Finite Group Databasebeta

← #20295217

Subgroups — #20295217

Order 6976.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#16812744C211yesmaximal, characteristic
2#16812683C212yesmaximal, complemented
3#16812680C211yesmaximal, characteristic, complemented
4#16812613C212yesmaximal, complemented
5#99×C109C211yesmaximal, characteristic, complemented
6#13182029—21no—
7#13182028—22no—
8C2×C22⋊D109—21no—
9C2×C22⋊D109C2212yes—
10D4×D109—22no—
11C22×Dic109C2211yescharacteristic, complemented
12C4⋊Dic109—22no—
13#13182008—21no—
14C22⋊Dic109—21no—
15C22⋊Dic109C2211yescharacteristic, complemented
16#13182006—22no—
17C2×D4×C109C2211yescharacteristic
18C22⋊C4×C109C2212yescomplemented
19C23×C109D411yescharacteristic
20C23×C109C2311yescharacteristic
21C22×D109—21no—
22C22×D109D412yes—
23D4×C109—22no—
24C2×C4×C109—22no—
25C2×C4×C109D411yescharacteristic
26C22⋊D109—22no—
27C22⋊D109—22no—
28C22⋊D109—22no—
29C2×Dic109—21no—
30C2×Dic109—22no—
31C2×Dic109—21no—
32C2×Dic109—21no—
33C2×Dic109D412yes—
34D436—21no—
35C4×D109—22no—
36Dic218—21no—
37C22×C109—21no—
38C22×C109—41no—
39C22×C109—41no—
40C22×C109C2×D412yes—
41C22×C109C2×D411yescharacteristic
42C2×D109—22no—
43C2×D109—42no—
44C2×D109—81no—
45C4×C109—21no—
46C4×C109—42no—
47Dic109—22no—
48Dic109—41no—
49Dic109—41no—
50C2×C109—21no—
51C2×C109—22no—
52C2×C109—41no—
53C2×C109#12011yescharacteristic
54D109—42no—
55C109#45811yescharacteristic, complemented, Sylow
56#458—1091nomaximal, Sylow
57H(5,𝔽2)—1091no—
58#123—1091no—
59#123—1092no—
60#99—1092no—
61#99D10911yescharacteristic, complemented
62#41—2182no—
63#41—2181no—
64C2×D4—1092no—
65C2×D4—2182no—
66C2×D4—2181no—
67C2×D4C2×D10911yescharacteristic
68C22×C4—1091no—
69C4⋊C4—2182no—
70C4⋊C4—2181no—
71C22⋊C4—1091no—
72C22⋊C4—2182no—
73C22⋊C4—2181no—
74C22⋊C4C2×D10912yescomplemented
75C23—1092no—
76C23—2181no—
77C23D43611yescharacteristic
78C23C22×D10911yescharacteristic
79Q8—2181no—
80D4—22no—
81D4—2181no—
82D4—2182no—
83D4—2182no—
84D4—2182no—
85C2×C4—22no—
86C2×C4—1092no—
87C2×C4—2182no—
88C2×C4—2182no—
89C2×C4—2181no—
90C2×C4—2181no—
91C2×C4—2181no—
92C2×C4D43611yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—8721no—
97C22—4362no—
98C22—2182no—
99C22D4×D10912yes—
100C22C2×D43611yescharacteristic
101C4—21no—
102C4—42no—
103C4—4361no—
104C4—4361no—
105C4—2182no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—4362no—
110C2#1681266911yescharacteristic
subgroupG/H
centerC2#16812669
commutatorC23×C109C23
FrattiniC23C22×D109
Fitting#99×C109C2
socleC2×C109#120
hypercenter#99D109
hypocenterC109#458
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #16812683C2semidirect
3. #16812680C2semidirect
4. #16812613C2semidirect
5. #99×C109C2semidirect
11. C22×Dic109C22semidirect
15. C22⋊Dic109C22semidirect
18. C22⋊C4×C109C22semidirect
55. C109#458semidirect
61. #99D109semidirect
74. C22⋊C4C2×D109semidirect