Finite Group Databasebeta

← #34109482

Subgroups — #34109482

Order 20096.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758109C211yesmaximal, characteristic, complemented
2#23758071C211yesmaximal, characteristic, complemented
3#23757819C211yesmaximal, characteristic, complemented
4#23757454C212yesmaximal, complemented
5#23757449C212yesmaximal, complemented
6#23757444C212yesmaximal, complemented
7#23757439C212yesmaximal, complemented
8C2×#19097698C211yesmaximal, characteristic, complemented
9C2×#19097697C211yesmaximal, characteristic, complemented
10#23757255C211yesmaximal, characteristic, complemented
11#575×C157C211yesmaximal, characteristic, complemented
12#19097819—21no—
13#19097818—21no—
14#19097818C2212yes—
15#19097817—21no—
16#19097817C2212yes—
17#19097816—21no—
18#19097816—22no—
19C2×#14578954—22no—
20C2×#14578954C2211yescharacteristic
21C2×#14578954C2211yescharacteristic
22#41×D157—22no—
23#41×D157—22no—
24#19097802C2212yescomplemented
25#19097801—21no—
26#19097797C2212yescomplemented
27#19097796—21no—
28#19097791—22no—
29C2×#14578938C2211yescharacteristic, complemented
30#19097769C2211yescharacteristic, complemented
31#19097767C2211yescharacteristic, complemented
32#19097760C2211yescharacteristic, complemented
33#19097758C2211yescharacteristic, complemented
34#19097743—22no—
35C4×C22⋊D157—22no—
36C2×C4×Dic157C2211yescharacteristic, complemented
37#19097698C2212yescomplemented
38#19097698C2212yescomplemented
39#19097697C2212yescomplemented
40#19097697C2212yescomplemented
41#19097689C2212yescomplemented
42#19097687C2212yescomplemented
43C2×#41×C157C2211yescharacteristic
44C2×#41×C157C2211yescharacteristic, complemented
45#137×C157—21no—
46#137×C157C2212yescomplemented
47AGL(1,ℤ/8)×C157—21no—
48AGL(1,ℤ/8)×C157C2212yescomplemented
49#135×C157—22no—
50C2×M16×C157C2211yescharacteristic, complemented
51#14578955—22no—
52#14578955—22no—
53#14578954—24no—
54#14578954—24no—
55#14578954—22no—
56#14578954—22no—
57#14578954—21no—
58#14578954—21no—
59#14578954—21no—
60#14578954—21no—
61#14578954—21no—
62#14578954—21no—
63#14578954D412yescomplemented
64#14578954C2312yes—
65#14578953—24no—
66#14578953—22no—
67#14578953—24no—
68#14578953—22no—
69C2×Dic314—22no—
70C2×Dic314D411yescharacteristic, complemented
71C2×Dic314C2311yescharacteristic
72Q8×D157—22no—
73Q8×D157—22no—
74C2×D628—22no—
75C2×D628D411yescharacteristic, complemented
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—24no—
78C2×C22⋊D157—22no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157D412yes—
81D4×D157—24no—
82D4×D157—22no—
83D4×D157—24no—
84D4×D157—22no—
85C22×Dic157—21no—
86C2×C4×D157—24no—
87C2×C4×D157—21no—
88C2×C4×D157D412yes—
89#14578944—22no—
90#14578942—22no—
91#14578941—22no—
92#14578939—22no—
93#14578938—21no—
94#14578938C2312yes—
95C2×#11929434—21no—
96#14578933—22no—
97C22⋊Dic157—22no—
98#14578931—22no—
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C4×Dic157C2311yescharacteristic
102#41×C157—21no—
103#41×C157—21no—
104#41×C157—21no—
105#41×C157—21no—
106#41×C157—21no—
107#41×C157—21no—
108#41×C157D412yescomplemented
109#41×C157C2312yes—
110C2×Q8×C157D411yescharacteristic, complemented
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157—21no—
113C2×D4×C157—21no—
114C2×D4×C157D411yescharacteristic, complemented
115C2×D4×C157C2311yescharacteristic
116C22×C4×C157—21no—
117C22×C4×C157—21no—
118C22×C4×C157C2311yescharacteristic
119Q16×C157—22no—
120SD16×C157—22no—
121SD16×C157—22no—
122D8×C157—22no—
123M16×C157—21no—
124M16×C157C2312yes—
125C2×C8×C157—21no—
126C22×D157—24no—
127C22×D157—21no—
128C22×D157C2×D412yes—
129Dic314—24no—
130Dic314—21no—
131Dic314—21no—
132Dic314—21no—
133Dic314C2×D412yes—
134D628—24no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628C2×D412yes—
139C22⋊D157—24no—
140C22⋊D157—24no—
141C22⋊D157—22no—
142C22⋊D157—22no—
143C22⋊D157—22no—
144C22⋊D157—22no—
145C22⋊D157—44no—
146C2×Dic157—24no—
147C2×Dic157—21no—
148C2×Dic157—21no—
149C2×Dic157—21no—
150C2×Dic157—42no—
151C2×Dic157C2×D412yes—
152C4×D157—24no—
153C4×D157—24no—
154C4×D157—22no—
155C4×D157—22no—
156C4×D157—22no—
157C4×D157—22no—
158#11929434—22no—
159C23×C157—21no—
160C23×C157—21no—
161C23×C157C2×D411yescharacteristic
162Q8×C157—21no—
163Q8×C157—21no—
164Q8×C157—21no—
165Q8×C157C2×D412yes—
166D4×C157—22no—
167D4×C157—22no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—21no—
171D4×C157—41no—
172D4×C157—41no—
173D4×C157C2×D412yes—
174C2×C4×C157—21no—
175C2×C4×C157—22no—
176C2×C4×C157—22no—
177C2×C4×C157—21no—
178C2×C4×C157—41no—
179C2×C4×C157—41no—
180C2×C4×C157C2×D411yescharacteristic
181C2×C4×C157C2×D411yescharacteristic
182C2×C4×C157C2×D411yescharacteristic
183C2×C4×C157C2×D411yescharacteristic
184C2×C4×C157C2×D411yescharacteristic
185C2×C4×C157C2411yescharacteristic
186C8×C157—22no—
187C22×C157—22no—
188C22×C157—41no—
189C22×C157—42no—
190C22×C157—41no—
191C22×C157—81no—
192C22×C157#12011yescharacteristic
193C22×C157#12011yescharacteristic
194C22×C157C22×D411yescharacteristic
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—42no—
198C2×D157—81no—
199C2×D157—84no—
200C4×C157—22no—
201C4×C157—41no—
202C4×C157#12011yescharacteristic
203C4×C157#12011yescharacteristic
204C4×C157C22×D411yescharacteristic
205C4×C157C22×D411yescharacteristic
206Dic157—22no—
207Dic157—22no—
208Dic157—22no—
209Dic157—41no—
210C2×C157—21no—
211C2×C157—21no—
212C2×C157—21no—
213C2×C157—42no—
214C2×C157—81no—
215C2×C157C2×#12011yescharacteristic
216D157—42no—
217D157—42no—
218C157#301811yescharacteristic, complemented, Sylow
219#3018—1571nomaximal, Sylow
220#585—1571no—
221#575—1571no—
222#575D15711yescharacteristic, complemented
223#532—1571no—
224#456—1572no—
225#455—1572no—
226#454—1572no—
227#453—1572no—
228C2×#104—1571no—
229C2×#104—1571no—
230#413—1571no—
2312−1+4—3141no—
2322−1+4—3141no—
2332−1+4—1572no—
234H(5,𝔽2)—3142no—
235H(5,𝔽2)—3141no—
236H(5,𝔽2)—3141no—
237H(5,𝔽2)—1572no—
238C2×#41—3142no—
239C2×#41—3142no—
240C2×#41—3142no—
241C2×#41—1571no—
242C2×#41—1571no—
243C2×#41C2×D15711yescharacteristic
244C2×#41C2×D15711yescharacteristic, complemented
245#137—21no—
246#137—3141no—
247#137—1572no—
248#137C2×D15712yescomplemented
249AGL(1,ℤ/8)—21no—
250AGL(1,ℤ/8)—3141no—
251AGL(1,ℤ/8)—1572no—
252AGL(1,ℤ/8)C2×D15712yescomplemented
253#135—22no—
254#135—3142no—
255C2×M16—1571no—
256C2×M16C2×D15711yescharacteristic, complemented
257C4⋊Q8—1571no—
258#127—1571no—
259#124—1571no—
260#124—1571no—
261#121—3142no—
262C4×D4—3142no—
263C2×C42—1571no—
264#104—1572no—
265#104—1572no—
266#104—1572no—
267#104—1572no—
268#101—1572no—
269#100—1572no—
270#41—21no—
271#41—21no—
272#41—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—3144no—
277#41—3144no—
278#41—3142no—
279#41—3142no—
280#41—3144no—
281#41—3142no—
282#41—3142no—
283#41—3142no—
284#41—3142no—
285#41—3144no—
286#41—3141no—
287#41—3141no—
288#41—3141no—
289#41—3141no—
290#41—3141no—
291#41—3141no—
292#41—1572no—
293#41—1572no—
294#41C22⋊D15712yescomplemented
295#41C22×D15712yes—
296C2×Q8—3142no—
297C2×Q8—3142no—
298C2×Q8—3142no—
299C2×Q8—1571no—
300C2×Q8—1571no—
301C2×Q8C22⋊D15711yescharacteristic, complemented
302C2×Q8C22×D15711yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—3142no—
306C2×D4—3144no—
307C2×D4—3142no—
308C2×D4—3144no—
309C2×D4—3144no—
310C2×D4—3141no—
311C2×D4—3142no—
312C2×D4—3142no—
313C2×D4—1572no—
314C2×D4—1571no—
315C2×D4—1571no—
316C2×D4C22⋊D15711yescharacteristic, complemented
317C2×D4C22×D15711yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—3144no—
321C22×C4—3141no—
322C22×C4—3141no—
323C22×C4—1572no—
324C22×C4C22×D15711yescharacteristic
325Q16—22no—
326Q16—3142no—
327SD16—22no—
328SD16—22no—
329SD16—3142no—
330SD16—3142no—
331D8—22no—
332D8—3142no—
333M16—21no—
334M16—3141no—
335M16—1572no—
336M16C22×D15712yes—
337C2×C8—21no—
338C2×C8—3141no—
339C4⋊C4—3142no—
340C22⋊C4—3142no—
341C22⋊C4—3142no—
342C42—3141no—
343C42—1571no—
344C42—1571no—
345C23—21no—
346C23—21no—
347C23—3144no—
348C23—3141no—
349C23—1572no—
350C23D4×D15711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—3144no—
355Q8—3141no—
356Q8—3141no—
357Q8—3141no—
358Q8—1572no—
359Q8C2×C22⋊D15712yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—3144no—
368D4—3144no—
369D4—3144no—
370D4—3142no—
371D4—3142no—
372D4—3142no—
373D4—3142no—
374D4—3141no—
375D4—3141no—
376D4—3141no—
377D4—6284no—
378D4—1572no—
379D4C2×C22⋊D15712yes—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—3144no—
387C2×C4—3144no—
388C2×C4—3144no—
389C2×C4—3141no—
390C2×C4—3141no—
391C2×C4—3142no—
392C2×C4—3142no—
393C2×C4—3142no—
394C2×C4—3142no—
395C2×C4—3141no—
396C2×C4—6282no—
397C2×C4—1572no—
398C2×C4D4×D15711yescharacteristic
399C2×C4D4×D15711yescharacteristic
400C2×C4D4×D15711yescharacteristic
401C2×C4C2×C22⋊D15711yescharacteristic
402C2×C4C2×C22⋊D15711yescharacteristic
403C2×C4C23×D15711yescharacteristic
404C8—22no—
405C8—3142no—
406C22—22no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—81no—
411C22—3142no—
412C22—3142no—
413C22—6282no—
414C22—12561no—
415C22—12564no—
416C22#1909773911yescharacteristic
417C22#1909773911yescharacteristic
418C22C2×D4×D15711yescharacteristic
419C4—22no—
420C4—41no—
421C4—3142no—
422C4—3142no—
423C4—3142no—
424C4—6281no—
425C4#1909773911yescharacteristic
426C4#1909773911yescharacteristic
427C4C2×D4×D15711yescharacteristic
428C4C22×C22⋊D15711yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—81no—
434C2—6282no—
435C2—6282no—
436C2C2×#1909773911yescharacteristic
subgroupG/H
centerC4#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#575×C157C2
socleC2×C157C2×#120
hypercenter#575D157
hypocenterC157#3018
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23758071C2semidirect
3. #23757819C2semidirect
4. #23757454C2semidirect
5. #23757449C2semidirect
6. #23757444C2semidirect
7. #23757439C2semidirect
8. C2×#19097698C2semidirect
9. C2×#19097697C2semidirect
10. #23757255C2semidirect
11. #575×C157C2semidirect
24. #19097802C22semidirect
26. #19097797C22semidirect
29. C2×#14578938C22semidirect
30. #19097769C22semidirect
31. #19097767C22semidirect
32. #19097760C22semidirect
33. #19097758C22semidirect
36. C2×C4×Dic157C22semidirect
37. #19097698C22semidirect
38. #19097698C22semidirect
39. #19097697C22semidirect
40. #19097697C22semidirect
41. #19097689C22semidirect
42. #19097687C22semidirect
44. C2×#41×C157C22semidirect
46. #137×C157C22semidirect
48. AGL(1,ℤ/8)×C157C22semidirect
50. C2×M16×C157C22semidirect
63. #14578954D4semidirect
70. C2×Dic314D4semidirect
75. C2×D628D4semidirect
108. #41×C157D4semidirect
110. C2×Q8×C157D4semidirect
114. C2×D4×C157D4semidirect
218. C157#3018semidirect
222. #575D157semidirect
244. C2×#41C2×D157semidirect
248. #137C2×D157semidirect
252. AGL(1,ℤ/8)C2×D157semidirect
256. C2×M16C2×D157semidirect
294. #41C22⋊D157semidirect
301. C2×Q8C22⋊D157semidirect
316. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237580712
#237578192
#237574542
#237574492
#237574442
#237574392
C2×#190976982
C2×#190976972
#237572552
#575×C1572
#3018157