Finite Group Databasebeta

← #27141047

Subgroups — #27141047

Order 12416.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261145C212yesmaximal, complemented
2#20261130C211yesmaximal, characteristic
3#20260477C212yesmaximal, complemented
4#20260475C211yesmaximal, characteristic
5#563×C97C211yesmaximal, characteristic
6C2×#13095501—21no—
7C2×#13095499—21no—
8#15690810C2211yescharacteristic
9#15690743—22no—
10#15690741—22no—
11#15690721C2211yescharacteristic
12#15690721C2212yes—
13C4⋊Q8×C97—21no—
14#126×C97—22no—
15#125×C97C2212yes—
16#124×C97C2211yescharacteristic
17#123×C97—21no—
18C22⋊Q8×C97—22no—
19C2×Dic194—21no—
20#13095501—42no—
21#13095499—42no—
22C2×#11715895—21no—
23C2×#11715895—22no—
24C4⋊Dic97—22no—
25C2×Q8×C97—21no—
26C2×Q8×C97D411yescharacteristic
27C2×D4×C97D411yescharacteristic
28C22×C4×C97—21no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97—22no—
31C4⋊C4×C97—22no—
32C4⋊C4×C97D412yes—
33C4⋊C4×C97D412yes—
34C22⋊C4×C97—22no—
35C22⋊C4×C97—21no—
36C22⋊C4×C97—22no—
37C42×C97—21no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40Q8×C97—41no—
41Q8×C97—81no—
42D4×C97—41no—
43C2×C4×C97—22no—
44C2×C4×C97—21no—
45C2×C4×C97—22no—
46C2×C4×C97—81no—
47C2×C4×C97C2×D412yes—
48C2×C4×C97C2×D411yescharacteristic
49C2×Dic97—41no—
50Dic194—41no—
51Dic194—41no—
52#11715895—24no—
53#11715895—22no—
54C22×C97—41no—
55C22×C97—42no—
56C22×C97#12011yescharacteristic
57C4×C97—21no—
58C4×C97—22no—
59C4×C97—42no—
60C4×C97—41no—
61C4×C97—42no—
62Dic97—81no—
63C2×C97—81no—
64C2×C97#45311yescharacteristic
65C2×C97#45612yes—
66#1660—971nomaximal, Sylow
67C97#166011yescharacteristic, complemented, Sylow
68#563D9711yescharacteristic
69#501—972no—
70#497—971no—
71#330—972no—
72#329—971no—
73C2×Q16—1941no—
74C2×SD16—1941no—
75C4⋊Q8—21no—
76C4⋊Q8—971no—
77#126—22no—
78#125C2×D9712yes—
79#124C2×D9711yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—1942no—
83#106—1942no—
84#97—972no—
85#97—971no—
86C2×Q8—21no—
87C2×Q8—1941no—
88C2×Q8C22⋊D9711yescharacteristic
89C2×D4C22⋊D9711yescharacteristic
90C22×C4—21no—
91Q16—3882no—
92SD16—3882no—
93C2×C8—1942no—
94C2×C8—1941no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—1942no—
99C4⋊C4C22⋊D9712yes—
100C4⋊C4C22⋊D9712yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D9711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—3881no—
110Q8—3881no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—3881no—
117C2×C4C2×C22⋊D9712yes—
118C2×C4C2×C22⋊D9711yescharacteristic
119C8—1942no—
120C8—1944no—
121C22—41no—
122C22—42no—
123C22#1569077911yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—7761no—
130C2—81no—
131C2#2026090311yescharacteristic
132C2#2026091812yes—
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#563×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#563D97
hypocenterC97#1660
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20261145C2semidirect
3. #20260477C2semidirect
67. C97#1660semidirect

Maximal subgroups

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

subgroupindex
#202611452
#202611302
#202604772
#202604752
#563×C972
#166097