Finite Group Databasebeta

← #34110602

Subgroups — #34110602

Order 20096.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19097804C212yesmaximal, complemented
2C2×#19097799C212yesmaximal, complemented
3#23757823C211yesmaximal, characteristic, complemented
4C2×#19097759C211yesmaximal, characteristic, complemented
5#23757667C212yesmaximal, complemented
6#23757662C212yesmaximal, complemented
7#23757657C214yesmaximal, complemented
8C4×M16×C157C211yesmaximal, characteristic, complemented
9C2×#14578954C2212yes—
10C22×Dic314C2211yescharacteristic
11C22×D628C2211yescharacteristic
12#19097804—28no—
13#19097799—28no—
14C2×Dic628C2214yescomplemented
15C2×#14578943C2214yes—
16C2×#14578943C2214yescomplemented
17C2×D1256C2214yes—
18#19097771C2211yescharacteristic, complemented
19#19097768C2211yescharacteristic, complemented
20#19097759—22no—
21#19097759C2212yes—
22#19097759C2212yes—
23#19097759C2212yescomplemented
24#19097745—22no—
25C4×D628—22no—
26C2×#14578931—22no—
27C2×M16×C157C2212yes—
28C2×C42×C157C2211yescharacteristic
29#97×C157C2212yescomplemented
30C4×C8×C157C2212yescomplemented
31C23×D157—21no—
32#14578954—24no—
33#14578954—24no—
34C2×Dic314—21no—
35C2×Dic314—24no—
36C2×Dic314C2312yes—
37C2×Dic314C2312yes—
38C2×D628—24no—
39C2×D628—22no—
40C2×D628—21no—
41C2×D628C2312yes—
42C2×D628C2312yes—
43C2×C22⋊D157—22no—
44C22×Dic157—21no—
45C2×C4×D157—22no—
46Dic628—28no—
47#14578943—28no—
48#14578943—28no—
49D1256—28no—
50C4⋊Dic157—22no—
51#14578931—24no—
52#14578931—24no—
53#14578931—24no—
54C22×C4×C157D412yes—
55C22×C4×C157C2311yescharacteristic
56M16×C157D418yescomplemented
57C2×C8×C157C2314yes—
58C42×C157D412yes—
59C42×C157C2311yescharacteristic
60C42×C157C2311yescharacteristic
61C22×D157—22no—
62C22×D157—22no—
63C22×D157—44no—
64C22×D157—42no—
65Dic314—22no—
66Dic314—24no—
67Dic314—22no—
68Dic314—42no—
69D628—24no—
70D628—22no—
71D628—22no—
72D628—44no—
73D628—42no—
74C22⋊D157—44no—
75C2×Dic157—22no—
76C2×Dic157—22no—
77C2×Dic157—42no—
78C4×D157—44no—
79C23×C157C2×D411yescharacteristic
80C2×C4×C157—42no—
81C2×C4×C157C2×D412yes—
82C2×C4×C157C2×D412yes—
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D414yes—
85C2×C4×C157C2411yescharacteristic
86C8×C157C2×D418yes—
87C22×C157—22no—
88C22×C157#12712yes—
89C22×C157C22×D411yescharacteristic
90C2×D157—42no—
91C2×D157—42no—
92C2×D157—44no—
93C2×D157—44no—
94C2×D157—84no—
95C4×C157—22no—
96C4×C157—22no—
97C4×C157#12712yes—
98C4×C157C22×D412yes—
99Dic157—42no—
100Dic157—42no—
101C2×C157—22no—
102C2×C157C2×#12711yescharacteristic
103C2×C157#57512yes—
104D157—82no—
105D157—82no—
106C157#315311yescharacteristic, complemented, Sylow
107#3153—1571nomaximal, Sylow
108C2×#137—1572no—
109C2×AGL(1,ℤ/8)—1572no—
110#532—1571no—
111C2×#124—1571no—
112#497—1572no—
113#496—1572no—
114#495—1574no—
115C4×M16D15711yescharacteristic, complemented
116C2×#41—1572no—
117C22×Q8—1571no—
118C22×D4—1571no—
119#137—3148no—
120AGL(1,ℤ/8)—3148no—
121C2×Q16—1574no—
122C2×SD16—1574no—
123C2×SD16—1574no—
124C2×D8—1574no—
125C2×M16C2×D15712yes—
126C4⋊Q8—1571no—
127#127—1571no—
128#124—3142no—
129#124—1572no—
130#124—1572no—
131#124—1572no—
132#121—3142no—
133C4×D4—3142no—
134C2×C22⋊C4—3142no—
135C2×C42C2×D15711yescharacteristic
136#97C2×D15712yescomplemented
137C4×C8C2×D15712yescomplemented
138C24—3141no—
139#41—3144no—
140#41—3144no—
141C2×Q8—3141no—
142C2×Q8—3144no—
143C2×Q8—1572no—
144C2×Q8—1572no—
145C2×D4—3142no—
146C2×D4—3144no—
147C2×D4—3142no—
148C2×D4—3141no—
149C2×D4—1572no—
150C2×D4—1572no—
151C22×C4—3141no—
152C22×C4—3142no—
153C22×C4D62812yes—
154C22×C4C22×D15711yescharacteristic
155Q16—3148no—
156SD16—3148no—
157SD16—3148no—
158D8—3148no—
159M16D62818yescomplemented
160C2×C8C22×D15714yes—
161C4⋊C4—3142no—
162C22⋊C4—3144no—
163C22⋊C4—3144no—
164C22⋊C4—3144no—
165C42D62812yes—
166C42C22×D15711yescharacteristic
167C42C22×D15711yescharacteristic
168C23—3142no—
169C23—3142no—
170C23—6284no—
171C23—6282no—
172C23C2×D62811yescharacteristic
173Q8—3142no—
174Q8—3144no—
175Q8—3142no—
176Q8—6282no—
177D4—3142no—
178D4—3144no—
179D4—3142no—
180D4—6284no—
181D4—6284no—
182D4—6282no—
183C2×C4—42no—
184C2×C4—3142no—
185C2×C4—3142no—
186C2×C4—6282no—
187C2×C4—6284no—
188C2×C4C2×D62812yes—
189C2×C4C2×D62812yes—
190C2×C4C2×D62811yescharacteristic
191C2×C4C2×D62814yes—
192C2×C4C23×D15711yescharacteristic
193C8C2×D62818yes—
194C22—22no—
195C22—6282no—
196C22—6282no—
197C22—6284no—
198C22—6284no—
199C22—12564no—
200C22#1909776812yes—
201C22C22×D62811yescharacteristic
202C4—22no—
203C4—22no—
204C4—6282no—
205C4—6282no—
206C4#1909776812yes—
207C4C22×D62812yes—
208C2—22no—
209C2—12562no—
210C2—12562no—
211C2C2×#1909776811yescharacteristic
212C2#2375807312yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×M16×C157C2
socleC22×C157C22×D4
hypercenterC4×M16D157
hypocenterC157#3153
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
2. C2×#19097799C2semidirect
3. #23757823C2semidirect
4. C2×#19097759C2semidirect
5. #23757667C2semidirect
6. #23757662C2semidirect
7. #23757657C2semidirect
8. C4×M16×C157C2semidirect
14. C2×Dic628C22semidirect
16. C2×#14578943C22semidirect
18. #19097771C22semidirect
19. #19097768C22semidirect
23. #19097759C22semidirect
29. #97×C157C22semidirect
30. C4×C8×C157C22semidirect
56. M16×C157D4semidirect
106. C157#3153semidirect
115. C4×M16D157semidirect
136. #97C2×D157semidirect
137. C4×C8C2×D157semidirect
159. M16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×#190977992
#237578232
C2×#190977592
#237576672
#237576622
#237576572
C4×M16×C1572
#3153157