Finite Group Databasebeta

← #22596524

Subgroups — #22596524

Order 9088.

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

normal subgroups only

This group has 372 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
1C71⋊#401C211yesmaximal, characteristic, complemented
2#17985936C211yesmaximal, characteristic
3#17985933C213yesmaximal, complemented
4#17985931C213yesmaximal, complemented
5#17985928C213yesmaximal
6#17985873C213yesmaximal
7#401×C71C211yesmaximal, characteristic, complemented
8C22×C4×D71C2211yescharacteristic
9C2×C4⋊Dic71C2213yes—
10C2×C4⋊Dic71C2213yes—
11C2×Dic71⋊C4C2213yes—
12C2×Dic71⋊C4C2213yes—
13C2×(C2×D71)⋊C4C2213yes—
14C2×(C2×D71)⋊C4C2213yes—
15C71⋊#95C2213yes—
16C71⋊#95C2213yes—
17C71⋊#95C2213yes—
18#95×C71C2213yes—
19#95×C71C2213yes—
20#95×C71C2211yescharacteristic
21C23×D71C2311yescharacteristic
22C22×Dic71C2311yescharacteristic
23C22×Dic71C2313yes—
24C22×Dic71C2313yes—
25C2×C4×D71—43no—
26C4⋊Dic71—43no—
27C4⋊Dic71—43no—
28Dic71⋊C4—43no—
29Dic71⋊C4—43no—
30(C2×D71)⋊C4—26no—
31(C2×D71)⋊C4—43no—
32C22×C4×C71C2311yescharacteristic
33C22×C4×C71C2313yes—
34C22×C4×C71C2313yes—
35C23×C71C2411yescharacteristic
36C22×D71—23no—
37C22×D71—23no—
38C22×D71#4112yes—
39C2×C4×C71—23no—
40C2×C4×C71—23no—
41C2×C4×C71—23no—
42C2×C4×C71—23no—
43C2×C4×C71—23no—
44C2×C4×C71—26no—
45C2×Dic71—23no—
46C2×Dic71—23no—
47C2×Dic71—23no—
48C2×Dic71—23no—
49C2×Dic71—23no—
50C2×Dic71—23no—
51C2×Dic71—23no—
52C4×D71—44no—
53C22×C71C2×#4111yescharacteristic
54C22×C71H(5,𝔽2)13yes—
55C22×C712−1+413yes—
56C2×D71—26no—
57C2×D71—41no—
58C2×D71—43no—
59C4×C71—43no—
60C4×C71—41no—
61C4×C71—43no—
62Dic71—43no—
63Dic71—41no—
64Dic71—43no—
65C2×C71#53913yes—
66C2×C71#56011yescharacteristic
67C2×C71#56313yes—
68D71—42no—
69#2834—711nomaximal, Sylow
70C71#283411yescharacteristic, complemented, Sylow
71#401—711no—
72#401D7111yescharacteristic, complemented
73#399—711no—
74#398—713no—
75#398—713no—
76#397—713no—
77#385—713no—
78C23×C4—711no—
79C2×C4⋊C4—713no—
80C2×C4⋊C4—713no—
81C2×C4⋊C4—713no—
82C2×C4⋊C4—713no—
83C2×C22⋊C4—713no—
84C2×C22⋊C4—713no—
85#95—713no—
86#95—713no—
87#95—713no—
88#95C2×D7113yes—
89#95C2×D7113yes—
90#95C2×D7111yescharacteristic
91C24—711no—
92C22×C4—2843no—
93C22×C4—713no—
94C22×C4—711no—
95C22×C4—713no—
96C22×C4C22×D7113yes—
97C22×C4C22×D7113yes—
98C22×C4C22×D7111yescharacteristic
99C4⋊C4—2843no—
100C4⋊C4—2843no—
101C4⋊C4—2843no—
102C4⋊C4—2843no—
103C22⋊C4—2843no—
104C22⋊C4—1426no—
105C23—712no—
106C23—1423no—
107C23—1423no—
108C23C23×D7111yescharacteristic
109C2×C4—23no—
110C2×C4—23no—
111C2×C4—23no—
112C2×C4—23no—
113C2×C4—23no—
114C2×C4—26no—
115C2×C4—2844no—
116C2×C4—1423no—
117C2×C4—1423no—
118C2×C4—1423no—
119C2×C4—1423no—
120C2×C4—1423no—
121C2×C4—1423no—
122C2×C4—1423no—
123C22—2841no—
124C22—2843no—
125C22—1426no—
126C22#41×D7111yescharacteristic
127C22#1455617413yes—
128C22#1455617513yes—
129C4—43no—
130C4—41no—
131C4—43no—
132C4—2843no—
133C4—2841no—
134C4—2843no—
135C2—2842no—
136C2#1798659013yes—
137C2#1798674011yescharacteristic
138C2#1798675113yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#401×C71C2
hypercenter#401D71
hypocenterC71#2834
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #401

Complements

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

NH ≅ G/Ntype
1. C71⋊#401C2semidirect
3. #17985933C2semidirect
4. #17985931C2semidirect
7. #401×C71C2semidirect
70. C71#2834semidirect
72. #401D71semidirect