirect
123. #488D71semidirect