Finite Group Databasebeta

← C2×#16811230 #20288078

Subgroups — C2×#16811230 #20288078

Order 6784.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811230C2116yesmaximal, direct product factor
2C2×#13133216C211yesmaximal, characteristic, complemented
3C2×#13133210C211yesmaximal, characteristic
4C2×#13133210C211yesmaximal, characteristic
5C2×#13133210C211yesmaximal, characteristic
6C2×#13133209C211yesmaximal, characteristic, complemented
7C2×#13133208C211yesmaximal, characteristic, complemented
8C2×#13133207C211yesmaximal, characteristic, complemented
9C2×#13133207C211yesmaximal, characteristic, complemented
10C2×#13133207C211yesmaximal, characteristic, complemented
11C2×#13133194C211yesmaximal, characteristic
12C2×#13133194C211yesmaximal, characteristic
13C2×C4×Dic106C211yesmaximal, characteristic, complemented
14C2×#13133182C211yesmaximal, characteristic, complemented
15C2×#13133173C211yesmaximal, characteristic
16C2×#126×C53C211yesmaximal, characteristic
17C22×Dic106C2211yescharacteristic
18C22×Dic106C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20#13133216C2218yescomplemented
21#13133210C2218yes—
22#13133210C2218yes—
23#13133210C2218yes—
24#13133209C2218yescomplemented
25#13133208C2218yescomplemented
26#13133207C2218yescomplemented
27#13133207C2218yescomplemented
28#13133207C2218yescomplemented
29#13133194C2218yes—
30#13133194C2218yes—
31C4×Dic106C2218yescomplemented
32#13133182C2218yescomplemented
33#13133173C2218yes—
34C2×C4⋊Dic53C2211yescharacteristic
35C2×C4⋊Dic53C2211yescharacteristic
36C2×C4⋊Dic53C2211yescharacteristic
37C2×C4⋊Dic53C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39C2×#11719919C2211yescharacteristic
40C2×#11719919C2211yescharacteristic
41C2×#11719919C2211yescharacteristic
42C2×#11719919C2211yescharacteristic
43C2×#11719919C2211yescharacteristic
44C2×#11719919C2211yescharacteristic
45C2×#11719919C2211yescharacteristic
46C2×#11719919C2211yescharacteristic
47C2×#11719919C2211yescharacteristic
48C2×#11719919C2211yescharacteristic
49C2×#11719919C2211yescharacteristic
50C2×#11719919C2211yescharacteristic
51C2×C22⋊Dic53C2211yescharacteristic
52C2×C22⋊Dic53C2211yescharacteristic
53C2×C22⋊Dic53C2211yescharacteristic
54C2×C4×Dic53C2211yescharacteristic
55C2×C4×Dic53C2211yescharacteristic
56C2×C4×Dic53C2211yescharacteristic
57C2×C4×Dic53C2211yescharacteristic
58C2×C4×Dic53C2211yescharacteristic
59#126×C53C2218yes—
60C2×C4⋊C4×C53C2211yescharacteristic
61C2×C4⋊C4×C53C2211yescharacteristic
62C2×C4⋊C4×C53C2211yescharacteristic
63C2×C22⋊C4×C53C2211yescharacteristic
64C2×C22⋊C4×C53C2211yescharacteristic
65C2×C22⋊C4×C53C2211yescharacteristic
66C2×C42×C53C2211yescharacteristic
67C24×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×Dic53—24no—
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2314yes—
83C22×Dic53C2311yescharacteristic
84C2×Dic106—24no—
85C2×Dic106—24no—
86C2×Dic106C2314yes—
87C2×Dic106C2314yes—
88C4⋊C4×C53C2314yes—
89C4⋊C4×C53C2314yes—
90C4⋊C4×C53C2314yes—
91C22⋊C4×C53C2314yes—
92C22⋊C4×C53C2314yes—
93C22⋊C4×C53C2314yes—
94C42×C53C2314yes—
95C22⋊Dic53C2314yes—
96C22⋊Dic53C2314yes—
97C22⋊Dic53C2314yes—
98C4⋊Dic53C2314yes—
99C4⋊Dic53C2314yes—
100C4⋊Dic53C2314yes—
101C4⋊Dic53C2314yes—
102C4⋊Dic53C2314yes—
103#11719919C2314yes—
104#11719919C2314yes—
105#11719919C2314yes—
106#11719919C2314yes—
107#11719919C2314yes—
108#11719919C2314yes—
109#11719919C2314yes—
110#11719919C2314yes—
111#11719919C2314yes—
112#11719919C2314yes—
113#11719919C2314yes—
114#11719919C2314yes—
115C4×Dic53C2314yes—
116C4×Dic53C2314yes—
117C4×Dic53C2314yes—
118C4×Dic53C2314yes—
119C4×Dic53C2314yes—
120C23×C53—22no—
121C23×C53—22no—
122C23×C53—22no—
123C23×C53C2412yes—
124C23×C53C2411yescharacteristic
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53C2412yes—
132C2×C4×C53C2412yes—
133C2×C4×C53C2412yes—
134C2×C4×C53C2412yes—
135C2×C4×C53C2412yes—
136C2×C4×C53C2412yes—
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—28no—
145C2×Dic53#4114yes—
146C2×Dic53C2412yes—
147C2×Dic53C2412yes—
148C2×Dic53C2412yes—
149C2×Dic53C2412yes—
150C2×Dic53C2412yes—
151C2×Dic53C2412yes—
152C2×Dic53C2412yes—
153C2×Dic53C2412yes—
154Dic106—28no—
155Dic106—28no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—44no—
160C22×C53C2×#4112yes—
161C22×C532−1+412yes—
162C22×C532−1+412yes—
163C22×C53C2511yescharacteristic
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—22no—
170Dic53—22no—
171Dic53—22no—
172Dic53—22no—
173Dic53—22no—
174Dic53—22no—
175Dic53—22no—
176Dic53—22no—
177Dic53C2×#4114yes—
178C2×#541—531nomaximal, Sylow
179C2×C53—42no—
180C2×C53#54114yescomplemented
181C2×C53C22×#4111yescharacteristic
182C2×C53C2×2−1+411yescharacteristic
183C2×C53C2×2−1+411yescharacteristic
184#541—5316no—
185C2×C4⋊Q8—531no—
186C2×#126—531no—
187C2×#126—531no—
188C2×#126—531no—
189C2×#126D5311yescharacteristic
190C2×#125—531no—
191C2×#125—531no—
192C2×#125—531no—
193C2×#125—531no—
194C2×#125—531no—
195C2×C22⋊Q8—531no—
196C2×C22⋊Q8—531no—
197C2×C4×Q8—531no—
198C2×C4×Q8—531no—
199C2×#117—531no—
200C53C2×#54111yescharacteristic, complemented, Sylow
201C22×Q8—531no—
202C22×Q8—531no—
203C23×C4—531no—
204C4⋊Q8—538no—
205#126—538no—
206#126—538no—
207#126—538no—
208#126C2×D5318yes—
209#125—538no—
210#125—538no—
211#125—538no—
212#125—538no—
213#125—538no—
214C22⋊Q8—538no—
215C22⋊Q8—538no—
216C4×Q8—538no—
217C4×Q8—538no—
218#117—538no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4C2×D5311yescharacteristic
237C2×C4⋊C4C2×D5311yescharacteristic
238C2×C4⋊C4C2×D5311yescharacteristic
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4C2×D5311yescharacteristic
243C2×C22⋊C4C2×D5311yescharacteristic
244C2×C22⋊C4C2×D5311yescharacteristic
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42C2×D5311yescharacteristic
251C24C22×D5311yescharacteristic
252C2×Q8—534no—
253C2×Q8—534no—
254C2×Q8—1064no—
255C2×Q8—1064no—
256C22×C4—531no—
257C22×C4—534no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—1064no—
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C4⋊C4—534no—
273C4⋊C4—534no—
274C4⋊C4—534no—
275C4⋊C4—534no—
276C4⋊C4—534no—
277C4⋊C4—534no—
278C4⋊C4—534no—
279C4⋊C4—534no—
280C4⋊C4—534no—
281C4⋊C4—534no—
282C4⋊C4—534no—
283C4⋊C4—534no—
284C4⋊C4—534no—
285C4⋊C4—534no—
286C4⋊C4—534no—
287C4⋊C4—534no—
288C4⋊C4—534no—
289C4⋊C4C22×D5314yes—
290C4⋊C4C22×D5314yes—
291C4⋊C4C22×D5314yes—
292C22⋊C4—534no—
293C22⋊C4—534no—
294C22⋊C4—534no—
295C22⋊C4C22×D5314yes—
296C22⋊C4C22×D5314yes—
297C22⋊C4C22×D5314yes—
298C42—534no—
299C42—534no—
300C42—534no—
301C42—534no—
302C42—534no—
303C42C22×D5314yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23C23×D5312yes—
308C23C23×D5311yescharacteristic
309Q8—1068no—
310Q8—1068no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—534no—
318C2×C4—532no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—532no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—532no—
325C2×C4—532no—
326C2×C4—1068no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4C23×D5312yes—
335C2×C4C23×D5312yes—
336C2×C4C23×D5312yes—
337C2×C4C23×D5312yes—
338C2×C4C23×D5312yes—
339C2×C4C23×D5312yes—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—44no—
344C22#41×D5312yes—
345C22#1313326412yes—
346C22#1313326412yes—
347C22C24×D5311yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—534no—
355C4—1062no—
356C4—1062no—
357C4—1062no—
358C4—1062no—
359C4—1062no—
360C4—1062no—
361C4—1062no—
362C2—42no—
363C2#1681123014yesdirect product factor
364C2C2×#41×D5311yescharacteristic
365C2C2×#1313326411yescharacteristic
366C2C2×#1313326411yescharacteristic
subgroupG/H
centerC23C23×D53
commutatorC22×C53C25
FrattiniC22C24×D53
FittingC2×#126×C53C2
socleC23×C53C24
hypercenterC2×#126D53
hypocenterC53C2×#541
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #16811230C2direct (N × H)
2. C2×#13133216C2semidirect
6. C2×#13133209C2semidirect
7. C2×#13133208C2semidirect
8. C2×#13133207C2semidirect
9. C2×#13133207C2semidirect
10. C2×#13133207C2semidirect
13. C2×C4×Dic106C2semidirect
14. C2×#13133182C2semidirect
20. #13133216C22semidirect
24. #13133209C22semidirect
25. #13133208C22semidirect
26. #13133207C22semidirect
27. #13133207C22semidirect
28. #13133207C22semidirect
31. C4×Dic106C22semidirect
32. #13133182C22semidirect
180. C2×C53#541semidirect
200. C53C2×#541semidirect
363. C2#16811230direct (N × H)