Finite Group Databasebeta

← C2×#20265784 #27184495

Subgroups — C2×#20265784 #27184495

Order 12928.

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#20265784C2116yesmaximal, direct product factor
2C2×#15692254C211yesmaximal, characteristic, complemented
3C2×#15692248C211yesmaximal, characteristic
4C2×#15692248C211yesmaximal, characteristic
5C2×#15692248C211yesmaximal, characteristic
6C2×#15692247C211yesmaximal, characteristic, complemented
7C2×#15692246C211yesmaximal, characteristic, complemented
8C2×#15692245C211yesmaximal, characteristic, complemented
9C2×#15692245C211yesmaximal, characteristic, complemented
10C2×#15692245C211yesmaximal, characteristic, complemented
11C2×#15692232C211yesmaximal, characteristic
12C2×#15692232C211yesmaximal, characteristic
13C2×C4×Dic202C211yesmaximal, characteristic, complemented
14C2×#15692220C211yesmaximal, characteristic, complemented
15C2×#15692211C211yesmaximal, characteristic
16C2×#126×C101C211yesmaximal, characteristic
17C22×Dic202C2211yescharacteristic
18C22×Dic202C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20#15692254C2218yescomplemented
21#15692248C2218yes—
22#15692248C2218yes—
23#15692248C2218yes—
24#15692247C2218yescomplemented
25#15692246C2218yescomplemented
26#15692245C2218yescomplemented
27#15692245C2218yescomplemented
28#15692245C2218yescomplemented
29#15692232C2218yes—
30#15692232C2218yes—
31C4×Dic202C2218yescomplemented
32#15692220C2218yescomplemented
33#15692211C2218yes—
34C2×C4⋊Dic101C2211yescharacteristic
35C2×C4⋊Dic101C2211yescharacteristic
36C2×C4⋊Dic101C2211yescharacteristic
37C2×C4⋊Dic101C2211yescharacteristic
38C2×C4⋊Dic101C2211yescharacteristic
39C2×#13108053C2211yescharacteristic
40C2×#13108053C2211yescharacteristic
41C2×#13108053C2211yescharacteristic
42C2×#13108053C2211yescharacteristic
43C2×#13108053C2211yescharacteristic
44C2×#13108053C2211yescharacteristic
45C2×#13108053C2211yescharacteristic
46C2×#13108053C2211yescharacteristic
47C2×#13108053C2211yescharacteristic
48C2×#13108053C2211yescharacteristic
49C2×#13108053C2211yescharacteristic
50C2×#13108053C2211yescharacteristic
51C2×C22⋊Dic101C2211yescharacteristic
52C2×C22⋊Dic101C2211yescharacteristic
53C2×C22⋊Dic101C2211yescharacteristic
54C2×C4×Dic101C2211yescharacteristic
55C2×C4×Dic101C2211yescharacteristic
56C2×C4×Dic101C2211yescharacteristic
57C2×C4×Dic101C2211yescharacteristic
58C2×C4×Dic101C2211yescharacteristic
59#126×C101C2218yes—
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
67C2×Dic202—24no—
68C2×Dic202—24no—
69C2×Dic202C2314yes—
70C2×Dic202C2314yes—
71C22×Dic101—24no—
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2314yes—
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C22×Dic101C2311yescharacteristic
80C22×Dic101C2311yescharacteristic
81C4⋊Dic101C2314yes—
82C4⋊Dic101C2314yes—
83C4⋊Dic101C2314yes—
84C4⋊Dic101C2314yes—
85C4⋊Dic101C2314yes—
86#13108053C2314yes—
87#13108053C2314yes—
88#13108053C2314yes—
89#13108053C2314yes—
90#13108053C2314yes—
91#13108053C2314yes—
92#13108053C2314yes—
93#13108053C2314yes—
94#13108053C2314yes—
95#13108053C2314yes—
96#13108053C2314yes—
97#13108053C2314yes—
98C22⋊Dic101C2314yes—
99C22⋊Dic101C2314yes—
100C22⋊Dic101C2314yes—
101C4×Dic101C2314yes—
102C4×Dic101C2314yes—
103C4×Dic101C2314yes—
104C4×Dic101C2314yes—
105C4×Dic101C2314yes—
106C24×C101C2311yescharacteristic
107C22×C4×C101C2311yescharacteristic
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C4⋊C4×C101C2314yes—
114C4⋊C4×C101C2314yes—
115C4⋊C4×C101C2314yes—
116C22⋊C4×C101C2314yes—
117C22⋊C4×C101C2314yes—
118C22⋊C4×C101C2314yes—
119C42×C101C2314yes—
120C23×C101—22no—
121C23×C101—22no—
122C23×C101—22no—
123C23×C101C2412yes—
124C23×C101C2411yescharacteristic
125C2×C4×C101—22no—
126C2×C4×C101—22no—
127C2×C4×C101—22no—
128C2×C4×C101—22no—
129C2×C4×C101—22no—
130C2×C4×C101—22no—
131C2×C4×C101C2412yes—
132C2×C4×C101C2412yes—
133C2×C4×C101C2412yes—
134C2×C4×C101C2412yes—
135C2×C4×C101C2412yes—
136C2×C4×C101C2412yes—
137C2×Dic101—22no—
138C2×Dic101—22no—
139C2×Dic101—22no—
140C2×Dic101—22no—
141C2×Dic101—22no—
142C2×Dic101—22no—
143C2×Dic101—22no—
144C2×Dic101—28no—
145C2×Dic101#4114yes—
146C2×Dic101C2412yes—
147C2×Dic101C2412yes—
148C2×Dic101C2412yes—
149C2×Dic101C2412yes—
150C2×Dic101C2412yes—
151C2×Dic101C2412yes—
152C2×Dic101C2412yes—
153C2×Dic101C2412yes—
154Dic202—28no—
155Dic202—28no—
156C22×C101—22no—
157C22×C101—22no—
158C22×C101—22no—
159C22×C101—44no—
160C22×C101C2×#4112yes—
161C22×C1012−1+412yes—
162C22×C1012−1+412yes—
163C22×C101C2511yescharacteristic
164C4×C101—22no—
165C4×C101—22no—
166C4×C101—22no—
167C4×C101—22no—
168C4×C101—22no—
169C4×C101—22no—
170Dic101—22no—
171Dic101—22no—
172Dic101—22no—
173Dic101—22no—
174Dic101—22no—
175Dic101—22no—
176Dic101—22no—
177Dic101C2×#4114yes—
178C2×C101—42no—
179C2×C101#54114yescomplemented
180C2×C101C22×#4111yescharacteristic
181C2×C101C2×2−1+411yescharacteristic
182C2×C101C2×2−1+411yescharacteristic
183C2×#541—1011nomaximal, Sylow
184C101C2×#54111yescharacteristic, complemented, Sylow
185#541—10116no—
186C2×C4⋊Q8—1011no—
187C2×#126—1011no—
188C2×#126—1011no—
189C2×#126—1011no—
190C2×#126D10111yescharacteristic
191C2×#125—1011no—
192C2×#125—1011no—
193C2×#125—1011no—
194C2×#125—1011no—
195C2×#125—1011no—
196C2×C22⋊Q8—1011no—
197C2×C22⋊Q8—1011no—
198C2×C4×Q8—1011no—
199C2×C4×Q8—1011no—
200C2×#117—1011no—
201C22×Q8—1011no—
202C22×Q8—1011no—
203C23×C4—1011no—
204C4⋊Q8—1018no—
205#126—1018no—
206#126—1018no—
207#126—1018no—
208#126C2×D10118yes—
209#125—1018no—
210#125—1018no—
211#125—1018no—
212#125—1018no—
213#125—1018no—
214C22⋊Q8—1018no—
215C22⋊Q8—1018no—
216C4×Q8—1018no—
217C4×Q8—1018no—
218#117—1018no—
219C2×C4⋊C4—1011no—
220C2×C4⋊C4—1011no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4—1011no—
225C2×C4⋊C4—1011no—
226C2×C4⋊C4—1011no—
227C2×C4⋊C4—1011no—
228C2×C4⋊C4—1011no—
229C2×C4⋊C4—1011no—
230C2×C4⋊C4—1011no—
231C2×C4⋊C4—1011no—
232C2×C4⋊C4—1011no—
233C2×C4⋊C4—1011no—
234C2×C4⋊C4—1011no—
235C2×C4⋊C4—1011no—
236C2×C4⋊C4C2×D10111yescharacteristic
237C2×C4⋊C4C2×D10111yescharacteristic
238C2×C4⋊C4C2×D10111yescharacteristic
239C2×C22⋊C4—1011no—
240C2×C22⋊C4—1011no—
241C2×C22⋊C4—1011no—
242C2×C22⋊C4C2×D10111yescharacteristic
243C2×C22⋊C4C2×D10111yescharacteristic
244C2×C22⋊C4C2×D10111yescharacteristic
245C2×C42—1011no—
246C2×C42—1011no—
247C2×C42—1011no—
248C2×C42—1011no—
249C2×C42—1011no—
250C2×C42C2×D10111yescharacteristic
251C24C22×D10111yescharacteristic
252C2×Q8—1014no—
253C2×Q8—1014no—
254C2×Q8—2024no—
255C2×Q8—2024no—
256C22×C4—1011no—
257C22×C4—1014no—
258C22×C4—1011no—
259C22×C4—1011no—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1011no—
263C22×C4—1011no—
264C22×C4—1011no—
265C22×C4—2024no—
266C22×C4C22×D10111yescharacteristic
267C22×C4C22×D10111yescharacteristic
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270C22×C4C22×D10111yescharacteristic
271C22×C4C22×D10111yescharacteristic
272C4⋊C4—1014no—
273C4⋊C4—1014no—
274C4⋊C4—1014no—
275C4⋊C4—1014no—
276C4⋊C4—1014no—
277C4⋊C4—1014no—
278C4⋊C4—1014no—
279C4⋊C4—1014no—
280C4⋊C4—1014no—
281C4⋊C4—1014no—
282C4⋊C4—1014no—
283C4⋊C4—1014no—
284C4⋊C4—1014no—
285C4⋊C4—1014no—
286C4⋊C4—1014no—
287C4⋊C4—1014no—
288C4⋊C4—1014no—
289C4⋊C4C22×D10114yes—
290C4⋊C4C22×D10114yes—
291C4⋊C4C22×D10114yes—
292C22⋊C4—1014no—
293C22⋊C4—1014no—
294C22⋊C4—1014no—
295C22⋊C4C22×D10114yes—
296C22⋊C4C22×D10114yes—
297C22⋊C4C22×D10114yes—
298C42—1014no—
299C42—1014no—
300C42—1014no—
301C42—1014no—
302C42—1014no—
303C42C22×D10114yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23C23×D10112yes—
308C23C23×D10111yescharacteristic
309Q8—2028no—
310Q8—2028no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—1014no—
318C2×C4—1012no—
319C2×C4—1012no—
320C2×C4—1012no—
321C2×C4—1012no—
322C2×C4—1012no—
323C2×C4—1012no—
324C2×C4—1012no—
325C2×C4—1012no—
326C2×C4—2028no—
327C2×C4—2022no—
328C2×C4—2022no—
329C2×C4—2022no—
330C2×C4—2022no—
331C2×C4—2022no—
332C2×C4—2022no—
333C2×C4—2022no—
334C2×C4C23×D10112yes—
335C2×C4C23×D10112yes—
336C2×C4C23×D10112yes—
337C2×C4C23×D10112yes—
338C2×C4C23×D10112yes—
339C2×C4C23×D10112yes—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—44no—
344C22#41×D10112yes—
345C22#1569230212yes—
346C22#1569230212yes—
347C22C24×D10111yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—1014no—
355C4—2022no—
356C4—2022no—
357C4—2022no—
358C4—2022no—
359C4—2022no—
360C4—2022no—
361C4—2022no—
362C2—42no—
363C2#2026578414yesdirect product factor
364C2C2×#41×D10111yescharacteristic
365C2C2×#1569230211yescharacteristic
366C2C2×#1569230211yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×#126×C101C2
socleC23×C101C24
prime radicalC24×C101C23
hypercenterC2×#126D101
hypocenterC101C2×#541
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20265784C2direct (N × H)
2. C2×#15692254C2semidirect
6. C2×#15692247C2semidirect
7. C2×#15692246C2semidirect
8. C2×#15692245C2semidirect
9. C2×#15692245C2semidirect
10. C2×#15692245C2semidirect
13. C2×C4×Dic202C2semidirect
14. C2×#15692220C2semidirect
20. #15692254C22semidirect
24. #15692247C22semidirect
25. #15692246C22semidirect
26. #15692245C22semidirect
27. #15692245C22semidirect
28. #15692245C22semidirect
31. C4×Dic202C22semidirect
32. #15692220C22semidirect
179. C2×C101#541semidirect
184. C101C2×#541semidirect
363. C2#20265784direct (N × H)

Maximal subgroups

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

subgroupindex
#202657842
C2×#156922542
C2×#156922482
C2×#156922472
C2×#156922462
C2×#156922452
C2×#156922322
C2×C4×Dic2022
C2×#156922202
C2×#156922112
C2×#126×C1012
C2×#541101