Finite Group Databasebeta

← #34109678

Subgroups — #34109678

Order 20096.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097794C211yesmaximal, characteristic, complemented
3#23757841C211yesmaximal, characteristic, complemented
4#23757755C211yesmaximal, characteristic, complemented
5#23757497C211yesmaximal, characteristic, complemented
6#23757491C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757473C211yesmaximal, characteristic, complemented
9#23757471C211yesmaximal, characteristic, complemented
10#23757469C211yesmaximal, characteristic, complemented
11#23757467C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757273C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#464×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097798—24no—
20#19097794—24no—
21C2×#14578941C2211yescharacteristic
22C2×#14578941C2211yescharacteristic, complemented
23C2×#14578941C2211yescharacteristic, complemented
24C2×D1256C2211yescharacteristic
25C2×D1256C2211yescharacteristic, complemented
26C2×#14578939C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097768C2211yescharacteristic
30#19097767C2211yescharacteristic
31#19097760C2211yescharacteristic
32#19097744—21no—
33#19097744—21no—
34#19097741—21no—
35#19097741—21no—
36Q8×Dic157C2211yescharacteristic, complemented
37C4×D628—21no—
38C4×D628C2211yescharacteristic, complemented
39#19097733—21no—
40#19097733C2211yescharacteristic, complemented
41#19097728—21no—
42#19097728C2211yescharacteristic
43C4⋊C4×D157—21no—
44C4⋊C4×D157C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097700C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic
52#19097691C2211yescharacteristic
53#19097691C2211yescharacteristic, complemented
54C2×Q16×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64#14578955—22no—
65#14578955D414yescomplemented
66C2×D628—21no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628C2311yescharacteristic
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76D4×D157—22no—
77D4×D157—22no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157C2311yescharacteristic
84#14578941—22no—
85#14578941—22no—
86#14578941—22no—
87D1256—22no—
88D1256—22no—
89#14578939—22no—
90#14578937—22no—
91C2×#11929434C2311yescharacteristic
92C2×#11929434C2311yescharacteristic
93C8×D157—22no—
94C4⋊Dic157—21no—
95C4⋊Dic157C2311yescharacteristic
96#14578933—21no—
97#14578933D411yescharacteristic
98#14578933D411yescharacteristic
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931D411yescharacteristic
104#14578931D411yescharacteristic
105C4×Dic157—21no—
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110Q16×C157—22no—
111SD16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C42×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157—21no—
122C22×D157—21no—
123C22×D157—42no—
124C22×D157—81no—
125C22×D157—81no—
126C22×D157C2×D411yescharacteristic
127D628—24no—
128D628—21no—
129D628—21no—
130D628—21no—
131D628—21no—
132D628—21no—
133D628—42no—
134D628—81no—
135D628—81no—
136D628C2×D412yes—
137C22⋊D157—42no—
138C22⋊D157—81no—
139C22⋊D157—81no—
140C2×Dic157—21no—
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C4×D157—24no—
145C4×D157—42no—
146C4×D157—81no—
147C4×D157—81no—
148C4×D157—81no—
149C4×D157C2×D412yes—
150C4×D157#4112yes—
151#11929434—21no—
152#11929434—21no—
153C23×C157—21no—
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157C2×D412yes—
158D4×C157—21no—
159D4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2411yescharacteristic
166C8×C157—21no—
167C8×C157—21no—
168C22×C157—41no—
169C22×C157—41no—
170C22×C157—41no—
171C22×C157C22×D411yescharacteristic
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—22no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—41no—
178C2×D157—41no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—84no—
184C2×D157#12112yes—
185C4×C157—22no—
186C4×C157—21no—
187C4×C157—21no—
188C4×C157—41no—
189C4×C157—41no—
190C4×C157C22×D411yescharacteristic
191C4×C157C2×#4111yescharacteristic
192Dic157—21no—
193Dic157—21no—
194Dic157—41no—
195Dic157#12112yes—
196C2×C157—81no—
197C2×C157C2×#12111yescharacteristic
198C2×C157#57611yescharacteristic
199C2×C157#57711yescharacteristic
200D157—22no—
201D157—42no—
202D157—81no—
203D157—81no—
204C157#304511yescharacteristic, complemented, Sylow
205#3045—1571nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1571no—
207C2×#135—1571no—
208#535—1571no—
209#520—1571no—
210#464—1571no—
211#464D15711yescharacteristic, complemented
212#463—1571no—
213#463—1571no—
214#460—1571no—
215#460—1571no—
216#459—1571no—
217#459—1571no—
218#424—1571no—
219#417—1571no—
220C2×#102—1571no—
221C2×#41—1571no—
222C2×#41—1571no—
223C22×D4—1571no—
224AGL(1,ℤ/8)—3144no—
225#135—3144no—
226C2×Q16C2×D15711yescharacteristic, complemented
227C2×SD16—1571no—
228C2×SD16—1571no—
229C2×SD16—1571no—
230C2×SD16C2×D15711yescharacteristic
231C2×D8—1571no—
232C2×D8—1571no—
233C2×D8—1571no—
234C2×M16—1571no—
235C22×C8—1571no—
236#127—1571no—
237#127—1571no—
238#124—1571no—
239#124C2×D15711yescharacteristic
240#121—3141no—
241#121—3141no—
242#120—3141no—
243#120—3141no—
244C4×Q8—1571no—
245C4×Q8C2×D15711yescharacteristic, complemented
246C4×D4—3141no—
247C4×D4—3141no—
248C4×D4—1571no—
249C4×D4—1571no—
250#117—3141no—
251#117—1571no—
252C2×C4⋊C4—3141no—
253C2×C4⋊C4—1571no—
254C4⋊C8—1571no—
255C4⋊C8—1571no—
256C4⋊C8—1571no—
257C4⋊C8C2×D15711yescharacteristic, complemented
258#103—1571no—
259#103C2×D15711yescharacteristic, complemented
260#102—1571no—
261#102—1571no—
262#102—1571no—
263#102—1571no—
264#102—1571no—
265#102C2×D15711yescharacteristic
266C24—3141no—
267#41—3142no—
268#41—3142no—
269#41—3142no—
270#41—1574no—
271C2×Q8C22×D15711yescharacteristic
272C2×Q8C22×D15711yescharacteristic
273C2×D4—3141no—
274C2×D4—3141no—
275C2×D4—3142no—
276C2×D4—3142no—
277C2×D4—3141no—
278C2×D4—3141no—
279C2×D4—3141no—
280C2×D4—3141no—
281C2×D4—3141no—
282C2×D4—1571no—
283C2×D4—1571no—
284C2×D4—1571no—
285C2×D4C22×D15711yescharacteristic
286C22×C4—3141no—
287C22×C4—3141no—
288C22×C4—3141no—
289C22×C4—3141no—
290C22×C4—3141no—
291C22×C4—1571no—
292Q16—22no—
293SD16—22no—
294SD16—3142no—
295SD16—3142no—
296SD16—3142no—
297D8—3142no—
298D8—3142no—
299D8—3142no—
300M16—3142no—
301C2×C8—3142no—
302C2×C8—1571no—
303C2×C8—1571no—
304C2×C8C22×D15711yescharacteristic
305C2×C8C22×D15711yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—3141no—
308C4⋊C4—3141no—
309C4⋊C4—1571no—
310C4⋊C4—1571no—
311C4⋊C4—1571no—
312C4⋊C4C22×D15711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—3141no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—1571no—
320C22⋊C4—1571no—
321C42—21no—
322C42—3141no—
323C42—1571no—
324C42C22×D15711yescharacteristic
325C23—21no—
326C23—3141no—
327C23—3141no—
328C23—3141no—
329C23—6282no—
330C23—1571no—
331C23—12561no—
332C23—12561no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8D4×D15712yes—
337D4—21no—
338D4—21no—
339D4—3144no—
340D4—3141no—
341D4—3141no—
342D4—3141no—
343D4—3141no—
344D4—3141no—
345D4—6282no—
346D4—6282no—
347D4—1572no—
348D4—12561no—
349D4—12561no—
350D4—12561no—
351D4—12561no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3144no—
356C2×C4—3141no—
357C2×C4—6282no—
358C2×C4—1571no—
359C2×C4—1571no—
360C2×C4—1571no—
361C2×C4—1572no—
362C2×C4—1572no—
363C2×C4—12561no—
364C2×C4—12561no—
365C2×C4—12561no—
366C2×C4D4×D15711yescharacteristic
367C2×C4D4×D15711yescharacteristic
368C2×C4C23×D15711yescharacteristic
369C8—21no—
370C8—21no—
371C8—3141no—
372C8—3141no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—3141no—
377C22—3141no—
378C22—3142no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22—6281no—
383C22—6281no—
384C22—6281no—
385C22—6281no—
386C22—6281no—
387C22—1572no—
388C22—12564no—
389C22C2×D4×D15711yescharacteristic
390C4—21no—
391C4—22no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—3141no—
396C4—3141no—
397C4—6281no—
398C4—1572no—
399C4C2×D4×D15711yescharacteristic
400C4#41×D15711yescharacteristic
401C2—81no—
402C2—3142no—
403C2—6282no—
404C2—12561no—
405C2—12561no—
406C2#121×D15711yescharacteristic
407C2#2375807511yescharacteristic
408C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#464×C157C2
socleC22×C157C22×D4
hypercenter#464D157
hypocenterC157#3045
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097794C2semidirect
3. #23757841C2semidirect
4. #23757755C2semidirect
5. #23757497C2semidirect
6. #23757491C2semidirect
7. #23757485C2semidirect
8. #23757473C2semidirect
9. #23757471C2semidirect
10. #23757469C2semidirect
11. #23757467C2semidirect
12. #23757298C2semidirect
13. #23757273C2semidirect
14. #102×D157C2semidirect
15. #464×C157C2semidirect
22. C2×#14578941C22semidirect
23. C2×#14578941C22semidirect
25. C2×D1256C22semidirect
36. Q8×Dic157C22semidirect
38. C4×D628C22semidirect
40. #19097733C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097700C22semidirect
47. #19097700C22semidirect
48. #19097695C22semidirect
50. #19097693C22semidirect
53. #19097691C22semidirect
54. C2×Q16×C157C22semidirect
57. C4×Q8×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
65. #14578955D4semidirect
204. C157#3045semidirect
211. #464D157semidirect
226. C2×Q16C2×D157semidirect
245. C4×Q8C2×D157semidirect
257. C4⋊C8C2×D157semidirect
259. #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×#190977942
#237578412
#237577552
#237574972
#237574912
#237574852
#237574732
#237574712
#237574692
#237574672
#237572982
#237572732
#102×D1572
#464×C1572
#3045157