Finite Group Databasebeta

← #27139766

Subgroups — #27139766

Order 12416.

140 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 37 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#20260504C211yesmaximal, characteristic
2#20260461C211yesmaximal, characteristic
3#395×C97C211yesmaximal, characteristic
4C2×#13095495—21no—
5C22⋊#11715895—21no—
6C22×Q8×C97C411yescharacteristic
7C2×C4⋊C4×C97—21no—
8C2×C4⋊C4×C97C2211yescharacteristic
9#95×C97—21no—
10#95×C97C411yescharacteristic
11#13095495—22no—
12#13095495—41no—
13C2×#11715895—41no—
14C2×#11715895—41no—
15C2×Q8×C97—42no—
16C2×Q8×C97—41no—
17C22×C4×C97—21no—
18C22×C4×C97—21no—
19C22×C4×C97C2×C411yescharacteristic
20C22×C4×C97D411yescharacteristic
21C22×C4×C97D411yescharacteristic
22C4⋊C4×C97—41no—
23C4⋊C4×C97—42no—
24C23×C97C22⋊C411yescharacteristic
25Q8×C97—42no—
26Q8×C97—42no—
27C2×C4×C97—21no—
28C2×C4×C97—21no—
29C2×C4×C97—22no—
30C2×C4×C97—22no—
31C2×C4×C97—21no—
32C2×C4×C97—21no—
33C2×C4×C97—41no—
34C2×C4×C97—41no—
35C2×C4×C97—41no—
36C2×C4×C97—41no—
37C2×C4×C97—41no—
38C2×C4×C97—41no—
39C2×C4×C97SD1611yescharacteristic
40C2×C4×C97Q1611yescharacteristic
41#11715895—42no—
42#11715895—81no—
43C22×C97—21no—
44C22×C97—21no—
45C22×C97#9911yescharacteristic
46C22×C97#10311yescharacteristic
47C22×C97#10411yescharacteristic
48C4×C97—21no—
49C4×C97—21no—
50C4×C97—42no—
51C4×C97—41no—
52C4×C97—42no—
53C4×C97—41no—
54C2×C97—21no—
55C2×C97—21no—
56C2×C97#32811yescharacteristic
57C2×C97AΣL(1,𝔽16)11yescharacteristic
58C2×C97#35611yescharacteristic
59#1354—971nomaximal, Sylow
60C97#135411yescharacteristic, complemented, Sylow
61#395D9711yescharacteristic
62#343—971no—
63#324—971no—
64C22×Q8Dic9711yescharacteristic
65C2×M16—1941no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×D9711yescharacteristic
68C22⋊C8—1941no—
69#95—21no—
70#95Dic9711yescharacteristic
71C2×Q8—42no—
72C2×Q8—41no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C2×Dic9711yescharacteristic
76C22×C4C22⋊D9711yescharacteristic
77C22×C4C22⋊D9711yescharacteristic
78M16—3881no—
79M16—1942no—
80C2×C8—3881no—
81C2×C8—3881no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C23C22⋊Dic9711yescharacteristic
85Q8—42no—
86Q8—42no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4#1309549811yescharacteristic
100C2×C4#1309550111yescharacteristic
101C8—7761no—
102C8—3882no—
103C22—21no—
104C22—21no—
105C22#1569072511yescharacteristic
106C22#1569073411yescharacteristic
107C22#1569073711yescharacteristic
108C4—21no—
109C4—21no—
110C4—42no—
111C4—41no—
112C4—42no—
113C4—41no—
114C2—21no—
115C2—21no—
116C2#2026047211yescharacteristic
117C2#2026052211yescharacteristic
118C2#2026053411yescharacteristic
subgroupG/H
centerC22#15690725
commutatorC22×C4×C97C2×C4
FrattiniC2×C4⋊C4C2×D97
Fitting#395×C97C2
socleC22×C97#99
prime radicalC23×C97C22⋊C4
cosocleC2×C4⋊C4×C97C22
hypercenter#395D97
hypocenterC97#1354
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
60. C97#1354semidirect

Maximal subgroups

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

subgroupindex
#202605042
#202604612
#395×C972
#135497