Finite Group Databasebeta

← #34109439

Subgroups — #34109439

Order 20096.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097816C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757851C211yesmaximal, characteristic, complemented
4#23757432C212yesmaximal, complemented
5#23757422C212yesmaximal, complemented
6#23757415C212yesmaximal, complemented
7#23757405C212yesmaximal, complemented
8#23757275C211yesmaximal, characteristic, complemented
9C2×#19097692C211yesmaximal, characteristic, complemented
10#23757241C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
12#19097816—24no—
13#19097816—24no—
14C2×#14578954C2211yescharacteristic
15C2×#14578954C2211yescharacteristic
16C2×#14578953—21no—
17C2×#14578953C2212yes—
18C22×C22⋊D157—22no—
19C2×D4×D157—21no—
20C2×D4×D157C2212yes—
21#19097791—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578942C2211yescharacteristic, complemented
24C2×#14578941C2211yescharacteristic, complemented
25C2×#14578939C2211yescharacteristic, complemented
26C22×#11929434C2211yescharacteristic, complemented
27#19097750—21no—
28#19097747C2212yescomplemented
29#19097745—21no—
30#19097743—22no—
31#19097742C2212yescomplemented
32C2×C4⋊Dic157C2211yescharacteristic, complemented
33#19097696C2212yescomplemented
34#19097693C2212yescomplemented
35#19097692C2212yescomplemented
36#19097692C2212yescomplemented
37#19097683C2212yescomplemented
38#19097683C2212yescomplemented
39C2×#41×C157C2211yescharacteristic, complemented
40C22×D4×C157C2211yescharacteristic
41AGL(1,ℤ/8)×C157—24no—
42C2×SD16×C157C2212yescomplemented
43C2×D8×C157C2212yescomplemented
44C2×M16×C157C2211yescharacteristic, complemented
45C23×D157—22no—
46#14578954—22no—
47#14578954—22no—
48#14578954—22no—
49#14578954D414yescomplemented
50#14578953—28no—
51#14578953—24no—
52#14578953—24no—
53C2×Dic314D411yescharacteristic
54C2×Dic314C2311yescharacteristic
55C2×D628D411yescharacteristic
56C2×D628C2311yescharacteristic
57C2×C22⋊D157—216no—
58C2×C22⋊D157—22no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—22no—
61C2×C22⋊D157—21no—
62C2×C22⋊D157D412yes—
63D4×D157—28no—
64D4×D157—24no—
65D4×D157—24no—
66C22×Dic157—22no—
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157C2312yes—
70#14578944—22no—
71#14578942—22no—
72#14578941—22no—
73#14578939—22no—
74C2×#11929434—22no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C4⋊Dic157—21no—
78C4⋊Dic157C2312yes—
79#14578933—22no—
80C22⋊Dic157—22no—
81#14578931—22no—
82C24×C157—21no—
83#41×C157—22no—
84#41×C157—22no—
85C2×Q8×C157C2311yescharacteristic
86C2×D4×C157—21no—
87C2×D4×C157—21no—
88C2×D4×C157—22no—
89C2×D4×C157D414yescomplemented
90C2×D4×C157C2311yescharacteristic
91C2×D4×C157C2312yes—
92C22×C4×C157—21no—
93C22×C4×C157C2311yescharacteristic
94SD16×C157—24no—
95D8×C157—24no—
96M16×C157—22no—
97C2×C8×C157C2312yes—
98C22×D157—21no—
99C22×D157—28no—
100C22×D157—82no—
101C22×D157—82no—
102C22×D157C2×D412yes—
103Dic314—22no—
104Dic314—21no—
105Dic314C2×D412yes—
106D628—22no—
107D628—21no—
108D628C2×D412yes—
109C22⋊D157—28no—
110C22⋊D157—28no—
111C22⋊D157—24no—
112C22⋊D157—24no—
113C22⋊D157—82no—
114C2×Dic157—28no—
115C2×Dic157—21no—
116C2×Dic157—22no—
117C2×Dic157—81no—
118C2×Dic157C2×D412yes—
119C4×D157—24no—
120C4×D157—22no—
121C4×D157—22no—
122#11929434—21no—
123#11929434—21no—
124C23×C157—22no—
125C23×C157—24no—
126C23×C157—21no—
127C23×C157—81no—
128C23×C157—81no—
129C23×C157C2×D411yescharacteristic
130Q8×C157—21no—
131Q8×C157—21no—
132D4×C157—24no—
133D4×C157—22no—
134D4×C157—21no—
135D4×C157—21no—
136D4×C157—42no—
137D4×C157C2×D414yes—
138C2×C4×C157—21no—
139C2×C4×C157—42no—
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D412yes—
142C2×C4×C157C2×D412yes—
143C2×C4×C157C2411yescharacteristic
144C8×C157—22no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157—24no—
148C22×C157—42no—
149C22×C157—42no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—84no—
154C22×C157#12012yes—
155C22×C157C22×D411yescharacteristic
156C2×D157—22no—
157C2×D157—22no—
158C2×D157—22no—
159C2×D157—22no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—88no—
163C4×C157—41no—
164C4×C157#12012yes—
165C4×C157C22×D411yescharacteristic
166C4×C157C22×D411yescharacteristic
167Dic157—22no—
168Dic157—22no—
169Dic157—42no—
170C2×C157—22no—
171C2×C157—44no—
172C2×C157—81no—
173C2×C157C2×#12011yescharacteristic
174C2×C157#57611yescharacteristic
175C2×C157#57711yescharacteristic
176D157—42no—
177D157—42no—
178C157#301211yescharacteristic, complemented, Sylow
179#3012—1571nomaximal, Sylow
180C2×H(5,𝔽2)—1571no—
181C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
182C2×#135—1571no—
183#537—1571no—
184#452—1572no—
185#450—1572no—
186#449—1572no—
187#447—1572no—
188#417—1571no—
189C2×#102—1571no—
190#408—1571no—
191H(5,𝔽2)—3144no—
192H(5,𝔽2)—3144no—
193C2×#41—3141no—
194C2×#41—1572no—
195C2×#41—1571no—
196C2×#41—1571no—
197C2×#41C2×D15711yescharacteristic, complemented
198C22×D4—3141no—
199C22×D4—3142no—
200C22×D4—1572no—
201C22×D4C2×D15711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—3144no—
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C2×SD16C2×D15712yescomplemented
208C2×D8—1571no—
209C2×D8C2×D15712yescomplemented
210C2×M16C2×D15711yescharacteristic, complemented
211C22×C8—1571no—
212C22⋊Q8—3141no—
213C22⋊Q8—1572no—
214#121—3141no—
215#121—3142no—
216#121—1572no—
217C2×C4⋊C4—1571no—
218#103—1572no—
219#102—1572no—
220#102—1572no—
221#102—1572no—
222C22⋊C8—1572no—
223C22⋊C8—1572no—
224C24—21no—
225C24—3142no—
226#41—22no—
227#41—22no—
228#41—3148no—
229#41—3144no—
230#41—3144no—
231#41—3142no—
232#41—3142no—
233#41—3142no—
234#41—1574no—
235C2×Q8—1571no—
236C2×Q8—1571no—
237C2×Q8C22×D15711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—3148no—
242C2×D4—3144no—
243C2×D4—31416no—
244C2×D4—3142no—
245C2×D4—3141no—
246C2×D4—3144no—
247C2×D4—3142no—
248C2×D4—3142no—
249C2×D4—1572no—
250C2×D4—1571no—
251C2×D4—1571no—
252C2×D4C22⋊D15714yescomplemented
253C2×D4C22×D15711yescharacteristic
254C2×D4C22×D15712yes—
255C22×C4—21no—
256C22×C4—3142no—
257C22×C4—3141no—
258C22×C4—3141no—
259C22×C4—1572no—
260C22×C4C22×D15711yescharacteristic
261Q16—3142no—
262SD16—24no—
263SD16—3142no—
264SD16—3142no—
265D8—24no—
266D8—3142no—
267M16—22no—
268C2×C8—3142no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15712yes—
272C4⋊C4—3142no—
273C4⋊C4—3141no—
274C4⋊C4—1572no—
275C22⋊C4—3142no—
276C22⋊C4—3142no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—3148no—
283C23—3141no—
284C23—1572no—
285C23—12562no—
286C23—12562no—
287C23D4×D15711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3142no—
291Q8—3141no—
292Q8—1572no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—3148no—
299D4—3148no—
300D4—3144no—
301D4—3144no—
302D4—3142no—
303D4—3141no—
304D4—1572no—
305D4—12562no—
306D4C2×C22⋊D15714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3148no—
310C2×C4—3141no—
311C2×C4—3144no—
312C2×C4—3142no—
313C2×C4—3142no—
314C2×C4—3142no—
315C2×C4—1572no—
316C2×C4—12561no—
317C2×C4D4×D15711yescharacteristic
318C2×C4D4×D15712yes—
319C2×C4C2×C22⋊D15712yes—
320C2×C4C23×D15711yescharacteristic
321C8—22no—
322C8—3141no—
323C8—3141no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—3142no—
334C22—3142no—
335C22—3142no—
336C22—3142no—
337C22—6281no—
338C22—6281no—
339C22—12568no—
340C22#1909773912yes—
341C22C2×D4×D15711yescharacteristic
342C4—41no—
343C4—3142no—
344C4—3142no—
345C4—6282no—
346C4#1909773912yes—
347C4C2×D4×D15711yescharacteristic
348C4C22×C22⋊D15711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—6282no—
353C2—6282no—
354C2C2×#1909773911yescharacteristic
355C2#2375807711yescharacteristic
356C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3012
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. C2×#19097791C2semidirect
3. #23757851C2semidirect
4. #23757432C2semidirect
5. #23757422C2semidirect
6. #23757415C2semidirect
7. #23757405C2semidirect
8. #23757275C2semidirect
9. C2×#19097692C2semidirect
10. #23757241C2semidirect
11. C2×AGL(1,ℤ/8)×C157C2semidirect
22. C2×#14578944C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578941C22semidirect
25. C2×#14578939C22semidirect
26. C22×#11929434C22semidirect
28. #19097747C22semidirect
31. #19097742C22semidirect
32. C2×C4⋊Dic157C22semidirect
33. #19097696C22semidirect
34. #19097693C22semidirect
35. #19097692C22semidirect
36. #19097692C22semidirect
37. #19097683C22semidirect
38. #19097683C22semidirect
39. C2×#41×C157C22semidirect
42. C2×SD16×C157C22semidirect
43. C2×D8×C157C22semidirect
44. C2×M16×C157C22semidirect
49. #14578954D4semidirect
89. C2×D4×C157D4semidirect
178. C157#3012semidirect
181. C2×AGL(1,ℤ/8)D157semidirect
197. C2×#41C2×D157semidirect
207. C2×SD16C2×D157semidirect
209. C2×D8C2×D157semidirect
210. C2×M16C2×D157semidirect
252. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190977912
#237578512
#237574322
#237574222
#237574152
#237574052
#237572752
C2×#190976922
#237572412
C2×AGL(1,ℤ/8)×C1572
#3012157