Finite Group Databasebeta

← #46834513

Subgroups — #46834513

Order 34688.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#31774074C211yesmaximal, characteristic
2#31773694C211yesmaximal, characteristic, complemented
3#329×C271C211yesmaximal, characteristic, complemented
4#22576205C411yescharacteristic
5#22576191—21no—
6#22576088—21no—
7#22576079—21no—
8#22576077C411yescharacteristic
9#124×C271C2211yescharacteristic
10#97×C271—21no—
11#16864654—21no—
12#16864653—41no—
13C2×C22⋊D271—41no—
14D4×D271—41no—
15#16864644—41no—
16#16864641—41no—
17#16864638—21no—
18#16864638—41no—
19C2×Q8×C271D411yescharacteristic
20C2×D4×C271C2×C411yescharacteristic
21C2×C8×C271—41no—
22C22⋊C4×C271—41no—
23C42×C271D411yescharacteristic
24C22×D271—41no—
25Dic542—21no—
26D1084—21no—
27C22⋊D271—42no—
28C22⋊D271—41no—
29C2×Dic271—41no—
30C4×D271—41no—
31C271⋊C8—41no—
32C271⋊C8—81no—
33C23×C271—21no—
34Q8×C271—41no—
35D4×C271—41no—
36C2×C4×C271—21no—
37C2×C4×C271—21no—
38C2×C4×C271C22⋊C411yescharacteristic
39C8×C271—42no—
40C22×C271—41no—
41C22×C271—81no—
42C22×C271#9911yescharacteristic
43C2×D271—41no—
44C2×D271—161no—
45C4×C271—21no—
46C4×C271—41no—
47C4×C271—41no—
48Dic271—41no—
49C2×C271—21no—
50C2×C271—81no—
51C2×C271AΣL(1,𝔽16)11yescharacteristic
52D271—81no—
53C271#140311yescharacteristic, complemented, Sylow
54#1403—2711nomaximal, Sylow
55#455—2711no—
56#355—2711no—
57#329D27111yescharacteristic, complemented
58H(5,𝔽2)—2711no—
59#137—5421no—
60#124C2×D27111yescharacteristic
61#104—5421no—
62#101—5421no—
63#100—2711no—
64#97—21no—
65#41—5421no—
66#41—10841no—
67C2×Q8D108411yescharacteristic
68C2×D4—10841no—
69C2×D4—10841no—
70C2×D4C4×D27111yescharacteristic
71Q16—10841no—
72SD16—10841no—
73M16—5421no—
74M16—10841no—
75C2×C8—41no—
76C22⋊C4—41no—
77C42C22⋊D27111yescharacteristic
78C23—21no—
79C23—10841no—
80Q8—41no—
81Q8—5421no—
82D4—41no—
83D4—5421no—
84D4—10842no—
85D4—10841no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—10841no—
89C2×C4—10841no—
90C2×C4(C2×D271)⋊C411yescharacteristic
91C8—42no—
92C8—10841no—
93C8—21681no—
94C22—41no—
95C22—81no—
96C22—10841no—
97C22—43361no—
98C22#2257607411yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—10841no—
103C2—21no—
104C2—81no—
105C2—21681no—
106C2#3177368511yescharacteristic
subgroupG/H
centerC2#31773685
commutatorC2×D4×C271C2×C4
Frattini#124C2×D271
Fitting#329×C271C2
socleC2×C271AΣL(1,𝔽16)
prime radical#22576205C4
cosocle#124×C271C22
hypercenter#329D271
hypocenterC271#1403
LCSG ▷ C2×D4×C271 ▷ C2×C4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31773694C2semidirect
3. #329×C271C2semidirect
53. C271#1403semidirect
57. #329D271semidirect

Maximal subgroups

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

subgroupindex
#317740742
#317736942
#329×C2712
#1403271