Finite Group Databasebeta

← #34101442

Subgroups — #34101442

Order 20096.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#23758073C211yesmaximal, characteristic
2#23757270C212yesmaximal, complemented
3#23757246C212yesmaximal, complemented
4#23757030C211yesmaximal, characteristic, complemented
5#424×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2212yes—
7#19097804—22no—
8#19097799—22no—
9#19097793—22no—
10#19097730C2212yes—
11#19097696—22no—
12#19097693—22no—
13#19097685—24no—
14C2×M16×C157C2211yescharacteristic
15C22×C8×C157C2211yescharacteristic
16#117×C157C2211yescharacteristic
17C4⋊C8×C157—22no—
18#14578954—22no—
19#14578954—22no—
20#14578954—22no—
21#14578954—22no—
22C2×Dic314D412yes—
23C2×D628D412yes—
24C2×C22⋊D157—22no—
25C2×C4×D157—22no—
26Dic628—22no—
27#14578943—24no—
28D1256—22no—
29C4⋊Dic157—22no—
30C22⋊Dic157—22no—
31C4×Dic157—22no—
32C22×C4×C157C2311yescharacteristic
33M16×C157—22no—
34C2×C8×C157—21no—
35C2×C8×C157—22no—
36C2×C8×C157D412yes—
37C4⋊C4×C157—21no—
38C22⋊C4×C157D412yes—
39C42×C157—21no—
40C22×D157—22no—
41Dic314—22no—
42Dic314—22no—
43D628—22no—
44D628—22no—
45C22⋊D157—42no—
46C22⋊D157—42no—
47C2×Dic157—22no—
48C2×Dic157—42no—
49C4×D157—42no—
50C4×D157—42no—
51C23×C157C2×D411yescharacteristic
52C2×C4×C157—22no—
53C2×C4×C157C2×D412yes—
54C2×C4×C157C2×D411yescharacteristic
55C2×C4×C157#4111yescharacteristic
56C2×C4×C157#4112yes—
57C8×C157—22no—
58C8×C157—41no—
59C22×C157—41no—
60C22×C157#12111yescharacteristic
61C22×C157#12112yes—
62C2×D157—42no—
63C2×D157—82no—
64C4×C157—42no—
65C4×C157#12011yescharacteristic
66C4×C157#12312yes—
67C4×C157#12411yescharacteristic
68Dic157—42no—
69Dic157—82no—
70C2×C157—22no—
71C2×C157—21no—
72C2×C157#39411yescharacteristic
73D157—82no—
74C157#205311yescharacteristic, complemented, Sylow
75#2053—1571nomaximal, Sylow
76#575—1571no—
77#424D15711yescharacteristic, complemented
78#416—1572no—
79#410—1572no—
80H(3,ℤ/4)—1571no—
81C2×#41—1572no—
82#137—3142no—
83AGL(1,ℤ/8)—3142no—
84#135—3142no—
85C2×M16C2×D15711yescharacteristic
86C22×C8C2×D15711yescharacteristic
87#117—1572no—
88#117C2×D15711yescharacteristic
89C4⋊C8—22no—
90#103—3142no—
91#102—3142no—
92#99—3144no—
93#41—3142no—
94#41—3142no—
95#41—3142no—
96#41—3142no—
97C2×Q8—1572no—
98C2×D4—3142no—
99C2×D4—1572no—
100C22×C4—3142no—
101C22×C4C22×D15711yescharacteristic
102Q16—3142no—
103SD16—3144no—
104D8—3142no—
105M16—22no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8C22⋊D15712yes—
109C4⋊C4—21no—
110C4⋊C4—3142no—
111C22⋊C4—3142no—
112C22⋊C4D62812yes—
113C42—21no—
114C42—3142no—
115C23—3142no—
116C23C2×D62811yescharacteristic
117Q8—3142no—
118Q8—3142no—
119D4—3142no—
120D4—3142no—
121D4—6282no—
122D4—6282no—
123C2×C4—22no—
124C2×C4—3142no—
125C2×C4—6282no—
126C2×C4—6282no—
127C2×C4—6282no—
128C2×C4D4×D15712yes—
129C2×C4C2×C22⋊D15711yescharacteristic
130C2×C4#1457895411yescharacteristic
131C2×C4#1457895512yes—
132C8—22no—
133C8—41no—
134C22—41no—
135C22—6282no—
136C22—12562no—
137C22#1909774511yescharacteristic
138C22#1909774612yes—
139C4—42no—
140C4—6282no—
141C4—12562no—
142C4#1909773911yescharacteristic
143C4#1909775612yes—
144C4#1909776011yescharacteristic
145C2—21no—
146C2—22no—
147C2—12562no—
148C2#2375719011yescharacteristic
subgroupG/H
centerC4#19097760
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#424×C157C2
socleC2×C157#394
prime radical#23758073C2
hypercenter#424D157
hypocenterC157#2053
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #23757270C2semidirect
3. #23757246C2semidirect
4. #23757030C2semidirect
5. #424×C157C2semidirect
74. C157#2053semidirect
77. #424D157semidirect

Maximal subgroups

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

subgroupindex
#237580732
#237572702
#237572462
#237570302
#424×C1572
#2053157