Finite Group Databasebeta

← #42122790

Subgroups — #42122790

Order 27712.

158 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 48 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#28357299C212yesmaximal, complemented
2#28357279C212yesmaximal, complemented
3#28357278C212yesmaximal, complemented
4C2×#20294097C211yesmaximal, characteristic
5C4⋊#13181884—22no—
6C22⋊#13181884C2214yes—
7C4×#13181884—22no—
8#20294097—22no—
9#20294097—22no—
10C2×Dic866C411yescharacteristic, complemented
11C2×D1732C2211yescharacteristic
12C2×C22⋊D433C411yescharacteristic, complemented
13C2×C22⋊D433C2211yescharacteristic
14C2×C4×D433C411yescharacteristic, complemented
15C2×C4×D433C2211yescharacteristic
16C22×C4×C433C411yescharacteristic, complemented
17C2×#13181884—24no—
18C22×D433C2×C411yescharacteristic
19C22×D433C2311yescharacteristic
20Dic866—22no—
21D1732—22no—
22C22⋊D433—22no—
23C22⋊D433—22no—
24C2×Dic433C2×C411yescharacteristic
25C2×Dic433C2×C411yescharacteristic
26C4×D433—21no—
27C4×D433—21no—
28C4×D433D412yes—
29C4×D433D412yes—
30C23×C433C2×C411yescharacteristic
31C2×C4×C433—21no—
32C2×C4×C433—21no—
33C2×C4×C433C2×C411yescharacteristic
34C2×C4×C433C2×C411yescharacteristic
35C22×C433—21no—
36C22×C433—41no—
37C22×C433C22×C411yescharacteristic
38C2×D433—21no—
39C2×D433—41no—
40C2×D433—41no—
41C2×D433C2×D412yes—
42#13181884—44no—
43C4×C433—21no—
44C4×C433C22⋊C411yescharacteristic
45C4×C433C22⋊C411yescharacteristic
46Dic433—21no—
47Dic433C22⋊C412yes—
48C2×C433—21no—
49C2×C433—41no—
50C2×C433C2×C22⋊C411yescharacteristic
51D433—22no—
52D433—41no—
53C433#41011yescharacteristic, complemented, Sylow
54#410—4331nomaximal, Sylow
55C2×#41—4331no—
56#117—4332no—
57#99—4332no—
58#99—4332no—
59#41—8662no—
60#41—8662no—
61C2×Q8—4331no—
62C2×D4—4331no—
63C2×D4—4331no—
64C2×D4—4331no—
65C22×C4—4331no—
66C22×C4—4331no—
67C22×C4#1318188411yescharacteristic, complemented
68C4⋊C4—8662no—
69C22⋊C4—4334no—
70C42—8662no—
71C23—4331no—
72C23—4331no—
73C23C2×#1318188411yescharacteristic
74Q8—8662no—
75D4—8662no—
76D4—8662no—
77D4—8662no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—8661no—
81C2×C4—8661no—
82C2×C4—8664no—
83C2×C4—4331no—
84C2×C4—4331no—
85C2×C4—4332no—
86C2×C4—4332no—
87C2×C4C2×#1318188411yescharacteristic
88C2×C4C2×#1318188411yescharacteristic
89C22—21no—
90C22—41no—
91C22—8661no—
92C22—4332no—
93C22—17321no—
94C22—17321no—
95C22C22×#1318188411yescharacteristic
96C4—21no—
97C4—8661no—
98C4—4332no—
99C4—17324no—
100C4C22⋊#1318188411yescharacteristic
101C4C22⋊#1318188411yescharacteristic
102C2—21no—
103C2—41no—
104C2—8662no—
105C2—17321no—
106C2C2×C22⋊#1318188411yescharacteristic
subgroupG/H
centerC4C22⋊#13181884
commutatorC22×C433C22×C4
FrattiniC22C22×#13181884
FittingC22×C4×C433C4
socleC2×C433C2×C22⋊C4
prime radicalC2×#20294097C2
cosocleC22×D433C23
hypercenterC22×C4#13181884
hypocenterC433#410
LCSG ▷ C22×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #28357299C2semidirect
2. #28357279C2semidirect
3. #28357278C2semidirect
10. C2×Dic866C4semidirect
12. C2×C22⋊D433C4semidirect
14. C2×C4×D433C4semidirect
16. C22×C4×C433C4semidirect
53. C433#410semidirect
67. C22×C4#13181884semidirect

Maximal subgroups

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

subgroupindex
#283572992
#283572792
#283572782
C2×#202940972
#410433