Finite Group Databasebeta

← #15756501

Subgroups — #15756501

Order 3328.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#13117757C211yesmaximal, characteristic
2#13114832C211yesmaximal, characteristic
3#13112738C212yesmaximal
4#13112695C212yesmaximal
5C13×#1461C211yesmaximal, characteristic
6C2×C4×Q8×C13C2211yescharacteristic
7C2×Q8×Dic13—21no—
8C13×#431C2212yes—
9C2×#88076—21no—
10#11718646—21no—
11#11718646—22no—
12#11718597C412yes—
13#11718597C2211yescharacteristic
14C4×C4⋊Dic13C412yes—
15C4×C4⋊Dic13C2211yescharacteristic
16#11718585C2212yes—
17C13×#324—22no—
18#11718122—21no—
19C22×Q8×C13D411yescharacteristic
20C4×Q8×C13—22no—
21C4×Q8×C13—22no—
22C2×C13×C4⋊C4D411yescharacteristic
23C2×C13×C4⋊C4D412yes—
24C2×C42×C13C2×C412yes—
25C2×C42×C13C2311yescharacteristic
26Q8×Dic13—44no—
27C2×C4⋊Dic13—22no—
28C2×C4⋊Dic13—22no—
29C2×C4⋊Dic13—21no—
30C2×C4⋊Dic13—21no—
31C2×C4×Dic13—22no—
32C2×C4×Dic13—21no—
33C2×#6099—22no—
34C13×C4⋊C8—22no—
35C13×C22⋊C8—22no—
36C4×C8×C13—22no—
37#88076—42no—
38#88071—22no—
39#88071—21no—
40#88071C2×C411yescharacteristic
41#88071C2×C412yes—
42#88071C2×C411yescharacteristic
43C4⋊#1055—22no—
44C4⋊#1055—22no—
45#8767—131nomaximal, Sylow
46C2×Q8×C13—22no—
47C2×Q8×C13—22no—
48C2×Q8×C13SD1612yes—
49C2×Q8×C13Q1612yes—
50C22×C4×C13C22⋊C412yes—
51C22×C4×C13C22⋊C412yes—
52C22×C4×C13C22×C411yescharacteristic
53C22×C4×C13C2×D411yescharacteristic
54C22×C4×C13C2×D411yescharacteristic
55C22×Dic13—21no—
56C22×Dic13—21no—
57C22×Dic13—22no—
58C2×C8×C13—42no—
59C13×C4⋊C4—22no—
60C13×C4⋊C4—24no—
61C42×C13—22no—
62C42×C13—22no—
63C42×C13#4112yes—
64C42×C13#4112yes—
65C4⋊Dic13—42no—
66C4⋊Dic13—44no—
67C4⋊Dic13—44no—
68C4⋊Dic13—42no—
69C4×Dic13—44no—
70C4×Dic13—42no—
71#6099—42no—
72#6099—42no—
73C2×#1055—42no—
74#2351—131no—
75#1801—131no—
76#1461D1311yescharacteristic
77#1303—132no—
78#1286—132no—
79C23×C13C2×C22⋊C411yescharacteristic
80Q8×C13—24no—
81Q8×C13—24no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—41no—
86C2×C4×C13—41no—
87C2×C4×C13—42no—
88C2×C4×C13#10314yes—
89C2×C4×C13#11711yescharacteristic
90C2×C4×C13#11711yescharacteristic
91C2×C4×C13#12312yes—
92C2×C4×C13#12312yes—
93C2×C4×C13C2×SD1611yescharacteristic
94C2×C4×C13C2×Q1611yescharacteristic
95C2×C4×C13AGL(1,ℤ/8)12yes—
96C2×Dic13—22no—
97C2×Dic13—41no—
98C2×Dic13—41no—
99C2×Dic13—41no—
100C2×Dic13—42no—
101C2×Dic13—41no—
102C2×Dic13—42no—
103C2×Dic13—42no—
104C2×Dic13—41no—
105C8×C13—82no—
106#1055—82no—
107C2×C4×Q8—261no—
108C2×C4×Q8C2×D1311yescharacteristic
109#431C2×D1312yes—
110#423—132no—
111C2×#101—261no—
112#389—261no—
113#389—262no—
114#382—132no—
115#382—131no—
116C4×C4⋊C4—132no—
117C4×C4⋊C4—131no—
118#343—261no—
119#324—22no—
120C22×C13—21no—
121C22×C13—21no—
122C22×C13#38111yescharacteristic
123C22×C13C2×#10311yescharacteristic
124C22×C13#41811yescharacteristic
125C4×C13—24no—
126C4×C13—22no—
127C4×C13—21no—
128C4×C13—21no—
129C4×C13—42no—
130C4×C13#48112yes—
131C4×C13#48412yes—
132Dic13—42no—
133Dic13—82no—
134Dic13—81no—
135C22×Q8D5211yescharacteristic
136C2×M16—262no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—524no—
140C2×C4⋊C4—262no—
141C2×C4⋊C4—262no—
142C2×C4⋊C4—261no—
143C2×C4⋊C4—261no—
144C2×C4⋊C4D5211yescharacteristic
145C2×C4⋊C4C22⋊D1312yes—
146C2×C42—262no—
147C2×C42—261no—
148C2×C42C4×D1312yes—
149C2×C42C22×D1311yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—262no—
152C4⋊C8—262no—
153#101—522no—
154C22⋊C8—22no—
155C4×C8—22no—
156#95—131no—
157#95—132no—
158#95—131no—
159#95—262no—
160#95—261no—
161C2×C13—21no—
162C2×C13—21no—
163C2×C13#182411yescharacteristic
164C2×C13#182711yescharacteristic
165C2×C13#183011yescharacteristic
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8#609512yes—
169C2×Q8Dic5212yes—
170C22×C4—261no—
171C22×C4—261no—
172C22×C4—262no—
173C22×C4#610312yes—
174C22×C4#610312yes—
175C22×C4C2×C4×D1311yescharacteristic
176C22×C4C2×D5211yescharacteristic
177C22×C4C2×C22⋊D1311yescharacteristic
178M16—522no—
179M16—522no—
180C2×C8—42no—
181C2×C8—522no—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—524no—
185C4⋊C4—522no—
186C4⋊C4—524no—
187C4⋊C4—522no—
188C42—22no—
189C42—22no—
190C42—524no—
191C42—522no—
192C42#612912yes—
193C42#612912yes—
194C13#876711yescharacteristic, complemented, Sylow
195C23C2×#610311yescharacteristic
196Q8—24no—
197Q8—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—262no—
205C2×C4—521no—
206C2×C4—521no—
207C2×C4—521no—
208C2×C4—522no—
209C2×C4—521no—
210C2×C4—522no—
211C2×C4—522no—
212C2×C4—521no—
213C2×C4#8805614yes—
214C2×C4#8813111yescharacteristic
215C2×C4#8813111yescharacteristic
216C2×C4#8814012yes—
217C2×C4C2×#609511yescharacteristic
218C2×C4C2×Dic5211yescharacteristic
219C2×C4#8816212yes—
220C2×C4#8818912yes—
221C8—82no—
222C8—1042no—
223C22—21no—
224C22—21no—
225C22#1171860411yescharacteristic
226C22C2×#8805611yescharacteristic
227C22#1171879211yescharacteristic
228C4—24no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—522no—
234C4—1042no—
235C4—1041no—
236C4#1171838312yes—
237C4#1171839612yes—
238C2—21no—
239C2—21no—
240C2#1311492711yescharacteristic
241C2#1311493711yescharacteristic
242C2#1311494811yescharacteristic
subgroupG/H
centerC22#11718604
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#1461C2
socleC22×C13#381
prime radicalC23×C13C2×C22⋊C4
cosocleC2×C42×C13C23
hypercenter#1461D13
hypocenterC13#8767
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
194. C13#8767semidirect

Maximal subgroups

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

subgroupindex
#131177572
#131148322
#131127382
#131126952
C13×#14612
#876713