Finite Group Databasebeta

← #27139738

Subgroups — #27139738

Order 12416.

168 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 35 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#20260652C211yesmaximal, characteristic
2#20260461C211yesmaximal, characteristic, complemented
3#343×C97C211yesmaximal, characteristic, complemented
4C22×D388C411yescharacteristic
5C2×#13095488—21no—
6C2×#13095488—21no—
7C22⋊#11715895—21no—
8#15690717C411yescharacteristic
9C2×M16×C97—21no—
10C2×C4⋊C4×C97C2211yescharacteristic
11C23×D97—21no—
12C2×D388—41no—
13C2×D388—42no—
14C22×Dic97—21no—
15C2×#11715895—41no—
16#13095488—42no—
17#13095488—41no—
18#13095488—41no—
19C22×C4×C97C2×C411yescharacteristic
20C22×C4×C97D411yescharacteristic
21C22×C4×C97D411yescharacteristic
22M16×C97—22no—
23M16×C97—41no—
24C2×C8×C97—41no—
25C4⋊C4×C97—41no—
26C23×C97C22⋊C411yescharacteristic
27C22×D97—22no—
28C22×D97—41no—
29C22×D97—41no—
30C22×D97—41no—
31C22×D97—41no—
32C22×D97—41no—
33C22×D97—41no—
34C2×C4×C97—21no—
35C2×C4×C97—21no—
36C2×C4×C97—21no—
37C2×C4×C97—21no—
38C2×C4×C97—41no—
39C2×C4×C97—41no—
40C2×C4×C97D811yescharacteristic
41C2×C4×C97SD1611yescharacteristic
42C2×Dic97—41no—
43C2×Dic97—41no—
44C2×Dic97—41no—
45D388—42no—
46D388—42no—
47C8×C97—42no—
48#11715895—81no—
49C22×C97—21no—
50C22×C97—21no—
51C22×C97#9911yescharacteristic
52C22×C97#10211yescharacteristic
53C22×C97#10411yescharacteristic
54C2×D97—42no—
55C2×D97—42no—
56C2×D97—42no—
57C2×D97—81no—
58C2×D97—81no—
59C2×D97—81no—
60C2×D97—81no—
61C4×C97—21no—
62C4×C97—21no—
63C4×C97—41no—
64C4×C97—41no—
65Dic97—81no—
66C2×C97—21no—
67C2×C97—21no—
68C2×C97#32711yescharacteristic
69C2×C97AΣL(1,𝔽16)11yescharacteristic
70C2×C97#35511yescharacteristic
71D97—82no—
72#1345—971nomaximal, Sylow
73C97#134511yescharacteristic, complemented, Sylow
74#394—971no—
75#343D9711yescharacteristic, complemented
76#324—971no—
77C22×D4—971no—
78C2×M16—21no—
79C2×C4⋊C4C2×D9711yescharacteristic
80C2×C22⋊C4—1941no—
81C2×C22⋊C4—1941no—
82C22⋊C8—1941no—
83#95—971no—
84C24—1941no—
85C2×D4—3881no—
86C2×D4—3882no—
87C22×C4—1941no—
88C22×C4C4×D9711yescharacteristic
89C22×C4D38811yescharacteristic
90C22×C4C22⋊D9711yescharacteristic
91M16—22no—
92M16—41no—
93C2×C8—41no—
94C2×C8—3881no—
95C4⋊C4—41no—
96C22⋊C4—3882no—
97C22⋊C4—3881no—
98C22⋊C4—3881no—
99C23—3881no—
100C23—3881no—
101C23—3881no—
102C23—3881no—
103C23—3881no—
104C23—3881no—
105C23—1942no—
106C23#1309548811yescharacteristic
107D4—3882no—
108D4—3882no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—3881no—
116C2×C4—3881no—
117C2×C4—3881no—
118C2×C4#1309549611yescharacteristic
119C2×C4#1309549811yescharacteristic
120C8—42no—
121C8—7761no—
122C22—21no—
123C22—21no—
124C22—7761no—
125C22—7761no—
126C22—7761no—
127C22—7761no—
128C22—3882no—
129C22—3882no—
130C22—3882no—
131C22#1569072411yescharacteristic
132C22#1569073011yescharacteristic
133C22#1569073811yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—7761no—
139C2—21no—
140C2—21no—
141C2—7762no—
142C2#2026046811yescharacteristic
143C2#2026052311yescharacteristic
144C2#2026053111yescharacteristic
subgroupG/H
centerC22#15690724
commutatorC22×C4×C97C2×C4
FrattiniC2×C4⋊C4C2×D97
Fitting#343×C97C2
socleC22×C97#99
prime radicalC22×D388C4
cosocleC2×C4⋊C4×C97C22
hypercenter#343D97
hypocenterC97#1345
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. #20260461C2semidirect
3. #343×C97C2semidirect
73. C97#1345semidirect
75. #343D97semidirect

Maximal subgroups

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

subgroupindex
#202606522
#202604612
#343×C972
#134597