Finite Group Databasebeta

← #31885295

Subgroups — #31885295

Order 18176.

134 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 27 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#22586511C211yesmaximal, characteristic
2#22586373C211yesmaximal, characteristic
3C71×#1658C211yesmaximal, characteristic
4#17985792—21no—
5#17985777—21no—
6#564×C71C411yescharacteristic, complemented
7#490×C71—21no—
8#490×C71C2211yescharacteristic
9#330×C71—21no—
10#330×C71C411yescharacteristic, complemented
11#14556068—41no—
12#14556043—41no—
13#127×C71D411yescharacteristic
14#125×C71—21no—
15#125×C71—41no—
16#125×C71—41no—
17#125×C71—41no—
18#125×C71C2×C411yescharacteristic
19#102×C71—41no—
20#102×C71—41no—
21#102×C71—41no—
22#97×C71—21no—
23#97×C71D411yescharacteristic
24C22⋊Dic71—81no—
25C71⋊C16—42no—
26C2×D4×C71—21no—
27C2×D4×C71—41no—
28C2×C8×C71—21no—
29C2×C8×C71—41no—
30C4⋊C4×C71—21no—
31C4⋊C4×C71—21no—
32C4⋊C4×C71—21no—
33C4⋊C4×C71—41no—
34C4⋊C4×C71—41no—
35C4⋊C4×C71—41no—
36C4⋊C4×C71—41no—
37C4⋊C4×C71—41no—
38C4⋊C4×C71—41no—
39C42×C71—41no—
40C42×C71C22⋊C411yescharacteristic
41C23×C71—41no—
42D4×C71—81no—
43D4×C71—82no—
44C2×C4×C71—21no—
45C2×C4×C71—41no—
46C2×C4×C71—41no—
47C2×C4×C71—41no—
48C2×C4×C71#9911yescharacteristic
49C2×Dic71—161no—
50C8×C71—41no—
51C8×C71—81no—
52C22×C71—81no—
53C22×C71—161no—
54C22×C71AΣL(1,𝔽16)11yescharacteristic
55C4×C71—21no—
56C4×C71—41no—
57C4×C71—81no—
58C4×C71—81no—
59C4×C71—81no—
60Dic71—321no—
61#7524—711nomaximal, Sylow
62C2×C71—21no—
63C2×C71—161no—
64C2×C71#140411yescharacteristic
65#1658D7111yescharacteristic
66#1412—711no—
67#1355—711no—
68C71#752411yescharacteristic, complemented, Sylow
69#564Dic7111yescharacteristic, complemented
70#490—21no—
71#490C2×D7111yescharacteristic
72#353—1421no—
73#347—1421no—
74#330—21no—
75#330Dic7111yescharacteristic, complemented
76#127C22⋊D7111yescharacteristic
77#125—21no—
78#125—41no—
79#125—41no—
80#125—41no—
81#125C2×Dic7111yescharacteristic
82M32—2841no—
83#102—41no—
84#102—41no—
85#102—41no—
86#99—2841no—
87#97—21no—
88#97C22⋊D7111yescharacteristic
89C2×D4—21no—
90C2×D4—41no—
91C2×C8—21no—
92C2×C8—41no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C22⋊C4—5681no—
103C42—41no—
104C42C22⋊Dic7111yescharacteristic
105C16—2842no—
106C23—41no—
107D4—81no—
108D4—82no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—11361no—
114C2×C4#1455604311yescharacteristic
115C8—41no—
116C8—81no—
117C22—81no—
118C22—161no—
119C22#1798578611yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—81no—
124C4—81no—
125C4—22721no—
126C2—21no—
127C2—161no—
128C2#2258649111yescharacteristic
subgroupG/H
centerC2#22586491
commutator#125×C71C2×C4
Frattini#490C2×D71
FittingC71×#1658C2
socleC2×C71#1404
prime radical#127×C71D4
cosocle#490×C71C22
hypercenter#1658D71
hypocenterC71#7524
LCSG ▷ #125×C71 ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1658

Complements

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

NH ≅ G/Ntype
6. #564×C71C4semidirect
10. #330×C71C4semidirect
68. C71#7524semidirect
69. #564Dic71semidirect
75. #330Dic71semidirect

Maximal subgroups

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

subgroupindex
#225865112
#225863732
C71×#16582
#752471