Finite Group Databasebeta

← #31885298

Subgroups — #31885298

Order 18176.

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

normal subgroups only

This group has 138 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#22586513C211yesmaximal, characteristic, complemented
2#22586375C211yesmaximal, characteristic, complemented
3C71×#1662C211yesmaximal, characteristic
4#17985798—21no—
5#17985777—21no—
6#562×C71C411yescharacteristic
7#491×C71C2211yescharacteristic
8#489×C71—21no—
9#330×C71C411yescharacteristic
10#329×C71—21no—
11#14556068—41no—
12#14556047—41no—
13C4⋊Q8×C71D411yescharacteristic
14#125×C71C2×C411yescharacteristic
15#124×C71—21no—
16#123×C71—41no—
17C22⋊Q8×C71—41no—
18#121×C71—41no—
19#103×C71—41no—
20#103×C71—41no—
21#102×C71—41no—
22#97×C71—21no—
23#97×C71D411yescharacteristic
24#11765849—81no—
25C71⋊C16—42no—
26C2×Q8×C71—21no—
27C2×Q8×C71—21no—
28C2×D4×C71—21no—
29C2×D4×C71—41no—
30C22×C4×C71—41no—
31C2×C8×C71—21no—
32C2×C8×C71—41no—
33C4⋊C4×C71—21no—
34C4⋊C4×C71—41no—
35C4⋊C4×C71—41no—
36C4⋊C4×C71—41no—
37C22⋊C4×C71—41no—
38C22⋊C4×C71—41no—
39C22⋊C4×C71—41no—
40C42×C71C22⋊C411yescharacteristic
41C23×C71—41no—
42Q8×C71—81no—
43Q8×C71—81no—
44D4×C71—81no—
45D4×C71—161no—
46C2×C4×C71—21no—
47C2×C4×C71—41no—
48C2×C4×C71—41no—
49C2×C4×C71—41no—
50C2×C4×C71—161no—
51C2×C4×C71#9911yescharacteristic
52C8×C71—41no—
53C8×C71—81no—
54C71⋊C8—161no—
55C22×C71—81no—
56C22×C71—81no—
57C22×C71—81no—
58C22×C71AΣL(1,𝔽16)11yescharacteristic
59C4×C71—21no—
60C4×C71—41no—
61C4×C71—81no—
62C4×C71—81no—
63C4×C71—81no—
64#7525—711nomaximal, Sylow
65C2×C71—21no—
66C2×C71—161no—
67C2×C71#140411yescharacteristic
68#1662D7111yescharacteristic
69#1413—711no—
70#1356—711no—
71C71#752511yescharacteristic, complemented, Sylow
72#562Dic7111yescharacteristic
73#491C2×D7111yescharacteristic
74#489—21no—
75#356—1421no—
76#347—1421no—
77#330Dic7111yescharacteristic
78#329—21no—
79C4⋊Q8C22⋊D7111yescharacteristic
80#125C2×Dic7111yescharacteristic
81#124—21no—
82#123—41no—
83C22⋊Q8—41no—
84#121—41no—
85M32—2841no—
86#103—41no—
87#103—41no—
88#102—41no—
89#101—2841no—
90#97—21no—
91#97C22⋊D7111yescharacteristic
92C2×Q8—21no—
93C2×Q8—21no—
94C2×D4—21no—
95C2×D4—41no—
96C22×C4—41no—
97M16—5681no—
98C2×C8—21no—
99C2×C8—41no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C22⋊Dic7111yescharacteristic
108C16—2842no—
109C23—41no—
110Q8—81no—
111Q8—81no—
112D4—81no—
113D4—161no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—161no—
119C2×C4#1455604311yescharacteristic
120C8—41no—
121C8—81no—
122C8—11361no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#1798578611yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—81no—
132C2—21no—
133C2—161no—
134C2#2258649111yescharacteristic
subgroupG/H
centerC2#22586491
commutator#125×C71C2×C4
Frattini#491C2×D71
FittingC71×#1662C2
socleC2×C71#1404
prime radical#562×C71C4
cosocle#491×C71C22
hypercenter#1662D71
hypocenterC71#7525
LCSG ▷ #125×C71 ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1662

Complements

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

NH ≅ G/Ntype
1. #22586513C2semidirect
2. #22586375C2semidirect
71. C71#7525semidirect

Maximal subgroups

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

subgroupindex
#225865132
#225863752
C71×#16622
#752571