Finite Group Databasebeta

← #27156718

Subgroups — #27156718

Order 12416.

168 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 30 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#20261615C211yesmaximal, characteristic, complemented
2#20261581C211yesmaximal, characteristic, complemented
3#20260649C211yesmaximal, characteristic
4C2×#13095520—21no—
5C22⋊#11715897—21no—
6C2×#13095490C2211yescharacteristic
7C2×C22⋊Dic97—21no—
8C2×C22⋊Dic97—21no—
9#15690717C411yescharacteristic
10C22×D4×C97C411yescharacteristic
11#13095520—22no—
12#13095520—41no—
13C2×#11715897—41no—
14C2×#11715897—41no—
15C22×Dic97—21no—
16C22×Dic97D411yescharacteristic
17C22×Dic97D411yescharacteristic
18#13095490—41no—
19C22⋊Dic97—42no—
20C22⋊Dic97—41no—
21C22⋊Dic97—41no—
22C24×C97—21no—
23C2×D4×C97—41no—
24C2×D4×C97—42no—
25C22×C4×C97C2×C411yescharacteristic
26C23×C97—22no—
27C23×C97—41no—
28C23×C97—41no—
29C23×C97—41no—
30C23×C97—41no—
31C23×C97—41no—
32C23×C97—41no—
33C23×C97C22⋊C411yescharacteristic
34D4×C97—42no—
35D4×C97—42no—
36C2×C4×C97—21no—
37C2×C4×C97—41no—
38C2×Dic97—21no—
39C2×Dic97—21no—
40C2×Dic97—21no—
41C2×Dic97—41no—
42C2×Dic97—41no—
43C2×Dic97—41no—
44C2×Dic97—41no—
45C2×Dic97D811yescharacteristic
46C2×Dic97SD1611yescharacteristic
47#11715897—42no—
48#11715897—81no—
49C22×C97—21no—
50C22×C97—21no—
51C22×C97—42no—
52C22×C97—42no—
53C22×C97—42no—
54C22×C97—81no—
55C22×C97—81no—
56C22×C97—81no—
57C22×C97—81no—
58C22×C97#9911yescharacteristic
59C22×C97#10211yescharacteristic
60C22×C97#10411yescharacteristic
61C4×C97—41no—
62Dic97—21no—
63Dic97—21no—
64Dic97—41no—
65Dic97—81no—
66C2×C97—21no—
67C2×C97—21no—
68C2×C97—82no—
69C2×C97#32711yescharacteristic
70C2×C97AΣL(1,𝔽16)11yescharacteristic
71C2×C97#35511yescharacteristic
72#1345—971nomaximal, Sylow
73C97#134511yescharacteristic, complemented, Sylow
74#394—971no—
75#343—971no—
76#324—971no—
77C22×D4#8760411yescharacteristic
78C2×M16—1941no—
79C2×C4⋊C4—971no—
80C2×C22⋊C4—1941no—
81C2×C22⋊C4—1941no—
82C22⋊C8—1941no—
83#95—971no—
84C24—21no—
85C2×D4—41no—
86C2×D4—42no—
87C22×C4—971no—
88C22×C4—971no—
89C22×C4—1941no—
90C22×C4C2×#8760411yescharacteristic
91M16—3881no—
92M16—1942no—
93C2×C8—3881no—
94C2×C8—3881no—
95C4⋊C4—3881no—
96C22⋊C4—3881no—
97C22⋊C4—3881no—
98C22⋊C4—3882no—
99C23—22no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C22⋊#8760411yescharacteristic
107D4—42no—
108D4—42no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—971no—
112C2×C4—971no—
113C2×C4—3881no—
114C2×C4—3881no—
115C2×C4—3881no—
116C2×C4—3881no—
117C2×C4—1941no—
118C2×C4—1941no—
119C2×C4—1941no—
120C8—7761no—
121C8—3882no—
122C22—21no—
123C22—21no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#1569086811yescharacteristic
132C22#1569087211yescharacteristic
133C22#1569087411yescharacteristic
134C4—41no—
135C4—7761no—
136C4—3881no—
137C4—1941no—
138C4—1941no—
139C2—21no—
140C2—21no—
141C2—82no—
142C2#2026158611yescharacteristic
143C2#2026163511yescharacteristic
144C2#2026164311yescharacteristic
subgroupG/H
centerC22#15690868
commutatorC22×C4×C97C2×C4
FrattiniC22×C4C2×#87604
Fitting, prime radicalC22×D4×C97C4
socleC22×C97#99
cosocleC2×#13095490C22
hypercenterC22×D4#87604
hypocenterC97#1345
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261615C2semidirect
2. #20261581C2semidirect
73. C97#1345semidirect

Maximal subgroups

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

subgroupindex
#202616152
#202615812
#202606492
#134597