Finite Group Databasebeta

← C2×#20265710 #27184397

Subgroups — C2×#20265710 #27184397

Order 12928.

1232 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265710C2116yesmaximal, direct product factor
2C2×#15692249C211yesmaximal, characteristic, complemented
3C2×#15692245C211yesmaximal, characteristic, complemented
4C2×#15692241C211yesmaximal, characteristic, complemented
5C2×#15692240C211yesmaximal, characteristic, complemented
6C2×#15692237C211yesmaximal, characteristic, complemented
7C2×#15692233C211yesmaximal, characteristic, complemented
8C2×#15692228C211yesmaximal, characteristic
9C2×#15692220C211yesmaximal, characteristic, complemented
10C2×#15692217C211yesmaximal, characteristic, complemented
11C2×#15692215C211yesmaximal, characteristic, complemented
12C2×#15692215C211yesmaximal, characteristic, complemented
13C2×#15692213C211yesmaximal, characteristic, complemented
14C2×#15692211C211yesmaximal, characteristic, complemented
15C2×C42×D101C211yesmaximal, characteristic, complemented
16C2×#126×C101C211yesmaximal, characteristic, complemented
17C22×Dic202C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×C22⋊D101C2211yescharacteristic
20C22×C22⋊D101C2211yescharacteristic
21C23×Dic101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic
23C22×C4×D101C2211yescharacteristic
24C22×C4×D101C2211yescharacteristic
25C22×C4×D101C2211yescharacteristic
26#15692249C2218yescomplemented
27#15692245C2218yescomplemented
28#15692241C2218yescomplemented
29#15692240C2218yescomplemented
30#15692237C2218yescomplemented
31#15692233C2218yescomplemented
32#15692228C2218yes—
33#15692220C2218yescomplemented
34#15692217C2218yescomplemented
35#15692215C2218yescomplemented
36#15692215C2218yescomplemented
37#15692213C2218yescomplemented
38#15692211C2218yescomplemented
39C2×C4⋊Dic101C2211yescharacteristic
40C2×#13108053C2211yescharacteristic
41C2×#13108053C2211yescharacteristic
42C2×#13108053C2211yescharacteristic
43C2×#13108053C2211yescharacteristic
44C2×#13108053C2211yescharacteristic
45C2×#13108053C2211yescharacteristic
46C2×C22⋊Dic101C2211yescharacteristic
47C2×#13108051C2211yescharacteristic
48C2×#13108051C2211yescharacteristic
49C2×#13108051C2211yescharacteristic
50C2×#13108051C2211yescharacteristic
51C2×#13108051C2211yescharacteristic
52C2×#13108051C2211yescharacteristic
53C2×C4×Dic101C2211yescharacteristic
54C2×C4×Dic101C2211yescharacteristic
55C2×C4×Dic101C2211yescharacteristic
56C2×C4×Dic101C2211yescharacteristic
57C2×C4×Dic101C2211yescharacteristic
58C42×D101C2218yescomplemented
59#126×C101C2218yescomplemented
60C2×C4⋊C4×C101C2211yescharacteristic
61C2×C4⋊C4×C101C2211yescharacteristic
62C2×C4⋊C4×C101C2211yescharacteristic
63C2×C22⋊C4×C101C2211yescharacteristic
64C2×C22⋊C4×C101C2211yescharacteristic
65C2×C22⋊C4×C101C2211yescharacteristic
66C2×C42×C101C2211yescharacteristic
67C23×D101C2311yescharacteristic
68C23×D101C2311yescharacteristic
69C2×Dic202—24no—
70C2×Dic202C2314yes—
71C2×D404—24no—
72C2×D404C2314yes—
73C2×C22⋊D101—24no—
74C2×C22⋊D101—24no—
75C2×C22⋊D101C2314yes—
76C2×C22⋊D101C2314yes—
77C22×Dic101—24no—
78C22×Dic101C2311yescharacteristic
79C22×Dic101C2311yescharacteristic
80C22×Dic101C2314yes—
81C22×Dic101C2311yescharacteristic
82C22×Dic101C2311yescharacteristic
83C22×Dic101C2311yescharacteristic
84C22×Dic101C2311yescharacteristic
85C2×C4×D101—24no—
86C2×C4×D101—24no—
87C2×C4×D101—24no—
88C2×C4×D101—24no—
89C2×C4×D101C2314yes—
90C2×C4×D101C2314yes—
91C2×C4×D101C2314yes—
92C2×C4×D101C2314yes—
93C4⋊Dic101C2314yes—
94#13108053C2314yes—
95#13108053C2314yes—
96#13108053C2314yes—
97#13108053C2314yes—
98#13108053C2314yes—
99#13108053C2314yes—
100C22⋊Dic101C2314yes—
101#13108051C2314yes—
102#13108051C2314yes—
103#13108051C2314yes—
104#13108051C2314yes—
105#13108051C2314yes—
106#13108051C2314yes—
107C4×Dic101C2314yes—
108C4×Dic101C2314yes—
109C4×Dic101C2314yes—
110C4×Dic101C2314yes—
111C4×Dic101C2314yes—
112C24×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117C22×C4×C101C2311yescharacteristic
118C22×C4×C101C2311yescharacteristic
119C4⋊C4×C101C2314yes—
120C4⋊C4×C101C2314yes—
121C4⋊C4×C101C2314yes—
122C22⋊C4×C101C2314yes—
123C22⋊C4×C101C2314yes—
124C22⋊C4×C101C2314yes—
125C42×C101C2314yes—
126C23×C101—22no—
127C23×C101—22no—
128C23×C101—22no—
129C23×C101C2412yes—
130C23×C101C2411yescharacteristic
131C22×D101—22no—
132C22×D101—22no—
133C22×D101—22no—
134C22×D101—22no—
135C22×D101—22no—
136C22×D101#4114yes—
137C22×D101C2412yes—
138C22×D101C2412yes—
139C2×C4×C101—22no—
140C2×C4×C101—22no—
141C2×C4×C101—22no—
142C2×C4×C101—22no—
143C2×C4×C101—22no—
144C2×C4×C101—22no—
145C2×C4×C101C2412yes—
146C2×C4×C101C2412yes—
147C2×C4×C101C2412yes—
148C2×C4×C101C2412yes—
149C2×C4×C101C2412yes—
150C2×C4×C101C2412yes—
151C22⋊D101—28no—
152C22⋊D101—28no—
153C2×Dic101—22no—
154C2×Dic101—22no—
155C2×Dic101—22no—
156C2×Dic101—22no—
157C2×Dic101—28no—
158C2×Dic101#4114yes—
159C2×Dic101#4114yes—
160C2×Dic101C2412yes—
161C2×Dic101C2412yes—
162C2×Dic101C2412yes—
163C2×Dic101C2412yes—
164C2×Dic101C2412yes—
165C2×Dic101C2412yes—
166D404—28no—
167C4×D101—28no—
168C4×D101—28no—
169C4×D101—28no—
170C4×D101—28no—
171Dic202—28no—
172C22×C101—22no—
173C22×C101—22no—
174C22×C101—22no—
175C22×C101—44no—
176C22×C101C2×#4112yes—
177C22×C101C2×#4112yes—
178C22×C101C2×#4112yes—
179C22×C101C2511yescharacteristic
180C2×D101—28no—
181C2×D101—22no—
182C2×D101—22no—
183C2×D101—22no—
184C2×D101—22no—
185C2×D101—22no—
186C2×D101—44no—
187C2×D101C2×#4114yes—
188C4×C101—22no—
189C4×C101—22no—
190C4×C101—22no—
191C4×C101—22no—
192C4×C101—22no—
193C4×C101—22no—
194Dic101—22no—
195Dic101—22no—
196Dic101—22no—
197Dic101—22no—
198Dic101C2×#4114yes—
199Dic101C2×#4114yes—
200C2×C101—42no—
201C2×C101#52914yescomplemented
202C2×C101C22×#4111yescharacteristic
203C2×C101C22×#4111yescharacteristic
204C2×C101C22×#4111yescharacteristic
205D101—24no—
206D101—42no—
207C2×#529—1011nomaximal, Sylow
208C101C2×#52911yescharacteristic, complemented, Sylow
209#529—10116no—
210C2×#126—1011no—
211C2×#126D10111yescharacteristic, complemented
212C2×#125—1011no—
213C2×#124—1011no—
214C2×#123—1011no—
215C2×#123—1011no—
216C2×C22⋊Q8—1011no—
217C2×#121—1011no—
218C2×C4×Q8—1011no—
219C2×C4×D4—1011no—
220C2×C4×D4—1011no—
221C2×C4×D4—1011no—
222C2×#117—1011no—
223C2×#117—1011no—
224C22×C42—1011no—
225C22×Q8—1011no—
226C22×D4—1011no—
227C22×D4—1011no—
228C22×D4—1011no—
229C23×C4—1011no—
230C23×C4—1011no—
231C23×C4—1011no—
232C23×C4—1011no—
233C23×C4—1011no—
234#126—1018no—
235#126C2×D10118yescomplemented
236#125—1018no—
237#124—1018no—
238#123—1018no—
239#123—1018no—
240C22⋊Q8—1018no—
241#121—1018no—
242C4×Q8—1018no—
243C4×D4—1018no—
244C4×D4—1018no—
245C4×D4—1018no—
246#117—1018no—
247#117—1018no—
248C2×C4⋊C4—1011no—
249C2×C4⋊C4—1011no—
250C2×C4⋊C4—1011no—
251C2×C4⋊C4—1011no—
252C2×C4⋊C4—1011no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4C2×D10111yescharacteristic
256C2×C4⋊C4C2×D10111yescharacteristic
257C2×C4⋊C4C2×D10111yescharacteristic
258C2×C22⋊C4—1011no—
259C2×C22⋊C4—1011no—
260C2×C22⋊C4—1011no—
261C2×C22⋊C4—1011no—
262C2×C22⋊C4—1011no—
263C2×C22⋊C4—1011no—
264C2×C22⋊C4—1011no—
265C2×C22⋊C4C2×D10111yescharacteristic
266C2×C22⋊C4C2×D10111yescharacteristic
267C2×C22⋊C4C2×D10111yescharacteristic
268C2×C42—1011no—
269C2×C42—1011no—
270C2×C42—1011no—
271C2×C42—1011no—
272C2×C42—1011no—
273C2×C42—1018no—
274C2×C42C2×D10111yescharacteristic
275C24—1011no—
276C24—1011no—
277C24C22×D10111yescharacteristic
278C2×Q8—1014no—
279C2×Q8—2024no—
280C2×D4—1014no—
281C2×D4—1014no—
282C2×D4—1014no—
283C2×D4—2024no—
284C2×D4—2024no—
285C2×D4—2024no—
286C22×C4—1011no—
287C22×C4—1011no—
288C22×C4—1014no—
289C22×C4—1011no—
290C22×C4—1011no—
291C22×C4—1011no—
292C22×C4—1011no—
293C22×C4—1014no—
294C22×C4—1014no—
295C22×C4—1014no—
296C22×C4—1014no—
297C22×C4—2024no—
298C22×C4—2024no—
299C22×C4—2024no—
300C22×C4—2024no—
301C22×C4—2024no—
302C22×C4C22×D10111yescharacteristic
303C22×C4C22×D10111yescharacteristic
304C22×C4C22×D10111yescharacteristic
305C22×C4C22×D10111yescharacteristic
306C22×C4C22×D10111yescharacteristic
307C22×C4C22×D10111yescharacteristic
308C4⋊C4—1014no—
309C4⋊C4—1014no—
310C4⋊C4—1014no—
311C4⋊C4—1014no—
312C4⋊C4—1014no—
313C4⋊C4—1014no—
314C4⋊C4—1014no—
315C4⋊C4C22×D10114yes—
316C4⋊C4C22×D10114yes—
317C4⋊C4C22×D10114yes—
318C22⋊C4—1014no—
319C22⋊C4—1014no—
320C22⋊C4—1014no—
321C22⋊C4—1014no—
322C22⋊C4—1014no—
323C22⋊C4—1014no—
324C22⋊C4—1014no—
325C22⋊C4C22×D10114yes—
326C22⋊C4C22×D10114yes—
327C22⋊C4C22×D10114yes—
328C42—1014no—
329C42—1014no—
330C42—1014no—
331C42—1014no—
332C42—1014no—
333C42C22×D10114yes—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—1014no—
338C23—1012no—
339C23—1012no—
340C23—2022no—
341C23—2022no—
342C23—2022no—
343C23—2022no—
344C23—2022no—
345C23C23×D10112yes—
346C23C23×D10111yescharacteristic
347Q8—2028no—
348D4—2028no—
349D4—2028no—
350D4—2028no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—1014no—
358C2×C4—1014no—
359C2×C4—1012no—
360C2×C4—1012no—
361C2×C4—1012no—
362C2×C4—1012no—
363C2×C4—1012no—
364C2×C4—1012no—
365C2×C4—2028no—
366C2×C4—2028no—
367C2×C4—2028no—
368C2×C4—2028no—
369C2×C4—2028no—
370C2×C4—2022no—
371C2×C4—2022no—
372C2×C4—2022no—
373C2×C4—2022no—
374C2×C4C23×D10112yes—
375C2×C4C23×D10112yes—
376C2×C4C23×D10112yes—
377C2×C4C23×D10112yes—
378C2×C4C23×D10112yes—
379C2×C4C23×D10112yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—44no—
384C22—1014no—
385C22—4044no—
386C22—2022no—
387C22—2022no—
388C22—2022no—
389C22—2022no—
390C22—2022no—
391C22—2028no—
392C22#41×D10112yes—
393C22#41×D10112yes—
394C22#41×D10112yes—
395C22C24×D10111yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—1014no—
403C4—1014no—
404C4—2022no—
405C4—2022no—
406C4—2022no—
407C4—2022no—
408C2—42no—
409C2—4042no—
410C2—2024no—
411C2#2026571014yesdirect product factor
412C2C2×#41×D10111yescharacteristic
413C2C2×#41×D10111yescharacteristic
414C2C2×#41×D10111yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#126×C101C2
socleC23×C101C24
prime radicalC2×#15692228C2
hypercenterC2×#126D101
hypocenterC101C2×#529
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20265710C2direct (N × H)
2. C2×#15692249C2semidirect
3. C2×#15692245C2semidirect
4. C2×#15692241C2semidirect
5. C2×#15692240C2semidirect
6. C2×#15692237C2semidirect
7. C2×#15692233C2semidirect
9. C2×#15692220C2semidirect
10. C2×#15692217C2semidirect
11. C2×#15692215C2semidirect
12. C2×#15692215C2semidirect
13. C2×#15692213C2semidirect
14. C2×#15692211C2semidirect
15. C2×C42×D101C2semidirect
16. C2×#126×C101C2semidirect
26. #15692249C22semidirect
27. #15692245C22semidirect
28. #15692241C22semidirect
29. #15692240C22semidirect
30. #15692237C22semidirect
31. #15692233C22semidirect
33. #15692220C22semidirect
34. #15692217C22semidirect
35. #15692215C22semidirect
36. #15692215C22semidirect
37. #15692213C22semidirect
38. #15692211C22semidirect
58. C42×D101C22semidirect
59. #126×C101C22semidirect
201. C2×C101#529semidirect
208. C101C2×#529semidirect
211. C2×#126D101semidirect
235. #126C2×D101semidirect
411. C2#20265710direct (N × H)

Maximal subgroups

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

subgroupindex
#202657102
C2×#156922492
C2×#156922452
C2×#156922412
C2×#156922402
C2×#156922372
C2×#156922332
C2×#156922282
C2×#156922202
C2×#156922172
C2×#156922152
C2×#156922132
C2×#156922112
C2×C42×D1012
C2×#126×C1012
C2×#529101