Finite Group Databasebeta

← #16828719

Subgroups — #16828719

Order 3712.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11745294C212yesmaximal, complemented
2#14271674C211yesmaximal, characteristic, complemented
3#14271314C214yesmaximal, complemented
4#14271297C214yesmaximal, complemented
5#14271168C212yesmaximal, complemented
6#14271127C211yesmaximal, characteristic, complemented
7C23×D4×C29C211yesmaximal, characteristic, complemented
8C25×C29—21no—
9C22×D4×C29—28no—
10C22×D4×C29—22no—
11C22×D4×C29—21no—
12C22×D4×C29C2214yes—
13C22×D4×C29C2212yes—
14C23×C4×C29C2211yescharacteristic
15C2×#90805C2211yescharacteristic, complemented
16#11745315—21no—
17C2×#90783C2214yescomplemented
18#11745294—28no—
19C2×#90782C2214yescomplemented
20#11745292C2212yescomplemented
21#11745291—21no—
22C4×C22⋊D29—22no—
23#11745288C2211yescharacteristic, complemented
24#11745286C2212yescomplemented
25C2×#90777C2212yescomplemented
26#11745180C2218yescomplemented
27C22⋊#6597C2214yescomplemented
28C24×C29—28no—
29C24×C29—24no—
30C24×C29—22no—
31C24×C29—48no—
32C24×C29D411yescharacteristic
33C2×D4×C29—28no—
34C2×D4×C29—216no—
35C2×D4×C29—216no—
36C2×D4×C29—24no—
37C2×D4×C29—22no—
38C2×D4×C29—24no—
39C2×D4×C29D418yescomplemented
40C2×D4×C29C2314yes—
41C22×C4×C29—24no—
42C22×C4×C29D412yes—
43C22×C4×C29D411yescharacteristic
44C22×C4×C29C2312yes—
45C22×C4×C29C2311yescharacteristic
46C2×C22⋊D29—22no—
47C2×C22⋊D29—21no—
48#90805—24no—
49C2×D116C2311yescharacteristic, complemented
50C2×C4×D29—21no—
51C2×Dic58C2311yescharacteristic, complemented
52C22⋊Dic29—22no—
53C22⋊Dic29—21no—
54#90783—28no—
55#90782—28no—
56#90781—22no—
57C4⋊Dic29C2312yescomplemented
58#90779—22no—
59C4×Dic29—21no—
60#90777—24no—
61C2×#6597C2314yescomplemented
62C23×C29—22no—
63C23×C29—24no—
64C23×C29—216no—
65C23×C29—28no—
66C23×C29—24no—
67C23×C29—432no—
68C23×C29—48no—
69C23×C29—48no—
70C23×C29—48no—
71C23×C29C2×D411yescharacteristic
72C23×C29C2×D412yes—
73C22×D29—21no—
74D4×C29—216no—
75D4×C29—28no—
76D4×C29—24no—
77D4×C29C2×D418yes—
78C2×C4×C29—22no—
79C2×C4×C29—24no—
80C2×C4×C29—24no—
81C2×C4×C29C2×D412yes—
82C2×C4×C29C2×D411yescharacteristic
83C2×C4×C29C2×D414yes—
84C2×C4×C29C2411yescharacteristic
85C22⋊D29—42no—
86C22⋊D29—44no—
87C2×Dic29—22no—
88C2×Dic29—21no—
89D116—22no—
90C4×D29—42no—
91Dic58—22no—
92#6597—24no—
93#3003—291nomaximal, Sylow
94C22×C29—28no—
95C22×C29—22no—
96C22×C29—22no—
97C22×C29—22no—
98C22×C29—28no—
99C22×C29—416no—
100C22×C29—432no—
101C22×C29—48no—
102C22×C29#12012yes—
103C22×C29AGL(1,ℤ/8)14yescomplemented
104C22×C29C22×D411yescharacteristic
105C2×D29—41no—
106C2×D29—42no—
107C4×C29—22no—
108C4×C29#12012yes—
109C4×C29C22×D412yes—
110Dic29—41no—
111Dic29—42no—
112C23×D4D2911yescharacteristic, complemented
113C2×AGL(1,ℤ/8)—292no—
114#525—291no—
115#450—294no—
116#447—294no—
117#418—292no—
118#407—291no—
119C2×C29—22no—
120C2×C29—24no—
121C2×C29—48no—
122C2×C29C2×#12011yescharacteristic
123C2×C29C2×AGL(1,ℤ/8)12yes—
124D29—81no—
125C25—21no—
126C2×#41—291no—
127C22×D4—28no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C2×D2914yes—
131C22×D4C2×D2912yes—
132C23×C4C2×D2911yescharacteristic
133AGL(1,ℤ/8)—588no—
134C2×SD16—294no—
135C2×D8—294no—
136C2×M16—292no—
137#123—581no—
138C22⋊Q8—292no—
139#121—292no—
140#120—581no—
141C4×D4—582no—
142#117—291no—
143#102—298no—
144C22⋊C8—294no—
145C29#300311yescharacteristic, complemented, Sylow
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D2911yescharacteristic
151#41—584no—
152C2×Q8—291no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—291no—
160C2×D4—582no—
161C2×D4—581no—
162C2×D4C22⋊D2918yescomplemented
163C2×D4C22×D2914yes—
164C22×C4—24no—
165C22×C4—581no—
166C22×C4C22⋊D2912yes—
167C22×C4C22⋊D2911yescharacteristic
168C22×C4C22×D2912yes—
169C22×C4C22×D2911yescharacteristic
170SD16—588no—
171D8—588no—
172M16—584no—
173C2×C8—294no—
174C4⋊C4—292no—
175C4⋊C4—582no—
176C22⋊C4—581no—
177C22⋊C4—582no—
178C22⋊C4—582no—
179C42—581no—
180C23—24no—
181C23—22no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—581no—
190C23C2×C22⋊D2912yes—
191C23C2×C22⋊D2911yescharacteristic
192Q8—582no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—582no—
197D4—1164no—
198D4—1162no—
199D4C2×C22⋊D2918yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—582no—
204C2×C4—581no—
205C2×C4—1162no—
206C2×C4C2×C22⋊D2912yes—
207C2×C4C2×C22⋊D2911yescharacteristic
208C2×C4C2×C22⋊D2914yes—
209C2×C4C23×D2911yescharacteristic
210C8—584no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—432no—
217C22—416no—
218C22—48no—
219C22—1161no—
220C22—1162no—
221C22#1174529414yescomplemented
222C22#1174531512yes—
223C22C22×C22⋊D2911yescharacteristic
224C4—22no—
225C4—1161no—
226C4—1162no—
227C4#1174531512yes—
228C4C22×C22⋊D2912yes—
229C2—24no—
230C2—22no—
231C2—48no—
232C2—2321no—
233C2C2×#1174531511yescharacteristic
234C2C2×#1174529412yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC23×D4×C29C2
socleC22×C29C22×D4
hypercenterC23×D4D29
hypocenterC29#3003
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11745294C2semidirect
2. #14271674C2semidirect
3. #14271314C2semidirect
4. #14271297C2semidirect
5. #14271168C2semidirect
6. #14271127C2semidirect
7. C23×D4×C29C2semidirect
15. C2×#90805C22semidirect
17. C2×#90783C22semidirect
19. C2×#90782C22semidirect
20. #11745292C22semidirect
23. #11745288C22semidirect
24. #11745286C22semidirect
25. C2×#90777C22semidirect
26. #11745180C22semidirect
27. C22⋊#6597C22semidirect
39. C2×D4×C29D4semidirect
49. C2×D116C23semidirect
51. C2×Dic58C23semidirect
57. C4⋊Dic29C23semidirect
61. C2×#6597C23semidirect
103. C22×C29AGL(1,ℤ/8)semidirect
112. C23×D4D29semidirect
145. C29#3003semidirect
162. C2×D4C22⋊D29semidirect
221. C22#11745294semidirect

Maximal subgroups

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

subgroupindex
C2×#117452942
#142716742
#142713142
#142712972
#142711682
#142711272
C23×D4×C292
#300329