Finite Group Databasebeta

← #27139881

Subgroups — #27139881

Order 12416.

122 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 25 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#20260528C211yesmaximal, characteristic, complemented
2#20260479C211yesmaximal, characteristic, complemented
3#453×C97C211yesmaximal, characteristic
4C4⋊#11715895—21no—
5#15690725—21no—
6H(5,𝔽2)×C97C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C97—21no—
8#127×C97C2211yescharacteristic
9#104×C97—21no—
10#100×C97C411yescharacteristic, complemented
11C2×#11715895—41no—
12C22⋊Dic97—41no—
13#41×C97—21no—
14#41×C97—41no—
15C2×D4×C97—41no—
16C2×D4×C97—41no—
17C2×D4×C97—41no—
18C2×D4×C97C2×C411yescharacteristic
19C2×D4×C97D411yescharacteristic
20SD16×C97—41no—
21D8×C97—41no—
22M16×C97—21no—
23C42×C97D411yescharacteristic
24C23×C97—21no—
25C23×C97—21no—
26C23×C97—41no—
27Q8×C97—21no—
28D4×C97—21no—
29D4×C97—41no—
30D4×C97—41no—
31D4×C97—42no—
32D4×C97—41no—
33D4×C97—81no—
34D4×C97—81no—
35C2×C4×C97—21no—
36C2×C4×C97—41no—
37C2×C4×C97—41no—
38C2×C4×C97C22⋊C411yescharacteristic
39C2×Dic97—81no—
40C8×C97—41no—
41#11715895—81no—
42C22×C97—41no—
43C22×C97—41no—
44C22×C97—41no—
45C22×C97—81no—
46C22×C97—81no—
47C22×C97—161no—
48C22×C97#9911yescharacteristic
49C4×C97—21no—
50C4×C97—21no—
51C4×C97—21no—
52C4×C97—41no—
53Dic97—161no—
54C2×C97—21no—
55C2×C97—81no—
56C2×C97—81no—
57C2×C97—81no—
58C2×C97AΣL(1,𝔽16)11yescharacteristic
59#1402—971nomaximal, Sylow
60C97#140211yescharacteristic, complemented, Sylow
61#453D9711yescharacteristic
62#353—971no—
63#331—971no—
64H(5,𝔽2)Dic9711yescharacteristic, complemented
65AGL(1,ℤ/8)—21no—
66#127C2×D9711yescharacteristic
67C4⋊C8—1941no—
68#104—21no—
69#100Dic9711yescharacteristic, complemented
70#99—1941no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4C2×Dic9711yescharacteristic
77C2×D4C22⋊D9711yescharacteristic
78SD16—41no—
79D8—41no—
80M16—21no—
81C2×C8—3881no—
82C22⋊C4—3881no—
83C42C22⋊D9711yescharacteristic
84C23—21no—
85C23—21no—
86C23—41no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—41no—
91D4—42no—
92D4—41no—
93D4—81no—
94D4—81no—
95C2×C4—21no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—7761no—
99C2×C4C22⋊Dic9711yescharacteristic
100C8—41no—
101C8—7761no—
102C22—41no—
103C22—41no—
104C22—41no—
105C22—81no—
106C22—81no—
107C22—161no—
108C22#1569072511yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—41no—
113C4—15521no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—81no—
118C2#2026052211yescharacteristic
subgroupG/H
centerC2#20260522
commutatorC2×D4×C97C2×C4
Frattini#127C2×D97
Fitting, prime radical#453×C97C2
socleC2×C97AΣL(1,𝔽16)
cosocle#127×C97C22
hypercenter#453D97
hypocenterC97#1402
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20260528C2semidirect
2. #20260479C2semidirect
6. H(5,𝔽2)×C97C4semidirect
10. #100×C97C4semidirect
60. C97#1402semidirect
64. H(5,𝔽2)Dic97semidirect
69. #100Dic97semidirect

Maximal subgroups

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

subgroupindex
#202605282
#202604792
#453×C972
#140297