Finite Group Databasebeta

← #34101180

Subgroups — #34101180

Order 20096.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19097799C212yesmaximal, complemented
2#23757854C211yesmaximal, characteristic, complemented
3C2×#19097697C212yesmaximal, complemented
4#23757174C211yesmaximal, characteristic, complemented
5#343×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2212yes—
7C22×D628C2211yescharacteristic
8#19097799—24no—
9#19097799—24no—
10C2×#14578943—22no—
11C2×D1256—22no—
12#19097751—22no—
13#19097746—22no—
14#19097745—22no—
15C2×#14578931—22no—
16C2×C4×Dic157C2211yescharacteristic
17#19097697—24no—
18C2×M16×C157C2212yes—
19C2×C4⋊C4×C157C2211yescharacteristic
20C23×D157—21no—
21#14578954—22no—
22#14578954—22no—
23#14578954—22no—
24#14578954—22no—
25C2×Dic314D412yes—
26C2×D628—22no—
27C2×D628—22no—
28C2×D628—22no—
29C2×D628—21no—
30C2×D628D412yes—
31C2×D628D412yescomplemented
32C2×C22⋊D157—22no—
33C22×Dic157—21no—
34C2×C4×D157—22no—
35#14578943—28no—
36D1256—28no—
37C4⋊Dic157—22no—
38#14578931—24no—
39#14578931—44no—
40C4×Dic157—22no—
41C22×C4×C157D412yes—
42C22×C4×C157C2311yescharacteristic
43M16×C157—22no—
44M16×C157D414yescomplemented
45C2×C8×C157—22no—
46C4⋊C4×C157—22no—
47C22×D157—22no—
48C22×D157—22no—
49C22×D157—42no—
50C22×D157—42no—
51C22×D157—41no—
52C22×D157—41no—
53Dic314—22no—
54Dic314—22no—
55D628—24no—
56D628—22no—
57D628—22no—
58D628—42no—
59D628—82no—
60C22⋊D157—42no—
61C22⋊D157—42no—
62C2×Dic157—22no—
63C2×Dic157—22no—
64C2×Dic157—41no—
65C2×Dic157—41no—
66C4×D157—42no—
67C4×D157—42no—
68C23×C157C2×D411yescharacteristic
69C2×C4×C157—22no—
70C2×C4×C157—42no—
71C2×C4×C157C2×D411yescharacteristic
72C2×C4×C157C2×D412yes—
73C2×C4×C157C2×D412yes—
74C2×C4×C157#4111yescharacteristic
75C8×C157—24no—
76C22×C157—21no—
77C22×C157—21no—
78C22×C157#12011yescharacteristic
79C22×C157#12111yescharacteristic
80C22×C157#12711yescharacteristic
81C2×D157—44no—
82C2×D157—42no—
83C2×D157—42no—
84C2×D157—42no—
85C2×D157—42no—
86C2×D157—82no—
87C2×D157—81no—
88C2×D157—81no—
89C4×C157—42no—
90C4×C157#12012yes—
91C4×C157#12112yes—
92Dic157—42no—
93Dic157—42no—
94C2×C157—21no—
95C2×C157—21no—
96C2×C157#39211yescharacteristic
97C2×C157#45311yescharacteristic
98C2×C157#45411yescharacteristic
99D157—82no—
100D157—82no—
101C157#200611yescharacteristic, complemented, Sylow
102#2006—1571nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1572no—
104#537—1571no—
105C2×#104—1572no—
106#390—1571no—
107#343D15711yescharacteristic, complemented
108C2×#41—1572no—
109C22×D4—1571no—
110AGL(1,ℤ/8)—3144no—
111AGL(1,ℤ/8)—3144no—
112C2×SD16—3142no—
113C2×D8—3142no—
114C2×M16C2×D15712yes—
115C22⋊Q8—3142no—
116#121—3142no—
117#121—3142no—
118C2×C4⋊C4C2×D15711yescharacteristic
119C2×C22⋊C4—3142no—
120C2×C42—1571no—
121#104—3144no—
122C24—3141no—
123#41—3142no—
124#41—3142no—
125#41—3142no—
126#41—3142no—
127C2×Q8—1572no—
128C2×D4—3142no—
129C2×D4—3142no—
130C2×D4—3142no—
131C2×D4—3142no—
132C2×D4—3141no—
133C2×D4—1572no—
134C2×D4—1572no—
135C22×C4—3141no—
136C22×C4—3142no—
137C22×C4D62812yes—
138C22×C4C22×D15711yescharacteristic
139SD16—3148no—
140D8—3148no—
141M16—22no—
142M16D62814yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—3142no—
146C22⋊C4—3144no—
147C22⋊C4—6284no—
148C42—3142no—
149C23—3142no—
150C23—3142no—
151C23—6282no—
152C23—6282no—
153C23—6281no—
154C23—6281no—
155C23C2×D62811yescharacteristic
156Q8—3142no—
157Q8—3142no—
158D4—3142no—
159D4—3144no—
160D4—3142no—
161D4—6282no—
162D4—6282no—
163D4—6282no—
164D4—12562no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—3142no—
168C2×C4—3142no—
169C2×C4—6281no—
170C2×C4—6281no—
171C2×C4—6282no—
172C2×C4—6282no—
173C2×C4D4×D15711yescharacteristic
174C2×C4D4×D15712yes—
175C2×C4C2×D62812yes—
176C2×C4#1457895511yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—6284no—
181C22—6282no—
182C22—6282no—
183C22—6282no—
184C22—6282no—
185C22—12562no—
186C22—12561no—
187C22—12561no—
188C22#1909774111yescharacteristic
189C22#1909774611yescharacteristic
190C22#1909776811yescharacteristic
191C4—42no—
192C4—6282no—
193C4—6282no—
194C4#1909774112yes—
195C4#1909774612yes—
196C2—21no—
197C2—21no—
198C2—12562no—
199C2—12562no—
200C2#2375718311yescharacteristic
201C2#2375744311yescharacteristic
202C2#2375744811yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#343×C157C2
socleC22×C157#120
hypercenter#343D157
hypocenterC157#2006
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. #23757854C2semidirect
3. C2×#19097697C2semidirect
4. #23757174C2semidirect
5. #343×C157C2semidirect
31. C2×D628D4semidirect
44. M16×C157D4semidirect
101. C157#2006semidirect
107. #343D157semidirect
142. M16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
#237578542
C2×#190976972
#237571742
#343×C1572
#2006157