Finite Group Databasebeta

← #34099440

Subgroups — #34099440

Order 20096.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#23758015C211yesmaximal, characteristic
2#23757635C212yesmaximal, complemented
3#23757013C211yesmaximal, characteristic, complemented
4#23757012C212yesmaximal, complemented
5#497×C157C211yesmaximal, characteristic, complemented
6#19097760—21no—
7#19097759C2212yes—
8#19097757—22no—
9#19097741—22no—
10#19097739—21no—
11#19097694—22no—
12#19097691—22no—
13#19097682C2212yescomplemented
14C2×Q16×C157—21no—
15C2×SD16×C157—21no—
16C4⋊Q8×C157C2211yescharacteristic, complemented
17#124×C157C2211yescharacteristic
18#97×C157C2211yescharacteristic, complemented
19C23×D157—21no—
20C2×Dic314D412yes—
21C2×D628—21no—
22C2×D628D412yes—
23C2×C22⋊D157—22no—
24C2×#11929434—22no—
25C22⋊Dic157—21no—
26#14578931—22no—
27#14578931—22no—
28#14578931—22no—
29C4×Dic157—21no—
30C2×Q8×C157—21no—
31C2×Q8×C157D411yescharacteristic
32C2×D4×C157D411yescharacteristic
33Q16×C157—42no—
34SD16×C157—42no—
35C2×C8×C157—21no—
36C4⋊C4×C157—22no—
37C22⋊C4×C157—22no—
38C42×C157C2311yescharacteristic
39C22×D157—22no—
40C22×D157—81no—
41C22×D157—82no—
42Dic314—42no—
43D628—42no—
44D628—81no—
45C22⋊D157—82no—
46C2×Dic157—22no—
47#11929434—42no—
48C23×C157—21no—
49Q8×C157—41no—
50Q8×C157—41no—
51Q8×C157—41no—
52D4×C157—41no—
53C2×C4×C157—21no—
54C2×C4×C157—41no—
55C2×C4×C157C2×D412yes—
56C2×C4×C157C2×D411yescharacteristic
57C8×C157—22no—
58C22×C157—41no—
59C22×C157—42no—
60C22×C157#12011yescharacteristic
61C2×D157—42no—
62C2×D157—42no—
63C2×D157—42no—
64C2×D157—84no—
65C4×C157—21no—
66C4×C157—22no—
67C4×C157—41no—
68C4×C157—81no—
69Dic157—42no—
70C2×C157—81no—
71C2×C157#45311yescharacteristic
72C2×C157#45512yes—
73D157—82no—
74C157#165911yescharacteristic, complemented, Sylow
75#1659—1571nomaximal, Sylow
76H(3,𝔽4)—1571no—
77#497D15711yescharacteristic, complemented
78#489—1572no—
79#329—1571no—
80#329—1572no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D15711yescharacteristic, complemented
84#124—3142no—
85#124—3141no—
86#124—1572no—
87#124C2×D15711yescharacteristic
88#120—3142no—
89#120—3141no—
90#103—3142no—
91#102—3142no—
92#97—1572no—
93#97C2×D15711yescharacteristic, complemented
94C24—3141no—
95C2×Q8—21no—
96C2×Q8—1572no—
97C2×Q8C22⋊D15711yescharacteristic
98C2×D4—3141no—
99C2×D4—3142no—
100C2×D4—1572no—
101C2×D4C22⋊D15711yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3142no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3142no—
109C22⋊C4—3141no—
110C22⋊C4—3142no—
111C22⋊C4—3142no—
112C42—3141no—
113C42C22×D15711yescharacteristic
114C23—21no—
115C23—3142no—
116C23—12561no—
117C23—12562no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—6282no—
122D4—41no—
123D4—6282no—
124D4—12561no—
125D4—12562no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3142no—
129C2×C4D4×D15712yes—
130C2×C4C2×C22⋊D15711yescharacteristic
131C8—22no—
132C8—6282no—
133C22—41no—
134C22—42no—
135C22—6282no—
136C22—6282no—
137C22—6282no—
138C22—12564no—
139C22#1909773911yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—6282no—
145C2—81no—
146C2—12562no—
147C2#2375744011yescharacteristic
148C2#2375744912yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#497×C157C2
socleC22×C157#120
prime radical#23758015C2
hypercenter#497D157
hypocenterC157#1659
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #23757635C2semidirect
3. #23757013C2semidirect
4. #23757012C2semidirect
5. #497×C157C2semidirect
13. #19097682C22semidirect
16. C4⋊Q8×C157C22semidirect
18. #97×C157C22semidirect
74. C157#1659semidirect
77. #497D157semidirect
83. C4⋊Q8C2×D157semidirect
93. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580152
#237576352
#237570132
#237570122
#497×C1572
#1659157