Finite Group Databasebeta

← #2104×D71 #32028439

Subgroups — #2104×D71 #32028439

Order 18176.

822 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 167 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⋊#2104C217yesmaximal, complemented
2#342×D71C217yesmaximal
3C71×#2104C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D71C2217yes—
5#17985766C2217yes—
6#17985765C22114yes—
7#342×C71C2217yes—
8C24×D71C2311yescharacteristic
9C22×C4×D71—27no—
10C2×C22⋊Dic71C2317yes—
11C2×(C2×D71)⋊C4—27no—
12C22⋊C4×D71—214no—
13C22⋊C4×D71—214no—
14C2×C22⋊C4×C71C2317yes—
15C23×D71—82no—
16C23×D71#4117yes—
17C23×D71#4117yes—
18C22×Dic71—27no—
19C2×C4×D71—47no—
20C2×C4×D71—47no—
21C2×C4×D71—814no—
22C22⋊Dic71—214no—
23(C2×D71)⋊C4—414no—
24C24×C71C2411yescharacteristic
25C22×C4×C71—27no—
26C22⋊C4×C71—214no—
27C23×C71—81no—
28C23×C71C2×#4117yes—
29C22×D71—414no—
30C22×D71—87no—
31C22×D71#12617yes—
32C22×D71#126114yes—
33C22×D71#12617yes—
34C2×C4×C71—47no—
35C2×C4×C71—87no—
36C2×Dic71—47no—
37C2×Dic71—87no—
38C4×D71—814no—
39C4×D71—814no—
40C22×C71—47no—
41C22×C71C2×#12617yes—
42C2×D71—214no—
43C2×D71—47no—
44C2×D71—414no—
45C2×D71#40117yes—
46C2×D71#40111yescharacteristic
47C4×C71—87no—
48Dic71—87no—
49C2×#2104—711nomaximal, Sylow
50C2×C71—27no—
51C2×C71C2×#40111yescharacteristic
52D71—27no—
53D71#210412yesdirect product factor
54#2104—717no—
55#2104D7111yescharacteristic, direct product factor
56C2×#342—717no—
57C71C2×#210411yescharacteristic, complemented, Sylow
58C22×C22⋊C4—717no—
59#342—7114no—
60#342—717no—
61#342C2×D7117yes—
62C25—711no—
63C23×C4—1427no—
64C2×C22⋊C4—717no—
65C2×C22⋊C4—14214no—
66C2×C22⋊C4—14214no—
67C2×C22⋊C4—1427no—
68C2×C22⋊C4C22×D7117yes—
69C24—5682no—
70C24—717no—
71C24—717no—
72C24C23×D7111yescharacteristic
73C22×C4—27no—
74C22×C4—2847no—
75C22×C4—2847no—
76C22×C4—56814no—
77C22×C4—1427no—
78C22⋊C4—214no—
79C22⋊C4—28414no—
80C22⋊C4—14214no—
81C23—81no—
82C23—28414no—
83C23—5687no—
84C23—7114no—
85C23—717no—
86C23—717no—
87C23#41×D7117yes—
88C2×C4—47no—
89C2×C4—87no—
90C2×C4—2847no—
91C2×C4—56814no—
92C2×C4—56814no—
93C2×C4—5687no—
94C22—47no—
95C22—2847no—
96C22—28414no—
97C22—711no—
98C22—717no—
99C22—14214no—
100C22#126×D7117yes—
101C4—87no—
102C4—5687no—
103C2—27no—
104C2—712no—
105C2—1427no—
106C2#401×D7111yescharacteristic
subgroupG/H
centerC2#401×D71
commutator, cosocleC24×C71C24
FrattiniC24C23×D71
FittingC71×#2104C2
socleC2×C71C2×#401
prime radicalC24×D71C23
hypercenter#2104D71
hypocenterC71C2×#2104
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C24 ◁ #2104

Complements

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

NH ≅ G/Ntype
1. C71⋊#2104C2semidirect
3. C71×#2104C2semidirect
53. D71#2104direct (N × H)
55. #2104D71direct (N × H)
57. C71C2×#2104semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C71⋊#21042
#342×D712
C71×#21042
C2×#210471