Finite Group Databasebeta

← #15760759

Subgroups — #15760759

Order 3328.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#13123860C211yesmaximal, characteristic
2#13114828C211yesmaximal, characteristic
3#13113265C212yesmaximal, complemented
4#13112694C212yesmaximal, complemented
5C13×#2580C211yesmaximal, characteristic
6C2×C4×Q8×C13C2211yescharacteristic
7C2×C4×D4×C13C2211yescharacteristic
8C13×#399C412yes—
9C13×#393C412yes—
10C13×#390—21no—
11C13×#389—21no—
12C13×#385—22no—
13C13×#382C2211yescharacteristic
14C2×#88073—21no—
15C2×C22⋊#1055—22no—
16D4×#1055—24no—
17#11718586C2212yes—
18#11718585C2212yes—
19Q8⋊#1055—24no—
20#11718122—21no—
21C22×Q8×C13C2×C411yescharacteristic
22C22×D4×C13D411yescharacteristic
23C23×C4×C13D412yes—
24C4×Q8×C13—22no—
25C4×Q8×C13—22no—
26C4×D4×C13—22no—
27C4×D4×C13—22no—
28C2×C13×C4⋊C4—21no—
29C2×C13×C4⋊C4—21no—
30C2×C13×C4⋊C4—22no—
31C2×C13×C4⋊C4C2×C411yescharacteristic
32C2×C13×C4⋊C4D411yescharacteristic
33C2×C13×C22⋊C4—22no—
34C2×C13×C22⋊C4—22no—
35C2×C13×C22⋊C4C2×C412yes—
36C2×C42×C13—21no—
37C2×C42×C13C2311yescharacteristic
38C2×#6099—22no—
39C22×#1055—24no—
40C13×#95—22no—
41C13×#95—21no—
42C13×#95C2×C412yes—
43#88073—42no—
44C22⋊#1055—24no—
45C22⋊#1055—22no—
46C4⋊#1055—22no—
47C4⋊#1055—22no—
48C4⋊#1055—22no—
49C4×#1055D414yes—
50#9483—131nomaximal, Sylow
51C24×C13C22⋊C412yes—
52C2×Q8×C13—22no—
53C2×Q8×C13—42no—
54C2×D4×C13—22no—
55C2×D4×C13—22no—
56C2×D4×C13SD1612yes—
57C2×D4×C13Q1612yes—
58C22×C4×C13—21no—
59C22×C4×C13—22no—
60C22×C4×C13—22no—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—82no—
64C22×C4×C13C22⋊C412yes—
65C22×C4×C13C22×C411yescharacteristic
66C22×C4×C13C2×D411yescharacteristic
67C22×C4×C13C2×D411yescharacteristic
68C13×C4⋊C4—22no—
69C13×C4⋊C4—22no—
70C13×C4⋊C4—41no—
71C13×C4⋊C4—41no—
72C13×C4⋊C4—42no—
73C13×C4⋊C4—82no—
74C13×C22⋊C4—24no—
75C13×C22⋊C4—44no—
76C13×C22⋊C4—82no—
77C42×C13—42no—
78C42×C13C2×D412yes—
79C42×C13#4112yes—
80#6099—42no—
81#6099—42no—
82C2×#1055—24no—
83C2×#1055—42no—
84C2×#1055—84no—
85#2972—131no—
86#2580D1311yescharacteristic
87#1800—131no—
88#1487—132no—
89#1286—132no—
90C23×C13—22no—
91C23×C13—22no—
92C23×C13—42no—
93C23×C13—42no—
94C23×C13#10314yes—
95C23×C13C2×C22⋊C411yescharacteristic
96Q8×C13—24no—
97Q8×C13—42no—
98D4×C13—24no—
99D4×C13—24no—
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13—41no—
103C2×C4×C13—41no—
104C2×C4×C13—44no—
105C2×C4×C13—41no—
106C2×C4×C13—41no—
107C2×C4×C13—42no—
108C2×C4×C13—42no—
109C2×C4×C13—41no—
110C2×C4×C13—42no—
111C2×C4×C13—41no—
112C2×C4×C13—41no—
113C2×C4×C13—42no—
114C2×C4×C13—82no—
115C2×C4×C13C4×D412yes—
116C2×C4×C13#12011yescharacteristic
117C2×C4×C13#12112yes—
118C2×C4×C13#12311yescharacteristic
119C2×C4×C13C2×SD1611yescharacteristic
120C2×C4×C13C2×Q1611yescharacteristic
121C2×C4×C13#13512yes—
122#1055—44no—
123#1055—82no—
124C2×C4×Q8C2×D1311yescharacteristic
125C2×C4×D4C2×D1311yescharacteristic
126C8×D4—264no—
127#431—132no—
128#423—132no—
129C2×#100—261no—
130C2×C22⋊C8—262no—
131#399Dic1312yes—
132#393Dic1312yes—
133#390—21no—
134#389—21no—
135#385—22no—
136#382C2×D1311yescharacteristic
137#343—261no—
138Q8⋊C8—264no—
139C22×C13—21no—
140C22×C13—21no—
141C22×C13—24no—
142C22×C13—44no—
143C22×C13—42no—
144C22×C13—42no—
145C22×C13—82no—
146C22×C13#38611yescharacteristic
147C22×C13C2×#10311yescharacteristic
148C22×C13#41611yescharacteristic
149C4×C13—22no—
150C4×C13—21no—
151C4×C13—21no—
152C4×C13—42no—
153C4×C13—42no—
154C4×C13—82no—
155C4×C13—81no—
156C4×C13#46512yes—
157C4×C13#46712yes—
158C22×Q8C2×Dic1311yescharacteristic
159C22×D4C22⋊D1311yescharacteristic
160C23×C4C22⋊D1312yes—
161C2×M16—262no—
162C22×C8—264no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic1311yescharacteristic
171C2×C4⋊C4C22⋊D1311yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C2×Dic1312yes—
175C2×C42—21no—
176C2×C42C22×D1311yescharacteristic
177C4⋊C8—262no—
178C4⋊C8—262no—
179C4⋊C8—262no—
180#100—522no—
181C22⋊C8—264no—
182C22⋊C8—262no—
183C4×C8—134no—
184#95—22no—
185#95—21no—
186#95C2×Dic1312yes—
187C2×C13—21no—
188C2×C13—21no—
189C2×C13—44no—
190C2×C13#189111yescharacteristic
191C2×C13#189211yescharacteristic
192C2×C13#191111yescharacteristic
193C24C22⋊Dic1312yes—
194C2×Q8—22no—
195C2×Q8—42no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4#610612yes—
199C2×D4#610712yes—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—82no—
206C22×C4C22⋊Dic1312yes—
207C22×C4C22×Dic1311yescharacteristic
208C22×C4C2×C22⋊D1311yescharacteristic
209C22×C4C2×C22⋊D1311yescharacteristic
210M16—522no—
211M16—522no—
212C2×C8—264no—
213C2×C8—522no—
214C2×C8—1044no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C22⋊C4—24no—
222C22⋊C4—44no—
223C22⋊C4—82no—
224C42—42no—
225C42D4×D1312yes—
226C42#612912yes—
227C13#948311yescharacteristic, complemented, Sylow
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#8807514yes—
233C23C2×C22⋊Dic1311yescharacteristic
234Q8—24no—
235Q8—42no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4D4×Dic1312yes—
254C2×C4#8818911yescharacteristic
255C2×C4#8819111yescharacteristic
256C2×C4#8819312yes—
257C2×C4C2×#610611yescharacteristic
258C2×C4C2×#610711yescharacteristic
259C2×C4#8820412yes—
260C8—524no—
261C8—1042no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1171888111yescharacteristic
270C22C2×#8807511yescharacteristic
271C22#1171889411yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#1171870912yes—
280C4#1171871212yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#1311525511yescharacteristic
285C2#1311526011yescharacteristic
286C2#1311536911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#2580C2
socleC22×C13#386
prime radicalC22×D4×C13D4
cosocleC2×C42×C13C23
hypercenter#2580D13
hypocenterC13#9483
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #13113265C2semidirect
4. #13112694C2semidirect
227. C13#9483semidirect

Maximal subgroups

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

subgroupindex
#131238602
#131148282
#131132652
#131126942
C13×#25802
#948313