Finite Group Databasebeta

← #34099219

Subgroups — #34099219

Order 20096.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757647C211yesmaximal, characteristic, complemented
2#23757468C212yesmaximal, complemented
3#23757022C211yesmaximal, characteristic, complemented
4D4⋊#11929434C212yesmaximal, complemented
5#555×C157C211yesmaximal, characteristic, complemented
6C2×#14578941—22no—
7C2×#14578939—22no—
8#19097768C2211yescharacteristic
9C4⋊#11929434C2212yes—
10#19097691—22no—
11C4×#11929434C2211yescharacteristic
12C4⋊Q8×C157C2211yescharacteristic
13#124×C157—22no—
14#121×C157—22no—
15C4×D4×C157C2212yes—
16#117×C157—22no—
17C2×D628—22no—
18C2×D628—21no—
19#14578941—44no—
20#14578939—44no—
21C2×#11929434—21no—
22C2×#11929434—22no—
23C2×Q8×C157D412yes—
24C2×D4×C157—22no—
25C2×D4×C157D412yes—
26C22×C4×C157—22no—
27C4⋊C4×C157—22no—
28C4⋊C4×C157D412yes—
29C22⋊C4×C157—24no—
30C22⋊C4×C157—22no—
31C42×C157—22no—
32C42×C157C2311yescharacteristic
33C22×D157—41no—
34D628—42no—
35D628—42no—
36D628—42no—
37#11929434—22no—
38#11929434—42no—
39C23×C157—22no—
40Q8×C157—42no—
41D4×C157—22no—
42D4×C157—22no—
43D4×C157—82no—
44C2×C4×C157—22no—
45C2×C4×C157—22no—
46C2×C4×C157—42no—
47C2×C4×C157—42no—
48C2×C4×C157C2×D411yescharacteristic
49C2×C4×C157C2×D412yes—
50C22×C157—42no—
51C22×C157—42no—
52C22×C157—42no—
53C22×C157#12011yescharacteristic
54C2×D157—82no—
55C2×D157—81no—
56C4×C157—21no—
57C4×C157—42no—
58C4×C157—42no—
59C4×C157#13512yes—
60C4×C157AGL(1,ℤ/8)12yes—
61C2×C157—82no—
62C2×C157#44912yes—
63C2×C157#45311yescharacteristic
64D157—161no—
65C157#162411yescharacteristic, complemented, Sylow
66#1624—1571nomaximal, Sylow
67#555D15711yescharacteristic, complemented
68#492—1571no—
69#459—1572no—
70#333—1571no—
71D4⋊C8—1572no—
72C2×SD16—3142no—
73C2×D8—3142no—
74C4⋊Q8C2×D15711yescharacteristic
75#127—1571no—
76#124—22no—
77#121—22no—
78C4×D4C2×D15712yes—
79#117—22no—
80C4⋊C8—1572no—
81#102—3142no—
82C4×C8—1571no—
83C2×Q8C22⋊D15712yes—
84C2×D4—22no—
85C2×D4—3142no—
86C2×D4—3141no—
87C2×D4C22⋊D15712yes—
88C22×C4—22no—
89SD16—6284no—
90D8—6284no—
91C2×C8—3141no—
92C2×C8—3142no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D15712yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D15711yescharacteristic
99C23—22no—
100C23—6281no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—6282no—
106D4—6282no—
107D4—6282no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D15711yescharacteristic
113C2×C4C2×C22⋊D15712yes—
114C8—3142no—
115C8—6282no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—12562no—
120C22—12561no—
121C22#1909774011yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1909779112yes—
126C4#1909779612yes—
127C2—82no—
128C2—25121no—
129C2#2375741812yes—
130C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#555×C157C2
socleC22×C157#120
hypercenter#555D157
hypocenterC157#1624
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #23757647C2semidirect
2. #23757468C2semidirect
3. #23757022C2semidirect
4. D4⋊#11929434C2semidirect
5. #555×C157C2semidirect
65. C157#1624semidirect
67. #555D157semidirect

Maximal subgroups

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

subgroupindex
#237576472
#237574682
#237570222
D4⋊#119294342
#555×C1572
#1624157