Finite Group Databasebeta

← C2×#13112781 #15772072

Subgroups — C2×#13112781 #15772072

Order 3328.

598 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#11718637C211yesmaximal, characteristic
2C2×#11718120C211yesmaximal, characteristic
3#13112781C214yesmaximal, direct product factor
4C2×C13×#324C211yesmaximal, characteristic
5C22×C13×C4⋊C4C2211yescharacteristic
6C22×C4⋊Dic13C411yescharacteristic
7C22×#6101—21no—
8C22×#6101—21no—
9C22×C4×Dic13C411yescharacteristic
10C2×C13×C22⋊C8—21no—
11C2×C22⋊#1055—21no—
12#11718637—21no—
13#11718637—21no—
14#11718637C412yes—
15#11718637C2212yes—
16C13×#324—21no—
17C13×#324C2212yes—
18#11718120—21no—
19#11718120C2212yes—
20C23×C4×C13C2×C411yescharacteristic
21C23×C4×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×Dic13—21no—
24C23×Dic13—21no—
25C22×C8×C13—41no—
26C2×C13×C4⋊C4—41no—
27C2×C13×C4⋊C4—41no—
28C2×C13×C4⋊C4C2×C411yescharacteristic
29C2×C13×C4⋊C4D411yescharacteristic
30C2×C13×C4⋊C4D411yescharacteristic
31C2×C13×C4⋊C4C2311yescharacteristic
32C2×C4⋊Dic13—21no—
33C2×C4⋊Dic13—41no—
34C2×C4⋊Dic13—41no—
35C2×C4⋊Dic13C2×C412yes—
36C2×#6101—22no—
37C2×#6101—22no—
38C2×#6101—22no—
39C2×#6101—22no—
40C2×#6101—44no—
41C2×#6101—44no—
42C2×C4×Dic13—21no—
43C2×C4×Dic13—22no—
44C2×C4×Dic13C814yescomplemented
45C2×C4×Dic13C2×C412yes—
46C22×#1055—41no—
47C13×C22⋊C8—22no—
48C13×C22⋊C8—41no—
49C22⋊#1055—22no—
50C22⋊#1055—41no—
51C2×#1325—131nomaximal, Sylow
52C24×C13C22⋊C411yescharacteristic
53C22×C4×C13—21no—
54C22×C4×C13—21no—
55C22×C4×C13—22no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—42no—
61C22×C4×C13—42no—
62C22×C4×C13C22⋊C411yescharacteristic
63C22×C4×C13C22⋊C411yescharacteristic
64C22×C4×C13C22⋊C411yescharacteristic
65C22×C4×C13C2×C812yes—
66C22×C4×C13M1612yes—
67C22×C4×C13C22×C411yescharacteristic
68C22×C4×C13C2×D411yescharacteristic
69C22×C4×C13C2×D411yescharacteristic
70C22×Dic13—24no—
71C22×Dic13—24no—
72C22×Dic13—24no—
73C22×Dic13—22no—
74C22×Dic13—22no—
75C22×Dic13—42no—
76C22×Dic13—42no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C2×C8×C13—42no—
80C2×C8×C13—82no—
81C13×C4⋊C4—41no—
82C13×C4⋊C4—41no—
83C13×C4⋊C4—41no—
84C13×C4⋊C4—41no—
85C4⋊Dic13—42no—
86C4⋊Dic13—42no—
87#6101—44no—
88#6101—44no—
89#6101—44no—
90#6101—44no—
91C4×Dic13—24no—
92C4×Dic13—22no—
93C4×Dic13C2×C814yescomplemented
94C2×#1055—42no—
95C2×#1055—82no—
96C2×#384—131no—
97C2×#324—131no—
98C2×#324D1311yescharacteristic
99#1325—134no—
100C23×C13—22no—
101C23×C13—22no—
102C23×C13C22⋊C812yes—
103C23×C13#9911yescharacteristic
104C23×C13#9911yescharacteristic
105C23×C13#10112yes—
106C23×C13C2×C22⋊C411yescharacteristic
107C2×C4×C13—22no—
108C2×C4×C13—22no—
109C2×C4×C13—22no—
110C2×C4×C13—22no—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—81no—
126C2×C4×C13—81no—
127C2×C4×C13C22⋊C812yes—
128C2×C4×C13C22×C811yescharacteristic
129C2×C4×C13C2×M1611yescharacteristic
130C2×Dic13—24no—
131C2×Dic13—216no—
132C2×Dic13—24no—
133C2×Dic13—24no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—82no—
138C2×Dic13—82no—
139C8×C13—82no—
140#1055—82no—
141C22×C4⋊C4—131no—
142C22×C4⋊C4—261no—
143C22×C4⋊C4—261no—
144C22×C4⋊C4C2×D1311yescharacteristic
145C22×C42—131no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8—261no—
148#384—132no—
149#384—132no—
150#384—261no—
151#384—261no—
152#324—21no—
153#324—132no—
154#324—261no—
155#324C2×D1312yes—
156C22×C13—24no—
157C22×C13—24no—
158C22×C13—21no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13#32412yes—
163C22×C13#32412yes—
164C22×C13#35312yes—
165C22×C13#35612yes—
166C22×C13C2×C22⋊C811yescharacteristic
167C22×C13C2×#9911yescharacteristic
168C22×C13C2×#10111yescharacteristic
169C4×C13—22no—
170C4×C13—21no—
171C4×C13—21no—
172C4×C13—41no—
173C4×C13—41no—
174C4×C13—41no—
175C4×C13—41no—
176Dic13—28no—
177Dic13—82no—
178C23×C4—261no—
179C23×C4—261no—
180C23×C4C4×D1311yescharacteristic
181C23×C4D5211yescharacteristic
182C23×C4C22⋊D1311yescharacteristic
183C22×C8—41no—
184C22×C8—521no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—132no—
188C2×C4⋊C4—261no—
189C2×C4⋊C4—262no—
190C2×C4⋊C4—262no—
191C2×C4⋊C4—262no—
192C2×C4⋊C4—262no—
193C2×C4⋊C4—521no—
194C2×C4⋊C4—521no—
195C2×C4⋊C4—524no—
196C2×C4⋊C4—524no—
197C2×C4⋊C4C4×D1311yescharacteristic
198C2×C4⋊C4D5211yescharacteristic
199C2×C4⋊C4C22⋊D1311yescharacteristic
200C2×C4⋊C4C22×D1311yescharacteristic
201C2×C42—134no—
202C2×C42—132no—
203C2×C42—262no—
204C2×C42—261no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C22⋊C8—262no—
208C22⋊C8—521no—
209C2×C13—21no—
210C2×C13—22no—
211C2×C13—21no—
212C2×C13#132514yescomplemented
213C2×C13C2×#32411yescharacteristic
214C2×C13C2×#35311yescharacteristic
215C2×C13C2×#35611yescharacteristic
216C24#610311yescharacteristic
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—264no—
227C22×C4—264no—
228C22×C4—262no—
229C22×C4—264no—
230C22×C4—262no—
231C22×C4—522no—
232C22×C4—522no—
233C22×C4—521no—
234C22×C4—521no—
235C22×C4C8×D1312yes—
236C22×C4#609412yes—
237C22×C4#610311yescharacteristic
238C22×C4#610311yescharacteristic
239C22×C4#610311yescharacteristic
240C22×C4C2×C4×D1311yescharacteristic
241C22×C4C2×D5211yescharacteristic
242C22×C4C2×C22⋊D1311yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C2×C8—522no—
246C2×C8—1042no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—522no—
252C4⋊C4—522no—
253C4⋊C4—524no—
254C4⋊C4—524no—
255C4⋊C4—524no—
256C4⋊C4—524no—
257C42—134no—
258C42—262no—
259C42—264no—
260C13C2×#132511yescharacteristic, complemented, Sylow
261C23—22no—
262C23—22no—
263C23#8804611yescharacteristic
264C23#8804611yescharacteristic
265C23#8806012yes—
266C23#8806412yes—
267C23C2×#610311yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—264no—
289C2×C4—264no—
290C2×C4—2616no—
291C2×C4—264no—
292C2×C4—522no—
293C2×C4—522no—
294C2×C4—522no—
295C2×C4—1042no—
296C2×C4—1042no—
297C2×C4#8806012yes—
298C2×C4C2×C8×D1311yescharacteristic
299C2×C4C2×#609411yescharacteristic
300C8—82no—
301C8—1042no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—24no—
308C22#1171806112yes—
309C22#1171806112yes—
310C22#1171806612yes—
311C22#1171807012yes—
312C22C2×#8804611yescharacteristic
313C22C2×#8806011yescharacteristic
314C22C2×#8806411yescharacteristic
315C4—21no—
316C4—21no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—268no—
323C4—1042no—
324C2—21no—
325C2—21no—
326C2—22no—
327C2#1311278114yesdirect product factor
328C2C2×#1171806111yescharacteristic
329C2C2×#1171806611yescharacteristic
330C2C2×#1171807011yescharacteristic
subgroupG/H
centerC23#88046
commutatorC2×C4×C13C22×C8
FrattiniC2×C4⋊C4C22×D13
FittingC2×C13×#324C2
socleC23×C13#99
prime radicalC24×C13C22⋊C4
cosocleC2×C13×C4⋊C4C23
hypercenterC2×#324D13
hypocenterC13C2×#1325
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #13112781C2direct (N × H)
44. C2×C4×Dic13C8semidirect
93. C4×Dic13C2×C8semidirect
212. C2×C13#1325semidirect
260. C13C2×#1325semidirect
327. C2#13112781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117186372
C2×#117181202
#131127812
C2×C13×#3242
C2×#132513