Finite Group Databasebeta

← #34001237

Subgroups — #34001237

Order 18688.

134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 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#22625071C211yesmaximal, characteristic
2#22624934C211yesmaximal, characteristic
3#22609233C211yesmaximal, characteristic
4#17989492—21no—
5#17989479—21no—
6#17988953—21no—
7#17988953C2211yescharacteristic
8#17988329—21no—
9#17988329C411yescharacteristic, complemented
10#564×C73C411yescharacteristic, complemented
11#14556665—41no—
12#14556650—41no—
13#14556588D411yescharacteristic
14#14556511—41no—
15#14556511—41no—
16#14556511—41no—
17#14556502—21no—
18#14556502D411yescharacteristic
19#125×C73—21no—
20#125×C73—41no—
21#125×C73—41no—
22#125×C73—41no—
23#125×C73C2×C411yescharacteristic
24C22⋊#64408—81no—
25#11924031—42no—
26C2×D292—21no—
27C2×D292—41no—
28C2×#10596831—21no—
29C2×#10596831—41no—
30C4⋊C4×C73—21no—
31C4⋊C4×C73—21no—
32C4⋊C4×C73—21no—
33C4⋊C4×C73—41no—
34C4⋊C4×C73—41no—
35C4⋊C4×C73—41no—
36C4⋊C4×C73—41no—
37C4⋊C4×C73—41no—
38C4⋊C4×C73—41no—
39C42×C73—41no—
40C42×C73C22⋊C411yescharacteristic
41C22×D73—41no—
42C2×#64408—161no—
43C2×C4×C73—21no—
44C2×C4×C73—41no—
45C2×C4×C73—41no—
46C2×C4×C73—41no—
47C2×C4×C73#9911yescharacteristic
48D292—81no—
49D292—82no—
50#10596831—41no—
51#10596831—81no—
52C22×C73AΣL(1,𝔽16)11yescharacteristic
53C2×D73—81no—
54C2×D73—161no—
55#64408—321no—
56C4×C73—21no—
57C4×C73—41no—
58C4×C73—81no—
59C4×C73—81no—
60C4×C73—81no—
61#7524—731nomaximal, Sylow
62C2×C73—21no—
63C2×C73#140411yescharacteristic
64D73—161no—
65#1658—731no—
66#1412—731no—
67#1355—731no—
68C73#752411yescharacteristic, complemented, Sylow
69#564#6440811yescharacteristic, complemented
70#490—731no—
71#490—1461no—
72#353—1461no—
73#347—1461no—
74#330—731no—
75#330—1461no—
76#127—731no—
77#125—21no—
78#125—41no—
79#125—41no—
80#125—41no—
81#125C2×#6440811yescharacteristic
82M32—2921no—
83#102—2921no—
84#102—2921no—
85#102—2921no—
86#99—2921no—
87#97—731no—
88#97—1461no—
89C2×D4—2921no—
90C2×D4—1461no—
91C2×C8—2921no—
92C2×C8—1461no—
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—5841no—
103C42—41no—
104C42C22⋊#6440811yescharacteristic
105C16—2922no—
106C23—2921no—
107D4—5841no—
108D4—5842no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—11681no—
114C2×C4#1455664911yescharacteristic
115C8—2921no—
116C8—5841no—
117C22—5841no—
118C22—11681no—
119C22#1798948711yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—81no—
124C4—81no—
125C4—23361no—
126C2—21no—
127C2—11681no—
128C2#2262505411yescharacteristic
subgroupG/H
centerC2#22625054
commutator#125×C73C2×C4
Frattini#125C2×#64408
Fitting#564×C73C4
socleC2×C73#1404
prime radical#14556588D4
cosocle#17988953C22
hypercenter#564#64408
hypocenterC73#7524
LCSG ▷ #125×C73 ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #125 ◁ #564

Complements

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

NH ≅ G/Ntype
9. #17988329C4semidirect
10. #564×C73C4semidirect
68. C73#7524semidirect
69. #564#64408semidirect

Maximal subgroups

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

subgroupindex
#226250712
#226249342
#226092332
#752473