Finite Group Databasebeta

← #16828729

Subgroups — #16828729

Order 3712.

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×Dic58C211yesmaximal, characteristic, complemented
2C2×#11745271C212yesmaximal, complemented
3#14271674C211yesmaximal, characteristic
4#14271321C214yesmaximal, complemented
5#14271304C214yesmaximal, complemented
6#14271173C212yesmaximal, complemented
7C29×#407C211yesmaximal, characteristic, complemented
8C23×C4×C29C2211yescharacteristic
9C23×Dic29—21no—
10C2×#90805C2211yescharacteristic
11C22×Dic58—22no—
12C22×Dic58—21no—
13C22×Dic58—28no—
14C22×Dic58C2214yes—
15C22×Dic58C2212yes—
16C2×M16×C29C2212yes—
17#11745315—21no—
18#11745292C2212yes—
19#11745291—21no—
20C4×C22⋊D29—22no—
21#11745288C2211yescharacteristic
22#11745286C2212yes—
23#11745271—28no—
24C2×Dic116C2214yescomplemented
25C2×#90773C2214yescomplemented
26C29×C22⋊C8C2214yes—
27#11745164C2218yescomplemented
28C24×C29D411yescharacteristic
29C22×C4×C29—24no—
30C22×C4×C29D412yes—
31C22×C4×C29D411yescharacteristic
32C22×C4×C29C2312yes—
33C22×C4×C29C2311yescharacteristic
34C2×C22⋊D29—22no—
35C2×C22⋊D29—21no—
36C22×Dic29—28no—
37C22×Dic29—24no—
38C22×Dic29—22no—
39#90805—24no—
40C2×D116C2311yescharacteristic
41C2×C4×D29—21no—
42C2×Dic58—28no—
43C2×Dic58—216no—
44C2×Dic58—24no—
45C2×Dic58—22no—
46C2×Dic58—24no—
47C2×Dic58—216no—
48C2×Dic58D418yescomplemented
49C2×Dic58C2311yescharacteristic
50C2×Dic58C2314yes—
51M16×C29—24no—
52C2×C8×C29C2314yes—
53C22⋊Dic29—22no—
54C22⋊Dic29—21no—
55#90781—22no—
56C4⋊Dic29C2312yes—
57#90779—22no—
58C4×Dic29—21no—
59Dic116—28no—
60#90773—28no—
61C23×C29—22no—
62C23×C29—24no—
63C23×C29C2×D411yescharacteristic
64C23×C29C2×D412yes—
65C22×D29—21no—
66C2×C4×C29—22no—
67C2×C4×C29—24no—
68C2×C4×C29—24no—
69C2×C4×C29C2×D412yes—
70C2×C4×C29C2×D411yescharacteristic
71C2×C4×C29C2×D414yes—
72C2×C4×C29C2411yescharacteristic
73C22⋊D29—42no—
74C22⋊D29—44no—
75C2×Dic29—216no—
76C2×Dic29—28no—
77C2×Dic29—24no—
78C2×Dic29—22no—
79C2×Dic29—21no—
80D116—22no—
81C4×D29—42no—
82Dic58—216no—
83Dic58—28no—
84Dic58—24no—
85Dic58—22no—
86Dic58C2×D418yes—
87C8×C29—24no—
88#3004—291nomaximal, Sylow
89C22×C29—28no—
90C22×C29—22no—
91C22×C29—22no—
92C22×C29—22no—
93C22×C29#12012yes—
94C22×C29#13714yescomplemented
95C22×C29C22×D411yescharacteristic
96C2×D29—41no—
97C2×D29—42no—
98C4×C29—22no—
99C4×C29#12012yes—
100C4×C29C22×D412yes—
101Dic29—28no—
102Dic29—41no—
103Dic29—42no—
104C23×Q8—291no—
105C2×#137—292no—
106#525—291no—
107#451—294no—
108#448—294no—
109#419—292no—
110#407D2911yescharacteristic, complemented
111C2×C29—22no—
112C2×C29—24no—
113C2×C29C2×#12011yescharacteristic
114C2×C29C2×#13712yes—
115D29—81no—
116C2×#41—291no—
117C22×Q8—292no—
118C22×Q8—294no—
119C22×Q8—588no—
120C22×Q8—582no—
121C22×Q8—581no—
122C23×C4—581no—
123C23×C4C2×D2911yescharacteristic
124#137—588no—
125C2×Q16—294no—
126C2×SD16—294no—
127C2×M16C2×D2912yes—
128#123—581no—
129C22⋊Q8—292no—
130#121—292no—
131#120—581no—
132C4×D4—582no—
133#117—291no—
134#103—298no—
135C22⋊C8C2×D2914yes—
136C29#300411yescharacteristic, complemented, Sylow
137C24D11611yescharacteristic
138#41—584no—
139C2×Q8—298no—
140C2×Q8—294no—
141C2×Q8—291no—
142C2×Q8—5816no—
143C2×Q8—588no—
144C2×Q8—5816no—
145C2×Q8—584no—
146C2×Q8—582no—
147C2×Q8—584no—
148C2×D4—291no—
149C2×D4—582no—
150C2×D4—581no—
151C22×C4—24no—
152C22×C4—588no—
153C22×C4—584no—
154C22×C4—582no—
155C22×C4—581no—
156C22×C4D11612yes—
157C22×C4D11611yescharacteristic
158C22×C4C22×D2912yes—
159C22×C4C22×D2911yescharacteristic
160Q16—588no—
161SD16—588no—
162M16—24no—
163C2×C8C22×D2914yes—
164C4⋊C4—292no—
165C4⋊C4—582no—
166C22⋊C4—581no—
167C22⋊C4—582no—
168C22⋊C4—582no—
169C42—581no—
170C23—22no—
171C23—24no—
172C23—581no—
173C23C2×D11611yescharacteristic
174C23C2×D11612yes—
175Q8—298no—
176Q8—5816no—
177Q8—584no—
178Q8—588no—
179Q8—582no—
180D4—582no—
181D4—1162no—
182D4—1164no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—5816no—
187C2×C4—584no—
188C2×C4—588no—
189C2×C4—582no—
190C2×C4—581no—
191C2×C4—1162no—
192C2×C4C2×D11612yes—
193C2×C4C2×D11611yescharacteristic
194C2×C4D4×D2914yes—
195C2×C4C23×D2911yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—1161no—
202C22—1162no—
203C22#1174524412yes—
204C22#1174527114yescomplemented
205C22C22×D11611yescharacteristic
206C4—22no—
207C4—588no—
208C4—1161no—
209C4—1162no—
210C4#1174524412yes—
211C4C2×D4×D2912yes—
212C2—22no—
213C2—24no—
214C2—2321no—
215C2C2×#1174524411yescharacteristic
216C2C2×#1174527112yes—
subgroupG/H
centerC22C22×D116
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#407C2
socleC22×C29C22×D4
prime radical#14271674C2
hypercenter#407D29
hypocenterC29#3004
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic58C2semidirect
2. C2×#11745271C2semidirect
4. #14271321C2semidirect
5. #14271304C2semidirect
6. #14271173C2semidirect
7. C29×#407C2semidirect
24. C2×Dic116C22semidirect
25. C2×#90773C22semidirect
27. #11745164C22semidirect
48. C2×Dic58D4semidirect
94. C22×C29#137semidirect
110. #407D29semidirect
136. C29#3004semidirect
204. C22#11745271semidirect

Maximal subgroups

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

subgroupindex
C23×Dic582
C2×#117452712
#142716742
#142713212
#142713042
#142711732
C29×#4072
#300429