Finite Group Databasebeta

← #27139880

Subgroups — #27139880

Order 12416.

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

normal subgroups only

This group has 122 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#20260902C211yesmaximal, characteristic
2#20260479C211yesmaximal, characteristic, complemented
3#353×C97C211yesmaximal, characteristic, complemented
4#15690855C411yescharacteristic, complemented
5#15690836—21no—
6C4⋊#11715895—21no—
7#15690738—21no—
8#15690727C411yescharacteristic, complemented
9#127×C97C2211yescharacteristic
10#99×C97—21no—
11#13095511—21no—
12#13095510—41no—
13C2×C22⋊D97—41no—
14D4×D97—41no—
15#13095499—41no—
16#13095496—41no—
17#13095495—21no—
18C2×#11715895—41no—
19C2×D4×C97—41no—
20C2×D4×C97C2×C411yescharacteristic
21C2×D4×C97D411yescharacteristic
22C22⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25C23×C97—21no—
26C22×D97—41no—
27D4×C97—41no—
28D4×C97—41no—
29D4×C97—81no—
30D4×C97—81no—
31C2×C4×C97—21no—
32C2×C4×C97—81no—
33C2×C4×C97C22⋊C411yescharacteristic
34C22⋊D97—42no—
35C22⋊D97—41no—
36C2×Dic97—41no—
37D388—21no—
38C4×D97—41no—
39Dic194—21no—
40#11715895—41no—
41#11715895—81no—
42C22×C97—41no—
43C22×C97—41no—
44C22×C97—81no—
45C22×C97—81no—
46C22×C97#9911yescharacteristic
47C2×D97—41no—
48C2×D97—161no—
49C4×C97—21no—
50C4×C97—21no—
51C4×C97—21no—
52C4×C97—161no—
53Dic97—41no—
54C2×C97—21no—
55C2×C97—81no—
56C2×C97—81no—
57C2×C97AΣL(1,𝔽16)11yescharacteristic
58D97—81no—
59#1402—971nomaximal, Sylow
60C97#140211yescharacteristic, complemented, Sylow
61#453—971no—
62#353D9711yescharacteristic, complemented
63#331—971no—
64H(5,𝔽2)—971no—
65AGL(1,ℤ/8)—1941no—
66#127C2×D9711yescharacteristic
67C4⋊C8—1941no—
68#104—1941no—
69#100—971no—
70#99—21no—
71#41—3881no—
72#41—1941no—
73C2×D4—41no—
74C2×D4—3881no—
75C2×D4—3881no—
76C2×D4C4×D9711yescharacteristic
77C2×D4C22⋊D9711yescharacteristic
78SD16—3881no—
79D8—3881no—
80M16—1941no—
81C2×C8—3881no—
82C22⋊C4—41no—
83C42D38811yescharacteristic
84C23—21no—
85C23—21no—
86C23—3881no—
87Q8—1941no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—3882no—
93D4—3881no—
94D4—1941no—
95C2×C4—21no—
96C2×C4—81no—
97C2×C4—3881no—
98C2×C4—3881no—
99C2×C4#1309548811yescharacteristic
100C8—7761no—
101C8—3881no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—15521no—
107C22—3881no—
108C22#1569072411yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—161no—
113C4—3881no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—7761no—
118C2#2026052111yescharacteristic
subgroupG/H
centerC2#20260521
commutatorC2×D4×C97C2×C4
Frattini#127C2×D97
Fitting#353×C97C2
socleC2×C97AΣL(1,𝔽16)
prime radical#20260902C2
cosocle#127×C97C22
hypercenter#353D97
hypocenterC97#1402
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20260479C2semidirect
3. #353×C97C2semidirect
4. #15690855C4semidirect
8. #15690727C4semidirect
60. C97#1402semidirect
62. #353D97semidirect

Maximal subgroups

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

subgroupindex
#202609022
#202604792
#353×C972
#140297