Finite Group Databasebeta

← #34109809

Subgroups — #34109809

Order 20096.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097777C211yesmaximal, characteristic, complemented
3#23757853C211yesmaximal, characteristic, complemented
4#23757839C211yesmaximal, characteristic, complemented
5#23757527C211yesmaximal, characteristic, complemented
6#23757524C211yesmaximal, characteristic, complemented
7#23757521C211yesmaximal, characteristic, complemented
8#23757516C211yesmaximal, characteristic, complemented
9#23757509C211yesmaximal, characteristic, complemented
10#23757508C211yesmaximal, characteristic, complemented
11#23757505C211yesmaximal, characteristic, complemented
12#23757312C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097798—24no—
20C2×#14578941C2211yescharacteristic
21C2×D1256C2211yescharacteristic
22C2×#14578939C2211yescharacteristic
23#19097777—24no—
24C2×#14578938C2211yescharacteristic, complemented
25C2×#14578937C2211yescharacteristic
26C2×#14578937C2211yescharacteristic, complemented
27C2×C8×D157C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic, complemented
29#19097767—21no—
30#19097757—21no—
31#19097749—21no—
32#19097747C2211yescharacteristic, complemented
33#19097746—21no—
34#19097746C2211yescharacteristic
35#19097746C2211yescharacteristic, complemented
36#19097745C2211yescharacteristic
37#19097745C2211yescharacteristic, complemented
38#19097744—21no—
39#19097744—21no—
40#19097742C2211yescharacteristic
41#19097741—21no—
42#19097739—21no—
43#19097729C2211yescharacteristic
44C4⋊C4×D157C2211yescharacteristic
45#19097705C2211yescharacteristic, complemented
46#19097704C2211yescharacteristic, complemented
47#19097702C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic
52#19097691C2211yescharacteristic
53#19097691C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic, complemented
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64#14578954—22no—
65#14578954—22no—
66C2×Dic314D411yescharacteristic
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628D411yescharacteristic
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157D411yescharacteristic
78C2×C22⋊D157D411yescharacteristic
79D4×D157—22no—
80D4×D157—22no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157C2311yescharacteristic
84C2×C4×D157C2311yescharacteristic
85#14578941—22no—
86D1256—22no—
87#14578939—22no—
88#14578938—22no—
89#14578937—22no—
90#14578937D414yescomplemented
91C2×#11929434C2311yescharacteristic
92C2×#11929434C2311yescharacteristic
93C8×D157—22no—
94C8×D157—22no—
95C4⋊Dic157C2311yescharacteristic
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
98#14578933—21no—
99C22⋊Dic157—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106C4×Dic157—21no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157—21no—
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111SD16×C157—22no—
112C2×C8×C157—22no—
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
115C4⋊C4×C157—21no—
116C4⋊C4×C157C2311yescharacteristic
117C4⋊C4×C157C2311yescharacteristic
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C22×D157—21no—
121C22×D157—21no—
122C22×D157—42no—
123C22×D157—81no—
124C22×D157—81no—
125C22×D157C2×D411yescharacteristic
126C22×D157C2×D411yescharacteristic
127Dic314—22no—
128D628—22no—
129D628—21no—
130D628—21no—
131D628—21no—
132D628—21no—
133D628—42no—
134D628—81no—
135D628—81no—
136C22⋊D157—22no—
137C22⋊D157—22no—
138C22⋊D157—42no—
139C22⋊D157—81no—
140C22⋊D157—81no—
141C22⋊D157—81no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157C2×D411yescharacteristic
145C2×Dic157C2×D411yescharacteristic
146C4×D157—21no—
147C4×D157—21no—
148C4×D157—42no—
149C4×D157—81no—
150C4×D157C2×D412yes—
151C4×D157#4112yes—
152#11929434—21no—
153#11929434C2×D412yes—
154C23×C157—21no—
155C23×C157C2×D411yescharacteristic
156Q8×C157—21no—
157Q8×C157—21no—
158D4×C157—21no—
159D4×C157—21no—
160D4×C157—81no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2411yescharacteristic
168C8×C157—21no—
169C8×C157C2×D412yes—
170C22×C157—21no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157C22×D411yescharacteristic
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—21no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C2×D157#12112yes—
190C4×C157—21no—
191C4×C157—41no—
192C4×C157—41no—
193C4×C157—41no—
194C4×C157C22×D411yescharacteristic
195C4×C157C2×#4111yescharacteristic
196Dic157—21no—
197Dic157—41no—
198Dic157—41no—
199Dic157#12112yes—
200C2×C157—41no—
201C2×C157—81no—
202C2×C157C2×#12111yescharacteristic
203C2×C157#57611yescharacteristic
204C2×C157#57711yescharacteristic
205D157—22no—
206D157—41no—
207D157—81no—
208D157—81no—
209C157#306211yescharacteristic, complemented, Sylow
210#3062—1571nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1571no—
212C2×#131—1571no—
213#537—1571no—
214#535—1571no—
215#469—1571no—
216#469—1571no—
217#468—1571no—
218#468—1571no—
219#466—1571no—
220#466—1571no—
221#465—1571no—
222#465D15711yescharacteristic, complemented
223#428—1571no—
224#416—1571no—
225C2×#102—1571no—
226C2×#41—1571no—
227C2×#41—1571no—
228C22×D4—1571no—
229AGL(1,ℤ/8)—3144no—
230C2×SD16—1571no—
231C2×SD16C2×D15711yescharacteristic
232C2×D8—1571no—
233C2×D8—1571no—
234#131—3144no—
235C2×M16—1571no—
236C2×M16—1571no—
237C2×M16—1571no—
238C22×C8—1571no—
239C22×C8—1571no—
240C22×C8C2×D15711yescharacteristic, complemented
241#127—3141no—
242#124—3141no—
243C22⋊Q8—3141no—
244C22⋊Q8—1571no—
245C22⋊Q8C2×D15711yescharacteristic, complemented
246#121—3141no—
247#121—3141no—
248#121—3141no—
249#121—1571no—
250#121—1571no—
251#121—1571no—
252#121—1571no—
253#121—1571no—
254#121C2×D15711yescharacteristic
255#120—3141no—
256#120—3141no—
257#117—1571no—
258C2×C4⋊C4—1571no—
259#107—1571no—
260#107—1571no—
261#106—1571no—
262#106C2×D15711yescharacteristic, complemented
263#103—1571no—
264#103C2×D15711yescharacteristic, complemented
265#102—1571no—
266#102—1571no—
267#102—1571no—
268#102—1571no—
269#102—1571no—
270#102C2×D15711yescharacteristic
271C24—3141no—
272#41—3142no—
273#41—3142no—
274#41—3142no—
275#41—3142no—
276C2×Q8—1571no—
277C2×Q8C22×D15711yescharacteristic
278C2×D4—21no—
279C2×D4—3141no—
280C2×D4—3141no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3142no—
286C2×D4—3142no—
287C2×D4—3141no—
288C2×D4—1571no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4—1571no—
292C2×D4—1571no—
293C2×D4C22×D15711yescharacteristic
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4—1571no—
298C22×C4C22×D15711yescharacteristic
299SD16—22no—
300SD16—3142no—
301D8—3142no—
302D8—3142no—
303M16—3142no—
304M16—3142no—
305M16—1574no—
306C2×C8—22no—
307C2×C8—3142no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—1571no—
317C4⋊C4—1571no—
318C4⋊C4C22×D15711yescharacteristic
319C4⋊C4C22×D15711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C42—3141no—
330C23—21no—
331C23—3141no—
332C23—3141no—
333C23—6282no—
334C23—1571no—
335C23—1571no—
336C23—12561no—
337C23—12561no—
338C23D4×D15711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—3142no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—3142no—
346D4—3142no—
347D4—3142no—
348D4—3141no—
349D4—3141no—
350D4—3141no—
351D4—3141no—
352D4—6282no—
353D4—6282no—
354D4—12561no—
355D4—12561no—
356D4—12561no—
357D4—12561no—
358D4—12561no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—6282no—
369C2×C4—1571no—
370C2×C4—1571no—
371C2×C4—1572no—
372C2×C4—1572no—
373C2×C4—12561no—
374C2×C4D4×D15711yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—21no—
377C8—3141no—
378C8—1572no—
379C8D4×D15712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3141no—
387C22—3141no—
388C22—3141no—
389C22—3141no—
390C22—3141no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22—6281no—
396C22—6281no—
397C22—1572no—
398C22—12564no—
399C22C2×D4×D15711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—3141no—
405C4—6281no—
406C4—6281no—
407C4—1572no—
408C4C2×D4×D15711yescharacteristic
409C4#41×D15711yescharacteristic
410C2—41no—
411C2—81no—
412C2—3142no—
413C2—6281no—
414C2—12561no—
415C2—12561no—
416C2#121×D15711yescharacteristic
417C2#2375807511yescharacteristic
418C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
hypercenter#465D157
hypocenterC157#3062
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097777C2semidirect
3. #23757853C2semidirect
4. #23757839C2semidirect
5. #23757527C2semidirect
6. #23757524C2semidirect
7. #23757521C2semidirect
8. #23757516C2semidirect
9. #23757509C2semidirect
10. #23757508C2semidirect
11. #23757505C2semidirect
12. #23757312C2semidirect
13. #23757268C2semidirect
14. #102×D157C2semidirect
15. #465×C157C2semidirect
24. C2×#14578938C22semidirect
26. C2×#14578937C22semidirect
28. C2×C8×D157C22semidirect
32. #19097747C22semidirect
35. #19097746C22semidirect
37. #19097745C22semidirect
45. #19097705C22semidirect
46. #19097704C22semidirect
47. #19097702C22semidirect
48. #19097695C22semidirect
50. #19097693C22semidirect
53. #19097691C22semidirect
55. C22×C8×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
90. #14578937D4semidirect
209. C157#3062semidirect
222. #465D157semidirect
240. C22×C8C2×D157semidirect
245. C22⋊Q8C2×D157semidirect
262. #106C2×D157semidirect
264. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977772
#237578532
#237578392
#237575272
#237575242
#237575212
#237575162
#237575092
#237575082
#237575052
#237573122
#237572682
#102×D1572
#465×C1572
#3062157