Finite Group Databasebeta

← C4×#19097795 #34108939

Subgroups — C4×#19097795 #34108939

Order 20096.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic
2C4×#14578953C211yesmaximal, characteristic, complemented
3C4×D4×D157C211yesmaximal, characteristic, complemented
4#23757385C211yesmaximal, characteristic, complemented
5#23757382C211yesmaximal, characteristic, complemented
6#23757372C211yesmaximal, characteristic, complemented
7#23757370C211yesmaximal, characteristic, complemented
8C4×#14578943C211yesmaximal, characteristic, complemented
9C4×#14578942C211yesmaximal, characteristic, complemented
10C4×#14578939C211yesmaximal, characteristic, complemented
11#23757314C211yesmaximal, characteristic, complemented
12#23757276C211yesmaximal, characteristic, complemented
13#23757271C211yesmaximal, characteristic, complemented
14C4×#14578937C211yesmaximal, characteristic, complemented
15C4×D8×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18C22×C4×D157—21no—
19#19097795C418yesdirect product factor
20C2×#14578943C2211yescharacteristic
21C2×#14578942C2211yescharacteristic
22C2×#14578939C2211yescharacteristic
23C2×#14578937C2211yescharacteristic
24C2×#14578937C2211yescharacteristic
25C4×Dic314C2211yescharacteristic, complemented
26#19097736C2211yescharacteristic, complemented
27C4×D628C2211yescharacteristic
28D4×Dic157C2211yescharacteristic
29D4×Dic157C2211yescharacteristic, complemented
30#19097733C2211yescharacteristic
31C4×C22⋊D157—21no—
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097731—21no—
35#19097729C2211yescharacteristic
36#19097727—21no—
37C4⋊C4×D157C2211yescharacteristic
38C22⋊C4×D157—21no—
39C2×C4×Dic157—21no—
40C42×D157C2211yescharacteristic
41#19097704C2211yescharacteristic, complemented
42#19097703C2211yescharacteristic, complemented
43#19097702C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097694C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic
50#19097681C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52C4×#11929434C2211yescharacteristic, complemented
53C2×D8×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55C4×D4×C157C2211yescharacteristic, complemented
56#107×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic, complemented
59C4×C8×C157C2211yescharacteristic, complemented
60C23×D157—21no—
61#14578953—22no—
62#14578953C2×C414yescomplemented
63C2×Dic314C2311yescharacteristic
64C2×D628C2311yescharacteristic
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67D4×D157—22no—
68D4×D157C2×C414yescomplemented
69C22×Dic157—21no—
70C22×Dic157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—44no—
76C2×C4×D157D411yescharacteristic
77C2×C4×D157D411yescharacteristic
78C2×C4×D157C2311yescharacteristic
79#14578943C2×C414yescomplemented
80#14578942C2×C414yescomplemented
81#14578939C2×C414yescomplemented
82#14578937—22no—
83#14578937C2×C414yescomplemented
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87C4⋊Dic157C2311yescharacteristic
88#14578933—21no—
89#14578933—21no—
90#14578933—21no—
91C22⋊Dic157—21no—
92C22⋊Dic157—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96C4×Dic157—21no—
97C4×Dic157—21no—
98C4×Dic157—21no—
99C4×Dic157D411yescharacteristic
100C4×Dic157D411yescharacteristic
101C4×Dic157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C2×D4×C157C2311yescharacteristic
104C22×C4×C157—21no—
105C22×C4×C157—21no—
106D8×C157C2×C414yescomplemented
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22⋊C4×C157—21no—
113C42×C157C2311yescharacteristic
114C22×D157—21no—
115C22×D157—24no—
116C22×D157—42no—
117C22×D157—42no—
118C22×D157C2×D411yescharacteristic
119Dic314—21no—
120Dic314C22×C412yescomplemented
121D628—21no—
122D628C22×C412yes—
123C22⋊D157—24no—
124C22⋊D157—24no—
125C2×Dic157—21no—
126C2×Dic157—24no—
127C2×Dic157—21no—
128C2×Dic157—21no—
129C2×Dic157—42no—
130C2×Dic157—42no—
131C2×Dic157C2×D411yescharacteristic
132C2×Dic157C2×D411yescharacteristic
133C2×Dic157C2×D411yescharacteristic
134C4×D157—22no—
135C4×D157—41no—
136C4×D157—42no—
137C4×D157—42no—
138C4×D157—42no—
139C4×D157C22×C412yes—
140C4×D157#4112yes—
141#11929434—21no—
142#11929434C22×C412yescomplemented
143C23×C157—21no—
144C23×C157—21no—
145D4×C157—21no—
146D4×C157—21no—
147D4×C157C22×C412yes—
148D4×C157C22×C412yescomplemented
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—42no—
152C2×C4×C157—42no—
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2411yescharacteristic
156C8×C157—21no—
157C8×C157C22×C412yescomplemented
158C22×C157—22no—
159C22×C157—22no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157—22no—
168C2×D157—48no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157C4×D412yes—
172C4×C157—21no—
173C4×C157—41no—
174C4×C157—41no—
175C4×C157AGL(1,ℤ/8)12yescomplemented
176C4×C157C23×C411yescharacteristic
177C4×C157C2×#4111yescharacteristic
178Dic157—22no—
179Dic157—21no—
180Dic157—21no—
181Dic157—41no—
182Dic157—41no—
183Dic157C4×D412yes—
184C2×C157—42no—
185C2×C157—42no—
186C2×C157C2×C4×D411yescharacteristic
187C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C157#57511yescharacteristic
189D157—22no—
190D157—42no—
191C157C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
192C4×AGL(1,ℤ/8)—1571nomaximal, Sylow
193C2×AGL(1,ℤ/8)—1571no—
194C4×#41—1571no—
195C2×C4×D4—1571no—
196#442—1571no—
197#442—1571no—
198#440—1571no—
199#440—1571no—
200C4×SD16—1571no—
201C4×SD16—1571no—
202C4×D8—1571no—
203C4×D8D15711yescharacteristic, complemented
204#428—1571no—
205#418—1571no—
206#417—1571no—
207C4×M16—1571no—
208C2×#41—1571no—
209C22×D4—1571no—
210C23×C4—3141no—
211AGL(1,ℤ/8)—1578no—
212C2×SD16—1571no—
213C2×SD16—1571no—
214C2×D8—1571no—
215C2×D8C2×D15711yescharacteristic
216C2×M16—1571no—
217C2×M16—1571no—
218C4×Q8—1571no—
219C4×Q8—1571no—
220C4×D4—3141no—
221C4×D4—3141no—
222C4×D4—3141no—
223C4×D4—3141no—
224C4×D4—1571no—
225C4×D4—1571no—
226C4×D4—1571no—
227C4×D4—1571no—
228C4×D4C2×D15711yescharacteristic
229C4×D4C2×D15711yescharacteristic, complemented
230#117—3141no—
231#117—1571no—
232C2×C4⋊C4—1571no—
233C2×C22⋊C4—3141no—
234C2×C42—3141no—
235C2×C42—1571no—
236#107—1571no—
237#107C2×D15711yescharacteristic, complemented
238#106—1571no—
239#106—1571no—
240#103—1571no—
241#103—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102C2×D15711yescharacteristic
247#102C2×D15711yescharacteristic, complemented
248#97—1571no—
249#97—1571no—
250C4×C8—1571no—
251C4×C8C2×D15711yescharacteristic, complemented
252C24—3141no—
253#41—3142no—
254#41—1574no—
255C2×Q8—1571no—
256C2×D4—3141no—
257C2×D4—3142no—
258C2×D4—3141no—
259C2×D4—1574no—
260C2×D4—1571no—
261C2×D4C22×D15711yescharacteristic
262C2×D4C22×D15711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4—3141no—
271C22×C4—6284no—
272C22×C4—1571no—
273C22×C4—1571no—
274C22×C4—1571no—
275SD16—1574no—
276SD16—1574no—
277D8—1574no—
278D8C4×D15714yescomplemented
279M16—3142no—
280M16—1574no—
281C2×C8—1571no—
282C2×C8—1571no—
283C2×C8C22×D15711yescharacteristic
284C2×C8C22×D15711yescharacteristic
285C4⋊C4—3141no—
286C4⋊C4—3141no—
287C4⋊C4—3141no—
288C4⋊C4—1571no—
289C4⋊C4—1571no—
290C4⋊C4C22×D15711yescharacteristic
291C4⋊C4C22×D15711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C42—3141no—
300C42—3141no—
301C42—3141no—
302C42—1571no—
303C42—1571no—
304C42—1571no—
305C42C22×D15711yescharacteristic
306C23—21no—
307C23—21no—
308C23—3144no—
309C23—3141no—
310C23—6282no—
311C23—6282no—
312C23—1571no—
313Q8—3141no—
314Q8—1572no—
315D4—21no—
316D4—21no—
317D4—3144no—
318D4—3144no—
319D4—3141no—
320D4—1572no—
321D4C2×C4×D15712yes—
322D4C2×C4×D15712yescomplemented
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—3141no—
328C2×C4—3144no—
329C2×C4—3142no—
330C2×C4—3141no—
331C2×C4—3141no—
332C2×C4—6281no—
333C2×C4—6282no—
334C2×C4—6282no—
335C2×C4—6282no—
336C2×C4—6282no—
337C2×C4—6282no—
338C2×C4—1571no—
339C2×C4—1571no—
340C2×C4—1571no—
341C2×C4—1572no—
342C2×C4—1572no—
343C2×C4D4×D15711yescharacteristic
344C2×C4D4×D15711yescharacteristic
345C2×C4C23×D15711yescharacteristic
346C8—21no—
347C8—3141no—
348C8—1572no—
349C8C2×C4×D15712yescomplemented
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3141no—
357C22—3141no—
358C22—3142no—
359C22—6288no—
360C22—6281no—
361C22—6281no—
362C22—1572no—
363C22C2×D4×D15711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3142no—
368C4—3141no—
369C4—3141no—
370C4—6281no—
371C4—6281no—
372C4—1572no—
373C4#1909779512yesdirect product factor
374C4C22×C4×D15711yescharacteristic
375C4#41×D15711yescharacteristic
376C2—42no—
377C2—42no—
378C2—3142no—
379C2—6282no—
380C2C4×D4×D15711yescharacteristic
381C2C2×#1909779511yescharacteristic
382C2#2375807011yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC4×C157C23×C4
FrattiniC2×C4C23×D157
FittingC4×D8×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097795C2
cosocleC2×C4×C157C24
hypercenterC4×D8D157
hypocenterC157C4×AGL(1,ℤ/8)
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. C4×#14578953C2semidirect
3. C4×D4×D157C2semidirect
4. #23757385C2semidirect
5. #23757382C2semidirect
6. #23757372C2semidirect
7. #23757370C2semidirect
8. C4×#14578943C2semidirect
9. C4×#14578942C2semidirect
10. C4×#14578939C2semidirect
11. #23757314C2semidirect
12. #23757276C2semidirect
13. #23757271C2semidirect
14. C4×#14578937C2semidirect
15. C4×D8×C157C2semidirect
19. #19097795C4direct (N × H)
25. C4×Dic314C22semidirect
26. #19097736C22semidirect
29. D4×Dic157C22semidirect
41. #19097704C22semidirect
42. #19097703C22semidirect
43. #19097702C22semidirect
44. #19097696C22semidirect
45. #19097694C22semidirect
48. #19097692C22semidirect
50. #19097681C22semidirect
51. #19097681C22semidirect
52. C4×#11929434C22semidirect
55. C4×D4×C157C22semidirect
56. #107×C157C22semidirect
58. #102×C157C22semidirect
59. C4×C8×C157C22semidirect
62. #14578953C2×C4semidirect
68. D4×D157C2×C4semidirect
79. #14578943C2×C4semidirect
80. #14578942C2×C4semidirect
81. #14578939C2×C4semidirect
83. #14578937C2×C4semidirect
106. D8×C157C2×C4semidirect
120. Dic314C22×C4semidirect
142. #11929434C22×C4semidirect
148. D4×C157C22×C4semidirect
157. C8×C157C22×C4semidirect
175. C4×C157AGL(1,ℤ/8)semidirect
191. C157C4×AGL(1,ℤ/8)semidirect
203. C4×D8D157semidirect
229. C4×D4C2×D157semidirect
237. #107C2×D157semidirect
247. #102C2×D157semidirect
251. C4×C8C2×D157semidirect
278. D8C4×D157semidirect
322. D4C2×C4×D157semidirect
349. C8C2×C4×D157semidirect
373. C4#19097795direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190977952
C4×#145789532
C4×D4×D1572
#237573852
#237573822
#237573722
#237573702
C4×#145789432
C4×#145789422
C4×#145789392
#237573142
#237572762
#237572712
C4×#145789372
C4×D8×C1572
C4×AGL(1,ℤ/8)157