Finite Group Databasebeta

← #27141029

Subgroups — #27141029

Order 12416.

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

normal subgroups only

This group has 230 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#20261125C212yesmaximal, complemented
2#20261101C211yesmaximal, characteristic, complemented
3#20260477C211yesmaximal, characteristic, complemented
4#20260475C212yesmaximal, complemented
5#562×C97C211yesmaximal, characteristic, complemented
6C2×#13095498—22no—
7C2×#13095496—22no—
8#15690807C2211yescharacteristic
9#15690730—22no—
10#15690721C2212yes—
11#15690721C2211yescharacteristic
12#125×C97C2211yescharacteristic
13#124×C97C2212yes—
14#123×C97—22no—
15C22⋊Q8×C97—22no—
16#121×C97—22no—
17C2×D388—22no—
18C2×D388—21no—
19#13095498—44no—
20#13095496—44no—
21C2×#11715895—22no—
22C2×#11715895—21no—
23C2×Q8×C97D412yes—
24C2×D4×C97—22no—
25C2×D4×C97D412yes—
26C22×C4×C97—22no—
27C4⋊C4×C97—22no—
28C4⋊C4×C97—22no—
29C4⋊C4×C97D412yes—
30C22⋊C4×C97—22no—
31C22⋊C4×C97—22no—
32C22⋊C4×C97—22no—
33C42×C97C2311yescharacteristic
34C23×C97—22no—
35C22×D97—41no—
36Q8×C97—42no—
37D4×C97—42no—
38D4×C97—82no—
39C2×C4×C97—22no—
40C2×C4×C97—22no—
41C2×C4×C97—82no—
42C2×C4×C97C2×D411yescharacteristic
43C2×C4×C97C2×D412yes—
44D388—42no—
45D388—42no—
46D388—42no—
47#11715895—24no—
48#11715895—41no—
49C22×C97—42no—
50C22×C97—42no—
51C22×C97—42no—
52C22×C97#12011yescharacteristic
53C2×D97—82no—
54C2×D97—81no—
55C4×C97—22no—
56C4×C97—21no—
57C4×C97—42no—
58C4×C97—42no—
59C2×C97—82no—
60C2×C97#45312yes—
61C2×C97#45411yescharacteristic
62D97—161no—
63#1656—971nomaximal, Sylow
64C97#165611yescharacteristic, complemented, Sylow
65#562D9711yescharacteristic, complemented
66#496—972no—
67#490—971no—
68#330—971no—
69#329—972no—
70C2×SD16—1942no—
71C2×D8—1942no—
72#127—971no—
73#125C2×D9711yescharacteristic
74#124C2×D9712yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—1942no—
79#97—971no—
80#97—972no—
81C2×Q8C22⋊D9712yes—
82C2×D4—22no—
83C2×D4—1941no—
84C2×D4—1942no—
85C2×D4C22⋊D9712yes—
86C22×C4—22no—
87SD16—3884no—
88D8—3884no—
89C2×C8—1941no—
90C2×C8—1942no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D9712yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D9711yescharacteristic
98C23—22no—
99C23—3881no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—3882no—
104D4—3882no—
105D4—3882no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D9711yescharacteristic
110C2×C4C2×C22⋊D9712yes—
111C8—3881no—
112C8—1944no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—7762no—
117C22—7761no—
118C22#1569077911yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—15521no—
125C2#2026090312yes—
126C2#2026090811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#562×C97C2
socleC22×C97#120
hypercenter#562D97
hypocenterC97#1656
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20261125C2semidirect
2. #20261101C2semidirect
3. #20260477C2semidirect
4. #20260475C2semidirect
5. #562×C97C2semidirect
64. C97#1656semidirect
65. #562D97semidirect

Maximal subgroups

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

subgroupindex
#202611252
#202611012
#202604772
#202604752
#562×C972
#165697