Finite Group Databasebeta

← #27157650

Subgroups — #27157650

Order 12416.

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

normal subgroups only

This group has 236 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×#15690871C212yesmaximal
2#20261616C214yesmaximal
3#20261212C211yesmaximal, characteristic
4C2×#13095520C2214yes—
5#15690871—22no—
6#15690871—22no—
7C22×Dic194C2211yescharacteristic
8C4×Dic194—22no—
9#15690775—22no—
10#15690769C411yescharacteristic
11#15690766C412yes—
12#15690766C2212yes—
13#117×C97C411yescharacteristic
14#13095520—24no—
15#13095520—24no—
16C2×#11715897—24no—
17C2×Dic194—24no—
18C2×Dic194D412yes—
19C2×Dic194D412yes—
20C22×Dic97C2×C411yescharacteristic
21C22×Dic97C2311yescharacteristic
22C4⋊Dic97—21no—
23#13095490—22no—
24#13095490—22no—
25C22⋊Dic97C2×C412yes—
26C4×Dic97—21no—
27C4×Dic97—22no—
28C4×Dic97—22no—
29C22×C4×C97C2×C411yescharacteristic
30C4⋊C4×C97—21no—
31C22⋊C4×C97C2×C412yes—
32C42×C97—21no—
33C23×C97C22×C411yescharacteristic
34C2×C4×C97—22no—
35C2×C4×C97—22no—
36C2×C4×C97C22⋊C411yescharacteristic
37C2×C4×C97C22⋊C411yescharacteristic
38C2×Dic97—22no—
39C2×Dic97—22no—
40C2×Dic97C22⋊C412yes—
41C2×Dic97C2×D412yes—
42C2×Dic97#4114yes—
43Dic194—24no—
44Dic194—24no—
45#11715897—44no—
46C22×C97—41no—
47C22×C97C2×C22⋊C411yescharacteristic
48C22×C97#11712yes—
49C4×C97—21no—
50C4×C97—21no—
51C4×C97—42no—
52Dic97—22no—
53Dic97—42no—
54Dic97#12314yes—
55C2×C97—21no—
56C2×C97—22no—
57C2×C97#38111yescharacteristic
58#1830—971nomaximal, Sylow
59C97#183011yescharacteristic, complemented, Sylow
60#519—971no—
61C2×#101—972no—
62#344—974no—
63C22×Q8—971no—
64C2×M16—974no—
65C4×Q8—1942no—
66C4×Q8—1942no—
67#117—971no—
68#117—972no—
69#117—972no—
70#117#8760411yescharacteristic
71#101—1942no—
72#101—1942no—
73C2×Q8—972no—
74C2×Q8—972no—
75C2×Q8—1944no—
76C22×C4—971no—
77C22×C4—971no—
78C22×C4C2×#8760411yescharacteristic
79M16—1944no—
80M16—1944no—
81C2×C8—1944no—
82C4⋊C4—21no—
83C4⋊C4—1942no—
84C4⋊C4—1942no—
85C4⋊C4—1941no—
86C22⋊C4—972no—
87C22⋊C4C2×#8760412yes—
88C42—21no—
89C42—1942no—
90C42—1942no—
91C42—1941no—
92C23C22×#8760411yescharacteristic
93Q8—1944no—
94Q8—1944no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—972no—
98C2×C4—974no—
99C2×C4—972no—
100C2×C4—1942no—
101C2×C4—1942no—
102C2×C4C22⋊#8760411yescharacteristic
103C2×C4C22⋊#8760411yescharacteristic
104C8—3884no—
105C22—41no—
106C22C2×C22⋊#8760411yescharacteristic
107C22#1569088912yes—
108C4—21no—
109C4—21no—
110C4—42no—
111C4—974no—
112C4—3882no—
113C4—1942no—
114C2—21no—
115C2—22no—
116C2#2026167811yescharacteristic
subgroupG/H
centerC2#20261678
commutator, prime radicalC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#117×C97C4
socleC2×C97#381
cosocleC22×Dic97C23
hypercenter#117#87604
hypocenterC97#1830
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ #117

Complements

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

NH ≅ G/Ntype
59. C97#1830semidirect

Maximal subgroups

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

subgroupindex
C2×#156908712
#202616162
#202612122
#183097