Finite Group Databasebeta

← #15765806

Subgroups — #15765806

Order 3328.

602 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 103 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#13125611C211yesmaximal, characteristic
2#13123175C212yesmaximal
3#13113409C212yesmaximal
4#13112704C211yesmaximal, characteristic
5C13×#1817C211yesmaximal, characteristic
6C22×C42×C13C2211yescharacteristic
7C2×#88178C2212yes—
8C2×C13×C22⋊C8C2211yescharacteristic
9C2×C13×C22⋊C8C2211yescharacteristic
10#11718873C2212yes—
11#11718872—22no—
12#11718871—22no—
13#11718777—22no—
14#11718594—24no—
15#11718590—22no—
16#11718589—22no—
17#11718400—22no—
18#11718383—22no—
19C13×#336—22no—
20#11718059—24no—
21C23×C4×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13C2311yescharacteristic
24C22×Dic26D412yes—
25C22×C8×C13—21no—
26C22×C8×C13D412yes—
27C2×C42×C13—21no—
28C2×C42×C13—22no—
29C2×C42×C13—22no—
30C2×C42×C13—21no—
31C2×C22⋊Dic13—22no—
32C2×C22⋊Dic13—22no—
33C2×C4⋊Dic13—22no—
34C2×C4⋊Dic13D412yes—
35#88178—22no—
36#88178—22no—
37#88178—22no—
38#88178—22no—
39C2×#6101—24no—
40C2×#6101—22no—
41C2×Dic52—22no—
42C13×C22⋊C8—22no—
43C13×C22⋊C8—22no—
44#88071—24no—
45#88071—24no—
46#88071—24no—
47#88071D414yes—
48#88058—22no—
49#88056—24no—
50#10325—131nomaximal, Sylow
51C24×C13C2×D411yescharacteristic
52C22×C4×C13—22no—
53C22×C4×C13—22no—
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13—22no—
58C22×C4×C13—42no—
59C22×C4×C13—42no—
60C22×C4×C13Q1612yescomplemented
61C22×C4×C13C2×D412yes—
62C22×C4×C13C2×D412yes—
63C22×C4×C13C2×D411yescharacteristic
64C22×C4×C13#4111yescharacteristic
65C22×Dic13—22no—
66C22×Dic13—24no—
67C22×Dic13—22no—
68C2×Dic26—22no—
69C2×Dic26—22no—
70C2×Dic26—44no—
71C2×C8×C13—22no—
72C2×C8×C13—42no—
73C2×C8×C13—42no—
74C2×C8×C13—41no—
75C42×C13—41no—
76C42×C13—42no—
77C42×C13—41no—
78C22⋊Dic13—42no—
79C22⋊Dic13—42no—
80C22⋊Dic13—42no—
81C22⋊Dic13—42no—
82C4⋊Dic13—22no—
83C4⋊Dic13—22no—
84C4⋊Dic13—44no—
85#6101—48no—
86#6101—42no—
87#6101—42no—
88Dic52—82no—
89#3189—131no—
90#2855—132no—
91#1817D1311yescharacteristic
92#1520—132no—
93#1289—131no—
94C23×C13—41no—
95C23×C13—41no—
96C23×C13#12012yes—
97C23×C13#12111yescharacteristic
98C23×C13C2×Q1611yescharacteristic
99C23×C13#13511yescharacteristic
100C23×C13#13712yes—
101C2×C4×C13—24no—
102C2×C4×C13—22no—
103C2×C4×C13—24no—
104C2×C4×C13—22no—
105C2×C4×C13—24no—
106C2×C4×C13—22no—
107C2×C4×C13—42no—
108C2×C4×C13—41no—
109C2×C4×C13—42no—
110C2×C4×C13—41no—
111C2×C4×C13—41no—
112C2×C4×C13—41no—
113C2×C4×C13—42no—
114C2×C4×C13—41no—
115C2×C4×C13#12011yescharacteristic
116C2×C4×C13#12112yes—
117C2×C4×C13#12711yescharacteristic
118C2×Dic13—44no—
119C2×Dic13—44no—
120C2×Dic13—44no—
121C2×Dic13—44no—
122C2×Dic13—42no—
123C2×Dic13—42no—
124C2×Dic13—44no—
125Dic26—44no—
126Dic26—82no—
127C8×C13—42no—
128C8×C13—81no—
129C2×C22⋊Q8—132no—
130C22×C42C2×D1311yescharacteristic
131#484—262no—
132#467—262no—
133#451—262no—
134C2×C22⋊C8C2×D1311yescharacteristic
135C2×C22⋊C8C2×D1311yescharacteristic
136#391—262no—
137#389—262no—
138#387—264no—
139#385—262no—
140#381—262no—
141#380—132no—
142#336—22no—
143#328—264no—
144C22×C13—22no—
145C22×C13—21no—
146C22×C13—22no—
147C22×C13—21no—
148C22×C13—41no—
149C22×C13—41no—
150C22×C13—42no—
151C22×C13#39211yescharacteristic
152C22×C13#45112yes—
153C22×C13#45212yes—
154C22×C13#46711yescharacteristic
155C22×C13#47011yescharacteristic
156C4×C13—21no—
157C4×C13—22no—
158C4×C13—42no—
159C4×C13—42no—
160C4×C13#45812yes—
161Dic13—84no—
162Dic13—82no—
163Dic13—82no—
164C22×Q8—132no—
165C23×C4D5211yescharacteristic
166C23×C4D5211yescharacteristic
167C23×C4C22×D1311yescharacteristic
168C2×Q16—262no—
169C22×C8—21no—
170C22×C8C22⋊D1312yes—
171C22⋊Q8—262no—
172C22⋊Q8—262no—
173C22⋊Q8—262no—
174C22⋊Q8—262no—
175C2×C4⋊C4—132no—
176C2×C4⋊C4—264no—
177C2×C4⋊C4—262no—
178C2×C4⋊C4—262no—
179C2×C22⋊C4—262no—
180C2×C22⋊C4—262no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185#107—262no—
186#103—264no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189#95—134no—
190#95—264no—
191#95—264no—
192#95—264no—
193C2×C13—22no—
194C2×C13—22no—
195C2×C13—21no—
196C2×C13—21no—
197C2×C13#200111yescharacteristic
198C2×C13#201011yescharacteristic
199C2×C13#202211yescharacteristic
200C24C2×D5211yescharacteristic
201C2×Q8—262no—
202C2×Q8—262no—
203C2×Q8—524no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—264no—
213C22×C4—262no—
214C22×C4—262no—
215C22×C4Dic5212yescomplemented
216C22×C4#612711yescharacteristic
217C22×C4D4×D1312yes—
218C22×C4D4×D1312yes—
219C22×C4C2×C22⋊D1311yescharacteristic
220Q16—1042no—
221C2×C8—22no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—41no—
225C4⋊C4—262no—
226C4⋊C4—262no—
227C4⋊C4—528no—
228C4⋊C4—524no—
229C4⋊C4—522no—
230C4⋊C4—522no—
231C22⋊C4—522no—
232C22⋊C4—522no—
233C22⋊C4—522no—
234C22⋊C4—522no—
235C42—42no—
236C42—41no—
237C42—41no—
238C13#1032511yescharacteristic, complemented, Sylow
239C23—41no—
240C23—41no—
241C23#8813612yes—
242C23#8815811yescharacteristic
243C23C2×Dic5211yescharacteristic
244C23#8816312yes—
245C23#8818411yescharacteristic
246Q8—524no—
247Q8—1042no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—524no—
263C2×C4—524no—
264C2×C4—524no—
265C2×C4—524no—
266C2×C4—522no—
267C2×C4—522no—
268C2×C4—524no—
269C2×C4#8813812yes—
270C2×C4#8819111yescharacteristic
271C2×C4#8819411yescharacteristic
272C8—42no—
273C8—81no—
274C22—22no—
275C22—21no—
276C22—22no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22#1171839412yes—
282C22#1171840012yes—
283C22#1171862111yescharacteristic
284C22#1171877711yescharacteristic
285C22#1171879811yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—1044no—
291C4—1042no—
292C4—1042no—
293C4#1171871412yes—
294C2—22no—
295C2—22no—
296C2—21no—
297C2—21no—
298C2#1311584611yescharacteristic
299C2#1311590311yescharacteristic
300C2#1311596711yescharacteristic
subgroupG/H
centerC22#11718621
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
FittingC13×#1817C2
socleC22×C13#392
prime radicalC24×C13C2×D4
hypercenter#1817D13
hypocenterC13#10325
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
60. C22×C4×C13Q16semidirect
215. C22×C4Dic52semidirect
238. C13#10325semidirect

Maximal subgroups

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

subgroupindex
#131256112
#131231752
#131134092
#131127042
C13×#18172
#1032513