Finite Group Databasebeta

← #16781620

Subgroups — #16781620

Order 3328.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#13130737C211yesmaximal, characteristic
2#13130733C211yesmaximal, characteristic
3#13130244C214yesmaximal
4C2×#11719402C211yesmaximal, characteristic
5C2×C13×#125C411yescharacteristic, complemented
6#11719402—28no—
7C2×#88152—22no—
8C2×#88151C2212yes—
9C2×#88144—22no—
10C2×#88141C412yescomplemented
11C2×#88126C2211yescharacteristic
12C2×#88123C411yescharacteristic, complemented
13#11719135C2212yes—
14#11719126C2212yes—
15#11718291—22no—
16#11718289—22no—
17#11718241—28no—
18C22×C4×D13C2311yescharacteristic
19C22×Dic26C2×C412yes—
20C2×C22⋊#261—22no—
21C2×#6122—22no—
22C13×#125—24no—
23C2×C13×C4⋊C4—22no—
24C2×C13×C4⋊C4C2×C412yes—
25C2×C42×C13C2×C411yescharacteristic
26C2×#6103—22no—
27C2×#6103D412yes—
28C2×C4⋊Dic13—22no—
29C2×C4⋊Dic13—22no—
30C2×#6101—22no—
31C2×#6101D412yes—
32C2×C4×Dic13—22no—
33C2×C4×Dic13C2×C411yescharacteristic
34#88152—216no—
35#88151—24no—
36#88151D418yes—
37#88144—216no—
38#88141—24no—
39#88141—24no—
40#88126—24no—
41#88123—24no—
42(C2×#261)⋊C4—24no—
43#88111—24no—
44#8521—131nomaximal, Sylow
45C23×D13C2×D411yescharacteristic
46C22×#261—42no—
47C22×C4×C13—22no—
48C22×C4×C13C22⋊C412yes—
49C22×C4×C13C22×C411yescharacteristic
50C22×Dic13—22no—
51C22×Dic13C22⋊C412yes—
52C22×Dic13C2×D411yescharacteristic
53C2×C4×D13—42no—
54C2×C4×D13C2×D414yes—
55C2×Dic26—22no—
56C2×Dic26—44no—
57C2×Dic26C22⋊C414yes—
58C22⋊#261—44no—
59#6122—44no—
60C2×#1057—42no—
61C13×C4⋊C4—28no—
62C13×C4⋊C4—22no—
63C13×C4⋊C4C22⋊C414yes—
64C42×C13—22no—
65#6103—24no—
66#6103—28no—
67C4⋊Dic13—28no—
68C4⋊Dic13—28no—
69#6101—28no—
70#6101—24no—
71C4×Dic13—28no—
72C4×Dic13—22no—
73C2×#541—131no—
74#1822—131no—
75#1820—131no—
76#1511—134no—
77C23×C13C2×C22⋊C411yescharacteristic
78C22×D13—22no—
79C22×D13—42no—
80C22×D13#12012yes—
81C2×#261—82no—
82C2×#261—84no—
83C2×C4×C13—22no—
84C2×C4×C13—24no—
85C2×C4×C13—22no—
86C2×C4×C13—44no—
87C2×C4×C13—42no—
88C2×C4×C13C2×C22⋊C412yes—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—24no—
92C2×Dic13—44no—
93C2×Dic13—42no—
94C2×Dic13#12012yes—
95C4×D13—44no—
96Dic26—44no—
97Dic26—44no—
98#1057—82no—
99#541—268no—
100C2×C4⋊Q8—131no—
101C2×#126—262no—
102C2×#125—262no—
103C2×#125#26111yescharacteristic, complemented
104C2×C22⋊Q8—132no—
105C2×C4×Q8—132no—
106C2×#117—131no—
107#407—132no—
108#381—132no—
109#343—262no—
110#342—262no—
111#328—268no—
112C22×C13—22no—
113C22×C13#37911yescharacteristic
114C22×C13#41012yes—
115C2×D13—24no—
116C2×D13—22no—
117C2×D13—82no—
118#261—162no—
119C4×C13—22no—
120C4×C13—42no—
121C4×C13—44no—
122Dic13—22no—
123Dic13—42no—
124Dic13—44no—
125C22×Q8—132no—
126C23×C4—131no—
127C2×M16—262no—
128C4⋊Q8—264no—
129#126—2616no—
130#125—24no—
131#125—2616no—
132C22⋊Q8—138no—
133C22⋊Q8—264no—
134C4×Q8—264no—
135C4×Q8—264no—
136#117—264no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4—262no—
140C2×C4⋊C4—262no—
141C2×C4⋊C4—262no—
142C2×C4⋊C4C2×#26112yes—
143C2×C22⋊C4—132no—
144C2×C22⋊C4—262no—
145C2×C22⋊C4—262no—
146C2×C42—131no—
147C2×C42—262no—
148C2×C42C2×#26111yescharacteristic
149C22⋊C8—264no—
150#95—264no—
151C2×C13—22no—
152C2×C13#178211yescharacteristic
153C2×C13#179612yes—
154D13—42no—
155C24—131no—
156C2×Q8—134no—
157C2×Q8—262no—
158C2×Q8—524no—
159C22×C4—22no—
160C22×C4—131no—
161C22×C4—134no—
162C22×C4—132no—
163C22×C4—262no—
164C22×C4—522no—
165C22×C4—522no—
166C22×C4C22⋊#26112yes—
167C22×C4C22×#26111yescharacteristic
168M16—524no—
169C2×C8—522no—
170C4⋊C4—28no—
171C4⋊C4—22no—
172C4⋊C4—264no—
173C4⋊C4—268no—
174C4⋊C4—268no—
175C4⋊C4—268no—
176C4⋊C4C22⋊#26114yes—
177C22⋊C4—264no—
178C22⋊C4—268no—
179C22⋊C4—524no—
180C42—22no—
181C42—268no—
182C42—262no—
183C13#852111yescharacteristic, complemented, Sylow
184C23—132no—
185C23—262no—
186C23—522no—
187C23C2×C22⋊#26111yescharacteristic
188Q8—524no—
189Q8—524no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—132no—
196C2×C4—262no—
197C2×C4—262no—
198C2×C4—264no—
199C2×C4—524no—
200C2×C4—524no—
201C2×C4—522no—
202C2×C4—1042no—
203C2×C4—1044no—
204C2×C4C2×C22⋊#26112yes—
205C8—1042no—
206C22—22no—
207C22—262no—
208C22—264no—
209C22—1042no—
210C22#1171915612yes—
211C22#1171917111yescharacteristic
212C4—22no—
213C4—42no—
214C4—44no—
215C4—262no—
216C4—524no—
217C4—522no—
218C4—2082no—
219C2—22no—
220C2—522no—
221C2#1313066711yescharacteristic
222C2#1313068912yes—
subgroupG/H
centerC22#11719171
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×#125C4
socleC22×C13#379
prime radicalC23×D13C2×D4
cosocleC22×C4×D13C23
hypercenterC2×#125#261
hypocenterC13#8521
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
5. C2×C13×#125C4semidirect
10. C2×#88141C4semidirect
12. C2×#88123C4semidirect
103. C2×#125#261semidirect
183. C13#8521semidirect

Maximal subgroups

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

subgroupindex
#131307372
#131307332
#131302442
C2×#117194022
#852113