Finite Group Databasebeta

← #15778111

Subgroups — #15778111

Order 3328.

512 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 125 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#13117549C211yesmaximal, characteristic
2C2×#11718134C211yesmaximal, characteristic
3#101×Dic13C212yesmaximal, complemented
4#13112962C212yesmaximal, complemented
5C2×C13×#355C211yesmaximal, characteristic
6C2×C13×#124C2211yescharacteristic
7C2×Q8×Dic13C2211yescharacteristic
8C2×D4×Dic13C411yescharacteristic
9C2×C13×#101C2212yes—
10#11718891C2211yescharacteristic
11C2×#88076C2212yes—
12#11718882C411yescharacteristic
13M16×Dic13—22no—
14#11718612—22no—
15#11718611—22no—
16#11718596C411yescharacteristic
17#11718593C411yescharacteristic
18C13×#355C414yescomplemented
19#11718146—22no—
20#11718134C414yescomplemented
21C22×Q8×C13C2311yescharacteristic
22C22×D4×C13C2×C411yescharacteristic
23C23×Dic13—21no—
24C2×C13×M16—22no—
25C13×#124—22no—
26C13×#124C2×C412yes—
27C13×#124C2×C412yes—
28C2×C13×C22⋊C4—22no—
29C2×C42×C13C2×C411yescharacteristic
30C2×C22⋊Dic13—21no—
31C2×C22⋊Dic13—22no—
32Q8×Dic13—42no—
33D4×Dic13—42no—
34C2×C4⋊Dic13—22no—
35C2×C4⋊Dic13C2×C411yescharacteristic
36C2×C4⋊Dic13C2×C411yescharacteristic
37C2×C4⋊Dic13D412yes—
38C2×C4×Dic13C2×C411yescharacteristic
39C2×C4×Dic13C2×C411yescharacteristic
40C2×C4×Dic13D412yes—
41C2×#6099—22no—
42C13×#101—22no—
43C13×#101C2×C414yescomplemented
44#88076—22no—
45#88076C2×C414yescomplemented
46#88071—22no—
47#88071—21no—
48#88071—21no—
49#88055—42no—
50C8×Dic13—42no—
51#12632—131nomaximal, Sylow
52C24×C13—21no—
53C2×Q8×C13—41no—
54C2×Q8×C13—41no—
55C2×Q8×C13C22×C411yescharacteristic
56C2×Q8×C13C22×C411yescharacteristic
57C2×Q8×C13#4112yes—
58C2×D4×C13—21no—
59C2×D4×C13—41no—
60C2×D4×C13—41no—
61C2×D4×C13C4212yes—
62C22×C4×C13—21no—
63C22×C4×C13C22×C411yescharacteristic
64C22×C4×C13C2×D412yes—
65C22×Dic13—21no—
66C22×Dic13—21no—
67C22×Dic13—42no—
68C22×Dic13—82no—
69C22×Dic13C22⋊C411yescharacteristic
70C22×Dic13C22⋊C412yes—
71C22×Dic13C22⋊C411yescharacteristic
72C13×M16—24no—
73C13×M16—42no—
74C2×C8×C13—42no—
75C13×C22⋊C4—24no—
76C13×C22⋊C4—24no—
77C42×C13—21no—
78C42×C13C4212yes—
79C22⋊Dic13—42no—
80C22⋊Dic13—82no—
81C4⋊Dic13—41no—
82C4⋊Dic13—42no—
83C4⋊Dic13—41no—
84C4⋊Dic13—82no—
85C4×Dic13—22no—
86C4×Dic13—22no—
87C4×Dic13—22no—
88C4×Dic13—41no—
89#6099—24no—
90#6099—42no—
91C2×#1055—42no—
92#2326—131no—
93C2×#355—131no—
94C2×#355D1311yescharacteristic
95C4×#101—132no—
96#1394—132no—
97C23×C13—22no—
98C23×C13—42no—
99C23×C13—42no—
100C23×C13—41no—
101C23×C13—41no—
102C23×C13C2×C22⋊C411yescharacteristic
103Q8×C13—41no—
104Q8×C13—41no—
105Q8×C13—42no—
106D4×C13—42no—
107D4×C13—42no—
108C2×C4×C13—22no—
109C2×C4×C13—21no—
110C2×C4×C13—22no—
111C2×C4×C13—21no—
112C2×C4×C13—22no—
113C2×C4×C13—41no—
114C2×C4×C13—41no—
115C2×C4×C13C2×C4211yescharacteristic
116C2×C4×C13#11711yescharacteristic
117C2×C4×C13C4×D412yes—
118C2×C4×C13C4×D412yes—
119C2×Dic13—21no—
120C2×Dic13—22no—
121C2×Dic13—41no—
122C2×Dic13—42no—
123C2×Dic13—41no—
124C2×Dic13—42no—
125C2×Dic13—41no—
126C2×Dic13—41no—
127C2×Dic13—41no—
128C2×Dic13—41no—
129C2×Dic13—84no—
130C2×Dic13#9912yes—
131C8×C13—44no—
132#1055—44no—
133C2×#124C2×D1311yescharacteristic
134C2×C4×Q8—131no—
135C2×C4×D4—131no—
136C2×#101—132no—
137C2×#101C2×D1312yes—
138C4×M16—262no—
139#391—131no—
140#390—131no—
141#388—262no—
142#385—262no—
143#384—131no—
144#376—131no—
145#355—134no—
146#355Dic1314yescomplemented
147#343—262no—
148C22×C13—21no—
149C22×C13—21no—
150C22×C13—44no—
151C22×C13—42no—
152C22×C13—82no—
153C22×C13—81no—
154C22×C13—81no—
155C22×C13C4×C22⋊C411yescharacteristic
156C22×C13C2×#9911yescharacteristic
157C22×C13#41011yescharacteristic
158C4×C13—22no—
159C4×C13—22no—
160C4×C13—21no—
161C4×C13—21no—
162C4×C13—42no—
163Dic13—22no—
164Dic13—42no—
165Dic13—41no—
166Dic13—81no—
167Dic13—81no—
168C22×Q8C22×D1311yescharacteristic
169C22×D4C4×D1311yescharacteristic
170C23×C4—261no—
171C2×M16—22no—
172C2×M16—262no—
173#124—22no—
174#124C4×D1312yes—
175#124C2×Dic1312yes—
176C4×Q8—522no—
177C4×D4—522no—
178C2×C4⋊C4—131no—
179C2×C4⋊C4—132no—
180C2×C4⋊C4—131no—
181C2×C4⋊C4—262no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—261no—
184C2×C22⋊C4—262no—
185C2×C42—131no—
186C2×C42—132no—
187C2×C42—131no—
188C2×C42C4×D1311yescharacteristic
189#101—22no—
190#101—134no—
191#101—262no—
192#101C2×Dic1314yescomplemented
193#97—522no—
194C4×C8—522no—
195#95—261no—
196#95—261no—
197#95—262no—
198C2×C13—21no—
199C2×C13—21no—
200C2×C13—82no—
201C2×C13C4×#9911yescharacteristic
202C2×C13#213311yescharacteristic
203C2×C13#213411yescharacteristic
204C24—21no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8C2×C4×D1311yescharacteristic
208C2×Q8#612912yes—
209C2×Q8C22×Dic1311yescharacteristic
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4C4×Dic1312yes—
214C22×C4—21no—
215C22×C4—131no—
216C22×C4—131no—
217C22×C4—132no—
218C22×C4—261no—
219C22×C4—261no—
220C22×C4—522no—
221C22×C4—1042no—
222C22×C4C2×C4×D1311yescharacteristic
223C22×C4D4×D1312yes—
224M16—24no—
225M16—42no—
226M16—264no—
227M16—522no—
228C2×C8—42no—
229C2×C8—522no—
230C4⋊C4—521no—
231C4⋊C4—522no—
232C4⋊C4—521no—
233C4⋊C4—1042no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—522no—
237C22⋊C4—1042no—
238C42—21no—
239C42—262no—
240C42—262no—
241C42—262no—
242C42—521no—
243C42C4×Dic1312yes—
244C13#1263211yescharacteristic, complemented, Sylow
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23C22⋊C4×D1311yescharacteristic
251Q8—41no—
252Q8—42no—
253Q8—41no—
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—132no—
264C2×C4—262no—
265C2×C4—261no—
266C2×C4—521no—
267C2×C4—522no—
268C2×C4—521no—
269C2×C4—521no—
270C2×C4—522no—
271C2×C4—521no—
272C2×C4—521no—
273C2×C4—521no—
274C2×C4—1044no—
275C2×C4#8813111yescharacteristic
276C2×C4#8813512yes—
277C2×C4C2×C4×Dic1311yescharacteristic
278C2×C4D4×Dic1312yes—
279C8—44no—
280C8—524no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—42no—
285C22—82no—
286C22—81no—
287C22—81no—
288C22#1171840111yescharacteristic
289C22D13×#9911yescharacteristic
290C22C22⋊C4×Dic1311yescharacteristic
291C4—22no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—42no—
296C4—262no—
297C4—522no—
298C4—521no—
299C4—1041no—
300C4—1041no—
301C2—21no—
302C2—21no—
303C2—82no—
304C2#99×Dic1311yescharacteristic
305C2#1311652211yescharacteristic
306C2#1311652611yescharacteristic
subgroupG/H
centerC22D13×#99
commutatorC2×C4×C13C2×C42
FrattiniC22×Q8C22×D13
FittingC2×C13×#355C2
socleC22×C13C2×#99
prime radicalC22×D4×C13C2×C4
cosocleC22×Q8×C13C23
hypercenterC2×#355D13
hypocenterC13#12632
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #101×Dic13C2semidirect
4. #13112962C2semidirect
18. C13×#355C4semidirect
20. #11718134C4semidirect
43. C13×#101C2×C4semidirect
45. #88076C2×C4semidirect
146. #355Dic13semidirect
192. #101C2×Dic13semidirect
244. C13#12632semidirect

Maximal subgroups

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

subgroupindex
#131175492
C2×#117181342
#101×Dic132
#131129622
C2×C13×#3552
#1263213