Finite Group Databasebeta

← #15774055

Subgroups — #15774055

Order 3328.

508 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 107 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#13114824C211yesmaximal, characteristic
2#13113154C211yesmaximal, characteristic
3#13112860C214yesmaximal, complemented
4C13×#2542C211yesmaximal, characteristic
5C2×C13×C4⋊Q8C411yescharacteristic, complemented
6C2×C13×C22⋊Q8—21no—
7C2×C13×C22⋊Q8—21no—
8C2×C4×D4×C13C2211yescharacteristic
9C13×#399C412yescomplemented
10C13×#399C2212yes—
11C13×#390—21no—
12C13×#384—21no—
13C4×C13×C4⋊C4C411yescharacteristic, complemented
14C2×#88074—21no—
15#11718873—21no—
16C2×C22⋊#1055—21no—
17#11718120—21no—
18#11718118C2212yes—
19#11718117C2212yes—
20C22×Q8×C13—21no—
21C22×D4×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13D411yescharacteristic
24C13×C4⋊Q8—41no—
25C13×C4⋊Q8—41no—
26C13×C22⋊Q8—44no—
27C13×C22⋊Q8—44no—
28C4×D4×C13—41no—
29C4×D4×C13—41no—
30C2×C13×C4⋊C4—21no—
31C2×C13×C4⋊C4—21no—
32C2×C13×C4⋊C4—22no—
33C2×C13×C4⋊C4—22no—
34C2×C13×C4⋊C4C2×C412yes—
35C2×C13×C4⋊C4D411yescharacteristic
36C2×C13×C22⋊C4—22no—
37C2×C13×C22⋊C4—22no—
38C2×C13×C22⋊C4C2×C411yescharacteristic
39C2×C13×C22⋊C4C2311yescharacteristic
40C2×C42×C13—21no—
41C2×C42×C13C2×C411yescharacteristic
42C2×C22⋊Dic13—22no—
43C2×C4⋊Dic13—41no—
44C22×#1055—41no—
45C13×#95C2×C412yes—
46#88074—42no—
47#88071—41no—
48C22⋊#1055—22no—
49C22⋊#1055—41no—
50#11901—131nomaximal, Sylow
51C24×C13C22⋊C411yescharacteristic
52C24×C13C2×D411yescharacteristic
53C2×Q8×C13—42no—
54C2×Q8×C13—42no—
55C2×Q8×C13—41no—
56C2×Q8×C13—41no—
57C2×D4×C13—41no—
58C2×D4×C13—41no—
59C2×D4×C13SD1612yes—
60C2×D4×C13Q1612yes—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—21no—
65C22×C4×C13—21no—
66C22×C4×C13—22no—
67C22×C4×C13—42no—
68C22×C4×C13—42no—
69C22×C4×C13C22⋊C411yescharacteristic
70C22×C4×C13C22⋊C411yescharacteristic
71C22×C4×C13C22⋊C411yescharacteristic
72C22×C4×C13C22×C411yescharacteristic
73C22×C4×C13C2×D411yescharacteristic
74C22×Dic13—42no—
75C13×C4⋊C4—24no—
76C13×C4⋊C4—41no—
77C13×C4⋊C4—42no—
78C13×C4⋊C4—44no—
79C13×C4⋊C4—44no—
80C13×C4⋊C4—42no—
81C13×C22⋊C4—41no—
82C13×C22⋊C4—41no—
83C13×C22⋊C4—44no—
84C13×C22⋊C4—44no—
85C42×C13—21no—
86C42×C13—21no—
87C42×C13—42no—
88C22⋊Dic13—42no—
89C22⋊Dic13—42no—
90C4⋊Dic13—82no—
91C2×#1055—42no—
92C2×#1055—82no—
93#2542D1311yescharacteristic
94#1799—131no—
95#1459—131no—
96#1352—134no—
97C23×C13—21no—
98C23×C13—21no—
99C23×C13—21no—
100C23×C13—21no—
101C23×C13—41no—
102C23×C13—41no—
103C23×C13—41no—
104C23×C13—41no—
105C23×C13#10312yes—
106C23×C13C2×C22⋊C411yescharacteristic
107C23×C13C2×SD1611yescharacteristic
108C23×C13C2×Q1611yescharacteristic
109Q8×C13—82no—
110Q8×C13—82no—
111D4×C13—42no—
112D4×C13—42no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—22no—
116C2×C4×C13—24no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—21no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13—41no—
127C2×C4×C13—42no—
128C2×C4×C13—42no—
129C2×C4×C13—42no—
130C2×C4×C13—81no—
131C2×C4×C13—81no—
132C2×C4×C13#9911yescharacteristic
133C2×C4×C13#9911yescharacteristic
134C2×C4×C13#10312yes—
135C2×Dic13—82no—
136C2×Dic13—82no—
137C2×Dic13—82no—
138#1055—82no—
139C2×C4⋊Q8Dic1311yescharacteristic, complemented
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—21no—
142C2×C4×D4C2×D1311yescharacteristic
143C2×#99—261no—
144C2×C22⋊C8—261no—
145#399Dic1312yescomplemented
146#399C2×D1312yes—
147#390—21no—
148#384—21no—
149#380—261no—
150C4×C4⋊C4Dic1311yescharacteristic, complemented
151#342—132no—
152#324—261no—
153#323—132no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—22no—
158C22×C13—21no—
159C22×C13—42no—
160C22×C13—41no—
161C22×C13—41no—
162C22×C13—41no—
163C22×C13—41no—
164C22×C13—41no—
165C22×C13—41no—
166C22×C13—81no—
167C22×C13—81no—
168C22×C13C2×#9911yescharacteristic
169C22×C13C2×#10311yescharacteristic
170C22×C13#42111yescharacteristic
171C4×C13—21no—
172C4×C13—22no—
173C4×C13—44no—
174C4×C13—41no—
175C4×C13—41no—
176C4×C13—41no—
177C4×C13—82no—
178C4×C13#35312yes—
179Dic13—162no—
180C22×Q8—21no—
181C22×D4C22⋊D1311yescharacteristic
182C23×C4C22⋊D1311yescharacteristic
183C23×C4C22⋊D1311yescharacteristic
184C22×C8—521no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—521no—
196C2×C4⋊C4C2×Dic1312yes—
197C2×C4⋊C4C22⋊D1311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—262no—
201C2×C22⋊C4C2×Dic1311yescharacteristic
202C2×C22⋊C4C22×D1311yescharacteristic
203C2×C42—21no—
204C2×C42C2×Dic1311yescharacteristic
205#99—522no—
206C22⋊C8—262no—
207C22⋊C8—521no—
208#95—521no—
209#95C2×Dic1312yes—
210C2×C13—21no—
211C2×C13—21no—
212C2×C13—41no—
213C2×C13—42no—
214C2×C13—41no—
215C2×C13#152011yescharacteristic
216C2×C13#212311yescharacteristic
217C2×C13C2×#35311yescharacteristic
218C24C22⋊Dic1311yescharacteristic
219C24C2×C22⋊D1311yescharacteristic
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4#610612yes—
227C2×D4#610712yes—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—522no—
237C22×C4C22⋊Dic1311yescharacteristic
238C22×C4C22⋊Dic1311yescharacteristic
239C22×C4C22⋊Dic1311yescharacteristic
240C22×C4C22×Dic1311yescharacteristic
241C22×C4C2×C22⋊D1311yescharacteristic
242C2×C8—522no—
243C2×C8—1042no—
244C4⋊C4—24no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—1042no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—522no—
256C22⋊C4—522no—
257C42—21no—
258C42—21no—
259C42—42no—
260C13#1190111yescharacteristic, complemented, Sylow
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#8807512yes—
270C23C2×#610611yescharacteristic
271C23C2×#610711yescharacteristic
272C23C2×C22⋊Dic1311yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—1042no—
297C2×C4—1042no—
298C2×C4—1042no—
299C2×C4#8807411yescharacteristic
300C2×C4#8807411yescharacteristic
301C2×C4#8807512yes—
302C8—1042no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#8807411yescharacteristic
318C22C2×#8807511yescharacteristic
319C22#1171889711yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—2082no—
328C4#1171813712yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#1311340811yescharacteristic
335C2#1311649011yescharacteristic
336C2C2×#1171813711yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC22×C4×C13C22×C4
FrattiniC2×C22⋊C4C22×D13
FittingC13×#2542C2
socleC22×C13C2×#99
prime radicalC22×D4×C13D4
cosocleC2×C13×C22⋊C4C23
hypercenter#2542D13
hypocenterC13#11901
LCSG ▷ C22×C4×C13 ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #13112860C2semidirect
5. C2×C13×C4⋊Q8C4semidirect
9. C13×#399C4semidirect
13. C4×C13×C4⋊C4C4semidirect
139. C2×C4⋊Q8Dic13semidirect
145. #399Dic13semidirect
150. C4×C4⋊C4Dic13semidirect
260. C13#11901semidirect

Maximal subgroups

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

subgroupindex
#131148242
#131131542
#131128602
C13×#25422
#1190113