Finite Group Databasebeta

← #15751659

Subgroups — #15751659

Order 3328.

536 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 95 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#13116070C212yesmaximal
2#13115923C211yesmaximal, characteristic
3#13112662C211yesmaximal, characteristic, complemented
4#13112659C212yesmaximal, complemented
5C13×#2734C211yesmaximal, characteristic, complemented
6C2×C13×C4⋊Q8—21no—
7C22×#6106—21no—
8C2×#88128C2211yescharacteristic
9C13×#398—21no—
10C13×#395—22no—
11C13×#391C2211yescharacteristic
12C13×#389C2212yes—
13C13×#387—22no—
14C2×#88049—21no—
15C2×#88049—22no—
16C2×#88049—22no—
17#11718116C2211yescharacteristic
18#11718116C2212yes—
19C22×Q8×C13—21no—
20C22×Q8×C13D411yescharacteristic
21C22×D52—21no—
22C22×D52—22no—
23C13×C4⋊Q8—42no—
24C13×C4⋊Q8—42no—
25C2×C13×C4⋊C4—22no—
26C2×C13×C4⋊C4—22no—
27C2×C13×C4⋊C4—21no—
28C2×C13×C4⋊C4D412yes—
29C2×C13×C4⋊C4D412yes—
30C2×C13×C4⋊C4D411yescharacteristic
31C2×C42×C13—21no—
32C2×C42×C13C2311yescharacteristic
33C2×#6106—42no—
34C2×#6106—42no—
35C2×#6106—42no—
36C22×#1055—21no—
37C22×#1055—22no—
38#88128—42no—
39C13×#95—21no—
40C13×#95—22no—
41C13×#95—22no—
42C13×#95—22no—
43#88049—42no—
44#88049—44no—
45#88049—44no—
46#7741—131nomaximal, Sylow
47C23×D13—41no—
48C2×Q8×C13—21no—
49C2×Q8×C13—42no—
50C2×Q8×C13—41no—
51C2×Q8×C13—41no—
52C2×Q8×C13—81no—
53C2×Q8×C13—81no—
54C2×Q8×C13D812yes—
55C22×C4×C13—22no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—22no—
59C22×C4×C13C2×D411yescharacteristic
60C22×C4×C13C2×D412yes—
61C2×D52—22no—
62C2×D52—24no—
63C2×D52—41no—
64C2×D52—42no—
65C2×D52—42no—
66C2×D52—41no—
67C2×D52—41no—
68C2×D52—42no—
69C2×D52—41no—
70C2×D52—42no—
71C2×D52—42no—
72C2×D52—41no—
73C13×C4⋊C4—22no—
74C13×C4⋊C4—41no—
75C13×C4⋊C4—42no—
76C13×C4⋊C4—44no—
77C13×C4⋊C4—42no—
78C13×C4⋊C4—42no—
79C13×C4⋊C4—81no—
80C13×C4⋊C4SD1614yescomplemented
81C42×C13—41no—
82C42×C13—42no—
83#6106—48no—
84C2×#1055—22no—
85C2×#1055—41no—
86C2×#1055—41no—
87C2×#1055—42no—
88C2×#1055—42no—
89C2×#1055—42no—
90#2734D1311yescharacteristic, complemented
91#2041—132no—
92#2014—131no—
93#1272—131no—
94#1271—132no—
95C23×C13#12011yescharacteristic
96C22×D13—42no—
97C22×D13—81no—
98C22×D13—82no—
99C22×D13—81no—
100C22×D13—82no—
101Q8×C13—42no—
102Q8×C13—81no—
103Q8×C13—84no—
104C2×C4×C13—21no—
105C2×C4×C13—22no—
106C2×C4×C13—22no—
107C2×C4×C13—21no—
108C2×C4×C13—22no—
109C2×C4×C13—24no—
110C2×C4×C13—42no—
111C2×C4×C13—41no—
112C2×C4×C13—42no—
113C2×C4×C13—41no—
114C2×C4×C13—42no—
115C2×C4×C13—41no—
116C2×C4×C13—42no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13C2×D811yescharacteristic
120C2×C4×C13C2×SD1612yes—
121C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C13#13712yes—
123D52—42no—
124D52—42no—
125D52—44no—
126D52—44no—
127D52—81no—
128D52—81no—
129D52—82no—
130D52—82no—
131#1055—42no—
132#1055—82no—
133C22×SD16—261no—
134C2×C4⋊Q8—21no—
135C2×#127—131no—
136C2×#102—261no—
137C2×#102—262no—
138C2×#102—262no—
139#398—21no—
140#395—22no—
141#391C2×D1311yescharacteristic
142#389C2×D1312yes—
143#387—22no—
144#336—131no—
145#336—132no—
146C22×C13#44711yescharacteristic
147C22×C13#44812yes—
148C22×C13#45311yescharacteristic
149C22×C13#45412yes—
150C22×C13B(4,𝔽2)11yescharacteristic
151C2×D13—82no—
152C2×D13—84no—
153C2×D13—161no—
154C2×D13—162no—
155C2×D13—161no—
156C4×C13—21no—
157C4×C13—21no—
158C4×C13—22no—
159C4×C13—22no—
160C4×C13—42no—
161C4×C13—44no—
162C4×C13—82no—
163C4×C13—81no—
164C22×Q8—21no—
165C22×Q8C22⋊D1311yescharacteristic
166C22×D4—261no—
167C22×D4—262no—
168C2×SD16—522no—
169C2×SD16—522no—
170C2×SD16—522no—
171C22×C8—261no—
172C22×C8—262no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175#127—522no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C22⋊D1312yes—
180C2×C4⋊C4C22⋊D1312yes—
181C2×C4⋊C4C22⋊D1311yescharacteristic
182C2×C42—21no—
183C2×C42C22×D1311yescharacteristic
184#102—522no—
185#102—524no—
186#102—524no—
187#95—21no—
188#95—22no—
189#95—22no—
190#95—22no—
191C2×C13#159511yescharacteristic
192C2×C13#159912yes—
193C2×C13#162311yescharacteristic
194C2×C13#162712yes—
195C2×C13#165711yescharacteristic
196D13—162no—
197C24—521no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8#610412yes—
205C2×D4—264no—
206C2×D4—262no—
207C2×D4—521no—
208C2×D4—521no—
209C2×D4—522no—
210C2×D4—522no—
211C2×D4—522no—
212C2×D4—521no—
213C2×D4—521no—
214C2×D4—521no—
215C2×D4—522no—
216C2×D4—522no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4C2×C22⋊D1311yescharacteristic
222C22×C4C2×C22⋊D1312yes—
223SD16—528no—
224C2×C8—262no—
225C2×C8—521no—
226C2×C8—521no—
227C2×C8—522no—
228C2×C8—522no—
229C2×C8—522no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—81no—
237C4⋊C4#610614yescomplemented
238C42—41no—
239C42—42no—
240C13#774111yescharacteristic, complemented, Sylow
241C23—522no—
242C23—1041no—
243C23—1042no—
244C23—1041no—
245C23—1042no—
246C23#8820711yescharacteristic
247Q8—42no—
248Q8—81no—
249Q8—84no—
250D4—522no—
251D4—522no—
252D4—524no—
253D4—524no—
254D4—1041no—
255D4—1041no—
256D4—1042no—
257D4—1042no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4C2×#610411yescharacteristic
274C2×C4#8818611yescharacteristic
275C2×C4C2×#610612yes—
276C2×C4#8819612yes—
277C8—522no—
278C8—1042no—
279C22—1042no—
280C22—1044no—
281C22—2081no—
282C22—2081no—
283C22—2082no—
284C22#1171887811yescharacteristic
285C22#1171888812yes—
286C22#1171890211yescharacteristic
287C22#1171890511yescharacteristic
288C22#1171890612yes—
289C4—21no—
290C4—21no—
291C4—22no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4—81no—
296C4—82no—
297C2—2082no—
298C2#1311375411yescharacteristic
299C2#1311377612yes—
300C2#1311391411yescharacteristic
301C2#1311393912yes—
302C2#1311413411yescharacteristic
subgroupG/H
centerC23#88207
commutator, cosocleC2×C42×C13C23
FrattiniC2×C42C22×D13
FittingC13×#2734C2
socleC23×C13#120
prime radicalC2×#88128C22
hypercenter#2734D13
hypocenterC13#7741
LCSG ▷ C2×C42×C13 ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
3. #13112662C2semidirect
4. #13112659C2semidirect
5. C13×#2734C2semidirect
80. C13×C4⋊C4SD16semidirect
90. #2734D13semidirect
237. C4⋊C4#6106semidirect
240. C13#7741semidirect

Maximal subgroups

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

subgroupindex
#131160702
#131159232
#131126622
#131126592
C13×#27342
#774113