Finite Group Databasebeta

← C2×#20265993 #27185898

Subgroups — C2×#20265993 #27185898

Order 12928.

1408 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
1C2×#15692303C211yesmaximal, characteristic, complemented
2C2×#15692302C211yesmaximal, characteristic, complemented
3#20265993C2116yesmaximal, direct product factor
4C2×#15692288C211yesmaximal, characteristic, complemented
5C2×#15692287C212yesmaximal, complemented
6C2×#15692284C212yesmaximal, complemented
7C2×#15692276C212yesmaximal, complemented
8C2×#15692275C211yesmaximal, characteristic, complemented
9C22×#13108064C211yesmaximal, characteristic, complemented
10C2×Q16×D101C212yesmaximal, complemented
11C2×#15692263C211yesmaximal, characteristic, complemented
12C2×#137×C101C211yesmaximal, characteristic, complemented
13#15692303—24no—
14#15692303C2218yescomplemented
15#15692302—24no—
16#15692302C2218yescomplemented
17C2×#13108075—21no—
18C2×#13108075C2212yes—
19C2×#13108074—21no—
20C2×#13108074—22no—
21C2×#13108074C2211yescharacteristic
22C2×#13108074C2211yescharacteristic
23C2×#13108073—22no—
24C2×#13108073C2212yes—
25C22×Dic202—21no—
26C22×Dic202C2211yescharacteristic
27C2×Q8×D101—21no—
28C2×Q8×D101C2212yes—
29C2×Q8×D101C2212yes—
30#15692288C2218yescomplemented
31#15692287C22116yescomplemented
32#15692284C22116yescomplemented
33#15692276C22116yescomplemented
34#15692275C2218yescomplemented
35C2×Dic404C2212yes—
36C2×#13108064C2211yescharacteristic
37C2×#13108064C2212yes—
38C2×#13108064C2212yes—
39C2×#13108064C2218yescomplemented
40Q16×D101C22116yescomplemented
41C2×#13108063C2212yes—
42C2×#13108062C2211yescharacteristic
43C2×#13108061C2211yescharacteristic
44C2×#13108059C2211yescharacteristic
45#15692263C2218yescomplemented
46C2×#13108057C2212yes—
47C22×#11717579C2211yescharacteristic
48C2×C8×D101C2212yes—
49C2×#41×C101C2211yescharacteristic
50C22×Q8×C101C2211yescharacteristic
51#137×C101C2218yescomplemented
52C2×Q16×C101C2212yes—
53C2×SD16×C101C2212yes—
54C2×M16×C101C2211yescharacteristic
55#13108075—28no—
56#13108075—24no—
57#13108075C2318yes—
58#13108074—22no—
59#13108074—216no—
60#13108074—28no—
61#13108074D414yes—
62#13108074C2314yes—
63#13108074C2314yes—
64#13108073—216no—
65#13108073—24no—
66#13108073C2318yes—
67C2×Dic202—21no—
68C2×Dic202—22no—
69C2×Dic202—22no—
70C2×Dic202—22no—
71C2×Dic202—28no—
72C2×Dic202D411yescharacteristic
73C2×Dic202C2314yes—
74C2×Dic202C2311yescharacteristic
75C2×Dic202C2312yes—
76Q8×D101—24no—
77Q8×D101—24no—
78Q8×D101—28no—
79Q8×D101C2318yes—
80Q8×D101C2318yes—
81C2×D404—22no—
82C2×D404D411yescharacteristic
83C2×D404C2311yescharacteristic
84C2×C22⋊D101—21no—
85C2×C22⋊D101—22no—
86C2×C22⋊D101D412yes—
87C22×Dic101—21no—
88C22×Dic101—22no—
89C2×C4×D101—22no—
90C2×C4×D101—22no—
91C2×C4×D101—21no—
92C2×C4×D101—22no—
93C2×C4×D101C2312yes—
94Dic404C2318yes—
95#13108064C2314yes—
96#13108064C2318yes—
97#13108064C2318yes—
98#13108063C2318yes—
99#13108062C2314yes—
100#13108061C2314yes—
101#13108059C2314yes—
102#13108057C2318yes—
103C2×#11717579C2311yescharacteristic
104C2×#11717579C2311yescharacteristic
105C2×#11717579C2314yes—
106C8×D101C2318yes—
107#41×C101—22no—
108#41×C101C2314yes—
109C2×Q8×C101—21no—
110C2×Q8×C101—22no—
111C2×Q8×C101C2311yescharacteristic
112C2×Q8×C101C2312yes—
113C2×Q8×C101C2314yes—
114C2×D4×C101—21no—
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101—21no—
117C22×C4×C101—21no—
118C22×C4×C101C2311yescharacteristic
119Q16×C101C2318yes—
120SD16×C101C2318yes—
121M16×C101C2314yes—
122C2×C8×C101C2312yes—
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101C2×D412yes—
127Q8×C101—22no—
128Q8×C101—24no—
129Q8×C101—21no—
130Q8×C101C2414yes—
131Q8×C101C2412yes—
132D4×C101—24no—
133D4×C101—21no—
134D4×C101C2412yes—
135C2×C4×C101—21no—
136C2×C4×C101—22no—
137C2×C4×C101—24no—
138C2×C4×C101—24no—
139C2×C4×C101C2×D411yescharacteristic
140C2×C4×C101C2×D412yes—
141C2×C4×C101C2411yescharacteristic
142C2×C4×C101C2412yes—
143C22⋊D101—24no—
144C22⋊D101—28no—
145C22⋊D101C2×D418yes—
146C2×Dic101—21no—
147C2×Dic101—22no—
148C2×Dic101—24no—
149C2×Dic101—28no—
150C2×Dic101C2×D412yes—
151D404—28no—
152D404—21no—
153D404C2×D414yes—
154D404C2412yes—
155C4×D101—28no—
156C4×D101—28no—
157C4×D101—24no—
158C4×D101—28no—
159C4×D101C2×D414yes—
160C4×D101C2414yes—
161Dic202—21no—
162Dic202—22no—
163Dic202—24no—
164Dic202—28no—
165Dic202—28no—
166Dic202C2×D414yes—
167Dic202C2412yes—
168Dic202C2414yes—
169C8×C101C2414yes—
170#11717579C2412yes—
171#11717579C2412yes—
172C22×C101—22no—
173C22×C101—22no—
174C22×C101—42no—
175C22×C101C22×D412yes—
176C22×C101C22×D411yescharacteristic
177C2×D101—24no—
178C2×D101—22no—
179C2×D101—42no—
180C2×D101C22×D414yes—
181C4×C101—22no—
182C4×C101—24no—
183C4×C101C22×D411yescharacteristic
184C4×C101C22×D412yes—
185C4×C101C2511yescharacteristic
186Dic101—22no—
187Dic101—24no—
188Dic101C22×D414yes—
189C2×C101—22no—
190C2×C101—42no—
191C2×C101#57812yescomplemented
192C2×C101C23×D411yescharacteristic
193D101—24no—
194D101—42no—
195C2×#578—1011nomaximal, Sylow
196C101C2×#57811yescharacteristic, complemented, Sylow
197C2×2−1+4—1011no—
198C2×2−1+4—1011no—
199#578—10116no—
200C2×#137—1012no—
201C2×#137—1012no—
202C2×#137—1011no—
203C2×#137D10111yescharacteristic, complemented
204C2×#135—1011no—
205C2×#135—1012no—
206C22×Q16—1011no—
207C22×Q16—1012no—
208C2×#131—1011no—
2092−1+4—1018no—
2102−1+4—1018no—
2112−1+4—2024no—
2122−1+4—2024no—
213C2×#41—1011no—
214C2×#41—1012no—
215C2×#41—1012no—
216C2×#41—1011no—
217C2×#41—2022no—
218C2×#41—2021no—
219C2×#41—2022no—
220C2×#41—2021no—
221C2×#41C2×D10111yescharacteristic
222C22×Q8—1012no—
223C22×Q8—1012no—
224C22×Q8—1011no—
225C22×Q8—2021no—
226C22×Q8—2021no—
227C22×Q8C2×D10111yescharacteristic
228#137—10116no—
229#137—10116no—
230#137—1018no—
231#137C2×D10118yescomplemented
232#135—1018no—
233#135—10116no—
234C2×Q16—1011no—
235C2×Q16—1012no—
236C2×Q16—1012no—
237C2×Q16—1018no—
238C2×Q16—10116no—
239C2×Q16—1012no—
240C2×Q16C2×D10112yes—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×SD16—1012no—
244C2×SD16C2×D10112yes—
245C2×D8—1011no—
246#131—1018no—
247C2×M16—1012no—
248C2×M16C2×D10111yescharacteristic
249C22×C8—1011no—
250C22×C8—1012no—
251#41—22no—
252#41—1014no—
253#41—1014no—
254#41—1018no—
255#41—1018no—
256#41—1014no—
257#41—20216no—
258#41—2028no—
259#41—2024no—
260#41—2024no—
261#41—20216no—
262#41—2028no—
263#41—2022no—
264#41C22×D10114yes—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—1011no—
268C2×Q8—1018no—
269C2×Q8—1018no—
270C2×Q8—1011no—
271C2×Q8—1012no—
272C2×Q8—1014no—
273C2×Q8—2022no—
274C2×Q8—2022no—
275C2×Q8—2028no—
276C2×Q8—2024no—
277C2×Q8—2024no—
278C2×Q8—2028no—
279C2×Q8—2021no—
280C2×Q8—2022no—
281C2×Q8C22×D10111yescharacteristic
282C2×Q8C22×D10112yes—
283C2×Q8C22×D10114yes—
284C2×D4—21no—
285C2×D4—1012no—
286C2×D4—1011no—
287C2×D4—1011no—
288C2×D4—2022no—
289C2×D4—2022no—
290C2×D4—2021no—
291C2×D4C22×D10111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1012no—
295C22×C4—2022no—
296C22×C4—2022no—
297C22×C4—2022no—
298C22×C4—2022no—
299C22×C4—2021no—
300C22×C4—2021no—
301C22×C4C22×D10111yescharacteristic
302Q16—1014no—
303Q16—1018no—
304Q16—1018no—
305Q16—1018no—
306Q16C22×D10118yes—
307SD16—1014no—
308SD16—1014no—
309SD16—1018no—
310SD16C22×D10118yes—
311D8—1014no—
312M16—1018no—
313M16C22×D10114yes—
314C2×C8—1011no—
315C2×C8—1011no—
316C2×C8—1014no—
317C2×C8—1018no—
318C2×C8C22×D10112yes—
319C23—21no—
320C23—1012no—
321C23—2021no—
322C23D4×D10111yescharacteristic
323Q8—24no—
324Q8—22no—
325Q8—21no—
326Q8—1014no—
327Q8—1012no—
328Q8—1014no—
329Q8—2028no—
330Q8—2028no—
331Q8—2024no—
332Q8—2021no—
333Q8—2022no—
334Q8C23×D10112yes—
335Q8C23×D10114yes—
336D4—24no—
337D4—21no—
338D4—1018no—
339D4—1014no—
340D4—1012no—
341D4—2028no—
342D4—2028no—
343D4—2024no—
344D4—2021no—
345D4C23×D10112yes—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—1012no—
351C2×C4—1014no—
352C2×C4—1014no—
353C2×C4—2028no—
354C2×C4—2028no—
355C2×C4—2028no—
356C2×C4—2028no—
357C2×C4—2024no—
358C2×C4—2021no—
359C2×C4—2022no—
360C2×C4—2024no—
361C2×C4D4×D10112yes—
362C2×C4D4×D10111yescharacteristic
363C2×C4C23×D10112yes—
364C2×C4C23×D10111yescharacteristic
365C8—1012no—
366C8—1012no—
367C8C23×D10114yes—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—1014no—
372C22—4042no—
373C22—2022no—
374C22—2024no—
375C22C2×D4×D10112yes—
376C22C2×D4×D10111yescharacteristic
377C4—22no—
378C4—24no—
379C4—1014no—
380C4—2022no—
381C4—2024no—
382C4C2×D4×D10112yes—
383C4C2×D4×D10111yescharacteristic
384C4C24×D10111yescharacteristic
385C2—22no—
386C2—42no—
387C2—4042no—
388C2—2024no—
389C2#2026599312yesdirect product factor
390C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
hypercenterC2×#137D101
hypocenterC101C2×#578
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#15692303C2semidirect
2. C2×#15692302C2semidirect
3. #20265993C2direct (N × H)
4. C2×#15692288C2semidirect
5. C2×#15692287C2semidirect
6. C2×#15692284C2semidirect
7. C2×#15692276C2semidirect
8. C2×#15692275C2semidirect
9. C22×#13108064C2semidirect
10. C2×Q16×D101C2semidirect
11. C2×#15692263C2semidirect
12. C2×#137×C101C2semidirect
14. #15692303C22semidirect
16. #15692302C22semidirect
30. #15692288C22semidirect
31. #15692287C22semidirect
32. #15692284C22semidirect
33. #15692276C22semidirect
34. #15692275C22semidirect
39. C2×#13108064C22semidirect
40. Q16×D101C22semidirect
45. #15692263C22semidirect
51. #137×C101C22semidirect
191. C2×C101#578semidirect
196. C101C2×#578semidirect
203. C2×#137D101semidirect
231. #137C2×D101semidirect
389. C2#20265993direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923032
C2×#156923022
#202659932
C2×#156922882
C2×#156922872
C2×#156922842
C2×#156922762
C2×#156922752
C22×#131080642
C2×Q16×D1012
C2×#156922632
C2×#137×C1012
C2×#578101