Finite Group Databasebeta

← #27139759

Subgroups — #27139759

Order 12416.

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

normal subgroups only

This group has 154 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#20260671C211yesmaximal, characteristic
2#20260459C211yesmaximal, characteristic
3#342×C97C211yesmaximal, characteristic
4C2×C4⋊Dic97C411yescharacteristic, complemented
5C2×#13095490—21no—
6C2×C22⋊Dic97—21no—
7C22⋊#11715895—21no—
8#15690717C411yescharacteristic, complemented
9C2×C22⋊C4×C97—21no—
10C2×C22⋊C4×C97C2211yescharacteristic
11C22×Dic97—21no—
12C22×Dic97—21no—
13C2×#11715895—41no—
14C4⋊Dic97—41no—
15#13095490—42no—
16C22⋊Dic97—42no—
17C24×C97D411yescharacteristic
18C22×C4×C97—41no—
19C22×C4×C97C2×C411yescharacteristic
20C22×C4×C97D411yescharacteristic
21C22⋊C4×C97—22no—
22C22⋊C4×C97—41no—
23C22⋊C4×C97—41no—
24C23×C97—21no—
25C23×C97—21no—
26C23×C97—41no—
27C23×C97—41no—
28C23×C97C22⋊C411yescharacteristic
29C23×C97SD1611yescharacteristic
30C23×C97Q1611yescharacteristic
31C2×C4×C97—21no—
32C2×C4×C97—21no—
33C2×C4×C97—41no—
34C2×C4×C97—41no—
35C2×C4×C97—42no—
36C2×C4×C97—81no—
37C2×C4×C97—81no—
38C2×Dic97—22no—
39C2×Dic97—41no—
40C2×Dic97—41no—
41C2×Dic97—41no—
42C2×Dic97—41no—
43C2×Dic97—41no—
44#11715895—81no—
45C22×C97—21no—
46C22×C97—21no—
47C22×C97—21no—
48C22×C97—21no—
49C22×C97—41no—
50C22×C97—41no—
51C22×C97—41no—
52C22×C97—41no—
53C22×C97—81no—
54C22×C97#9911yescharacteristic
55C22×C97#10311yescharacteristic
56C22×C97#10411yescharacteristic
57C4×C97—41no—
58C4×C97—41no—
59C4×C97—82no—
60Dic97—42no—
61Dic97—81no—
62C2×C97—21no—
63C2×C97—21no—
64C2×C97—41no—
65C2×C97—41no—
66C2×C97#32811yescharacteristic
67C2×C97#35211yescharacteristic
68C2×C97#35311yescharacteristic
69#1352—971nomaximal, Sylow
70C97#135211yescharacteristic, complemented, Sylow
71#399—971no—
72#342D9711yescharacteristic
73#323—971no—
74C2×C4⋊C4—971no—
75C2×C4⋊C4—1941no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—1941no—
78C2×C22⋊C4C2×D9711yescharacteristic
79C22⋊C8—1941no—
80#95—971no—
81C24C22⋊D9711yescharacteristic
82C22×C4—41no—
83C22×C4—1941no—
84C22×C4—1941no—
85C22×C4C4×D9711yescharacteristic
86C22×C4D38811yescharacteristic
87C2×C8—3881no—
88C4⋊C4—3881no—
89C4⋊C4—3882no—
90C22⋊C4—22no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—3882no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23#1309548811yescharacteristic
99C23#1309549911yescharacteristic
100C23#1309550111yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4—3881no—
109C2×C4—3881no—
110C2×C4—3881no—
111C2×C4—3881no—
112C2×C4—3881no—
113C2×C4—1942no—
114C8—7761no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#1569072411yescharacteristic
125C22#1569073311yescharacteristic
126C22#1569073811yescharacteristic
127C4—41no—
128C4—41no—
129C4—82no—
130C4—7761no—
131C4—3882no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2026047111yescharacteristic
137C2#2026052411yescharacteristic
138C2#2026052711yescharacteristic
subgroupG/H
centerC22#15690724
commutatorC22×C4×C97C2×C4
FrattiniC2×C22⋊C4C2×D97
Fitting#342×C97C2
socleC22×C97#99
prime radicalC24×C97D4
cosocleC2×C22⋊C4×C97C22
hypercenter#342D97
hypocenterC97#1352
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic97C4semidirect
8. #15690717C4semidirect
70. C97#1352semidirect

Maximal subgroups

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

subgroupindex
#202606712
#202604592
#342×C972
#135297