Finite Group Databasebeta

← #28362115

Subgroups — #28362115

Order 13952.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#20295400C211yesmaximal, characteristic, complemented
2#20295221C212yesmaximal, complemented
3#20294775C211yesmaximal, characteristic, complemented
4D4⋊#11724784C212yesmaximal, complemented
5#555×C109C211yesmaximal, characteristic, complemented
6C2×#13182016—22no—
7C2×#13182014—22no—
8#16812696C2211yescharacteristic
9C4⋊#11724784C2212yes—
10#16812619—22no—
11C4×#11724784C2211yescharacteristic
12C4⋊Q8×C109C2211yescharacteristic
13#124×C109—22no—
14#121×C109—22no—
15C4×D4×C109C2212yes—
16#117×C109—22no—
17C2×D436—22no—
18C2×D436—21no—
19#13182016—44no—
20#13182014—44no—
21C2×#11724784—21no—
22C2×#11724784—22no—
23C2×Q8×C109D412yes—
24C2×D4×C109—22no—
25C2×D4×C109D412yes—
26C22×C4×C109—22no—
27C4⋊C4×C109—22no—
28C4⋊C4×C109D412yes—
29C22⋊C4×C109—24no—
30C22⋊C4×C109—22no—
31C42×C109—22no—
32C42×C109C2311yescharacteristic
33C23×C109—22no—
34C22×D109—41no—
35Q8×C109—42no—
36D4×C109—22no—
37D4×C109—22no—
38D4×C109—82no—
39C2×C4×C109—22no—
40C2×C4×C109—22no—
41C2×C4×C109—42no—
42C2×C4×C109—42no—
43C2×C4×C109C2×D411yescharacteristic
44C2×C4×C109C2×D412yes—
45D436—42no—
46D436—42no—
47D436—42no—
48#11724784—22no—
49#11724784—42no—
50C22×C109—42no—
51C22×C109—42no—
52C22×C109—42no—
53C22×C109#12011yescharacteristic
54C2×D109—82no—
55C2×D109—81no—
56C4×C109—21no—
57C4×C109—42no—
58C4×C109—42no—
59C4×C109#13512yes—
60C4×C109AGL(1,ℤ/8)12yes—
61C2×C109—82no—
62C2×C109#44912yes—
63C2×C109#45311yescharacteristic
64D109—161no—
65#1624—1091nomaximal, Sylow
66C109#162411yescharacteristic, complemented, Sylow
67#555D10911yescharacteristic, complemented
68#492—1091no—
69#459—1092no—
70#333—1091no—
71D4⋊C8—1092no—
72C2×SD16—2182no—
73C2×D8—2182no—
74C4⋊Q8C2×D10911yescharacteristic
75#127—1091no—
76#124—22no—
77#121—22no—
78C4×D4C2×D10912yes—
79#117—22no—
80C4⋊C8—1092no—
81#102—2182no—
82C4×C8—1091no—
83C2×Q8C22⋊D10912yes—
84C2×D4—22no—
85C2×D4—2182no—
86C2×D4—2181no—
87C2×D4C22⋊D10912yes—
88C22×C4—22no—
89SD16—4364no—
90D8—4364no—
91C2×C8—2182no—
92C2×C8—2181no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D10912yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D10911yescharacteristic
99C23—22no—
100C23—4361no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—4362no—
106D4—4362no—
107D4—4362no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D10911yescharacteristic
113C2×C4C2×C22⋊D10912yes—
114C8—4362no—
115C8—2182no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—8722no—
120C22—8721no—
121C22#1681266811yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1681271912yes—
126C4#1681272412yes—
127C2—82no—
128C2—17441no—
129C2#2029517112yes—
130C2#2029519411yescharacteristic
subgroupG/H
centerC22#16812668
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#555×C109C2
socleC22×C109#120
hypercenter#555D109
hypocenterC109#1624
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20295400C2semidirect
2. #20295221C2semidirect
3. #20294775C2semidirect
4. D4⋊#11724784C2semidirect
5. #555×C109C2semidirect
66. C109#1624semidirect
67. #555D109semidirect

Maximal subgroups

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

subgroupindex
#202954002
#202952212
#202947752
D4⋊#117247842
#555×C1092
#1624109