Finite Group Databasebeta

← #34109373

Subgroups — #34109373

Order 20096.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic314C211yesmaximal, characteristic, complemented
2C2×#19097804C212yesmaximal, complemented
3#23757784C211yesmaximal, characteristic
4#23757431C214yesmaximal, complemented
5#23757414C214yesmaximal, complemented
6#23757283C212yesmaximal, complemented
7#407×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314—22no—
10C22×Dic314—21no—
11C22×Dic314—28no—
12C22×Dic314C2214yes—
13C22×Dic314C2212yes—
14C23×Dic157—21no—
15#19097804—28no—
16C2×Dic628C2214yescomplemented
17C2×#14578943C2214yescomplemented
18#19097754—21no—
19#19097750C2212yes—
20#19097745C2212yes—
21#19097740—21no—
22C4×C22⋊D157—22no—
23#19097730C2211yescharacteristic
24#19097696C2218yescomplemented
25C23×C4×C157C2211yescharacteristic
26C2×M16×C157C2212yes—
27C22⋊C8×C157C2214yes—
28#14578954—24no—
29C2×Dic314—28no—
30C2×Dic314—216no—
31C2×Dic314—216no—
32C2×Dic314—24no—
33C2×Dic314—24no—
34C2×Dic314—22no—
35C2×Dic314D418yescomplemented
36C2×Dic314C2311yescharacteristic
37C2×Dic314C2314yes—
38C2×D628C2311yescharacteristic
39C2×C22⋊D157—22no—
40C2×C22⋊D157—21no—
41C22×Dic157—28no—
42C22×Dic157—22no—
43C22×Dic157—24no—
44C2×C4×D157—21no—
45Dic628—28no—
46#14578943—28no—
47C4⋊Dic157C2312yes—
48#14578933—22no—
49C22⋊Dic157—21no—
50C22⋊Dic157—22no—
51#14578931—22no—
52C4×Dic157—21no—
53C24×C157D411yescharacteristic
54C22×C4×C157—24no—
55C22×C4×C157D412yes—
56C22×C4×C157D411yescharacteristic
57C22×C4×C157C2312yes—
58C22×C4×C157C2311yescharacteristic
59M16×C157—24no—
60C2×C8×C157C2314yes—
61C22×D157—21no—
62Dic314—216no—
63Dic314—28no—
64Dic314—24no—
65Dic314—22no—
66Dic314C2×D418yes—
67D628—22no—
68C22⋊D157—42no—
69C22⋊D157—44no—
70C2×Dic157—216no—
71C2×Dic157—28no—
72C2×Dic157—24no—
73C2×Dic157—21no—
74C2×Dic157—22no—
75C4×D157—42no—
76C23×C157—22no—
77C23×C157—24no—
78C23×C157C2×D411yescharacteristic
79C23×C157C2×D412yes—
80C2×C4×C157—22no—
81C2×C4×C157—24no—
82C2×C4×C157—24no—
83C2×C4×C157C2×D414yes—
84C2×C4×C157C2×D412yes—
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2411yescharacteristic
87C8×C157—24no—
88C22×C157—28no—
89C22×C157—22no—
90C22×C157—22no—
91C22×C157—22no—
92C22×C157#12012yes—
93C22×C157#13714yescomplemented
94C22×C157C22×D411yescharacteristic
95C2×D157—41no—
96C2×D157—42no—
97C4×C157—22no—
98C4×C157#12012yes—
99C4×C157C22×D412yes—
100Dic157—28no—
101Dic157—41no—
102Dic157—42no—
103C2×C157—22no—
104C2×C157—24no—
105C2×C157C2×#12011yescharacteristic
106C2×C157C2×#13712yes—
107D157—81no—
108C157#300411yescharacteristic, complemented, Sylow
109#3004—1571nomaximal, Sylow
110C23×Q8—1571no—
111C2×#137—1572no—
112#525—1571no—
113#451—1574no—
114#448—1574no—
115#419—1572no—
116#407D15711yescharacteristic, complemented
117C2×#41—1571no—
118C22×Q8—3148no—
119C22×Q8—3142no—
120C22×Q8—3141no—
121C22×Q8—1574no—
122C22×Q8—1572no—
123C23×C4—3141no—
124C23×C4C2×D15711yescharacteristic
125#137—3148no—
126C2×Q16—1574no—
127C2×SD16—1574no—
128C2×M16C2×D15712yes—
129#123—3141no—
130C22⋊Q8—1572no—
131#121—1572no—
132#120—3141no—
133C4×D4—3142no—
134#117—1571no—
135#103—1578no—
136C22⋊C8C2×D15714yes—
137C24D62811yescharacteristic
138#41—3144no—
139C2×Q8—3148no—
140C2×Q8—31416no—
141C2×Q8—31416no—
142C2×Q8—3144no—
143C2×Q8—3144no—
144C2×Q8—3142no—
145C2×Q8—1578no—
146C2×Q8—1571no—
147C2×Q8—1574no—
148C2×D4—3142no—
149C2×D4—3141no—
150C2×D4—1571no—
151C22×C4—24no—
152C22×C4—3148no—
153C22×C4—3144no—
154C22×C4—3142no—
155C22×C4—3141no—
156C22×C4D62812yes—
157C22×C4D62811yescharacteristic
158C22×C4C22×D15712yes—
159C22×C4C22×D15711yescharacteristic
160Q16—3148no—
161SD16—3148no—
162M16—24no—
163C2×C8C22×D15714yes—
164C4⋊C4—3142no—
165C4⋊C4—1572no—
166C22⋊C4—3141no—
167C22⋊C4—3142no—
168C22⋊C4—3142no—
169C42—3141no—
170C23—22no—
171C23—24no—
172C23—3141no—
173C23C2×D62811yescharacteristic
174C23C2×D62812yes—
175Q8—31416no—
176Q8—3148no—
177Q8—3144no—
178Q8—3142no—
179Q8—1578no—
180D4—3142no—
181D4—6282no—
182D4—6284no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—31416no—
187C2×C4—3148no—
188C2×C4—3144no—
189C2×C4—3142no—
190C2×C4—3141no—
191C2×C4—6282no—
192C2×C4D4×D15714yes—
193C2×C4C2×D62812yes—
194C2×C4C2×D62811yescharacteristic
195C2×C4C23×D15711yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—6281no—
202C22—6282no—
203C22#1909774112yes—
204C22#1909780414yescomplemented
205C22C22×D62811yescharacteristic
206C4—22no—
207C4—3148no—
208C4—6281no—
209C4—6282no—
210C4#1909774112yes—
211C4C2×D4×D15712yes—
212C2—22no—
213C2—24no—
214C2—12561no—
215C2C2×#1909774111yescharacteristic
216C2C2×#1909780412yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#407×C157C2
socleC22×C157C22×D4
prime radical#23757784C2
hypercenter#407D157
hypocenterC157#3004
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic314C2semidirect
2. C2×#19097804C2semidirect
4. #23757431C2semidirect
5. #23757414C2semidirect
6. #23757283C2semidirect
7. #407×C157C2semidirect
16. C2×Dic628C22semidirect
17. C2×#14578943C22semidirect
24. #19097696C22semidirect
35. C2×Dic314D4semidirect
93. C22×C157#137semidirect
108. C157#3004semidirect
116. #407D157semidirect
204. C22#19097804semidirect

Maximal subgroups

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

subgroupindex
C23×Dic3142
C2×#190978042
#237577842
#237574312
#237574142
#237572832
#407×C1572
#3004157