Finite Group Databasebeta

← #15756490

Subgroups — #15756490

Order 3328.

498 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 131 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#13114828C211yesmaximal, characteristic
2#13113157C211yesmaximal, characteristic
3#13112739C212yesmaximal, complemented
4#13112694C212yesmaximal, complemented
5C13×#2349C211yesmaximal, characteristic
6C2×C4×D4×C13—21no—
7C2×C4×D4×C13C2211yescharacteristic
8C13×#389—21no—
9C13×#385—22no—
10C13×#382C2211yescharacteristic
11C13×#380C412yes—
12C4×C13×C4⋊C4C2211yescharacteristic
13C2×C13×#95C412yes—
14C2×#88073—21no—
15#11718586C2212yes—
16#11718585C2212yes—
17#11718122—21no—
18#11718117—22no—
19C22×D4×C13D411yescharacteristic
20C23×C4×C13—22no—
21C23×C4×C13C2×C412yes—
22C4×D4×C13—22no—
23C4×D4×C13—22no—
24C4×D4×C13—44no—
25C2×C13×C4⋊C4—21no—
26C2×C13×C4⋊C4—21no—
27C2×C13×C4⋊C4—22no—
28C2×C13×C4⋊C4D411yescharacteristic
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4D412yes—
31C2×C42×C13—21no—
32C2×C42×C13C2311yescharacteristic
33C2×#6099—22no—
34C13×#95—22no—
35C13×#95—21no—
36C13×#95C2×C411yescharacteristic
37C13×#95C2×C411yescharacteristic
38C13×#95C2×C412yes—
39#88073—42no—
40C22⋊#1055—22no—
41C4⋊#1055—22no—
42C4⋊#1055—22no—
43C4⋊#1055—22no—
44C4×#1055—22no—
45#8765—131nomaximal, Sylow
46C24×C13C22⋊C412yes—
47C2×D4×C13—22no—
48C2×D4×C13—22no—
49C2×D4×C13SD1612yes—
50C2×D4×C13Q1612yes—
51C22×C4×C13—21no—
52C22×C4×C13—22no—
53C22×C4×C13—22no—
54C22×C4×C13—22no—
55C22×C4×C13—21no—
56C22×C4×C13—44no—
57C22×C4×C13—44no—
58C22×C4×C13—82no—
59C22×C4×C13—82no—
60C22×C4×C13C22⋊C412yes—
61C22×C4×C13C22×C411yescharacteristic
62C22×C4×C13C2×D411yescharacteristic
63C22×C4×C13C2×D411yescharacteristic
64C13×C4⋊C4—22no—
65C13×C4⋊C4—42no—
66C13×C4⋊C4—42no—
67C13×C4⋊C4—82no—
68C13×C22⋊C4—24no—
69C13×C22⋊C4—44no—
70C42×C13—42no—
71C42×C13#4112yes—
72C42×C13#4112yes—
73#6099—42no—
74#6099—42no—
75C2×#1055—42no—
76C2×#1055—42no—
77#2349D1311yescharacteristic
78#1800—131no—
79#1460—131no—
80#1303—132no—
81#1286—132no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13—42no—
85C23×C13—42no—
86C23×C13#10314yes—
87C23×C13C2×C22⋊C411yescharacteristic
88D4×C13—24no—
89D4×C13—24no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—41no—
93C2×C4×C13—41no—
94C2×C4×C13—41no—
95C2×C4×C13—41no—
96C2×C4×C13—42no—
97C2×C4×C13—41no—
98C2×C4×C13—41no—
99C2×C4×C13—42no—
100C2×C4×C13—42no—
101C2×C4×C13—41no—
102C2×C4×C13—42no—
103C2×C4×C13—82no—
104C2×C4×C13—82no—
105C2×C4×C13—88no—
106C2×C4×C13#11711yescharacteristic
107C2×C4×C13#11711yescharacteristic
108C2×C4×C13#12312yes—
109C2×C4×C13#12312yes—
110C2×C4×C13C2×SD1611yescharacteristic
111C2×C4×C13C2×Q1611yescharacteristic
112C2×C4×C13AGL(1,ℤ/8)12yes—
113#1055—82no—
114#1055—82no—
115C2×C4×D4—21no—
116C2×C4×D4C2×D1311yescharacteristic
117#431—132no—
118#423—132no—
119C2×#100—261no—
120#389—21no—
121#385—22no—
122#382C2×D1311yescharacteristic
123#380Dic1312yes—
124C4×C4⋊C4C2×D1311yescharacteristic
125C2×#95Dic1312yes—
126#343—261no—
127#323—262no—
128C22×C13—21no—
129C22×C13—21no—
130C22×C13—24no—
131C22×C13—44no—
132C22×C13—44no—
133C22×C13—42no—
134C22×C13—42no—
135C22×C13#38111yescharacteristic
136C22×C13C2×#10311yescharacteristic
137C22×C13#41811yescharacteristic
138C4×C13—22no—
139C4×C13—21no—
140C4×C13—21no—
141C4×C13—42no—
142C4×C13—42no—
143C4×C13—81no—
144C4×C13—82no—
145C4×C13#48112yes—
146C4×C13#48412yes—
147C22×D4C22⋊D1311yescharacteristic
148C23×C4—22no—
149C23×C4C2×Dic1312yes—
150C2×M16—262no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊D1311yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4C22⋊D1312yes—
160C2×C42—21no—
161C2×C42C22×D1311yescharacteristic
162C4⋊C8—262no—
163C4⋊C8—262no—
164C4⋊C8—262no—
165#100—522no—
166C22⋊C8—262no—
167C4×C8—262no—
168#95—22no—
169#95—21no—
170#95C2×Dic1311yescharacteristic
171#95C2×Dic1312yes—
172#95C2×Dic1311yescharacteristic
173C2×C13—21no—
174C2×C13—21no—
175C2×C13—44no—
176C2×C13#182411yescharacteristic
177C2×C13#182711yescharacteristic
178C2×C13#182911yescharacteristic
179C24C22⋊Dic1312yes—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4#610612yes—
183C2×D4#610712yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4C22⋊Dic1312yes—
194C22×C4C22×Dic1311yescharacteristic
195C22×C4C2×C22⋊D1311yescharacteristic
196C22×C4C2×C22⋊D1311yescharacteristic
197M16—522no—
198M16—522no—
199C2×C8—522no—
200C2×C8—522no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C22⋊C4—24no—
206C22⋊C4—44no—
207C42—42no—
208C42#612712yes—
209C42#612712yes—
210C13#876511yescharacteristic, complemented, Sylow
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#8807514yes—
216C23C2×C22⋊Dic1311yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#8818011yescharacteristic
236C2×C4#8818011yescharacteristic
237C2×C4#8818312yes—
238C2×C4#8818312yes—
239C2×C4#8818612yes—
240C2×C4C2×#610611yescharacteristic
241C2×C4C2×#610711yescharacteristic
242C8—1042no—
243C8—1042no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#1171887211yescharacteristic
252C22#1171887611yescharacteristic
253C22C2×#8807511yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—81no—
260C4—82no—
261C4#1171862912yes—
262C4#1171863112yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#1311492611yescharacteristic
267C2#1311494011yescharacteristic
268C2#1311494711yescharacteristic
subgroupG/H
centerC22#11718872
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#2349C2
socleC22×C13#381
prime radicalC22×D4×C13D4
cosocleC2×C42×C13C23
hypercenter#2349D13
hypocenterC13#8765
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #13112739C2semidirect
4. #13112694C2semidirect
210. C13#8765semidirect

Maximal subgroups

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

subgroupindex
#131148282
#131131572
#131127392
#131126942
C13×#23492
#876513