Finite Group Databasebeta

← #28379311

Subgroups — #28379311

Order 13952.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#20296005C212yesmaximal, complemented
2#20295930C214yesmaximal, complemented
3#20295593C211yesmaximal, characteristic
4#16812778—22no—
5#16812758C2214yescomplemented
6#16812757—22no—
7C2×#13182029C2211yescharacteristic
8#16812695C2212yes—
9#16812686C412yescomplemented
10#16812670—22no—
11#16812668—22no—
12#16812658C411yescharacteristic, complemented
13C22×D4×C109C411yescharacteristic, complemented
14C4⋊#89137—42no—
15C22⋊#89137—24no—
16C4×#89137—42no—
17#13182029—42no—
18C2×Dic218C2×C411yescharacteristic
19C2×D436C2311yescharacteristic
20C2×C22⋊D109—24no—
21C2×C22⋊D109D412yes—
22C2×C4×D109D412yes—
23C4⋊Dic109—21no—
24C22⋊Dic109—21no—
25C22⋊Dic109—24no—
26C4×Dic109C2×C412yes—
27C24×C109—21no—
28C2×D4×C109—21no—
29C2×D4×C109—42no—
30C2×D4×C109C2×C412yes—
31C22×C4×C109C2×C411yescharacteristic
32C23×C109—22no—
33C23×C109—42no—
34C23×C109—82no—
35C23×C109C22⋊C411yescharacteristic
36C22×D109C2×D412yes—
37C2×#89137—44no—
38D4×C109—42no—
39D4×C109—42no—
40C2×C4×C109—22no—
41C2×C4×C109C22⋊C411yescharacteristic
42C2×C4×C109C22×C411yescharacteristic
43C22⋊D109—42no—
44C22⋊D109—84no—
45C2×Dic109—22no—
46C2×Dic109C22⋊C412yes—
47D436—41no—
48C4×D109—22no—
49C4×D109—22no—
50Dic218—41no—
51C22×C109—21no—
52C22×C109—41no—
53C22×C109—44no—
54C22×C109—42no—
55C22×C109—84no—
56C22×C109C2×C22⋊C411yescharacteristic
57C2×D109—22no—
58C2×D109—42no—
59#89137—84no—
60C4×C109—21no—
61C4×C109#9912yes—
62Dic109—22no—
63Dic109—42no—
64C2×C109—21no—
65C2×C109—41no—
66C2×C109—82no—
67C2×C109C2×#9911yescharacteristic
68D109—42no—
69#2128—1091nomaximal, Sylow
70C109#212811yescharacteristic, complemented, Sylow
71#535—1091no—
72#410—1092no—
73#353—1094no—
74C2×#41—1091no—
75C22×D4#8913711yescharacteristic, complemented
76#127—1092no—
77#124—1092no—
78#121—2182no—
79#120—2182no—
80#117—1091no—
81#117—2182no—
82#99—1094no—
83#99—2182no—
84C24—21no—
85#41—4362no—
86C2×Q8—1091no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—1092no—
90C2×D4—1091no—
91C2×D4—2184no—
92C2×D4C2×#8913712yes—
93C22×C4—1092no—
94C22×C4C2×#8913711yescharacteristic
95C4⋊C4—4362no—
96C4⋊C4—2181no—
97C22⋊C4—2184no—
98C22⋊C4—2184no—
99C22⋊C4—2181no—
100C42—1092no—
101C42—4362no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1092no—
106C23C22⋊#8913711yescharacteristic
107Q8—4361no—
108D4—42no—
109D4—42no—
110D4—8724no—
111D4—4361no—
112D4—4362no—
113C2×C4—22no—
114C2×C4—1092no—
115C2×C4—4364no—
116C2×C4—2182no—
117C2×C4—2182no—
118C2×C4—2182no—
119C2×C4C22⋊#8913711yescharacteristic
120C2×C4C22×#8913711yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—4362no—
127C22—2182no—
128C22C2×C22⋊#8913711yescharacteristic
129C4—21no—
130C4—8724no—
131C4—4362no—
132C4—2182no—
133C4#1681275712yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—4362no—
138C2C2×#1681275711yescharacteristic
subgroupG/H
centerC2C2×#16812757
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22×D4×C109C4
socleC2×C109C2×#99
prime radical#20295593C2
cosocleC2×D436C23
hypercenterC22×D4#89137
hypocenterC109#2128
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20296005C2semidirect
2. #20295930C2semidirect
5. #16812758C22semidirect
9. #16812686C4semidirect
12. #16812658C4semidirect
13. C22×D4×C109C4semidirect
70. C109#2128semidirect
75. C22×D4#89137semidirect

Maximal subgroups

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

subgroupindex
#202960052
#202959302
#202955932
#2128109