Finite Group Databasebeta

← #16828873

Subgroups — #16828873

Order 3712.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#14271999C211yesmaximal, characteristic
2C2×#11745271C212yesmaximal, complemented
3#14271734C211yesmaximal, characteristic, complemented
4#14271348C214yesmaximal, complemented
5#14271343C214yesmaximal, complemented
6#14271179C212yesmaximal, complemented
7C29×#413C211yesmaximal, characteristic, complemented
8C2×#41×C29C2211yescharacteristic
9#11745365—22no—
10#11745364—22no—
11#41×D29—24no—
12#11745362C2212yes—
13#11745359—22no—
14#11745359C2212yes—
15C2×#90805—22no—
16C2×#90805C2212yes—
17C22×Dic58C2211yescharacteristic, complemented
18C2×M16×C29C2212yescomplemented
19#11745306C2212yescomplemented
20#11745298C2212yescomplemented
21#11745297—22no—
22#11745288C2211yescharacteristic, complemented
23#11745286—22no—
24#11745271—24no—
25#11745271C2218yescomplemented
26C2×Dic116—22no—
27C2×#90773—22no—
28C29×#101C2212yescomplemented
29C29×#100C2212yescomplemented
30#11745173C2218yescomplemented
31#41×C29—22no—
32#41×C29—22no—
33C2×Q8×C29C2311yescharacteristic
34C2×D4×C29D412yes—
35C2×D4×C29C2311yescharacteristic
36C22×C4×C29D412yes—
37C22×C4×C29C2311yescharacteristic
38C2×C22⋊D29—24no—
39C2×C22⋊D29—22no—
40C22×Dic29—21no—
41#90809—24no—
42Q8×D29—24no—
43#90807—28no—
44#90807—24no—
45D4×D29—28no—
46D4×D29—24no—
47#90805—24no—
48#90805—24no—
49#90805—22no—
50#90805—24no—
51#90805—22no—
52#90805D414yes—
53#90805C2314yes—
54C2×D116—22no—
55C2×D116D412yes—
56C2×C4×D29—24no—
57C2×C4×D29—22no—
58C2×Dic58—22no—
59C2×Dic58—22no—
60C2×Dic58—22no—
61C2×Dic58—21no—
62C2×Dic58D412yes—
63C2×Dic58C2312yes—
64M16×C29—22no—
65M16×C29C2314yes—
66C2×C8×C29—22no—
67C22⋊Dic29—24no—
68C22⋊Dic29—21no—
69C4⋊Dic29—21no—
70#90779—24no—
71C4×Dic29C2312yes—
72Dic116—28no—
73#90773—28no—
74C23×C29C2×D412yes—
75C23×C29C2×D411yescharacteristic
76C22×D29—24no—
77C22×D29—22no—
78Q8×C29—22no—
79D4×C29—24no—
80D4×C29—22no—
81C2×C4×C29—22no—
82C2×C4×C29—22no—
83C2×C4×C29C2×D412yes—
84C2×C4×C29C2×D411yescharacteristic
85C2×C4×C29C2×D412yes—
86C2×C4×C29C2×D412yes—
87C2×C4×C29C2411yescharacteristic
88C22⋊D29—28no—
89C22⋊D29—24no—
90C22⋊D29—24no—
91C2×Dic29—24no—
92C2×Dic29—22no—
93C2×Dic29—22no—
94C2×Dic29—22no—
95C2×Dic29—42no—
96D116—24no—
97D116—22no—
98D116C2×D414yes—
99C4×D29—28no—
100C4×D29—24no—
101C4×D29—24no—
102Dic58—24no—
103Dic58—22no—
104Dic58—24no—
105Dic58—41no—
106Dic58—41no—
107Dic58C2×D414yes—
108C8×C29—24no—
109#3022—291nomaximal, Sylow
110C22×C29—22no—
111C22×C29—41no—
112C22×C29—42no—
113C22×C29#12012yes—
114C22×C29C22×D411yescharacteristic
115C2×D29—24no—
116C2×D29—82no—
117C2×D29—84no—
118C4×C29—22no—
119C4×C29#12011yescharacteristic
120C4×C29#12011yescharacteristic
121C4×C29C22×D412yes—
122Dic29—24no—
123Dic29—42no—
124Dic29—42no—
125#585—291no—
126C2×#137—292no—
127#536—291no—
128#456—294no—
129#455—294no—
130#421—292no—
131#413D2911yescharacteristic, complemented
132C2×C29—21no—
133C2×C29—22no—
134C2×C29—42no—
135C2×C29C2×#12011yescharacteristic
136D29—44no—
1372−1+4—292no—
1382−1+4—582no—
139H(5,𝔽2)—292no—
140H(5,𝔽2)—582no—
141H(5,𝔽2)—582no—
142C2×#41—292no—
143C2×#41—584no—
144C2×#41—582no—
145C2×#41C2×D2911yescharacteristic
146C22×Q8—291no—
147#137—298no—
148#137—584no—
149C2×Q16—582no—
150C2×SD16—582no—
151C2×M16C2×D2912yescomplemented
152C4⋊Q8—292no—
153#124—292no—
154#123—582no—
155C22⋊Q8—582no—
156#117—291no—
157#104—298no—
158#101C2×D2912yescomplemented
159#100C2×D2912yescomplemented
160C29#302211yescharacteristic, complemented, Sylow
161#41—22no—
162#41—22no—
163#41—294no—
164#41—294no—
165#41—584no—
166#41—584no—
167#41—584no—
168#41—588no—
169#41—584no—
170#41—582no—
171#41—584no—
172#41—582no—
173C2×Q8—292no—
174C2×Q8—292no—
175C2×Q8—582no—
176C2×Q8—584no—
177C2×Q8—581no—
178C2×Q8—582no—
179C2×Q8—582no—
180C2×Q8C22×D2911yescharacteristic
181C2×D4—292no—
182C2×D4—582no—
183C2×D4—584no—
184C2×D4—588no—
185C2×D4—582no—
186C2×D4—584no—
187C2×D4D11612yes—
188C2×D4C22×D2911yescharacteristic
189C22×C4—584no—
190C22×C4—582no—
191C22×C4—581no—
192C22×C4D11612yes—
193C22×C4C22×D2911yescharacteristic
194Q16—588no—
195SD16—588no—
196M16—22no—
197M16C22×D2914yes—
198C2×C8—22no—
199C4⋊C4—581no—
200C4⋊C4—584no—
201C22⋊C4—581no—
202C22⋊C4—584no—
203C42—292no—
204C23—584no—
205C23—582no—
206C23C2×D11612yes—
207C23C2×D11611yescharacteristic
208Q8—22no—
209Q8—294no—
210Q8—584no—
211Q8—582no—
212Q8—584no—
213Q8—1161no—
214Q8—1161no—
215D4—24no—
216D4—22no—
217D4—294no—
218D4—584no—
219D4—588no—
220D4—584no—
221D4—584no—
222D4—582no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—588no—
226C2×C4—584no—
227C2×C4—582no—
228C2×C4—584no—
229C2×C4—584no—
230C2×C4—582no—
231C2×C4—582no—
232C2×C4—1162no—
233C2×C4C2×D11612yes—
234C2×C4C2×D11611yescharacteristic
235C2×C4D4×D2912yes—
236C2×C4D4×D2912yes—
237C2×C4C23×D2911yescharacteristic
238C8—24no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—584no—
243C22—2322no—
244C22—2324no—
245C22#1174524412yes—
246C22C22×D11611yescharacteristic
247C4—22no—
248C4—584no—
249C4—1162no—
250C4—1162no—
251C4#1174524411yescharacteristic
252C4#1174524411yescharacteristic
253C4C2×D4×D2912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—1164no—
258C2C2×#1174524411yescharacteristic
subgroupG/H
centerC2C2×#11745244
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#413C2
socleC2×C29C2×#120
prime radical#14271999C2
hypercenter#413D29
hypocenterC29#3022
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. C2×#11745271C2semidirect
3. #14271734C2semidirect
4. #14271348C2semidirect
5. #14271343C2semidirect
6. #14271179C2semidirect
7. C29×#413C2semidirect
17. C22×Dic58C22semidirect
18. C2×M16×C29C22semidirect
19. #11745306C22semidirect
20. #11745298C22semidirect
22. #11745288C22semidirect
25. #11745271C22semidirect
28. C29×#101C22semidirect
29. C29×#100C22semidirect
30. #11745173C22semidirect
131. #413D29semidirect
151. C2×M16C2×D29semidirect
158. #101C2×D29semidirect
159. #100C2×D29semidirect
160. C29#3022semidirect

Maximal subgroups

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

subgroupindex
#142719992
C2×#117452712
#142717342
#142713482
#142713432
#142711792
C29×#4132
#302229