Finite Group Databasebeta

← #34102209

Subgroups — #34102209

Order 20096.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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#23757709C212yesmaximal, complemented
2#23757391C211yesmaximal, characteristic, complemented
3#23757094C211yesmaximal, characteristic, complemented
4#23757086C212yesmaximal, complemented
5#493×C157C211yesmaximal, characteristic, complemented
6#19097793C2211yescharacteristic
7#19097777—21no—
8#19097715—22no—
9#19097712—22no—
10#19097711C2212yes—
11#19097707C2211yescharacteristic
12#19097699—21no—
13C2×D8×C157—21no—
14C2×D8×C157C2212yes—
15#127×C157—21no—
16C4×C8×C157C2211yescharacteristic
17#14578954—21no—
18Dic628D411yescharacteristic, complemented
19#14578943—21no—
20D1256D411yescharacteristic, complemented
21#14578938—21no—
22#14578937—41no—
23C8×D157—41no—
24#14578929—22no—
25C2×D4×C157—22no—
26C2×D4×C157—42no—
27D8×C157—22no—
28D8×C157—22no—
29D8×C157—41no—
30D8×C157—41no—
31C2×C8×C157D411yescharacteristic
32C2×C8×C157C2311yescharacteristic
33C42×C157D411yescharacteristic
34Dic314—21no—
35D628—21no—
36C22⋊D157—41no—
37C4×D157—41no—
38#11929434—41no—
39C23×C157—42no—
40D4×C157—42no—
41D4×C157—42no—
42D4×C157—84no—
43C2×C4×C157—21no—
44C2×C4×C157C2×D411yescharacteristic
45C8×C157D811yescharacteristic
46C8×C157D811yescharacteristic
47C8×C157C2×D411yescharacteristic
48C8×C157#4111yescharacteristic
49C22×C157—84no—
50C22×C157—82no—
51C22×C157AGL(1,ℤ/8)11yescharacteristic
52C2×D157—41no—
53C4×C157—22no—
54C4×C157#12111yescharacteristic
55C4×C157C2×D811yescharacteristic
56Dic157—41no—
57C2×C157—21no—
58C2×C157—162no—
59C2×C157#45911yescharacteristic
60D157—81no—
61C157#221311yescharacteristic, complemented, Sylow
62#2213—1571nomaximal, Sylow
63#509—1572no—
64#493D15711yescharacteristic, complemented
65#443—1571no—
66#364—1571no—
67#361—1572no—
68#135—1571no—
69C2×D8—21no—
70C2×D8C2×D15712yes—
71#131—3141no—
72#127—21no—
73SD32—3142no—
74D16—3142no—
75M32—1572no—
76#108—1571no—
77#104—3141no—
78C4×C8C2×D15711yescharacteristic
79#41—3141no—
80C2×D4—22no—
81C2×D4—42no—
82Q16—1571no—
83SD16—3141no—
84D8—22no—
85D8—22no—
86D8—41no—
87D8—41no—
88D8—1571no—
89M16—3141no—
90M16—6281no—
91C2×C8—6281no—
92C2×C8C22⋊D15711yescharacteristic
93C2×C8C22×D15711yescharacteristic
94C42C22⋊D15711yescharacteristic
95C16—3142no—
96C23—42no—
97Q8—3141no—
98D4—42no—
99D4—42no—
100D4—84no—
101D4—3141no—
102D4—6281no—
103C2×C4—21no—
104C2×C4—6281no—
105C2×C4C2×C22⋊D15711yescharacteristic
106C8—6281no—
107C8#1457893911yescharacteristic
108C8#1457893911yescharacteristic
109C8D4×D15711yescharacteristic
110C8#1457895311yescharacteristic
111C22—84no—
112C22—82no—
113C22—6281no—
114C22#1909779711yescharacteristic
115C4—22no—
116C4—6281no—
117C4#1909774211yescharacteristic
118C4C2×#1457893911yescharacteristic
119C2—21no—
120C2—162no—
121C2—12561no—
122C2#2375746511yescharacteristic
subgroupG/H
centerC2#23757465
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#493×C157C2
socleC2×C157#459
hypercenter#493D157
hypocenterC157#2213
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #23757709C2semidirect
2. #23757391C2semidirect
3. #23757094C2semidirect
4. #23757086C2semidirect
5. #493×C157C2semidirect
18. Dic628D4semidirect
20. D1256D4semidirect
61. C157#2213semidirect
64. #493D157semidirect

Maximal subgroups

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

subgroupindex
#237577092
#237573912
#237570942
#237570862
#493×C1572
#2213157