Finite Group Databasebeta

← #27158774

Subgroups — #27158774

Order 12416.

158 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 27 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#20261820C211yesmaximal, characteristic
2#20261820C211yesmaximal, characteristic
3#20261682C211yesmaximal, characteristic
4C22⋊#11715906—21no—
5C22⋊#11715906—21no—
6C23×#87604C411yescharacteristic
7C2×C4⋊#87604—21no—
8C2×C22⋊#87604C2211yescharacteristic
9(C2×#87604)⋊C4—21no—
10C22⋊C4×D97C411yescharacteristic
11C2×#11715906—41no—
12C2×#11715906—41no—
13C22×#87604—21no—
14C22×#87604—42no—
15C22×#87604—41no—
16C22×#87604D411yescharacteristic
17C22×#87604D411yescharacteristic
18C4⋊#87604—42no—
19C22⋊#87604—41no—
20C23×D97C2×C411yescharacteristic
21C2×C4×D97—21no—
22C22⋊Dic97C811yescharacteristic, complemented
23#13095488—21no—
24C22⋊C4×C97C811yescharacteristic, complemented
25C23×C97C2×C811yescharacteristic
26C22×D97—21no—
27C22×D97—21no—
28C22×D97—41no—
29C22×D97—41no—
30C22×D97C22⋊C411yescharacteristic
31C22×D97M1611yescharacteristic
32C2×#87604—22no—
33C2×#87604—21no—
34C2×#87604—21no—
35C2×#87604—41no—
36C2×#87604—42no—
37C2×#87604—41no—
38C2×#87604—41no—
39C2×#87604—41no—
40C2×#87604—42no—
41#11715906—81no—
42#11715906—81no—
43C2×C4×C97—21no—
44C2×Dic97—21no—
45C4×D97—41no—
46C4×D97—41no—
47C22×C97—21no—
48C22×C97—41no—
49C22×C97C22⋊C811yescharacteristic
50C2×D97—21no—
51C2×D97—21no—
52C2×D97—21no—
53C2×D97—42no—
54C2×D97—41no—
55C2×D97—41no—
56C2×D97—41no—
57C2×D97#9911yescharacteristic
58C2×D97#10111yescharacteristic
59#87604—41no—
60#87604—42no—
61#87604—41no—
62C4×C97—41no—
63Dic97—41no—
64C2×C97—21no—
65C2×C97—41no—
66C2×C97#32411yescharacteristic
67D97—21no—
68D97—41no—
69D97AΣL(1,𝔽16)11yescharacteristic
70D97#35211yescharacteristic
71#1326—971nomaximal, Sylow
72C97#132611yescharacteristic, complemented, Sylow
73#385—971no—
74#323—971no—
75#323—971no—
76C23×C4—971no—
77C2×C4⋊C4—1941no—
78C2×C22⋊C4—971no—
79C2×C22⋊C4—971no—
80C22⋊C8—1941no—
81C22⋊C8—1941no—
82#95—1941no—
83C24—971no—
84C22×C4—971no—
85C22×C4—971no—
86C22×C4—3881no—
87C22×C4—3882no—
88C22×C4—1941no—
89C22×C4—1941no—
90C2×C8—3881no—
91C2×C8—3881no—
92C4⋊C4—3882no—
93C22⋊C4—971no—
94C22⋊C4—3881no—
95C22⋊C4—1941no—
96C22⋊C4#1171590611yescharacteristic, complemented
97C23—971no—
98C23—971no—
99C23—3881no—
100C23—3881no—
101C23—1941no—
102C23—1941no—
103C23C2×#1171590611yescharacteristic
104C2×C4—21no—
105C2×C4—3881no—
106C2×C4—3881no—
107C2×C4—3881no—
108C2×C4—3882no—
109C2×C4—3881no—
110C2×C4—3882no—
111C2×C4—3881no—
112C2×C4—3881no—
113C2×C4—1941no—
114C2×C4—1941no—
115C2×C4—1942no—
116C2×C4—1941no—
117C8—7761no—
118C8—7761no—
119C22—21no—
120C22—41no—
121C22—971no—
122C22—971no—
123C22—3881no—
124C22—3881no—
125C22—3881no—
126C22—3882no—
127C22—1941no—
128C22—1941no—
129C22—1941no—
130C22C22⋊#1171590611yescharacteristic
131C4—41no—
132C4—3881no—
133C4—3881no—
134C4—3881no—
135C4—3882no—
136C2—21no—
137C2—41no—
138C2—971no—
139C2—971no—
140C2—3881no—
141C2—1941no—
142C2#2026182111yescharacteristic
subgroupG/H
centerC2#20261821
commutatorC23×C97C2×C8
FrattiniC23C2×#11715906
FittingC22⋊C4×C97C8
socleC2×C97#324
prime radicalC23×D97C2×C4
cosocleC2×C22⋊#87604C22
hypercenterC22⋊C4#11715906
hypocenterC97#1326
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
22. C22⋊Dic97C8semidirect
24. C22⋊C4×C97C8semidirect
72. C97#1326semidirect
96. C22⋊C4#11715906semidirect

Maximal subgroups

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

subgroupindex
#202618202
#202616822
#132697