Finite Group Databasebeta

← #15778108

Subgroups — #15778108

Order 3328.

512 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 113 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#13117550C211yesmaximal, characteristic
2C2×#11718069C212yesmaximal
3#101×Dic13C212yesmaximal, complemented
4#13112963C211yesmaximal, characteristic, complemented
5C13×#1394C211yesmaximal, characteristic, complemented
6C2×#88201C2211yescharacteristic
7C2×Q8×Dic13C2211yescharacteristic
8C2×#88147C411yescharacteristic
9C2×C13×#101C2212yes—
10C13×#391C2211yescharacteristic
11C2×#88076C2212yes—
12M16×Dic13—22no—
13#11718637C411yescharacteristic
14#11718600C411yescharacteristic
15#11718593C411yescharacteristic
16#11718330—22no—
17#11718329—22no—
18#11718146—22no—
19#11718069C418yescomplemented
20C22×Q8×C13C2311yescharacteristic
21C22×D52C2×C411yescharacteristic
22C22×C4×D13—21no—
23C2×C13×M16—22no—
24C2×C13×C4⋊C4C2×C411yescharacteristic
25C2×C42×C13C2×C411yescharacteristic
26#88201—21no—
27#88201—21no—
28#88201C2×C414yes—
29Q8×Dic13—42no—
30C2×#6103—22no—
31C2×#6103—21no—
32C2×#6103—22no—
33C2×C4⋊Dic13C2×C411yescharacteristic
34C2×C4⋊Dic13D412yes—
35C2×#6101—22no—
36C2×C4×Dic13C2×C411yescharacteristic
37C2×C4×Dic13C2×C411yescharacteristic
38C2×C4×Dic13D412yes—
39C2×#6099—22no—
40#88147—42no—
41C13×#101—22no—
42C13×#101C2×C414yescomplemented
43C13×#95—22no—
44#88076—22no—
45#88076C2×C414yescomplemented
46#88071—21no—
47#88071—21no—
48#88055—42no—
49C8×Dic13—42no—
50#12632—131nomaximal, Sylow
51C23×D13—21no—
52C2×Q8×C13—41no—
53C2×Q8×C13—41no—
54C2×Q8×C13C22×C412yes—
55C2×Q8×C13#4111yescharacteristic
56C2×Q8×C13#4111yescharacteristic
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13C22×C411yescharacteristic
60C22×C4×C13C2×D412yes—
61C22×Dic13—21no—
62C22×Dic13C22⋊C411yescharacteristic
63C22×Dic13C22⋊C412yes—
64C22×Dic13C22⋊C411yescharacteristic
65C2×D52—21no—
66C2×D52—41no—
67C2×D52—41no—
68C2×D52C4212yes—
69C2×C4×D13—42no—
70C2×C4×D13—82no—
71C13×M16—24no—
72C13×M16—42no—
73C2×C8×C13—42no—
74C13×C4⋊C4—41no—
75C42×C13—41no—
76#6103—24no—
77#6103—24no—
78#6103—42no—
79#6103—82no—
80C4⋊Dic13—41no—
81C4⋊Dic13—42no—
82#6101—82no—
83C4×Dic13—21no—
84C4×Dic13—22no—
85C4×Dic13—22no—
86C4×Dic13—22no—
87C4×Dic13C4212yes—
88#6099—24no—
89#6099—42no—
90C2×#1055—42no—
91#2326—131no—
92C2×#355—132no—
93C4×#101—132no—
94#1394—131no—
95#1394D1311yescharacteristic, complemented
96C23×C13C2×C22⋊C411yescharacteristic
97C22×D13—22no—
98C22×D13—42no—
99C22×D13—42no—
100C22×D13—41no—
101C22×D13—41no—
102Q8×C13—42no—
103Q8×C13—41no—
104Q8×C13—41no—
105C2×C4×C13—22no—
106C2×C4×C13—21no—
107C2×C4×C13—22no—
108C2×C4×C13—21no—
109C2×C4×C13—42no—
110C2×C4×C13—41no—
111C2×C4×C13—41no—
112C2×C4×C13—41no—
113C2×C4×C13—41no—
114C2×C4×C13C2×C4211yescharacteristic
115C2×C4×C13#11711yescharacteristic
116C2×C4×C13C4×D412yes—
117C2×C4×C13C4×D412yes—
118C2×Dic13—22no—
119C2×Dic13—21no—
120C2×Dic13—22no—
121C2×Dic13—41no—
122C2×Dic13—41no—
123C2×Dic13—41no—
124C2×Dic13—42no—
125C2×Dic13—41no—
126C2×Dic13#9912yes—
127D52—42no—
128D52—42no—
129C4×D13—84no—
130C8×C13—44no—
131#1055—44no—
132C2×#124—131no—
133C2×C4×Q8—131no—
134C2×C4×D4—131no—
135C2×#101—132no—
136C2×#101C2×D1312yes—
137C4×M16—262no—
138#391C2×D1311yescharacteristic
139#390—131no—
140#388—262no—
141#385—262no—
142#384—131no—
143#376—131no—
144#355—138no—
145#343—262no—
146C22×C13—21no—
147C22×C13—21no—
148C22×C13C4×C22⋊C411yescharacteristic
149C22×C13C2×#9911yescharacteristic
150C22×C13#41011yescharacteristic
151C2×D13—44no—
152C2×D13—42no—
153C2×D13—82no—
154C2×D13—81no—
155C2×D13—81no—
156C4×C13—22no—
157C4×C13—22no—
158C4×C13—21no—
159C4×C13—21no—
160C4×C13—81no—
161C4×C13—81no—
162Dic13—22no—
163Dic13—42no—
164Dic13—41no—
165Dic13—42no—
166C22×Q8C22×D1311yescharacteristic
167C22×D4—131no—
168C23×C4—261no—
169C2×M16—22no—
170C2×M16—262no—
171#124—134no—
172#124—261no—
173#124—261no—
174C4×Q8—522no—
175C4×D4—522no—
176C2×C4⋊C4—131no—
177C2×C4⋊C4—132no—
178C2×C4⋊C4—262no—
179C2×C4⋊C4C4×D1311yescharacteristic
180C2×C22⋊C4—262no—
181C2×C22⋊C4—261no—
182C2×C22⋊C4—262no—
183C2×C42—131no—
184C2×C42—131no—
185C2×C42—132no—
186C2×C42C4×D1311yescharacteristic
187#101—22no—
188#101—134no—
189#101—262no—
190#101C4×D1314yescomplemented
191#97—522no—
192C4×C8—522no—
193#95—22no—
194#95—261no—
195#95—261no—
196C2×C13—21no—
197C2×C13—21no—
198C2×C13C4×#9911yescharacteristic
199C2×C13#213311yescharacteristic
200C2×C13#213411yescharacteristic
201D13—82no—
202C24—261no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C4×D1312yes—
206C2×Q8#612911yescharacteristic
207C2×Q8#613111yescharacteristic
208C2×D4—132no—
209C2×D4—261no—
210C2×D4—521no—
211C2×D4—521no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—131no—
215C22×C4—131no—
216C22×C4—132no—
217C22×C4—261no—
218C22×C4—522no—
219C22×C4—1042no—
220C22×C4C2×C4×D1311yescharacteristic
221C22×C4D4×D1312yes—
222M16—24no—
223M16—42no—
224M16—264no—
225M16—522no—
226C2×C8—42no—
227C2×C8—522no—
228C4⋊C4—41no—
229C4⋊C4—521no—
230C4⋊C4—522no—
231C4⋊C4—1042no—
232C22⋊C4—264no—
233C22⋊C4—264no—
234C22⋊C4—522no—
235C22⋊C4—1042no—
236C42—41no—
237C42—132no—
238C42—262no—
239C42—262no—
240C42—262no—
241C42—261no—
242C13#1263211yescharacteristic, complemented, Sylow
243C23—262no—
244C23—522no—
245C23—522no—
246C23—521no—
247C23—521no—
248C23C22⋊C4×D1311yescharacteristic
249Q8—42no—
250Q8—41no—
251Q8—41no—
252D4—522no—
253D4—522no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—132no—
264C2×C4—262no—
265C2×C4—261no—
266C2×C4—262no—
267C2×C4—521no—
268C2×C4—521no—
269C2×C4—521no—
270C2×C4—522no—
271C2×C4—521no—
272C2×C4—1044no—
273C2×C4C42×D1311yescharacteristic
274C2×C4#8813512yes—
275C2×C4#8814611yescharacteristic
276C2×C4#8814712yes—
277C8—44no—
278C8—524no—
279C22—21no—
280C22—21no—
281C22—524no—
282C22—522no—
283C22—1042no—
284C22—1041no—
285C22—1041no—
286C22#1171832711yescharacteristic
287C22#1171840111yescharacteristic
288C22D13×#9911yescharacteristic
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—81no—
294C4—81no—
295C4—262no—
296C4—522no—
297C4—522no—
298C4—521no—
299C2—21no—
300C2—21no—
301C2—1042no—
302C2#1311462311yescharacteristic
303C2#1311652411yescharacteristic
304C2#1311652611yescharacteristic
subgroupG/H
centerC22D13×#99
commutatorC2×C4×C13C2×C42
FrattiniC22×Q8C22×D13
FittingC13×#1394C2
socleC22×C13C2×#99
prime radicalC22×D52C2×C4
cosocleC22×Q8×C13C23
hypercenter#1394D13
hypocenterC13#12632
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #101×Dic13C2semidirect
4. #13112963C2semidirect
5. C13×#1394C2semidirect
19. #11718069C4semidirect
42. C13×#101C2×C4semidirect
45. #88076C2×C4semidirect
95. #1394D13semidirect
190. #101C4×D13semidirect
242. C13#12632semidirect

Maximal subgroups

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

subgroupindex
#131175502
C2×#117180692
#101×Dic132
#131129632
C13×#13942
#1263213