Finite Group Databasebeta

← #27140372

Subgroups — #27140372

Order 12416.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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#20261291C211yesmaximal, characteristic
2#20260790C212yesmaximal
3#20260477C211yesmaximal, characteristic, complemented
4#20260476C212yesmaximal, complemented
5#330×C97C211yesmaximal, characteristic, complemented
6#15690809C412yes—
7#15690802C2211yescharacteristic
8#15690788—22no—
9#15690775—22no—
10#15690768C412yes—
11C42×D97C2211yescharacteristic
12#15690739—22no—
13#15690722—22no—
14#15690721C2212yes—
15#125×C97C2211yescharacteristic
16#97×C97C2212yes—
17C2×Dic194—22no—
18C2×C4×D97C2×C411yescharacteristic
19C2×C4×D97D412yes—
20C2×#11715895—22no—
21C4⋊Dic97C2×C412yes—
22#13095490—22no—
23#13095490—22no—
24#13095488—22no—
25C4×Dic97—22no—
26C4×Dic97C2×C411yescharacteristic
27C4×Dic97D412yes—
28C2×C8×C97—22no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97C2×C412yes—
31C42×C97C2311yescharacteristic
32C22×D97C22⋊C411yescharacteristic
33C2×C4×C97—22no—
34C2×C4×C97C22×C411yescharacteristic
35C2×C4×C97C2×D412yes—
36C2×Dic97—22no—
37C2×Dic97C22⋊C412yes—
38C2×Dic97C22⋊C411yescharacteristic
39C4×D97—22no—
40C4×D97—42no—
41Dic194—44no—
42C8×C97—42no—
43#11715895—42no—
44C22×C97C2×C22⋊C411yescharacteristic
45C2×D97—21no—
46C2×D97—22no—
47C4×C97—21no—
48C4×C97—22no—
49C4×C97—42no—
50Dic97—22no—
51Dic97—42no—
52Dic97#10112yes—
53C2×C97C2×#10111yescharacteristic
54C2×C97#42112yes—
55D97—41no—
56#1537—971nomaximal, Sylow
57C97#153711yescharacteristic, complemented, Sylow
58#533—971no—
59#432—972no—
60#330—971no—
61#330—972no—
62#330D9711yescharacteristic, complemented
63C4⋊Q8—972no—
64#125—971no—
65#125C2×D9711yescharacteristic
66C22⋊Q8—1942no—
67C4×Q8—1942no—
68#117—972no—
69C2×C42—971no—
70C4⋊C8—1942no—
71C22⋊C8—1942no—
72#97—972no—
73#97C2×D9712yes—
74C2×Q8—1942no—
75C22×C4—972no—
76C22×C4—971no—
77C2×C8—22no—
78C2×C8—1942no—
79C4⋊C4—22no—
80C4⋊C4—972no—
81C4⋊C4—1942no—
82C4⋊C4—1942no—
83C4⋊C4C4×D9712yes—
84C22⋊C4—1942no—
85C42—971no—
86C42—972no—
87C42—1942no—
88C42C22×D9711yescharacteristic
89C23—971no—
90Q8—3884no—
91C2×C4—22no—
92C2×C4—971no—
93C2×C4—972no—
94C2×C4—3882no—
95C2×C4—1942no—
96C2×C4—1942no—
97C2×C4C2×C4×D9711yescharacteristic
98C2×C4D4×D9712yes—
99C8—42no—
100C8—3882no—
101C22—1941no—
102C22—1942no—
103C22C22⋊C4×D9711yescharacteristic
104C4—21no—
105C4—22no—
106C4—42no—
107C4—972no—
108C4—3882no—
109C4—1942no—
110C2—3881no—
111C2#101×D9711yescharacteristic
112C2#2026075012yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#330×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#330D97
hypocenterC97#1537
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #20260477C2semidirect
4. #20260476C2semidirect
5. #330×C97C2semidirect
57. C97#1537semidirect
62. #330D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
#202607902
#202604772
#202604762
#330×C972
#153797