Finite Group Databasebeta

← #15751492

Subgroups — #15751492

Order 3328.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#13123188C211yesmaximal, characteristic
2#13114598C211yesmaximal, characteristic
3#13112661C214yesmaximal, complemented
4C13×#1817C211yesmaximal, characteristic
5C22×C42×C13C2211yescharacteristic
6C2×#88178C412yes—
7C2×#88123C411yescharacteristic
8C2×C13×C22⋊C8—21no—
9#11718873—21no—
10#11718873—21no—
11C2×C22⋊#1055—21no—
12#11718596C411yescharacteristic
13#11718590C2212yes—
14#11718589—21no—
15#11718589—21no—
16C2×#88043—21no—
17C13×#336—21no—
18C13×#336C2212yes—
19#11718116C2212yes—
20C23×C4×C13C2×C411yescharacteristic
21C23×C4×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C22×Dic26C2×C412yes—
24C22×C8×C13—22no—
25C2×C42×C13—42no—
26C2×C42×C13C2×C411yescharacteristic
27C2×C42×C13D411yescharacteristic
28C2×C42×C13D411yescharacteristic
29C2×C42×C13C2311yescharacteristic
30C2×C22⋊Dic13—22no—
31C2×C4⋊Dic13—21no—
32C2×C4⋊Dic13—21no—
33C2×C4⋊Dic13—21no—
34C2×C4⋊Dic13—21no—
35C2×C4⋊Dic13C2×C412yes—
36#88178—44no—
37C2×#6101—22no—
38C22×#1055—22no—
39#88123—42no—
40C13×C22⋊C8—42no—
41#88071—22no—
42#88071—22no—
43C22⋊#1055—24no—
44#88043—42no—
45#7707—131nomaximal, Sylow
46C24×C13C22⋊C411yescharacteristic
47C22×C4×C13—22no—
48C22×C4×C13—22no—
49C22×C4×C13—22no—
50C22×C4×C13—22no—
51C22×C4×C13—22no—
52C22×C4×C13—22no—
53C22×C4×C13—41no—
54C22×C4×C13—41no—
55C22×C4×C13—41no—
56C22×C4×C13C22⋊C411yescharacteristic
57C22×C4×C13C22⋊C411yescharacteristic
58C22×C4×C13C22⋊C411yescharacteristic
59C22×C4×C13C22×C411yescharacteristic
60C22×C4×C13C2×D411yescharacteristic
61C22×C4×C13C2×D411yescharacteristic
62C22×Dic13—22no—
63C22×Dic13—22no—
64C2×Dic26—24no—
65C2×Dic26—42no—
66C2×Dic26—42no—
67C2×C8×C13—42no—
68C2×C8×C13—42no—
69C2×C8×C13—42no—
70C42×C13—22no—
71C42×C13—22no—
72C42×C13—41no—
73C42×C13—41no—
74C22⋊Dic13—44no—
75C4⋊Dic13—42no—
76C4⋊Dic13—42no—
77C4⋊Dic13—42no—
78C4⋊Dic13—42no—
79C4⋊Dic13—42no—
80#6101—44no—
81C2×#1055—24no—
82C2×#1055—42no—
83C2×#1055—42no—
84#2857—131no—
85#1817D1311yescharacteristic
86#1749—131no—
87#1272—134no—
88C23×C13—21no—
89C23×C13—21no—
90C23×C13—21no—
91C23×C13—21no—
92C23×C13—21no—
93C23×C13—21no—
94C23×C13#10012yes—
95C23×C13C2×C22⋊C411yescharacteristic
96C2×C4×C13—21no—
97C2×C4×C13—21no—
98C2×C4×C13—21no—
99C2×C4×C13—21no—
100C2×C4×C13—21no—
101C2×C4×C13—21no—
102C2×C4×C13—21no—
103C2×C4×C13—21no—
104C2×C4×C13—21no—
105C2×C4×C13—21no—
106C2×C4×C13—21no—
107C2×C4×C13—21no—
108C2×C4×C13—21no—
109C2×C4×C13—44no—
110C2×C4×C13—44no—
111C2×C4×C13—44no—
112C2×C4×C13#9912yes—
113C2×C4×C13#10412yes—
114C2×C4×C13#13112yes—
115C2×C4×C13#13712yes—
116C2×C4×C13#13712yes—
117C2×Dic13—42no—
118C2×Dic13—42no—
119C2×Dic13—42no—
120C2×Dic13—42no—
121C2×Dic13—42no—
122C2×Dic13—42no—
123Dic26—84no—
124C8×C13—82no—
125#1055—44no—
126C2×C4⋊Q8—131no—
127C2×C22⋊Q8—132no—
128C22×C42C2×D1311yescharacteristic
129C2×C22⋊C8—21no—
130C2×C22⋊C8—261no—
131C2×#97—261no—
132#391—132no—
133#389—261no—
134#389—261no—
135#384—131no—
136#380—261no—
137#380—261no—
138#336—21no—
139#336—132no—
140#336C2×D1312yes—
141C22×C13—22no—
142C22×C13—22no—
143C22×C13—22no—
144C22×C13—41no—
145C22×C13—41no—
146C22×C13—41no—
147C22×C13—41no—
148C22×C13#40811yescharacteristic
149C22×C13C2×#9911yescharacteristic
150C22×C13C2×#10011yescharacteristic
151C22×C13#41911yescharacteristic
152C22×C13#41911yescharacteristic
153C22×C13C2×#10411yescharacteristic
154C22×C13#42111yescharacteristic
155C4×C13—22no—
156C4×C13—22no—
157C4×C13—22no—
158C4×C13—22no—
159C4×C13—22no—
160C4×C13—41no—
161Dic13—82no—
162Dic13—82no—
163C22×Q8—132no—
164C23×C4C4×D1311yescharacteristic
165C23×C4D5211yescharacteristic
166C23×C4C22⋊D1311yescharacteristic
167C22×C8—22no—
168C22×C8—262no—
169C4⋊Q8—522no—
170C22⋊Q8—524no—
171C2×C4⋊C4—132no—
172C2×C4⋊C4—262no—
173C2×C4⋊C4—261no—
174C2×C4⋊C4—261no—
175C2×C4⋊C4—261no—
176C2×C4⋊C4—261no—
177C2×C22⋊C4—262no—
178C2×C42—42no—
179C2×C42C4×D1311yescharacteristic
180C2×C42D5211yescharacteristic
181C2×C42C22⋊D1311yescharacteristic
182C2×C42C22×D1311yescharacteristic
183C22⋊C8—42no—
184C22⋊C8—264no—
185#97—522no—
186#95—262no—
187#95—262no—
188C2×C13—42no—
189C2×C13#147111yescharacteristic
190C2×C13#148511yescharacteristic
191C2×C13#149711yescharacteristic
192C2×C13#150811yescharacteristic
193C2×C13#151111yescharacteristic
194C2×C13#153011yescharacteristic
195C2×C13#155811yescharacteristic
196C24#610311yescharacteristic
197C2×Q8—264no—
198C2×Q8—522no—
199C2×Q8—522no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—262no—
210C22×C4—262no—
211C22×C4#610311yescharacteristic
212C22×C4#610311yescharacteristic
213C22×C4#610311yescharacteristic
214C22×C4C2×C4×D1311yescharacteristic
215C22×C4C2×D5211yescharacteristic
216C22×C4C2×C22⋊D1311yescharacteristic
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—264no—
221C2×C8—522no—
222C2×C8—522no—
223C4⋊C4—522no—
224C4⋊C4—524no—
225C4⋊C4—522no—
226C4⋊C4—522no—
227C4⋊C4—522no—
228C4⋊C4—522no—
229C22⋊C4—524no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—41no—
234C13#770711yescharacteristic, complemented, Sylow
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#8806312yes—
242C23C2×#610311yescharacteristic
243Q8—1044no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—522no—
261C2×C4—522no—
262C2×C4—522no—
263C2×C4—522no—
264C2×C4—522no—
265C2×C4—522no—
266C2×C4#8804612yes—
267C2×C4#8806512yes—
268C2×C4#8815512yes—
269C2×C4#8816312yes—
270C2×C4#8820512yes—
271C8—82no—
272C8—524no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22C2×#8804611yescharacteristic
281C22#1171866511yescharacteristic
282C22#1171866611yescharacteristic
283C22#1171877111yescharacteristic
284C22#1171878111yescharacteristic
285C22C2×#8806311yescharacteristic
286C22C2×#8806511yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—1042no—
294C4—1042no—
295C2—42no—
296C2#1311319511yescharacteristic
297C2#1311325611yescharacteristic
298C2#1311331011yescharacteristic
299C2#1311335711yescharacteristic
300C2#1311336811yescharacteristic
301C2#1311344811yescharacteristic
302C2#1311356111yescharacteristic
subgroupG/H
centerC23C2×#6103
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#1817C2
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC2×C42×C13C23
hypercenter#1817D13
hypocenterC13#7707
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #13112661C2semidirect
234. C13#7707semidirect

Maximal subgroups

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

subgroupindex
#131231882
#131145982
#131126612
C13×#18172
#770713