Finite Group Databasebeta

← C2×#20265779 #27184489

Subgroups — C2×#20265779 #27184489

Order 12928.

1260 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265779C2116yesmaximal, direct product factor
2C2×#15692251C211yesmaximal, characteristic, complemented
3C2×#15692245C211yesmaximal, characteristic, complemented
4C2×#15692240C211yesmaximal, characteristic, complemented
5C2×#15692239C211yesmaximal, characteristic, complemented
6C2×#15692237C211yesmaximal, characteristic, complemented
7C2×#15692230C211yesmaximal, characteristic, complemented
8C2×#15692229C211yesmaximal, characteristic, complemented
9C2×#15692228C211yesmaximal, characteristic, complemented
10C2×#15692228C211yesmaximal, characteristic, complemented
11C2×#15692228C211yesmaximal, characteristic, complemented
12C2×#15692227C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
14C2×#15692215C211yesmaximal, characteristic, complemented
15C2×#15692213C211yesmaximal, characteristic, complemented
16C2×#123×C101C211yesmaximal, characteristic, complemented
17C22×D404C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×D404C2211yescharacteristic
20C22×C22⋊D101C2211yescharacteristic
21C22×C22⋊D101C2211yescharacteristic
22C22×C22⋊D101C2211yescharacteristic
23C22×C22⋊D101C2211yescharacteristic
24C22×C22⋊D101C2211yescharacteristic
25C22×C22⋊D101C2211yescharacteristic
26C23×Dic101C2211yescharacteristic
27C22×C4×D101C2211yescharacteristic
28C22×C4×D101C2211yescharacteristic
29C22×C4×D101C2211yescharacteristic
30#15692251C2218yescomplemented
31#15692245C2218yescomplemented
32#15692240C2218yescomplemented
33#15692239C2218yescomplemented
34#15692237C2218yescomplemented
35#15692230C2218yescomplemented
36#15692229C2218yescomplemented
37#15692228C2218yescomplemented
38#15692228C2218yescomplemented
39#15692228C2218yescomplemented
40#15692227C2218yescomplemented
41C4×C22⋊D101C2218yescomplemented
42#15692215C2218yescomplemented
43#15692213C2218yescomplemented
44C2×C4⋊Dic101C2211yescharacteristic
45C2×C4⋊Dic101C2211yescharacteristic
46C2×#13108053C2211yescharacteristic
47C2×#13108053C2211yescharacteristic
48C2×#13108053C2211yescharacteristic
49C2×#13108053C2211yescharacteristic
50C2×C22⋊Dic101C2211yescharacteristic
51C2×#13108051C2211yescharacteristic
52C2×#13108051C2211yescharacteristic
53C2×#13108051C2211yescharacteristic
54C2×#13108051C2211yescharacteristic
55C2×#13108051C2211yescharacteristic
56C2×#13108051C2211yescharacteristic
57C2×C4×Dic101C2211yescharacteristic
58C2×C4×Dic101C2211yescharacteristic
59C22×D4×C101C2211yescharacteristic
60C23×C4×C101C2211yescharacteristic
61#123×C101C2218yescomplemented
62C2×C4⋊C4×C101C2211yescharacteristic
63C2×C4⋊C4×C101C2211yescharacteristic
64C2×C22⋊C4×C101C2211yescharacteristic
65C2×C22⋊C4×C101C2211yescharacteristic
66C2×C22⋊C4×C101C2211yescharacteristic
67C23×D101C2311yescharacteristic
68C23×D101C2311yescharacteristic
69C23×D101C2311yescharacteristic
70C2×D404—42no—
71C2×D404—42no—
72C2×D404—42no—
73C2×D404C2314yes—
74C2×D404C2314yes—
75C2×D404C2314yes—
76C2×C22⋊D101—24no—
77C2×C22⋊D101—24no—
78C2×C22⋊D101—42no—
79C2×C22⋊D101—42no—
80C2×C22⋊D101—42no—
81C2×C22⋊D101—42no—
82C2×C22⋊D101C2314yes—
83C2×C22⋊D101C2314yes—
84C2×C22⋊D101C2314yes—
85C2×C22⋊D101C2314yes—
86C2×C22⋊D101C2314yes—
87C2×C22⋊D101C2314yes—
88C22×Dic101—42no—
89C22×Dic101C2311yescharacteristic
90C22×Dic101C2311yescharacteristic
91C22×Dic101C2311yescharacteristic
92C22×Dic101C2311yescharacteristic
93C22×Dic101C2314yes—
94C22×Dic101C2311yescharacteristic
95C2×C4×D101—24no—
96C2×C4×D101—42no—
97C2×C4×D101—42no—
98C2×C4×D101C2314yes—
99C2×C4×D101C2314yes—
100C2×C4×D101C2314yes—
101C4⋊Dic101C2314yes—
102C4⋊Dic101C2314yes—
103#13108053C2314yes—
104#13108053C2314yes—
105#13108053C2314yes—
106#13108053C2314yes—
107C22⋊Dic101C2314yes—
108#13108051C2314yes—
109#13108051C2314yes—
110#13108051C2314yes—
111#13108051C2314yes—
112#13108051C2314yes—
113#13108051C2314yes—
114C4×Dic101C2314yes—
115C4×Dic101C2314yes—
116C24×C101C2311yescharacteristic
117C24×C101C2311yescharacteristic
118C2×D4×C101—42no—
119C2×D4×C101C2314yes—
120C22×C4×C101—42no—
121C22×C4×C101C2311yescharacteristic
122C22×C4×C101C2311yescharacteristic
123C22×C4×C101C2311yescharacteristic
124C22×C4×C101C2311yescharacteristic
125C22×C4×C101C2311yescharacteristic
126C22×C4×C101C2314yes—
127C4⋊C4×C101C2314yes—
128C4⋊C4×C101C2314yes—
129C22⋊C4×C101C2314yes—
130C22⋊C4×C101C2314yes—
131C22⋊C4×C101C2314yes—
132C23×C101—22no—
133C23×C101—22no—
134C23×C101—22no—
135C23×C101—22no—
136C23×C101—22no—
137C23×C101—22no—
138C23×C101C2412yes—
139C23×C101C2412yes—
140C23×C101C2411yescharacteristic
141C22×D101—22no—
142C22×D101—22no—
143C22×D101—22no—
144C22×D101—22no—
145C22×D101—22no—
146C22×D101—22no—
147C22×D101—22no—
148C22×D101—22no—
149C22×D101—22no—
150C22×D101C2412yes—
151C22×D101C2412yes—
152C22×D101C2412yes—
153D4×C101—44no—
154C2×C4×C101—22no—
155C2×C4×C101—22no—
156C2×C4×C101—22no—
157C2×C4×C101—22no—
158C2×C4×C101—22no—
159C2×C4×C101—44no—
160C2×C4×C101C2412yes—
161C2×C4×C101C2412yes—
162C2×C4×C101C2412yes—
163C2×C4×C101C2412yes—
164C2×C4×C101C2412yes—
165C22⋊D101—28no—
166C22⋊D101—28no—
167C22⋊D101—44no—
168C22⋊D101—44no—
169C22⋊D101—44no—
170C22⋊D101—44no—
171C2×Dic101—22no—
172C2×Dic101—22no—
173C2×Dic101—22no—
174C2×Dic101—22no—
175C2×Dic101—44no—
176C2×Dic101#4114yes—
177C2×Dic101C2412yes—
178C2×Dic101C2412yes—
179C2×Dic101C2412yes—
180C2×Dic101C2412yes—
181C2×Dic101C2412yes—
182D404—44no—
183D404—44no—
184D404—44no—
185C4×D101—28no—
186C4×D101—44no—
187C4×D101—44no—
188C22×C101—22no—
189C22×C101—22no—
190C22×C101—22no—
191C22×C101—22no—
192C22×C101—22no—
193C22×C101—22no—
194C22×C101—44no—
195C22×C101—44no—
196C22×C101C2×#4112yes—
197C22×C101H(5,𝔽2)12yes—
198C22×C101H(5,𝔽2)12yes—
199C22×C101C2511yescharacteristic
200C2×D101—22no—
201C2×D101—22no—
202C2×D101—22no—
203C2×D101—22no—
204C2×D101—22no—
205C2×D101—22no—
206C2×D101—22no—
207C2×D101—22no—
208C2×D101—22no—
209C2×D101—44no—
210C2×D101—44no—
211C2×D101—44no—
212C4×C101—22no—
213C4×C101—22no—
214C4×C101—22no—
215C4×C101—22no—
216C4×C101—22no—
217Dic101—22no—
218Dic101—22no—
219Dic101—22no—
220Dic101—22no—
221Dic101C2×#4114yes—
222C2×C101—42no—
223C2×C101—42no—
224C2×C101#54014yescomplemented
225C2×C101C22×#4111yescharacteristic
226C2×C101C2×H(5,𝔽2)11yescharacteristic
227C2×C101C2×H(5,𝔽2)11yescharacteristic
228D101—42no—
229D101—42no—
230D101—42no—
231C2×#540—1011nomaximal, Sylow
232C101C2×#54011yescharacteristic, complemented, Sylow
233#540—10116no—
234C2×#127—1011no—
235C2×#125—1011no—
236C2×#123—1011no—
237C2×#123—1011no—
238C2×#123—1011no—
239C2×#123D10111yescharacteristic, complemented
240C2×#121—1011no—
241C2×#121—1011no—
242C2×#121—1011no—
243C2×#121—1011no—
244C2×#121—1011no—
245C2×#121—1011no—
246C2×C4×D4—1011no—
247C2×C4×D4—1011no—
248C2×#117—1011no—
249C22×D4—1011no—
250C22×D4—1011no—
251C22×D4—1011no—
252C22×D4—1011no—
253C22×D4—1011no—
254C22×D4—1011no—
255C22×D4—1011no—
256C22×D4—1011no—
257C22×D4—1011no—
258C22×D4C2×D10111yescharacteristic
259C23×C4—1011no—
260C23×C4—1011no—
261C23×C4—1011no—
262C23×C4—1011no—
263C23×C4C2×D10111yescharacteristic
264#127—1018no—
265#125—1018no—
266#123—1018no—
267#123—1018no—
268#123—1018no—
269#123C2×D10118yescomplemented
270#121—1018no—
271#121—1018no—
272#121—1018no—
273#121—1018no—
274#121—1018no—
275#121—1018no—
276C4×D4—1018no—
277C4×D4—1018no—
278#117—1018no—
279C2×C4⋊C4—1011no—
280C2×C4⋊C4—1011no—
281C2×C4⋊C4—1011no—
282C2×C4⋊C4—1011no—
283C2×C4⋊C4—1011no—
284C2×C4⋊C4—1011no—
285C2×C4⋊C4C2×D10111yescharacteristic
286C2×C4⋊C4C2×D10111yescharacteristic
287C2×C22⋊C4—1011no—
288C2×C22⋊C4—1011no—
289C2×C22⋊C4—1011no—
290C2×C22⋊C4—1011no—
291C2×C22⋊C4—1011no—
292C2×C22⋊C4—1011no—
293C2×C22⋊C4—1011no—
294C2×C22⋊C4C2×D10111yescharacteristic
295C2×C22⋊C4C2×D10111yescharacteristic
296C2×C22⋊C4C2×D10111yescharacteristic
297C2×C42—1011no—
298C2×C42—1011no—
299C24—1011no—
300C24—1011no—
301C24—1011no—
302C24C22×D10111yescharacteristic
303C24C22×D10111yescharacteristic
304C2×D4—42no—
305C2×D4—1014no—
306C2×D4—1014no—
307C2×D4—1014no—
308C2×D4—1014no—
309C2×D4—1014no—
310C2×D4—1014no—
311C2×D4—1014no—
312C2×D4—1014no—
313C2×D4—1014no—
314C2×D4—4042no—
315C2×D4—4042no—
316C2×D4—4042no—
317C2×D4—4042no—
318C2×D4—4042no—
319C2×D4—4042no—
320C2×D4—4042no—
321C2×D4—2024no—
322C2×D4—2024no—
323C2×D4C22×D10114yes—
324C22×C4—42no—
325C22×C4—1011no—
326C22×C4—1014no—
327C22×C4—1011no—
328C22×C4—1011no—
329C22×C4—1011no—
330C22×C4—1011no—
331C22×C4—1014no—
332C22×C4—1014no—
333C22×C4—1014no—
334C22×C4—4042no—
335C22×C4—4042no—
336C22×C4—4042no—
337C22×C4—2024no—
338C22×C4C22×D10111yescharacteristic
339C22×C4C22×D10111yescharacteristic
340C22×C4C22×D10111yescharacteristic
341C22×C4C22×D10111yescharacteristic
342C22×C4C22×D10111yescharacteristic
343C22×C4C22×D10114yes—
344C4⋊C4—1014no—
345C4⋊C4—1014no—
346C4⋊C4—1014no—
347C4⋊C4—1014no—
348C4⋊C4—1014no—
349C4⋊C4—1014no—
350C4⋊C4C22×D10114yes—
351C4⋊C4C22×D10114yes—
352C22⋊C4—1014no—
353C22⋊C4—1014no—
354C22⋊C4—1014no—
355C22⋊C4—1014no—
356C22⋊C4—1014no—
357C22⋊C4—1014no—
358C22⋊C4—1014no—
359C22⋊C4C22×D10114yes—
360C22⋊C4C22×D10114yes—
361C22⋊C4C22×D10114yes—
362C42—1014no—
363C42—1014no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—1012no—
371C23—1012no—
372C23—1012no—
373C23—2022no—
374C23—2022no—
375C23—2022no—
376C23—2022no—
377C23—2022no—
378C23—2022no—
379C23—2022no—
380C23—2022no—
381C23—2022no—
382C23C23×D10112yes—
383C23C23×D10112yes—
384C23C23×D10111yescharacteristic
385D4—44no—
386D4—4044no—
387D4—4044no—
388D4—4044no—
389D4—4044no—
390D4—4044no—
391D4—4044no—
392D4—4044no—
393D4—2028no—
394D4—2028no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—1014no—
402C2×C4—1012no—
403C2×C4—1012no—
404C2×C4—1012no—
405C2×C4—1012no—
406C2×C4—1012no—
407C2×C4—4044no—
408C2×C4—4044no—
409C2×C4—4044no—
410C2×C4—2028no—
411C2×C4—2022no—
412C2×C4—2022no—
413C2×C4—2022no—
414C2×C4—2022no—
415C2×C4C23×D10112yes—
416C2×C4C23×D10112yes—
417C2×C4C23×D10112yes—
418C2×C4C23×D10112yes—
419C2×C4C23×D10112yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—4044no—
429C22—4044no—
430C22—4044no—
431C22—2022no—
432C22—2022no—
433C22—2022no—
434C22—2022no—
435C22—2022no—
436C22—2022no—
437C22—2022no—
438C22—2022no—
439C22—2022no—
440C22#41×D10112yes—
441C22#1569230012yes—
442C22#1569230112yes—
443C22C24×D10111yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—1014no—
450C4—2022no—
451C4—2022no—
452C4—2022no—
453C4—2022no—
454C2—42no—
455C2—42no—
456C2—4042no—
457C2—4042no—
458C2—4042no—
459C2#2026577914yesdirect product factor
460C2C2×#41×D10111yescharacteristic
461C2C2×#1569230011yescharacteristic
462C2C2×#1569230111yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#123×C101C2
socleC23×C101C24
hypercenterC2×#123D101
hypocenterC101C2×#540
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #20265779C2direct (N × H)
2. C2×#15692251C2semidirect
3. C2×#15692245C2semidirect
4. C2×#15692240C2semidirect
5. C2×#15692239C2semidirect
6. C2×#15692237C2semidirect
7. C2×#15692230C2semidirect
8. C2×#15692229C2semidirect
9. C2×#15692228C2semidirect
10. C2×#15692228C2semidirect
11. C2×#15692228C2semidirect
12. C2×#15692227C2semidirect
13. C2×C4×C22⋊D101C2semidirect
14. C2×#15692215C2semidirect
15. C2×#15692213C2semidirect
16. C2×#123×C101C2semidirect
30. #15692251C22semidirect
31. #15692245C22semidirect
32. #15692240C22semidirect
33. #15692239C22semidirect
34. #15692237C22semidirect
35. #15692230C22semidirect
36. #15692229C22semidirect
37. #15692228C22semidirect
38. #15692228C22semidirect
39. #15692228C22semidirect
40. #15692227C22semidirect
41. C4×C22⋊D101C22semidirect
42. #15692215C22semidirect
43. #15692213C22semidirect
61. #123×C101C22semidirect
224. C2×C101#540semidirect
232. C101C2×#540semidirect
239. C2×#123D101semidirect
269. #123C2×D101semidirect
459. C2#20265779direct (N × H)

Maximal subgroups

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

subgroupindex
#202657792
C2×#156922512
C2×#156922452
C2×#156922402
C2×#156922392
C2×#156922372
C2×#156922302
C2×#156922292
C2×#156922282
C2×#156922272
C2×C4×C22⋊D1012
C2×#156922152
C2×#156922132
C2×#123×C1012
C2×#540101