Finite Group Databasebeta

← #28378155

Subgroups — #28378155

Order 13952.

112 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 20 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#20295932C211yesmaximal, characteristic, complemented
2#20295882C211yesmaximal, characteristic, complemented
3#20295199C211yesmaximal, characteristic
4#16812758—21no—
5#16812754—21no—
6#16812724—21no—
7#16812688C2211yescharacteristic
8#16812626—21no—
9#16812618C411yescharacteristic, complemented
102−1+4×C109C411yescharacteristic, complemented
11C2×#11724786—41no—
12C22⋊#89137—41no—
13C2×D436D411yescharacteristic
14#13182016—41no—
15#13182014—41no—
16#13182013—21no—
17#13182006—41no—
18C4×Dic109D411yescharacteristic
19#41×C109—21no—
20#41×C109—41no—
21#41×C109—41no—
22C2×Q8×C109—41no—
23C2×Q8×C109C2×C411yescharacteristic
24C22×D109—21no—
25C2×#89137—81no—
26Q8×C109—21no—
27Q8×C109—41no—
28Q8×C109—41no—
29D4×C109—21no—
30D4×C109—41no—
31D4×C109—41no—
32C2×C4×C109—21no—
33C2×C4×C109—41no—
34C2×C4×C109—42no—
35C2×C4×C109C22⋊C411yescharacteristic
36C2×Dic109—21no—
37D436—41no—
38#11724786—42no—
39#11724784—41no—
40C22×C109—41no—
41C22×C109#9911yescharacteristic
42C2×D109—41no—
43C2×D109—81no—
44#89137—161no—
45C4×C109—21no—
46C4×C109—41no—
47C4×C109—41no—
48Dic109—41no—
49C2×C109—21no—
50C2×C109—81no—
51C2×C109AΣL(1,𝔽16)11yescharacteristic
52D109—81no—
53#1404—1091nomaximal, Sylow
54C109#140411yescharacteristic, complemented, Sylow
55#454—1091no—
56#354—1091no—
57#329—1091no—
582−1+4#8913711yescharacteristic, complemented
59AGL(1,ℤ/8)—2181no—
60#124—1091no—
61#104—2181no—
62#101—1091no—
63#99—2181no—
64#97—2181no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×#8913711yescharacteristic
70C2×D4—1091no—
71SD16—4361no—
72D8—4361no—
73M16—2181no—
74C2×C8—4361no—
75C22⋊C4—4361no—
76C22⋊C4—4361no—
77C42—1091no—
78C23—2181no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—4361no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—8721no—
90C2×C4—2181no—
91C2×C4C22⋊#8913711yescharacteristic
92C8—4361no—
93C8—4362no—
94C22—41no—
95C22—8721no—
96C22—4361no—
97C22#1681275711yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—4361no—
102C4—17441no—
103C2—21no—
104C2—81no—
105C2—8721no—
106C2#2029592611yescharacteristic
subgroupG/H
centerC2#20295926
commutatorC2×Q8×C109C2×C4
FrattiniC2×Q8C2×#89137
Fitting2−1+4×C109C4
socleC2×C109AΣL(1,𝔽16)
prime radical#20295199C2
cosocle#16812688C22
hypercenter2−1+4#89137
hypocenterC109#1404
LCSG ▷ C2×Q8×C109 ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #20295932C2semidirect
2. #20295882C2semidirect
9. #16812618C4semidirect
10. 2−1+4×C109C4semidirect
54. C109#1404semidirect
58. 2−1+4#89137semidirect

Maximal subgroups

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

subgroupindex
#202959322
#202958822
#202951992
#1404109