Finite Group Databasebeta

← #34101481

Subgroups — #34101481

Order 20096.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#23757904C211yesmaximal, characteristic
2#23757313C212yesmaximal
3#23757042C211yesmaximal, characteristic
4#23757033C212yesmaximal
5#384×C157C211yesmaximal, characteristic
6C2×#14578938C2212yes—
7#19097770—22no—
8#19097762—22no—
9#19097750—22no—
10#19097730C2212yes—
11#19097704—22no—
12#19097702—22no—
13C2×C4⋊C4×C157—22no—
14C2×C4⋊C4×C157C2211yescharacteristic
15C2×C4⋊C4×C157C2211yescharacteristic
16C2×C42×C157C2211yescharacteristic
17C2×Dic314—22no—
18#14578938—22no—
19#14578938Q814yescomplemented
20C2×#11929434—22no—
21C4⋊Dic157—22no—
22C4⋊Dic157D412yes—
23C4⋊Dic157D412yes—
24#14578933—24no—
25C22⋊Dic157—22no—
26C4×Dic157—22no—
27C22×C4×C157—21no—
28C22×C4×C157—21no—
29C22×C4×C157D412yes—
30C22×C4×C157C2311yescharacteristic
31C4⋊C4×C157—22no—
32C4⋊C4×C157—22no—
33C4⋊C4×C157—44no—
34C42×C157—21no—
35C42×C157Q812yescomplemented
36Dic314—82no—
37C2×Dic157—22no—
38C2×Dic157—22no—
39#11929434—24no—
40C23×C157C2×D411yescharacteristic
41C2×C4×C157—22no—
42C2×C4×C157—22no—
43C2×C4×C157—22no—
44C2×C4×C157—22no—
45C2×C4×C157—42no—
46C2×C4×C157—42no—
47C2×C4×C157—41no—
48C2×C4×C157C2×D412yes—
49C2×C4×C157C2×Q812yes—
50C2×C4×C157C2×Q811yescharacteristic
51C2×C4×C157#4111yescharacteristic
52C22×C157—21no—
53C22×C157—21no—
54C22×C157#12011yescharacteristic
55C22×C157C22⋊Q811yescharacteristic
56C22×C157C4⋊Q811yescharacteristic
57C4×C157—24no—
58C4×C157—42no—
59C4×C157—81no—
60C4×C157C22⋊Q812yes—
61C4×C157C4⋊Q812yes—
62Dic157—42no—
63Dic157—42no—
64C2×C157—21no—
65C2×C157—21no—
66C2×C157#39511yescharacteristic
67C2×C157#45311yescharacteristic
68C2×C157#45611yescharacteristic
69C157#206011yescharacteristic, complemented, Sylow
70#2060—1571nomaximal, Sylow
71#544—1571no—
72#428—1572no—
73#384D15711yescharacteristic
74#343—1571no—
75#339—1572no—
76C2×M16—1572no—
77C4⋊Q8—3142no—
78#125—3142no—
79C22⋊Q8—3142no—
80#117—1572no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2×D15711yescharacteristic
83C2×C4⋊C4C2×D15711yescharacteristic
84C2×C42C2×D15711yescharacteristic
85#107—3142no—
86#106—3142no—
87C2×Q8—3142no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D15712yes—
91C22×C4C22×D15711yescharacteristic
92M16—3142no—
93M16—1574no—
94C2×C8—3142no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—3142no—
99C4⋊C4—3144no—
100C4⋊C4—1572no—
101C4⋊C4—1572no—
102C22⋊C4—3142no—
103C42—21no—
104C42—3142no—
105C42Dic31412yescomplemented
106C23C2×C22⋊D15711yescharacteristic
107Q8—12562no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—3142no—
116C2×C4—3142no—
117C2×C4D4×D15712yes—
118C2×C4Q8×D15712yes—
119C2×C4C2×Dic31411yescharacteristic
120C2×C4#1457895411yescharacteristic
121C8—3144no—
122C22—21no—
123C22—21no—
124C22#1909773911yescharacteristic
125C22#1909775011yescharacteristic
126C22#1909776911yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—6282no—
131C4—6282no—
132C4#1909774912yes—
133C4#1909777012yes—
134C2—21no—
135C2—21no—
136C2#2375719311yescharacteristic
137C2#2375744011yescharacteristic
138C2#2375745511yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#384×C157C2
socleC22×C157#120
prime radicalC23×C157C2×D4
hypercenter#384D157
hypocenterC157#2060
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
19. #14578938Q8semidirect
35. C42×C157Q8semidirect
69. C157#2060semidirect
105. C42Dic314semidirect

Maximal subgroups

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

subgroupindex
#237579042
#237573132
#237570422
#237570332
#384×C1572
#2060157