Finite Group Databasebeta

← #27157649

Subgroups — #27157649

Order 12416.

272 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#15690870C212yesmaximal
2#20261616C214yesmaximal, complemented
3#20261208C211yesmaximal, characteristic
4C2×#13095520C2214yes—
5#15690870—22no—
6#15690870—22no—
7C22×C22⋊D97C2211yescharacteristic
8C4×C22⋊D97—22no—
9#15690770—22no—
10#15690766C2212yes—
11C2×C22⋊Dic97C411yescharacteristic
12C22⋊C4×D97C412yes—
13C2×C22⋊C4×C97C411yescharacteristic
14#13095520—24no—
15#13095520—24no—
16C2×#11715897—24no—
17C23×D97C2×C411yescharacteristic
18C2×C22⋊D97—24no—
19C2×C22⋊D97D412yes—
20C2×C22⋊D97D412yes—
21C22×Dic97—21no—
22C22×Dic97C2311yescharacteristic
23C2×C4×D97—22no—
24#13095490—22no—
25C22⋊Dic97—21no—
26C22⋊Dic97C2×C412yes—
27#13095488—22no—
28C4×Dic97—22no—
29C24×C97C2×C411yescharacteristic
30C22×C4×C97—21no—
31C22⋊C4×C97—21no—
32C22⋊C4×C97C2×C412yes—
33C23×C97—22no—
34C23×C97—81no—
35C23×C97C22⋊C411yescharacteristic
36C23×C97C22⋊C411yescharacteristic
37C23×C97C22×C411yescharacteristic
38C22×D97—22no—
39C22×D97—81no—
40C22×D97C22⋊C412yes—
41C2×C4×C97—22no—
42C2×C4×C97—81no—
43C22⋊D97—24no—
44C22⋊D97—24no—
45C2×Dic97—22no—
46C2×Dic97—81no—
47C2×Dic97C2×D412yes—
48C2×Dic97#4114yes—
49C4×D97—82no—
50#11715897—44no—
51C22×C97—21no—
52C22×C97—21no—
53C22×C97—41no—
54C22×C97—41no—
55C22×C97—41no—
56C22×C97—82no—
57C22×C97C2×C22⋊C411yescharacteristic
58C22×C97#11712yes—
59C2×D97—22no—
60C2×D97—42no—
61C2×D97—82no—
62C4×C97—42no—
63Dic97—42no—
64Dic97#12314yes—
65C2×C97—21no—
66C2×C97—22no—
67C2×C97—41no—
68C2×C97—41no—
69C2×C97#38111yescharacteristic
70D97—42no—
71#1829—971nomaximal, Sylow
72C97#182911yescharacteristic, complemented, Sylow
73#518—971no—
74C2×#100—972no—
75#344—974no—
76C22×D4—971no—
77C2×M16—974no—
78C4×D4—1942no—
79C4×D4—1942no—
80#117—972no—
81C2×C22⋊C4—971no—
82C2×C22⋊C4—972no—
83C2×C22⋊C4#8760411yescharacteristic
84#100—1942no—
85#100—1942no—
86C24—971no—
87C24C2×#8760411yescharacteristic
88C2×D4—972no—
89C2×D4—972no—
90C2×D4—1944no—
91C22×C4—21no—
92C22×C4—971no—
93C22×C4—1942no—
94C22×C4—1941no—
95M16—1944no—
96M16—1944no—
97C2×C8—1944no—
98C4⋊C4—1942no—
99C22⋊C4—21no—
100C22⋊C4—972no—
101C22⋊C4—1942no—
102C22⋊C4—1941no—
103C22⋊C4C2×#8760412yes—
104C42—1942no—
105C23—22no—
106C23—81no—
107C23—7761no—
108C23—972no—
109C23—1942no—
110C23C22⋊#8760411yescharacteristic
111C23C22⋊#8760411yescharacteristic
112C23C22×#8760411yescharacteristic
113D4—1944no—
114D4—1944no—
115C2×C4—22no—
116C2×C4—81no—
117C2×C4—7762no—
118C2×C4—7761no—
119C2×C4—972no—
120C2×C4—974no—
121C2×C4—1942no—
122C8—3884no—
123C22—21no—
124C22—21no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—82no—
129C22—7762no—
130C22—3882no—
131C22—1942no—
132C22C2×C22⋊#8760411yescharacteristic
133C22#1569088912yes—
134C4—42no—
135C4—974no—
136C4—3882no—
137C2—21no—
138C2—22no—
139C2—41no—
140C2—41no—
141C2—3882no—
142C2#2026167811yescharacteristic
subgroupG/H
centerC2#20261678
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC2×C97#381
prime radicalC22×C22⋊D97C22
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1829
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20261616C2semidirect
72. C97#1829semidirect

Maximal subgroups

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

subgroupindex
C2×#156908702
#202616162
#202612082
#182997