Finite Group Databasebeta

← C2×#20265859 #27184636

Subgroups — C2×#20265859 #27184636

Order 12928.

1124 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265859C2116yesmaximal, direct product factor
2C2×#15692250C211yesmaximal, characteristic, complemented
3C2×#15692250C211yesmaximal, characteristic, complemented
4C2×#15692245C211yesmaximal, characteristic, complemented
5C2×#15692245C211yesmaximal, characteristic, complemented
6C2×#15692240C211yesmaximal, characteristic
7C2×#15692240C211yesmaximal, characteristic
8C2×#15692233C211yesmaximal, characteristic, complemented
9C2×#15692233C211yesmaximal, characteristic, complemented
10C2×#15692220C211yesmaximal, characteristic, complemented
11C2×#15692217C211yesmaximal, characteristic
12C2×#15692213C211yesmaximal, characteristic, complemented
13C2×#15692212C211yesmaximal, characteristic, complemented
14C2×#15692212C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
16C2×#125×C101C211yesmaximal, characteristic, complemented
17C22×Dic202C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic
23#15692250C2218yescomplemented
24#15692250C2218yescomplemented
25#15692245C2218yescomplemented
26#15692245C2218yescomplemented
27#15692240C2218yes—
28#15692240C2218yes—
29#15692233C2218yescomplemented
30#15692233C2218yescomplemented
31#15692220C2218yescomplemented
32#15692217C2218yes—
33#15692213C2218yescomplemented
34#15692212C2218yescomplemented
35#15692212C2218yescomplemented
36C2×C4⋊Dic101C2211yescharacteristic
37C2×C4⋊Dic101C2211yescharacteristic
38C2×#13108053C2211yescharacteristic
39C2×#13108053C2211yescharacteristic
40C2×#13108053C2211yescharacteristic
41C2×#13108053C2211yescharacteristic
42C2×#13108053C2211yescharacteristic
43C2×#13108053C2211yescharacteristic
44C2×#13108053C2211yescharacteristic
45C2×#13108053C2211yescharacteristic
46C4⋊C4×D101C2218yescomplemented
47C2×#13108051C2211yescharacteristic
48C2×#13108051C2211yescharacteristic
49C2×#13108051C2211yescharacteristic
50C2×#13108051C2211yescharacteristic
51C2×#13108051C2211yescharacteristic
52C2×#13108051C2211yescharacteristic
53C2×#13108051C2211yescharacteristic
54C2×#13108051C2211yescharacteristic
55C2×C4×Dic101C2211yescharacteristic
56C2×C4×Dic101C2211yescharacteristic
57C2×C4×Dic101C2211yescharacteristic
58C2×C4×Dic101C2211yescharacteristic
59#125×C101C2218yescomplemented
60C2×C4⋊C4×C101C2211yescharacteristic
61C2×C4⋊C4×C101C2211yescharacteristic
62C2×C4⋊C4×C101C2211yescharacteristic
63C2×C4⋊C4×C101C2211yescharacteristic
64C2×C4⋊C4×C101C2211yescharacteristic
65C2×C4⋊C4×C101C2211yescharacteristic
66C2×C42×C101C2211yescharacteristic
67C23×D101C2311yescharacteristic
68C23×D101C2311yescharacteristic
69C2×Dic202—24no—
70C2×Dic202C2314yes—
71C2×D404—24no—
72C2×D404C2314yes—
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C2×C4×D101—24no—
80C2×C4×D101—24no—
81C2×C4×D101—24no—
82C2×C4×D101—24no—
83C2×C4×D101C2314yes—
84C2×C4×D101C2314yes—
85C2×C4×D101C2314yes—
86C2×C4×D101C2314yes—
87C4⋊Dic101C2314yes—
88C4⋊Dic101C2314yes—
89#13108053C2314yes—
90#13108053C2314yes—
91#13108053C2314yes—
92#13108053C2314yes—
93#13108053C2314yes—
94#13108053C2314yes—
95#13108053C2314yes—
96#13108053C2314yes—
97#13108051C2314yes—
98#13108051C2314yes—
99#13108051C2314yes—
100#13108051C2314yes—
101#13108051C2314yes—
102#13108051C2314yes—
103#13108051C2314yes—
104#13108051C2314yes—
105C4×Dic101C2314yes—
106C4×Dic101C2314yes—
107C4×Dic101C2314yes—
108C4×Dic101C2314yes—
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C4⋊C4×C101C2314yes—
117C4⋊C4×C101C2314yes—
118C4⋊C4×C101C2314yes—
119C4⋊C4×C101C2314yes—
120C4⋊C4×C101C2314yes—
121C4⋊C4×C101C2314yes—
122C42×C101C2314yes—
123C23×C101C2411yescharacteristic
124C22×D101—22no—
125C22×D101—22no—
126C22×D101—22no—
127C22×D101—22no—
128C22×D101—22no—
129C22×D101#4114yes—
130C22×D101C2412yes—
131C22×D101C2412yes—
132C2×C4×C101—22no—
133C2×C4×C101—22no—
134C2×C4×C101—22no—
135C2×C4×C101—22no—
136C2×C4×C101—22no—
137C2×C4×C101—22no—
138C2×C4×C101—22no—
139C2×C4×C101C2412yes—
140C2×C4×C101C2412yes—
141C2×C4×C101C2412yes—
142C2×C4×C101C2412yes—
143C2×C4×C101C2412yes—
144C2×C4×C101C2412yes—
145C2×C4×C101C2412yes—
146C2×Dic101—22no—
147C2×Dic101—22no—
148C2×Dic101—22no—
149C2×Dic101—22no—
150C2×Dic101—22no—
151C2×Dic101#4114yes—
152C2×Dic101C2412yes—
153C2×Dic101C2412yes—
154C2×Dic101C2412yes—
155C2×Dic101C2412yes—
156C2×Dic101C2412yes—
157C2×Dic101C2412yes—
158D404—28no—
159C4×D101—28no—
160C4×D101—28no—
161C4×D101—28no—
162C4×D101—28no—
163Dic202—28no—
164C22×C101C2×#4112yes—
165C22×C101C2×#4112yes—
166C22×C1012−1+412yes—
167C22×C101C2511yescharacteristic
168C2×D101—28no—
169C2×D101—22no—
170C2×D101—22no—
171C2×D101—22no—
172C2×D101—22no—
173C2×D101—22no—
174C2×D101—44no—
175C2×D101C2×#4114yes—
176C4×C101—22no—
177C4×C101—22no—
178C4×C101—22no—
179C4×C101—22no—
180C4×C101—22no—
181C4×C101—22no—
182C4×C101—22no—
183Dic101—22no—
184Dic101—22no—
185Dic101—22no—
186Dic101—22no—
187Dic101—22no—
188Dic101C2×#4114yes—
189C2×C101#55214yescomplemented
190C2×C101C22×#4111yescharacteristic
191C2×C101C22×#4111yescharacteristic
192C2×C101C2×2−1+411yescharacteristic
193D101—24no—
194D101—42no—
195C2×#552—1011nomaximal, Sylow
196C101C2×#55211yescharacteristic, complemented, Sylow
197#552—10116no—
198C2×#126—1011no—
199C2×#126—1011no—
200C2×#125—1011no—
201C2×#125—1011no—
202C2×#125D10111yescharacteristic, complemented
203C2×#123—1011no—
204C2×#123—1011no—
205C2×C22⋊Q8—1011no—
206C2×C22⋊Q8—1011no—
207C2×C4×Q8—1011no—
208C2×C4×D4—1011no—
209C2×#117—1011no—
210C2×#117—1011no—
211C2×#117—1011no—
212C22×C4⋊C4—1011no—
213C22×Q8—1011no—
214C22×D4—1011no—
215C23×C4—1011no—
216C23×C4—1011no—
217C23×C4—1011no—
218C23×C4—1011no—
219#126—1018no—
220#126—1018no—
221#125—1018no—
222#125—1018no—
223#125C2×D10118yescomplemented
224#123—1018no—
225#123—1018no—
226C22⋊Q8—1018no—
227C22⋊Q8—1018no—
228C4×Q8—1018no—
229C4×D4—1018no—
230#117—1018no—
231#117—1018no—
232#117—1018no—
233C2×C4⋊C4—1011no—
234C2×C4⋊C4—1011no—
235C2×C4⋊C4—1011no—
236C2×C4⋊C4—1011no—
237C2×C4⋊C4—1011no—
238C2×C4⋊C4—1011no—
239C2×C4⋊C4—1011no—
240C2×C4⋊C4—1011no—
241C2×C4⋊C4—1011no—
242C2×C4⋊C4—1011no—
243C2×C4⋊C4—1018no—
244C2×C4⋊C4C2×D10111yescharacteristic
245C2×C4⋊C4C2×D10111yescharacteristic
246C2×C4⋊C4C2×D10111yescharacteristic
247C2×C4⋊C4C2×D10111yescharacteristic
248C2×C4⋊C4C2×D10111yescharacteristic
249C2×C4⋊C4C2×D10111yescharacteristic
250C2×C22⋊C4—1011no—
251C2×C22⋊C4—1011no—
252C2×C22⋊C4—1011no—
253C2×C22⋊C4—1011no—
254C2×C22⋊C4—1011no—
255C2×C22⋊C4—1011no—
256C2×C22⋊C4—1011no—
257C2×C22⋊C4—1011no—
258C2×C42—1011no—
259C2×C42—1011no—
260C2×C42—1011no—
261C2×C42—1011no—
262C2×C42C2×D10111yescharacteristic
263C24—1011no—
264C24—1011no—
265C2×Q8—1014no—
266C2×Q8—2024no—
267C2×D4—1014no—
268C2×D4—2024no—
269C22×C4—1011no—
270C22×C4—1011no—
271C22×C4—1011no—
272C22×C4—1011no—
273C22×C4—1011no—
274C22×C4—1011no—
275C22×C4—1014no—
276C22×C4—1014no—
277C22×C4—1014no—
278C22×C4—1014no—
279C22×C4—2024no—
280C22×C4—2024no—
281C22×C4—2024no—
282C22×C4—2024no—
283C22×C4C22×D10111yescharacteristic
284C22×C4C22×D10111yescharacteristic
285C22×C4C22×D10111yescharacteristic
286C22×C4C22×D10111yescharacteristic
287C22×C4C22×D10111yescharacteristic
288C22×C4C22×D10111yescharacteristic
289C22×C4C22×D10111yescharacteristic
290C4⋊C4—1014no—
291C4⋊C4—1014no—
292C4⋊C4—1014no—
293C4⋊C4—1014no—
294C4⋊C4—1014no—
295C4⋊C4—1014no—
296C4⋊C4—1014no—
297C4⋊C4—1014no—
298C4⋊C4—1014no—
299C4⋊C4—1014no—
300C4⋊C4C22×D10114yes—
301C4⋊C4C22×D10114yes—
302C4⋊C4C22×D10114yes—
303C4⋊C4C22×D10114yes—
304C4⋊C4C22×D10114yes—
305C4⋊C4C22×D10114yes—
306C22⋊C4—1014no—
307C22⋊C4—1014no—
308C22⋊C4—1014no—
309C22⋊C4—1014no—
310C22⋊C4—1014no—
311C22⋊C4—1014no—
312C22⋊C4—1014no—
313C22⋊C4—1014no—
314C42—1014no—
315C42—1014no—
316C42—1014no—
317C42—1014no—
318C42C22×D10114yes—
319C23—1014no—
320C23—1012no—
321C23—1012no—
322C23—2022no—
323C23—2022no—
324C23—2022no—
325C23—2022no—
326C23—2022no—
327C23C23×D10111yescharacteristic
328Q8—2028no—
329D4—2028no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—1014no—
338C2×C4—1012no—
339C2×C4—1012no—
340C2×C4—1012no—
341C2×C4—1012no—
342C2×C4—1012no—
343C2×C4—1012no—
344C2×C4—2028no—
345C2×C4—2028no—
346C2×C4—2028no—
347C2×C4—2028no—
348C2×C4—2022no—
349C2×C4—2022no—
350C2×C4—2022no—
351C2×C4—2022no—
352C2×C4—2022no—
353C2×C4C23×D10112yes—
354C2×C4C23×D10112yes—
355C2×C4C23×D10112yes—
356C2×C4C23×D10112yes—
357C2×C4C23×D10112yes—
358C2×C4C23×D10112yes—
359C2×C4C23×D10112yes—
360C22—1014no—
361C22—4044no—
362C22—2022no—
363C22—2022no—
364C22—2022no—
365C22—2022no—
366C22—2022no—
367C22—2028no—
368C22#41×D10112yes—
369C22#41×D10112yes—
370C22#1569230312yes—
371C22C24×D10111yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—1014no—
380C4—2022no—
381C4—2022no—
382C4—2022no—
383C4—2022no—
384C4—2022no—
385C2—4042no—
386C2—2024no—
387C2#2026585914yesdirect product factor
388C2C2×#41×D10111yescharacteristic
389C2C2×#41×D10111yescharacteristic
390C2C2×#1569230311yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#125×C101C2
socleC23×C101C24
prime radicalC22×D404C22
hypercenterC2×#125D101
hypocenterC101C2×#552
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #20265859C2direct (N × H)
2. C2×#15692250C2semidirect
3. C2×#15692250C2semidirect
4. C2×#15692245C2semidirect
5. C2×#15692245C2semidirect
8. C2×#15692233C2semidirect
9. C2×#15692233C2semidirect
10. C2×#15692220C2semidirect
12. C2×#15692213C2semidirect
13. C2×#15692212C2semidirect
14. C2×#15692212C2semidirect
15. C2×C4⋊C4×D101C2semidirect
16. C2×#125×C101C2semidirect
23. #15692250C22semidirect
24. #15692250C22semidirect
25. #15692245C22semidirect
26. #15692245C22semidirect
29. #15692233C22semidirect
30. #15692233C22semidirect
31. #15692220C22semidirect
33. #15692213C22semidirect
34. #15692212C22semidirect
35. #15692212C22semidirect
46. C4⋊C4×D101C22semidirect
59. #125×C101C22semidirect
189. C2×C101#552semidirect
196. C101C2×#552semidirect
202. C2×#125D101semidirect
223. #125C2×D101semidirect
387. C2#20265859direct (N × H)

Maximal subgroups

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

subgroupindex
#202658592
C2×#156922502
C2×#156922452
C2×#156922402
C2×#156922332
C2×#156922202
C2×#156922172
C2×#156922132
C2×#156922122
C2×C4⋊C4×D1012
C2×#125×C1012
C2×#552101