Finite Group Databasebeta

← #27158732

Subgroups — #27158732

Order 12416.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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#20261815C215yesmaximal, complemented
2#20261814C2110yesmaximal, complemented
3C2×#15690898C215yesmaximal, complemented
4D97⋊#131C2110yesmaximal, complemented
5#20261572C211yesmaximal, characteristic
6#15690898C22110yescomplemented
7#15690898C22110yescomplemented
8#15690898C22120yescomplemented
9#15690897C22110yescomplemented
10#15690897C22130yescomplemented
11C2×#13095521C22115yescomplemented
12D97⋊M16C22130yescomplemented
13C2×#13095519C22115yescomplemented
14#15690858C415yes—
15#15690856C4110yes—
16C2×#13095512C2215yes—
17#41×D97C22110yes—
182−1+4×C97C411yescharacteristic
19#13095521C23110yes—
20#13095521C23130yes—
21#13095520C23120yes—
22#13095519C23110yes—
23#13095519C23130yes—
24C2×#11715897C2315yes—
25C2×#11715897C23115yes—
26#13095512C2×C4120yes—
27#13095512C23110yes—
28#13095511C2×C4130yes—
29#13095510C23110yes—
30Q8×D97C2×C4110yes—
31C2×D388C2×C4115yes—
32D4×D97C2×C4130yes—
33C2×C4×D97C23115yes—
34#41×C97C2×C4110yes—
35C2×Q8×C97C2×C415yes—
36C22×D97C22×C4115yes—
37Q8×C97C22×C4110yes—
38D4×C97C22×C4110yes—
39C2×C4×C97C22×C4115yes—
40C22⋊D97C22×C4120yes—
41C2×Dic97C2415yes—
42D388C22×C4130yes—
43C4×D97C22×C4130yes—
44C4×D97C24110yes—
45Dic194C22×C4110yes—
46#11715897C2411yescharacteristic
47#11715897C24110yes—
48#11715897C2415yes—
49C22×C97C23×C415yes—
50C2×D97—415no—
51C2×D97C23×C4110yes—
52C4×C97C23×C4110yes—
53Dic97C23×C415yes—
54Dic97C2511yescharacteristic
55C2×C97—25no—
56C2×C97C24×C411yescharacteristic
57D97—210no—
58#3573—971nomaximal, Sylow
59C97#357311yescharacteristic, complemented, Sylow
60#585—971no—
61#568—9710no—
62#568—975no—
63C2×#131—9710no—
64C2×#131—975no—
652−1+4—975no—
662−1+4#8760411yescharacteristic
67H(5,𝔽2)—9710no—
68C2×#41—9710no—
69C2×#41—975no—
70#131—9710no—
71#131—9710no—
72#131—9730no—
73#131—9710no—
74#131—9720no—
75C2×M16—9730no—
76C2×M16—9715no—
77C22×C8—9715no—
78#41—9720no—
79#41—9710no—
80#41—9710no—
81#41—9730no—
82#41C2×#87604110yes—
83C2×Q8—9710no—
84C2×Q8C2×#8760415yes—
85C2×D4—9730no—
86C2×D4—9715no—
87C22×C4—9715no—
88M16—9720no—
89M16—9710no—
90M16—9730no—
91C2×C8—975no—
92C2×C8—9715no—
93C2×C8—9710no—
94C2×C8—9730no—
95C23—9715no—
96Q8—9710no—
97Q8C22×#87604110yes—
98D4—9720no—
99D4—9730no—
100D4C22×#87604110yes—
101C2×C4—975no—
102C2×C4—9710no—
103C2×C4—9730no—
104C2×C4C22×#87604115yes—
105C8—971no—
106C8—975no—
107C8—9710no—
108C22—9710no—
109C22—38815no—
110C22C23×#8760415yes—
111C4—971no—
112C4—975no—
113C4C23×#87604110yes—
114C2—25no—
115C2—19410no—
116C2C24×#8760411yescharacteristic
subgroupG/H
center, FrattiniC2C24×#87604
commutator, socleC2×C97C24×C4
Fitting2−1+4×C97C4
prime radical#20261572C2
cosocleDic97C25
hypercenter2−1+4#87604
hypocenterC97#3573
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #20261815C2semidirect
2. #20261814C2semidirect
3. C2×#15690898C2semidirect
4. D97⋊#131C2semidirect
6. #15690898C22semidirect
7. #15690898C22semidirect
8. #15690898C22semidirect
9. #15690897C22semidirect
10. #15690897C22semidirect
11. C2×#13095521C22semidirect
12. D97⋊M16C22semidirect
13. C2×#13095519C22semidirect
59. C97#3573semidirect

Maximal subgroups

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

subgroupindex
#202618152
#202618142
C2×#156908982
D97⋊#1312
#202615722
#357397