Finite Group Databasebeta

← C2×#5779 #87196

Subgroups — C2×#5779 #87196

Order 384.

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×#1010C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C2×C3×#137C211yesmaximal, characteristic, complemented
4C2×#950C211yesmaximal, characteristic, complemented
5C22×#216C211yesmaximal, characteristic, complemented
6#5779C2116yesmaximal, direct product factor
7C2×#918C212yesmaximal, complemented
8C2×D3×Q16C212yesmaximal, complemented
9C2×#916C212yesmaximal, complemented
10C2×#915C212yesmaximal, complemented
11C2×#909C211yesmaximal, characteristic, complemented
12C2×#907C211yesmaximal, characteristic, complemented
13C2×#578—31nomaximal, Sylow
14C2×C3×#41C2211yescharacteristic
15C22×C3×Q8C2211yescharacteristic
16#1010—24no—
17#1010C2218yescomplemented
18#1007—24no—
19#1007C2218yescomplemented
20C2×#239—21no—
21C2×#239C2212yes—
22C2×D3×Q8—21no—
23C2×D3×Q8C2212yes—
24C2×D3×Q8C2212yes—
25C2×#237—22no—
26C2×#237C2212yes—
27C2×#235—22no—
28C2×#235—21no—
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic
31C22×Dic6—21no—
32C22×Dic6C2211yescharacteristic
33C3×#137C2218yescomplemented
34C2×C3×Q16C2212yes—
35C2×C3×SD16C2212yes—
36C2×C3×M16C2211yescharacteristic
37#950C2218yescomplemented
38C2×#216C2212yes—
39C2×#216C2212yes—
40C2×#216C2211yescharacteristic
41C2×#216C2218yescomplemented
42C2×#215C2211yescharacteristic
43C2×#214C2211yescharacteristic
44C2×#213C2211yescharacteristic
45C22×C3⋊C8C2211yescharacteristic
46#918C22116yescomplemented
47D3×Q16C22116yescomplemented
48#916C22116yescomplemented
49#915C22116yescomplemented
50#909C2218yescomplemented
51#907C2218yescomplemented
52C2×Dic12C2212yes—
53C2×#204C2212yes—
54C2×ΓL(1,𝔽25)C2212yes—
55C2×D3×C8C2212yes—
56C2×2−1+4—31no—
57C2×2−1+4—31no—
58#578—316no—
59C2×#137—32no—
60C2×#137—32no—
61C2×#137—31no—
62C2×#137D311yescharacteristic, complemented
63C2×#135—31no—
64C2×#135—32no—
65C22×Q16—31no—
66C22×Q16—32no—
67C2×#131—31no—
68C3×#41—22no—
69C3×#41C2314yes—
70C2×C3×Q8—21no—
71C2×C3×Q8—22no—
72C2×C3×Q8C2311yescharacteristic
73C2×C3×Q8C2312yes—
74C2×C3×Q8C2314yes—
75C2×C3×D4—21no—
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—22no—
82C2×#67D412yes—
83C22×Dic3—21no—
84C22×Dic3—22no—
85#239—28no—
86#239—24no—
87#239C2318yes—
88D3×Q8—24no—
89D3×Q8—24no—
90D3×Q8—28no—
91D3×Q8C2318yes—
92D3×Q8C2318yes—
93#237—216no—
94#237—24no—
95#237C2318yes—
96#235—22no—
97#235—216no—
98#235—28no—
99#235D414yes—
100#235C2314yes—
101#235C2314yes—
102C2×D12—22no—
103C2×D12D411yescharacteristic
104C2×D12C2311yescharacteristic
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—21no—
108C2×C4×D3—22no—
109C2×C4×D3C2312yes—
110C2×Dic6—21no—
111C2×Dic6—22no—
112C2×Dic6—22no—
113C2×Dic6—22no—
114C2×Dic6—28no—
115C2×Dic6D411yescharacteristic
116C2×Dic6C2312yes—
117C2×Dic6C2311yescharacteristic
118C2×Dic6C2314yes—
119C3×Q16C2318yes—
120C3×SD16C2318yes—
121C3×M16C2314yes—
122C2×C3×C8C2312yes—
123#216C2318yes—
124#216C2318yes—
125#216C2314yes—
126#215C2314yes—
127#214C2314yes—
128#213C2314yes—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2314yes—
132Dic12C2318yes—
133#204C2318yes—
134ΓL(1,𝔽25)C2318yes—
135D3×C8C2318yes—
1362−1+4—38no—
1372−1+4—38no—
1382−1+4—64no—
1392−1+4—64no—
140C2×#41—31no—
141C2×#41—32no—
142C2×#41—32no—
143C2×#41—31no—
144C2×#41—62no—
145C2×#41—61no—
146C2×#41—62no—
147C2×#41—61no—
148C2×#41C2×D311yescharacteristic
149C22×Q8—32no—
150C22×Q8—32no—
151C22×Q8—31no—
152C22×Q8—61no—
153C22×Q8—61no—
154C22×Q8C2×D311yescharacteristic
155#137—316no—
156#137—316no—
157#137—38no—
158#137C2×D318yescomplemented
159#135—38no—
160#135—316no—
161C2×Q16—31no—
162C2×Q16—32no—
163C2×Q16—32no—
164C2×Q16—38no—
165C2×Q16—316no—
166C2×Q16—32no—
167C2×Q16C2×D312yes—
168C2×SD16—31no—
169C2×SD16—31no—
170C2×SD16—32no—
171C2×SD16C2×D312yes—
172C2×D8—31no—
173#131—38no—
174C2×M16—32no—
175C2×M16C2×D311yescharacteristic
176C22×C8—31no—
177C22×C8—32no—
178C23×C3—21no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C22×D3C2×D412yes—
182C3×Q8—21no—
183C3×Q8—22no—
184C3×Q8—24no—
185C3×Q8C2412yes—
186C3×Q8C2414yes—
187C3×D4—24no—
188C3×D4—21no—
189C3×D4C2412yes—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D412yes—
196C2×C3×C4C2411yescharacteristic
197C2×C3×C4C2412yes—
198#67—24no—
199#67—28no—
200#67C2×D418yes—
201C2×Dic3—21no—
202C2×Dic3—22no—
203C2×Dic3—24no—
204C2×Dic3—28no—
205C2×Dic3C2×D412yes—
206D12—28no—
207D12—21no—
208D12C2×D414yes—
209D12C2412yes—
210C4×D3—28no—
211C4×D3—28no—
212C4×D3—24no—
213C4×D3—28no—
214C4×D3C2×D414yes—
215C4×D3C2414yes—
216Dic6—24no—
217Dic6—22no—
218Dic6—21no—
219Dic6—28no—
220Dic6—28no—
221Dic6C2×D414yes—
222Dic6C2414yes—
223Dic6C2412yes—
224C3×C8C2414yes—
225C3⋊C8C2412yes—
226C3⋊C8C2412yes—
227#41—22no—
228#41—34no—
229#41—34no—
230#41—38no—
231#41—38no—
232#41—34no—
233#41—616no—
234#41—68no—
235#41—64no—
236#41—64no—
237#41—616no—
238#41—68no—
239#41—62no—
240#41C22×D314yes—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—31no—
244C2×Q8—38no—
245C2×Q8—38no—
246C2×Q8—31no—
247C2×Q8—32no—
248C2×Q8—34no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—68no—
252C2×Q8—64no—
253C2×Q8—64no—
254C2×Q8—68no—
255C2×Q8—61no—
256C2×Q8—62no—
257C2×Q8C22×D311yescharacteristic
258C2×Q8C22×D312yes—
259C2×Q8C22×D314yes—
260C2×D4—21no—
261C2×D4—32no—
262C2×D4—31no—
263C2×D4—31no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—61no—
267C2×D4C22×D311yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—32no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4C22×D311yescharacteristic
278Q16—34no—
279Q16—38no—
280Q16—38no—
281Q16—38no—
282Q16C22×D318yes—
283SD16—34no—
284SD16—34no—
285SD16—38no—
286SD16C22×D318yes—
287D8—34no—
288M16—38no—
289M16C22×D314yes—
290C2×C8—31no—
291C2×C8—31no—
292C2×C8—34no—
293C2×C8—38no—
294C2×C8C22×D312yes—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—42no—
298C22×C3C22×D412yes—
299C22×C3C22×D411yescharacteristic
300C2×D3—24no—
301C2×D3—22no—
302C2×D3—42no—
303C2×D3C22×D414yes—
304C3×C4—22no—
305C3×C4—24no—
306C3×C4C22×D411yescharacteristic
307C3×C4C22×D412yes—
308C3×C4C2511yescharacteristic
309Dic3—22no—
310Dic3—24no—
311Dic3C22×D414yes—
312C23—21no—
313C23—32no—
314C23—61no—
315C23D3×D411yescharacteristic
316Q8—24no—
317Q8—22no—
318Q8—21no—
319Q8—34no—
320Q8—32no—
321Q8—34no—
322Q8—68no—
323Q8—68no—
324Q8—64no—
325Q8—61no—
326Q8—62no—
327Q8C23×D312yes—
328Q8C23×D314yes—
329D4—24no—
330D4—21no—
331D4—38no—
332D4—34no—
333D4—32no—
334D4—68no—
335D4—68no—
336D4—64no—
337D4—61no—
338D4C23×D312yes—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—32no—
344C2×C4—34no—
345C2×C4—34no—
346C2×C4—68no—
347C2×C4—68no—
348C2×C4—68no—
349C2×C4—68no—
350C2×C4—64no—
351C2×C4—61no—
352C2×C4—62no—
353C2×C4—64no—
354C2×C4D3×D412yes—
355C2×C4D3×D411yescharacteristic
356C2×C4C23×D312yes—
357C2×C4C23×D311yescharacteristic
358C8—32no—
359C8—32no—
360C8C23×D314yes—
361C2×C3—22no—
362C2×C3—42no—
363C2×C3#57812yescomplemented
364C2×C3C23×D411yescharacteristic
365D3—24no—
366D3—42no—
367C22—22no—
368C22—22no—
369C22—34no—
370C22—42no—
371C22—64no—
372C22—62no—
373C22—122no—
374C22C2×D3×D412yes—
375C22C2×D3×D411yescharacteristic
376C4—22no—
377C4—24no—
378C4—34no—
379C4—62no—
380C4—64no—
381C4C2×D3×D412yes—
382C4C2×D3×D411yescharacteristic
383C4C24×D311yescharacteristic
384C3C2×#57811yescharacteristic, complemented, Sylow
385C2—22no—
386C2—42no—
387C2—64no—
388C2—122no—
389C2#577912yesdirect product factor
390C2C22×D3×D411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
hypercenterC2×#137D3
hypocenterC3C2×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#1007C2semidirect
3. C2×C3×#137C2semidirect
4. C2×#950C2semidirect
5. C22×#216C2semidirect
6. #5779C2direct (N × H)
7. C2×#918C2semidirect
8. C2×D3×Q16C2semidirect
9. C2×#916C2semidirect
10. C2×#915C2semidirect
11. C2×#909C2semidirect
12. C2×#907C2semidirect
17. #1010C22semidirect
19. #1007C22semidirect
33. C3×#137C22semidirect
37. #950C22semidirect
41. C2×#216C22semidirect
46. #918C22semidirect
47. D3×Q16C22semidirect
48. #916C22semidirect
49. #915C22semidirect
50. #909C22semidirect
51. #907C22semidirect
62. C2×#137D3semidirect
158. #137C2×D3semidirect
363. C2×C3#578semidirect
384. C3C2×#578semidirect
389. C2#5779direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#10102
C2×#10072
C2×C3×#1372
C2×#9502
C22×#2162
#57792
C2×#9182
C2×D3×Q162
C2×#9162
C2×#9152
C2×#9092
C2×#9072
C2×#5783