Finite Group Databasebeta

← #34101360

Subgroups — #34101360

Order 20096.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#23757854C211yesmaximal, characteristic
2C2×#19097699C212yesmaximal, complemented
3#23757272C212yesmaximal, complemented
4#23757042C211yesmaximal, characteristic, complemented
5#384×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2212yes—
7C2×#14578938C2212yes—
8#19097751—22no—
9#19097746—22no—
10#19097745—22no—
11#19097699—24no—
12#19097694—22no—
13#19097691—22no—
14C2×C4⋊C4×C157—22no—
15C2×C4⋊C4×C157C2211yescharacteristic
16C2×C4⋊C4×C157C2211yescharacteristic
17C2×C42×C157C2211yescharacteristic
18#14578954—22no—
19#14578954—22no—
20#14578954—22no—
21#14578954—22no—
22C2×Dic314D412yes—
23C2×D628—22no—
24C2×D628D412yes—
25C2×C22⋊D157—22no—
26C2×C4×D157—22no—
27#14578938—22no—
28#14578938—22no—
29C2×#11929434—22no—
30C4⋊Dic157—22no—
31#14578931—24no—
32C22×C4×C157—21no—
33C22×C4×C157—21no—
34C22×C4×C157D412yes—
35C22×C4×C157C2311yescharacteristic
36C4⋊C4×C157—22no—
37C4⋊C4×C157—22no—
38C4⋊C4×C157—44no—
39C42×C157—21no—
40C42×C157D412yescomplemented
41C22×D157—22no—
42Dic314—22no—
43Dic314—22no—
44D628—22no—
45D628—22no—
46D628—82no—
47C22⋊D157—42no—
48C22⋊D157—42no—
49C2×Dic157—22no—
50C4×D157—42no—
51C4×D157—42no—
52#11929434—42no—
53C23×C157C2×D411yescharacteristic
54C2×C4×C157—22no—
55C2×C4×C157—22no—
56C2×C4×C157—22no—
57C2×C4×C157—22no—
58C2×C4×C157—42no—
59C2×C4×C157—42no—
60C2×C4×C157—41no—
61C2×C4×C157C2×D412yes—
62C2×C4×C157C2×D411yescharacteristic
63C2×C4×C157#4111yescharacteristic
64C2×C4×C157#4112yes—
65C22×C157—21no—
66C22×C157—21no—
67C22×C157#12011yescharacteristic
68C22×C157#12111yescharacteristic
69C22×C157#12411yescharacteristic
70C2×D157—42no—
71C2×D157—42no—
72C2×D157—42no—
73C4×C157—24no—
74C4×C157—42no—
75C4×C157—81no—
76C4×C157#12112yes—
77C4×C157#12312yes—
78Dic157—42no—
79C2×C157—21no—
80C2×C157—21no—
81C2×C157#39411yescharacteristic
82C2×C157#45311yescharacteristic
83C2×C157#45611yescharacteristic
84D157—82no—
85C157#203911yescharacteristic, complemented, Sylow
86#2039—1571nomaximal, Sylow
87#537—1571no—
88C2×#104—1572no—
89#417—1572no—
90#384D15711yescharacteristic, complemented
91#343—1571no—
92C2×#41—1572no—
93C2×M16—1572no—
94C22⋊Q8—3142no—
95#121—3142no—
96#121—3142no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D15711yescharacteristic
99C2×C4⋊C4C2×D15711yescharacteristic
100C2×C42C2×D15711yescharacteristic
101#104—3144no—
102#103—3142no—
103#102—3142no—
104#41—3142no—
105#41—3142no—
106#41—3142no—
107#41—3142no—
108C2×Q8—1572no—
109C2×D4—3142no—
110C2×D4—3142no—
111C2×D4—1572no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3142no—
115C22×C4C22⋊D15712yes—
116C22×C4C22×D15711yescharacteristic
117M16—3142no—
118M16—3142no—
119C2×C8—3142no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—3142no—
124C22⋊C4—3144no—
125C42—21no—
126C42D62812yescomplemented
127C23—3142no—
128C23C2×C22⋊D15711yescharacteristic
129Q8—3142no—
130Q8—3142no—
131D4—3142no—
132D4—3142no—
133D4—6282no—
134D4—6282no—
135D4—12562no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—3142no—
144C2×C4—6282no—
145C2×C4—6282no—
146C2×C4D4×D15712yes—
147C2×C4C2×D62811yescharacteristic
148C2×C4#1457895411yescharacteristic
149C2×C4#1457895512yes—
150C8—6282no—
151C22—21no—
152C22—21no—
153C22—6282no—
154C22—6282no—
155C22—6282no—
156C22#1909773911yescharacteristic
157C22#1909774511yescharacteristic
158C22#1909776011yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—6282no—
163C4#1909774612yes—
164C4#1909775612yes—
165C2—21no—
166C2—21no—
167C2—12562no—
168C2#2375719011yescharacteristic
169C2#2375744011yescharacteristic
170C2#2375745511yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#384×C157C2
socleC22×C157#120
prime radical#23757854C2
hypercenter#384D157
hypocenterC157#2039
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
2. C2×#19097699C2semidirect
3. #23757272C2semidirect
4. #23757042C2semidirect
5. #384×C157C2semidirect
40. C42×C157D4semidirect
85. C157#2039semidirect
90. #384D157semidirect
126. C42D628semidirect

Maximal subgroups

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

subgroupindex
#237578542
C2×#190976992
#237572722
#237570422
#384×C1572
#2039157