Finite Group Databasebeta

← #27139736

Subgroups — #27139736

Order 12416.

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

normal subgroups only

This group has 146 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#20260502C211yesmaximal, characteristic
2#20260459C211yesmaximal, characteristic
3#397×C97C211yesmaximal, characteristic
4C2×C22⋊Dic97—21no—
5C22⋊#11715895—21no—
6C2×C4⋊C4×C97C411yescharacteristic, complemented
7C2×C22⋊C4×C97—21no—
8C2×C22⋊C4×C97C2211yescharacteristic
9#95×C97—21no—
10#95×C97C411yescharacteristic, complemented
11C22×Dic97—41no—
12C2×#11715895—41no—
13C22⋊Dic97—22no—
14C22⋊Dic97—41no—
15C24×C97D411yescharacteristic
16C22×C4×C97—21no—
17C22×C4×C97—21no—
18C22×C4×C97C2×C411yescharacteristic
19C22×C4×C97D411yescharacteristic
20C4⋊C4×C97—41no—
21C22⋊C4×C97—41no—
22C22⋊C4×C97—41no—
23C22⋊C4×C97—41no—
24C23×C97—21no—
25C23×C97—21no—
26C23×C97—41no—
27C23×C97—41no—
28C23×C97C22⋊C411yescharacteristic
29C23×C97D811yescharacteristic
30C23×C97SD1611yescharacteristic
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×C97—41no—
40C2×C4×C97—41no—
41C2×Dic97—42no—
42C2×Dic97—81no—
43C2×Dic97—81no—
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#10211yescharacteristic
56C22×C97#10411yescharacteristic
57C4×C97—41no—
58C4×C97—41no—
59C4×C97—81no—
60C4×C97—81no—
61Dic97—82no—
62C2×C97—21no—
63C2×C97—21no—
64C2×C97—41no—
65C2×C97—41no—
66C2×C97#32711yescharacteristic
67C2×C97#35211yescharacteristic
68C2×C97#35411yescharacteristic
69#1344—971nomaximal, Sylow
70C97#134411yescharacteristic, complemented, Sylow
71#397D9711yescharacteristic
72#342—971no—
73#323—971no—
74C2×C4⋊C4Dic9711yescharacteristic, complemented
75C2×C22⋊C4—21no—
76C2×C22⋊C4—1941no—
77C2×C22⋊C4C2×D9711yescharacteristic
78C22⋊C8—1941no—
79#95—21no—
80#95Dic9711yescharacteristic, complemented
81C24C22⋊D9711yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—3881no—
85C22×C4C2×Dic9711yescharacteristic
86C22×C4C22⋊D9711yescharacteristic
87C2×C8—3881no—
88C4⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—3881no—
93C22⋊C4—1942no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C22⋊Dic9711yescharacteristic
99C23#1309549611yescharacteristic
100C23#1309549911yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—7761no—
112C2×C4—7761no—
113C2×C4—3882no—
114C8—7761no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#1569072511yescharacteristic
125C22#1569073111yescharacteristic
126C22#1569073711yescharacteristic
127C4—41no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—7762no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2026046911yescharacteristic
137C2#2026052511yescharacteristic
138C2#2026053011yescharacteristic
subgroupG/H
centerC22#15690725
commutatorC22×C4×C97C2×C4
FrattiniC2×C22⋊C4C2×D97
Fitting#397×C97C2
socleC22×C97#99
prime radicalC24×C97D4
cosocleC2×C22⋊C4×C97C22
hypercenter#397D97
hypocenterC97#1344
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
6. C2×C4⋊C4×C97C4semidirect
10. #95×C97C4semidirect
70. C97#1344semidirect
74. C2×C4⋊C4Dic97semidirect
80. #95Dic97semidirect

Maximal subgroups

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

subgroupindex
#202605022
#202604592
#397×C972
#134497