Finite Group Databasebeta

← #27157225

Subgroups — #27157225

Order 12416.

238 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261758C212yesmaximal, complemented
2#20261590C214yesmaximal, complemented
3#20261283C211yesmaximal, characteristic
4#15690877—22no—
5#15690867—22no—
6#15690865C2214yes—
7C2×#13095510C412yes—
8#15690810C411yescharacteristic
9#15690797C2212yes—
10#15690781—22no—
11D4×Dic97—22no—
12C42×D97C2211yescharacteristic
13#127×C97C411yescharacteristic
14C2×#11715897—24no—
15#13095510—44no—
16C2×Dic194C2×C412yes—
17C2×C22⋊D97—22no—
18C22×Dic97—22no—
19C2×C4×D97C2×C411yescharacteristic
20C2×C4×D97D412yes—
21C4⋊Dic97—22no—
22C22⋊Dic97—24no—
23C4×Dic97D412yes—
24C4×Dic97C2311yescharacteristic
25C2×D4×C97—22no—
26C2×D4×C97C2×C412yes—
27C42×C97C2×C411yescharacteristic
28C23×C97—22no—
29C22×D97C22⋊C411yescharacteristic
30D4×C97—42no—
31D4×C97—44no—
32C2×C4×C97C22⋊C412yes—
33C2×C4×C97C22×C411yescharacteristic
34C22⋊D97—44no—
35C2×Dic97—22no—
36C2×Dic97—44no—
37C2×Dic97C22⋊C411yescharacteristic
38C2×Dic97C2×D412yes—
39C4×D97—22no—
40C4×D97—42no—
41Dic194—42no—
42#11715897—44no—
43C22×C97—42no—
44C22×C97—44no—
45C22×C97C2×C22⋊C411yescharacteristic
46C2×D97—21no—
47C2×D97—22no—
48C4×C97—21no—
49C4×C97—22no—
50Dic97—22no—
51Dic97—42no—
52Dic97#10012yes—
53C2×C97—82no—
54C2×C97C2×#10011yescharacteristic
55C2×C97#42112yes—
56D97—41no—
57#1536—971nomaximal, Sylow
58C97#153611yescharacteristic, complemented, Sylow
59#532—971no—
60#432—972no—
61#329—974no—
62C2×#41—972no—
63C4⋊Q8—971no—
64#127#8760411yescharacteristic
65#124—972no—
66#121—1942no—
67C4×D4—1942no—
68C2×C42—971no—
69C4⋊C8—1942no—
70C22⋊C8—1942no—
71#97—974no—
72#41—3884no—
73C2×Q8—972no—
74C2×D4—22no—
75C2×D4—1942no—
76C2×D4C2×#8760412yes—
77C22×C4—972no—
78C22×C4—971no—
79C22×C4—1942no—
80C2×C8—1944no—
81C4⋊C4—1942no—
82C22⋊C4—1944no—
83C42—972no—
84C42—971no—
85C42C2×#8760411yescharacteristic
86C23—22no—
87C23—971no—
88Q8—3882no—
89D4—42no—
90D4—44no—
91D4—3884no—
92C2×C4—971no—
93C2×C4—972no—
94C2×C4—3882no—
95C2×C4—3884no—
96C2×C4—1942no—
97C2×C4—1942no—
98C2×C4C22⋊#8760412yes—
99C2×C4C22×#8760411yescharacteristic
100C8—3884no—
101C22—42no—
102C22—44no—
103C22—1941no—
104C22—1942no—
105C22C2×C22⋊#8760411yescharacteristic
106C4—21no—
107C4—22no—
108C4—972no—
109C4—3882no—
110C4—1942no—
111C2—82no—
112C2—3881no—
113C2D97⋊#10011yescharacteristic
114C2#2026173312yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC4×Dic97C23
hypercenter#127#87604
hypocenterC97#1536
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261758C2semidirect
2. #20261590C2semidirect
58. C97#1536semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202615902
#202612832
#153697