Finite Group Databasebeta

← #15760674

Subgroups — #15760674

Order 3328.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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#13117422C211yesmaximal, characteristic, complemented
2#13114760C211yesmaximal, characteristic, complemented
3#13113365C212yesmaximal
4#13112689C212yesmaximal, complemented
5C13×#2436C211yesmaximal, characteristic
6C23×Q8×C13C2211yescharacteristic
7C2×C13×#124C412yescomplemented
8C2×C13×#121C2211yescharacteristic
9C2×#88180—21no—
10C13×#393—22no—
11C13×#391C412yescomplemented
12C13×#386—21no—
13C13×#386—21no—
14C4×C13×C22⋊C4C2211yescharacteristic
15#11718891—24no—
16C2×#88076—22no—
17#11718872—21no—
18#11718872C2212yes—
19#11718868C2212yes—
20#11718129—24no—
21C22×Q8×C13—22no—
22C22×Q8×C13—22no—
23C22×Q8×C13—44no—
24C22×Q8×C13C2×C412yes—
25C22×Q8×C13D412yes—
26C22×D4×C13—21no—
27C22×D4×C13C2×C411yescharacteristic
28C23×C4×C13D411yescharacteristic
29C23×C4×C13D411yescharacteristic
30C23×C4×C13C2311yescharacteristic
31C13×#124—44no—
32C13×#121—22no—
33C13×#121—22no—
34C2×C13×C4⋊C4C2×C411yescharacteristic
35C2×C13×C22⋊C4—21no—
36C2×C13×C22⋊C4—22no—
37C2×C13×C22⋊C4—22no—
38C2×C13×C22⋊C4—21no—
39C2×C42×C13C2×C412yes—
40C2×C22⋊Dic13—22no—
41#88180—44no—
42C2×C4⋊Dic13—22no—
43C2×C4×Dic13—22no—
44C2×#6099—22no—
45C13×#95—22no—
46C13×#95—22no—
47C13×#95—21no—
48#88076—44no—
49#88071—24no—
50#88071—22no—
51#88071D414yes—
52C22⋊#1055—22no—
53C22⋊#1055—22no—
54#9466—131nomaximal, Sylow
55C24×C13—21no—
56C24×C13C2×D411yescharacteristic
57C2×Q8×C13—24no—
58C2×Q8×C13—24no—
59C2×Q8×C13—24no—
60C2×Q8×C13—24no—
61C2×Q8×C13—42no—
62C2×Q8×C13—42no—
63C2×Q8×C13—44no—
64C2×Q8×C13—44no—
65C2×Q8×C13—48no—
66C2×D4×C13—22no—
67C2×D4×C13—42no—
68C2×D4×C13—44no—
69C2×D4×C13—41no—
70C2×D4×C13—41no—
71C22×C4×C13—21no—
72C22×C4×C13—21no—
73C22×C4×C13—24no—
74C22×C4×C13—21no—
75C22×C4×C13—21no—
76C22×C4×C13—21no—
77C22×C4×C13—22no—
78C22×C4×C13—42no—
79C22×C4×C13—42no—
80C22×C4×C13C22⋊C412yes—
81C22×C4×C13C22⋊C412yes—
82C22×C4×C13C22×C411yescharacteristic
83C22×C4×C13C2×D412yes—
84C22×C4×C13C2×D411yescharacteristic
85C22×C4×C13#4112yes—
86C22×Dic13—24no—
87C13×C4⋊C4—22no—
88C13×C22⋊C4—44no—
89C13×C22⋊C4—42no—
90C13×C22⋊C4—44no—
91C13×C22⋊C4—42no—
92C42×C13—42no—
93C22⋊Dic13—44no—
94C4⋊Dic13—44no—
95C4×Dic13—44no—
96#6099—44no—
97C2×#1055—42no—
98#2436D1311yescharacteristic
99#2310—131no—
100#1784—131no—
101#1510—132no—
102#1284—132no—
103C23×C13—22no—
104C23×C13—22no—
105C23×C13—41no—
106C23×C13—42no—
107C23×C13—42no—
108C23×C13—41no—
109C23×C13—42no—
110C23×C13C2×C22⋊C411yescharacteristic
111C23×C13#12011yescharacteristic
112C23×C13#12311yescharacteristic
113Q8×C13—48no—
114Q8×C13—48no—
115D4×C13—82no—
116D4×C13—82no—
117D4×C13—82no—
118C2×C4×C13—22no—
119C2×C4×C13—22no—
120C2×C4×C13—24no—
121C2×C4×C13—22no—
122C2×C4×C13—22no—
123C2×C4×C13—22no—
124C2×C4×C13—22no—
125C2×C4×C13—22no—
126C2×C4×C13—22no—
127C2×C4×C13—22no—
128C2×C4×C13—42no—
129C2×C4×C13—42no—
130C2×C4×C13—42no—
131C2×C4×C13—42no—
132C2×C4×C13—42no—
133C2×C4×C13—42no—
134C2×C4×C13—42no—
135C2×C4×C13—41no—
136C2×C4×C13—42no—
137C2×C4×C13#9914yes—
138C2×C4×C13C4×D412yes—
139C2×C4×C13#12112yes—
140C2×Dic13—44no—
141C2×Dic13—44no—
142C2×Dic13—44no—
143#1055—82no—
144C23×Q8C2×D1311yescharacteristic
145C2×#124Dic1312yescomplemented
146C2×#121C2×D1311yescharacteristic
147C2×#117—261no—
148C2×#101—262no—
149#407—132no—
150#393—22no—
151#391—264no—
152#391Dic1312yescomplemented
153#386—21no—
154#386—21no—
155#381—132no—
156#381—261no—
157C4×C22⋊C4C2×D1311yescharacteristic
158#327—264no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—21no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—44no—
167C22×C13—84no—
168C22×C13—81no—
169C22×C13—81no—
170C22×C13—81no—
171C22×C13#38611yescharacteristic
172C22×C13C2×#9912yes—
173C22×C13#45412yes—
174C4×C13—22no—
175C4×C13—24no—
176C4×C13—22no—
177C4×C13—42no—
178C4×C13—82no—
179C4×C13—81no—
180Dic13—84no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—44no—
184C22×Q8C2×Dic1312yes—
185C22×Q8C22⋊D1312yes—
186C22×D4—21no—
187C22×D4C2×Dic1311yescharacteristic
188C23×C4C22⋊D1311yescharacteristic
189C23×C4C22⋊D1311yescharacteristic
190C23×C4C22×D1311yescharacteristic
191C2×M16—262no—
192#124—44no—
193#121—22no—
194#121—22no—
195#117—524no—
196C2×C4⋊C4—262no—
197C2×C4⋊C4C2×Dic1311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—262no—
203C2×C42—262no—
204C2×C42C2×Dic1312yes—
205#101—524no—
206C22⋊C8—262no—
207C22⋊C8—262no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—134no—
212#95—264no—
213#95—262no—
214C2×C13—22no—
215C2×C13—22no—
216C2×C13—41no—
217C2×C13—161no—
218C2×C13#188811yescharacteristic
219C2×C13#189811yescharacteristic
220C2×C13#191211yescharacteristic
221C24—21no—
222C24C2×C22⋊D1311yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—264no—
247C22×C4C22⋊Dic1312yes—
248C22×C4C22⋊Dic1312yes—
249C22×C4D4×D1312yes—
250C22×C4#612912yes—
251C22×C4C22×Dic1311yescharacteristic
252C22×C4C2×C22⋊D1311yescharacteristic
253M16—524no—
254C2×C8—522no—
255C4⋊C4—22no—
256C4⋊C4—524no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—524no—
262C42—42no—
263C42—524no—
264C13#946611yescharacteristic, complemented, Sylow
265C23—22no—
266C23—22no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23#8818911yescharacteristic
273C23#8819111yescharacteristic
274C23C2×C22⋊Dic1311yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—82no—
278D4—82no—
279D4—82no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—524no—
300C2×C4—524no—
301C2×C4—524no—
302C2×C4#8807414yes—
303C2×C4D4×Dic1312yes—
304C2×C4#8819312yes—
305C8—1042no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—84no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#1171872312yes—
319C22C2×#8807412yes—
320C22#1171888111yescharacteristic
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—82no—
326C4—81no—
327C4—1044no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2#1311524011yescharacteristic
333C2#1311529011yescharacteristic
334C2#1311537411yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2436C2
socleC22×C13#386
prime radicalC2×C13×#121C22
cosocleC23×C4×C13C23
hypercenter#2436D13
hypocenterC13#9466
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. #13117422C2semidirect
2. #13114760C2semidirect
4. #13112689C2semidirect
7. C2×C13×#124C4semidirect
11. C13×#391C4semidirect
145. C2×#124Dic13semidirect
152. #391Dic13semidirect
264. C13#9466semidirect

Maximal subgroups

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

subgroupindex
#131174222
#131147602
#131133652
#131126892
C13×#24362
#946613