Finite Group Databasebeta

← #27139889

Subgroups — #27139889

Order 12416.

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

normal subgroups only

This group has 106 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#20260917C211yesmaximal, characteristic
2#20260484C211yesmaximal, characteristic, complemented
3#356×C97C211yesmaximal, characteristic, complemented
4#15690858C411yescharacteristic
5#15690841—21no—
6C4⋊#11715895—21no—
7#15690738—21no—
8#15690729C411yescharacteristic
9C4⋊Q8×C97C2211yescharacteristic
10#101×C97—21no—
11#13095512—41no—
12#13095511—21no—
13#13095511—41no—
14Q8×D97—41no—
15#13095501—41no—
16#13095498—41no—
17#13095495—21no—
18C2×#11715895—41no—
19C2×Q8×C97C2×C411yescharacteristic
20C2×Q8×C97D411yescharacteristic
21M16×C97—41no—
22C4⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24Q8×C97—41no—
25Q8×C97—41no—
26C2×C4×C97—21no—
27C2×C4×C97—21no—
28C2×C4×C97—21no—
29C2×C4×C97C22⋊C411yescharacteristic
30C22⋊D97—41no—
31D388—21no—
32D388—41no—
33C4×D97—41no—
34C4×D97—42no—
35Dic194—21no—
36Dic194—41no—
37C8×C97—81no—
38#11715895—41no—
39#11715895—81no—
40C22×C97#9911yescharacteristic
41C2×D97—41no—
42C4×C97—21no—
43C4×C97—21no—
44C4×C97—21no—
45C4×C97—41no—
46C4×C97—41no—
47Dic97—41no—
48C2×C97—21no—
49C2×C97AΣL(1,𝔽16)11yescharacteristic
50D97—81no—
51#1405—971nomaximal, Sylow
52C97#140511yescharacteristic, complemented, Sylow
53#456—971no—
54#356D9711yescharacteristic, complemented
55#333—971no—
562−1+4—971no—
57#137—1941no—
58C4⋊Q8C2×D9711yescharacteristic
59C4⋊C8—1941no—
60#104—1941no—
61#101—21no—
62#101—971no—
63#41—3881no—
64#41—3881no—
65#41—1941no—
66C2×Q8—3881no—
67C2×Q8C4×D9711yescharacteristic
68C2×Q8C22⋊D9711yescharacteristic
69Q16—3881no—
70SD16—3881no—
71M16—41no—
72M16—1941no—
73C2×C8—3881no—
74C4⋊C4—41no—
75C42D38811yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—3881no—
79Q8—1941no—
80D4—3881no—
81D4—3881no—
82D4—1941no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—3881no—
87C2×C4—3882no—
88C2×C4#1309548811yescharacteristic
89C8—81no—
90C8—7761no—
91C8—3881no—
92C22—3881no—
93C22#1569072411yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—3881no—
100C2—21no—
101C2—7761no—
102C2#2026052111yescharacteristic
subgroupG/H
centerC2#20260521
commutatorC2×Q8×C97C2×C4
FrattiniC4⋊Q8C2×D97
Fitting#356×C97C2
socleC2×C97AΣL(1,𝔽16)
prime radical#15690858C4
cosocleC4⋊Q8×C97C22
hypercenter#356D97
hypocenterC97#1405
LCSG ▷ C2×Q8×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #20260484C2semidirect
3. #356×C97C2semidirect
52. C97#1405semidirect
54. #356D97semidirect

Maximal subgroups

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

subgroupindex
#202609172
#202604842
#356×C972
#140597