Finite Group Databasebeta

← #34110388

Subgroups — #34110388

Order 20096.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757885C211yesmaximal, characteristic, complemented
2#23757614C212yesmaximal, complemented
3#23757573C212yesmaximal, complemented
4#23757478C212yesmaximal, complemented
5#23757424C212yesmaximal
6#23757373C212yesmaximal, complemented
7C4×#14578942C212yesmaximal, complemented
8#23757334C211yesmaximal, characteristic, complemented
9#518×C157C211yesmaximal, characteristic
10C2×#14578942C2212yes—
11C2×#14578942C2212yes—
12#19097771C2211yescharacteristic, complemented
13#19097764—22no—
14#19097763C2211yescharacteristic, complemented
15#19097761—22no—
16#19097750C2212yes—
17C4×Dic314C2212yescomplemented
18#19097704C2212yescomplemented
19#19097702C2212yescomplemented
20C4⋊#11929434C2212yescomplemented
21#19097694C2212yescomplemented
22#19097694C2212yescomplemented
23#19097692C2212yes—
24#19097692C2212yes—
25C22⋊#11929434C2212yes—
26#19097682C2211yescharacteristic, complemented
27C4×#11929434C2211yescharacteristic, complemented
28C22×D4×C157C2211yescharacteristic
29C4×D4×C157—22no—
30C4×D4×C157C2212yes—
31C4×D4×C157C2212yes—
32#117×C157C2211yescharacteristic
33#117×C157C2211yescharacteristic
34C2×C22⋊C4×C157—22no—
35C2×Dic314C2312yes—
36#14578942—24no—
37#14578942—24no—
38C2×#11929434C2312yes—
39C2×#11929434C2312yes—
40C4⋊Dic157—22no—
41C4⋊Dic157C2312yes—
42#14578933—22no—
43#14578933—22no—
44#14578933—22no—
45C22⋊Dic157—22no—
46C4×Dic157—22no—
47C24×C157—21no—
48C2×D4×C157—24no—
49C2×D4×C157—21no—
50C2×D4×C157C2312yes—
51C22×C4×C157—22no—
52C22×C4×C157—22no—
53C22×C4×C157C2311yescharacteristic
54C4⋊C4×C157D412yes—
55C4⋊C4×C157C2312yes—
56C22⋊C4×C157—21no—
57C22⋊C4×C157—22no—
58C22⋊C4×C157—22no—
59C22⋊C4×C157D412yes—
60C42×C157—21no—
61C42×C157C2311yescharacteristic
62C42×C157C2311yescharacteristic
63Dic314—22no—
64Dic314—22no—
65C2×Dic157—22no—
66C2×Dic157—22no—
67#11929434—22no—
68#11929434—22no—
69C23×C157—24no—
70C23×C157—22no—
71C23×C157—82no—
72C23×C157C2×D411yescharacteristic
73D4×C157—24no—
74D4×C157—22no—
75D4×C157#4114yes—
76C2×C4×C157—22no—
77C2×C4×C157—22no—
78C2×C4×C157—82no—
79C2×C4×C157—82no—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D412yes—
82C2×C4×C157C2×D412yes—
83C2×C4×C157C2411yescharacteristic
84C22×C157—21no—
85C22×C157—24no—
86C22×C157—22no—
87C22×C157—42no—
88C22×C157—42no—
89C22×C157—84no—
90C22×C157C22×D411yescharacteristic
91C4×C157—21no—
92C4×C157—22no—
93C4×C157—22no—
94C4×C157—42no—
95C4×C157C2×#4112yes—
96Dic157—42no—
97Dic157—42no—
98C2×C157—41no—
99C2×C157—44no—
100C2×C157#52511yescharacteristic
101C2×C157#57712yes—
102C157#312311yescharacteristic, complemented, Sylow
103#3123—1571nomaximal, Sylow
104#541—1571no—
105#518D15711yescharacteristic
106#485—1572no—
107#478—1572no—
108#461—1572no—
109#450—1572no—
110#440—1572no—
111C4×SD16—1572no—
112#433—1571no—
113C22×D4C2×D15711yescharacteristic
114C2×SD16—1572no—
115C2×SD16—1572no—
116C4⋊Q8—1571no—
117#126—3142no—
118#125—3142no—
119#125—1571no—
120C22⋊Q8—1572no—
121C4×Q8—1572no—
122C4×D4—22no—
123C4×D4C2×D15712yes—
124C4×D4C2×D15712yes—
125#117C2×D15711yescharacteristic
126#117C2×D15711yescharacteristic
127C2×C22⋊C4—22no—
128#107—1572no—
129#106—1572no—
130C4⋊C8—1572no—
131#103—1572no—
132#103—1572no—
133#102—1572no—
134#102—1572no—
135C22⋊C8—1572no—
136#97—1571no—
137C4×C8—1571no—
138C24—21no—
139C2×Q8—1572no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D15712yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D15711yescharacteristic
146SD16—3144no—
147SD16—3144no—
148C2×C8—1572no—
149C2×C8—1572no—
150C4⋊C4—3142no—
151C4⋊C4—3142no—
152C4⋊C4—3142no—
153C4⋊C4—3142no—
154C4⋊C4—1572no—
155C4⋊C4C22⋊D15712yes—
156C4⋊C4C22×D15712yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—3142no—
161C22⋊C4C22⋊D15712yes—
162C42—21no—
163C42—3142no—
164C42C22×D15711yescharacteristic
165C42C22×D15711yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D15711yescharacteristic
170Q8—3142no—
171Q8—3142no—
172D4—24no—
173D4—22no—
174D4#1457895414yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—3142no—
180C2×C4—3142no—
181C2×C4C2×C22⋊D15711yescharacteristic
182C2×C4C2×C22⋊D15712yes—
183C2×C4C2×C22⋊D15712yes—
184C2×C4C23×D15711yescharacteristic
185C8—3142no—
186C8—3142no—
187C22—22no—
188C22—21no—
189C22—24no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D15711yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—6282no—
199C4—6282no—
200C4C2×#1457895412yes—
201C2—41no—
202C2—44no—
203C2#2375778411yescharacteristic
204C2#2375808412yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#518×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenter#518D157
hypocenterC157#3123
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #23757885C2semidirect
2. #23757614C2semidirect
3. #23757573C2semidirect
4. #23757478C2semidirect
6. #23757373C2semidirect
7. C4×#14578942C2semidirect
8. #23757334C2semidirect
12. #19097771C22semidirect
14. #19097763C22semidirect
17. C4×Dic314C22semidirect
18. #19097704C22semidirect
19. #19097702C22semidirect
20. C4⋊#11929434C22semidirect
21. #19097694C22semidirect
22. #19097694C22semidirect
26. #19097682C22semidirect
27. C4×#11929434C22semidirect
102. C157#3123semidirect

Maximal subgroups

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

subgroupindex
#237578852
#237576142
#237575732
#237574782
#237574242
#237573732
C4×#145789422
#237573342
#518×C1572
#3123157