Finite Group Databasebeta

← #27140395

Subgroups — #27140395

Order 12416.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261283C211yesmaximal, characteristic
2#20260787C212yesmaximal, complemented
3#20260483C212yesmaximal, complemented
4#20260479C211yesmaximal, characteristic, complemented
5#331×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C412yes—
7#15690810C2211yescharacteristic
8#15690797C412yes—
9#15690781—22no—
10D4×Dic97—22no—
11C42×D97C2211yescharacteristic
12C4⋊#11715895C2212yes—
13#15690722—22no—
14C8×Dic97—22no—
15#127×C97C2211yescharacteristic
16C4⋊C8×C97C2212yes—
17#13095510—44no—
18C2×Dic194C2×C412yes—
19C2×C22⋊D97—22no—
20C22×Dic97—22no—
21C2×C4×D97C2×C411yescharacteristic
22C2×C4×D97D412yes—
23C2×#11715895—22no—
24C4⋊Dic97—22no—
25C22⋊Dic97—24no—
26C4×Dic97C2×C411yescharacteristic
27C4×Dic97D412yes—
28C2×D4×C97—22no—
29C2×D4×C97C2×C412yes—
30C2×C8×C97—22no—
31C42×C97C2311yescharacteristic
32C23×C97—22no—
33C22×D97C22⋊C411yescharacteristic
34D4×C97—42no—
35D4×C97—42no—
36D4×C97—42no—
37C2×C4×C97C22×C411yescharacteristic
38C2×C4×C97C2×D412yes—
39C22⋊D97—44no—
40C2×Dic97—22no—
41C2×Dic97—44no—
42C2×Dic97C22⋊C412yes—
43C2×Dic97C22⋊C411yescharacteristic
44C4×D97—22no—
45C4×D97—22no—
46C4×D97—22no—
47Dic194—42no—
48C8×C97—42no—
49#11715895—42no—
50C22×C97—42no—
51C22×C97—44no—
52C22×C97C2×C22⋊C411yescharacteristic
53C2×D97—21no—
54C2×D97—22no—
55C4×C97—21no—
56C4×C97#13512yes—
57C4×C97#13512yes—
58Dic97—22no—
59Dic97—42no—
60Dic97#10012yes—
61C2×C97—82no—
62C2×C97C2×#10011yescharacteristic
63C2×C97#41612yes—
64D97—41no—
65#1543—971nomaximal, Sylow
66C97#154311yescharacteristic, complemented, Sylow
67#532—971no—
68#431—972no—
69#333—972no—
70#331—971no—
71#331D9711yescharacteristic, complemented
72C2×#41—972no—
73C4⋊Q8—971no—
74#127C2×D9711yescharacteristic
75#124—972no—
76#121—1942no—
77C4×D4—1942no—
78C2×C42—971no—
79C4⋊C8—972no—
80C4⋊C8C2×D9712yes—
81C22⋊C8—1942no—
82C4×C8—1942no—
83#41—3884no—
84C2×Q8—972no—
85C2×D4—22no—
86C2×D4—1942no—
87C2×D4C4×D9712yes—
88C22×C4—971no—
89C22×C4—972no—
90C22×C4—1942no—
91C2×C8—22no—
92C2×C8—1942no—
93C4⋊C4—1942no—
94C22⋊C4—1944no—
95C42—971no—
96C42—972no—
97C42C22×D9711yescharacteristic
98C23—22no—
99C23—971no—
100Q8—3882no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—3884no—
105C2×C4—971no—
106C2×C4—972no—
107C2×C4—3884no—
108C2×C4—1942no—
109C2×C4—1942no—
110C2×C4—1942no—
111C2×C4—1942no—
112C2×C4C2×C4×D9711yescharacteristic
113C2×C4D4×D9712yes—
114C8—42no—
115C8—3882no—
116C22—42no—
117C22—44no—
118C22—1941no—
119C22—1942no—
120C22C22⋊C4×D9711yescharacteristic
121C4—21no—
122C4—972no—
123C4—3882no—
124C4—1942no—
125C4#1569082912yes—
126C4#1569083112yes—
127C2—82no—
128C2—3881no—
129C2#100×D9711yescharacteristic
130C2#2026072812yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC42×C97C23
hypercenter#331D97
hypocenterC97#1543
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20260787C2semidirect
3. #20260483C2semidirect
4. #20260479C2semidirect
5. #331×C97C2semidirect
66. C97#1543semidirect
71. #331D97semidirect

Maximal subgroups

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

subgroupindex
#202612832
#202607872
#202604832
#202604792
#331×C972
#154397