Finite Group Databasebeta

← #34100683

Subgroups — #34100683

Order 20096.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757837C211yesmaximal, characteristic
2#23757257C211yesmaximal, characteristic, complemented
3C2×#19097687C212yesmaximal, complemented
4#23757045C212yesmaximal, complemented
5#568×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic
7C2×#14578938C2212yes—
8#19097767—21no—
9#19097760—21no—
10#19097745—22no—
11#19097739C414yes—
12#19097730C2211yescharacteristic
13#19097690—21no—
14#19097688—21no—
15#19097687—24no—
16C2×#41×C157C2211yescharacteristic
17#131×C157—24no—
18C2×M16×C157—22no—
19C2×M16×C157C2212yes—
20C23×D157C2×C412yes—
21C2×D628—22no—
22C2×D628—21no—
23C2×D628—21no—
24C2×D628—22no—
25C2×C22⋊D157—24no—
26#14578938—22no—
27#14578938—22no—
28C2×#11929434—22no—
29C4⋊Dic157—21no—
30C22⋊Dic157C2×C412yes—
31#14578931—24no—
32C4×Dic157—21no—
33#41×C157—22no—
34#41×C157—22no—
35C2×Q8×C157D411yescharacteristic
36C2×D4×C157C2×C412yes—
37C2×D4×C157D411yescharacteristic
38C22×C4×C157D412yes—
39C22×C4×C157C2311yescharacteristic
40M16×C157—24no—
41M16×C157—22no—
42M16×C157D414yescomplemented
43C2×C8×C157—24no—
44C2×C8×C157—22no—
45C22×D157—22no—
46C22×D157—24no—
47C22×D157—82no—
48D628—41no—
49D628—42no—
50D628—41no—
51C22⋊D157—84no—
52C2×Dic157—22no—
53#11929434—42no—
54C23×C157C22⋊C412yes—
55C23×C157C22×C411yescharacteristic
56Q8×C157—22no—
57D4×C157—24no—
58D4×C157—22no—
59C2×C4×C157—22no—
60C2×C4×C157—22no—
61C2×C4×C157C22⋊C412yes—
62C2×C4×C157C2×D412yes—
63C2×C4×C157C2×D411yescharacteristic
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157#4112yes—
66C8×C157—24no—
67C22×C157—22no—
68C22×C157—41no—
69C22×C157—42no—
70C22×C157C2×C22⋊C411yescharacteristic
71C22×C157C4×D412yes—
72C2×D157—44no—
73C2×D157—42no—
74C2×D157—84no—
75C4×C157—22no—
76C4×C157#12011yescharacteristic
77C4×C157#12112yes—
78C4×C157#12311yescharacteristic
79Dic157—42no—
80C2×C157—21no—
81C2×C157—22no—
82C2×C157—42no—
83C2×C157#38611yescharacteristic
84D157—82no—
85C157#191411yescharacteristic, complemented, Sylow
86#1914—1571nomaximal, Sylow
87#568D15711yescharacteristic, complemented
88#535—1571no—
89#413—1571no—
90C2×#100—1572no—
91#344—1572no—
92C2×#41C2×D15711yescharacteristic
93C22×D4—1571no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1572no—
97C2×M16C2×D15712yes—
98#127—3141no—
99#124—3141no—
100#121—3142no—
101#120—1574no—
102#117—1571no—
103#101—3141no—
104#100—3141no—
105#100—3144no—
106C24—1572no—
107#41—22no—
108#41—22no—
109C2×Q8D62811yescharacteristic
110C2×D4—3144no—
111C2×D4—3142no—
112C2×D4—3141no—
113C2×D4—3141no—
114C2×D4—3142no—
115C2×D4C4×D15712yes—
116C2×D4D62811yescharacteristic
117C22×C4C22⋊D15712yes—
118C22×C4C22×D15711yescharacteristic
119M16—24no—
120M16—22no—
121M16—3142no—
122M16—3142no—
123M16C22⋊D15714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—3142no—
127C4⋊C4—3141no—
128C22⋊C4—3144no—
129C22⋊C4—1572no—
130C42—3141no—
131C23—3142no—
132C23—3144no—
133C23—12562no—
134C23#1457893112yes—
135C23C2×C4×D15711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—6281no—
140D4—6282no—
141D4—6281no—
142D4—12564no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3142no—
146C2×C4#1457893112yes—
147C2×C4C2×C22⋊D15712yes—
148C2×C4C2×C22⋊D15711yescharacteristic
149C2×C4C2×D62811yescharacteristic
150C2×C4#1457895412yes—
151C8—24no—
152C8—6282no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—6284no—
157C22—6282no—
158C22—12564no—
159C22C2×#1457893111yescharacteristic
160C22C4×C22⋊D15712yes—
161C4—22no—
162C4—6282no—
163C4#1909774011yescharacteristic
164C4#1909774512yes—
165C4#1909775411yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—12562no—
170C2#2375715211yescharacteristic
subgroupG/H
centerC2#23757152
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radical#23757837C2
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1914
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757257C2semidirect
3. C2×#19097687C2semidirect
4. #23757045C2semidirect
5. #568×C157C2semidirect
42. M16×C157D4semidirect
85. C157#1914semidirect
87. #568D157semidirect
123. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237572572
C2×#190976872
#237570452
#568×C1572
#1914157