Finite Group Databasebeta

← #34109359

Subgroups — #34109359

Order 20096.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14578955C211yesmaximal, characteristic, complemented
2C2×#19097794C212yesmaximal, complemented
3#23757786C211yesmaximal, characteristic, complemented
4#23757434C212yesmaximal, complemented
5#23757423C211yesmaximal, characteristic, complemented
6#23757421C212yesmaximal, complemented
7#23757412C211yesmaximal, characteristic, complemented
8#23757409C211yesmaximal, characteristic, complemented
9#23757268C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic, complemented
11#451×C157C211yesmaximal, characteristic, complemented
12C2×#14578955—28no—
13C2×#14578955—21no—
14C2×#14578955—21no—
15C2×#14578955C2212yes—
16C2×#14578955C2212yes—
17C2×#14578955C2211yescharacteristic, complemented
18C22×D628—21no—
19C22×D628C2211yescharacteristic
20C22×C4×D157—21no—
21C22×C4×D157C2211yescharacteristic
22#19097794—28no—
23C2×#14578941C2212yescomplemented
24C2×#14578941C2212yescomplemented
25C2×D1256C2212yescomplemented
26C2×C8×D157C2212yescomplemented
27#19097756—21no—
28#19097747C2211yescharacteristic, complemented
29#19097746C2211yescharacteristic, complemented
30#19097745C2211yescharacteristic
31#19097741—21no—
32#19097733—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic, complemented
35#19097695C2212yescomplemented
36#19097693C2212yescomplemented
37#19097691C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic
39#19097683C2212yescomplemented
40C22×Q8×C157C2211yescharacteristic
41C2×Q16×C157C2212yescomplemented
42C22⋊Q8×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic
45C23×D157—21no—
46C23×D157D411yescharacteristic
47#14578955—28no—
48#14578955—28no—
49#14578955—24no—
50#14578955—24no—
51#14578955—24no—
52#14578955—24no—
53#14578955D418yescomplemented
54C2×D628—28no—
55C2×D628—22no—
56C2×D628—22no—
57C2×D628—24no—
58C2×D628—21no—
59C2×D628—21no—
60C2×D628—21no—
61C2×D628C2311yescharacteristic
62C2×D628C2312yes—
63C2×C22⋊D157—21no—
64C22×Dic157D411yescharacteristic
65C2×C4×D157—28no—
66C2×C4×D157—22no—
67C2×C4×D157—22no—
68C2×C4×D157—24no—
69C2×C4×D157—21no—
70C2×C4×D157D411yescharacteristic
71C2×C4×D157D411yescharacteristic
72C2×C4×D157C2311yescharacteristic
73C2×C4×D157C2311yescharacteristic
74#14578941—24no—
75#14578941—24no—
76D1256—24no—
77C2×#11929434C2312yes—
78C8×D157—24no—
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578931—21no—
82#14578931—21no—
83#14578931—21no—
84#14578931—21no—
85C4×Dic157—21no—
86C2×Q8×C157—24no—
87C2×Q8×C157—21no—
88C2×Q8×C157C2311yescharacteristic
89C2×Q8×C157C2312yes—
90C22×C4×C157—21no—
91C22×C4×C157C2311yescharacteristic
92Q16×C157—24no—
93C2×C8×C157C2312yes—
94C4⋊C4×C157—21no—
95C4⋊C4×C157C2311yescharacteristic
96C22⋊C4×C157—21no—
97C22×D157—24no—
98C22×D157—22no—
99C22×D157—22no—
100C22×D157—22no—
101C22×D157—21no—
102C22×D157—41no—
103C22×D157—42no—
104C22×D157—42no—
105C22×D157C2×D411yescharacteristic
106C22×D157C2×D411yescharacteristic
107D628—28no—
108D628—24no—
109D628—28no—
110D628—22no—
111D628—22no—
112D628—42no—
113D628—42no—
114D628C2×D414yes—
115C22⋊D157—42no—
116C2×Dic157—22no—
117C2×Dic157—21no—
118C2×Dic157C2×D411yescharacteristic
119C2×Dic157C2×D411yescharacteristic
120C4×D157—28no—
121C4×D157—22no—
122C4×D157—22no—
123C4×D157—28no—
124C4×D157—22no—
125C4×D157—42no—
126C4×D157C2×D414yes—
127#11929434—22no—
128C23×C157C2×D411yescharacteristic
129Q8×C157—24no—
130Q8×C157—22no—
131Q8×C157—22no—
132Q8×C157C2×D414yes—
133C2×C4×C157—24no—
134C2×C4×C157—22no—
135C2×C4×C157—22no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157C2×D411yescharacteristic
139C2×C4×C157C2411yescharacteristic
140C8×C157—22no—
141C22×C157—21no—
142C22×C157—21no—
143C22×C157#13512yescomplemented
144C22×C157C22×D411yescharacteristic
145C2×D157—21no—
146C2×D157—21no—
147C2×D157—21no—
148C2×D157—24no—
149C2×D157—21no—
150C2×D157—21no—
151C2×D157—44no—
152C2×D157—48no—
153C2×D157—42no—
154C2×D157—42no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C2×D157#12012yes—
159C4×C157—24no—
160C4×C157—21no—
161C4×C157—41no—
162C4×C157—41no—
163C4×C157C22×D412yes—
164Dic157—21no—
165Dic157—41no—
166Dic157#12012yes—
167C2×C157—22no—
168C2×C157C2×#12011yescharacteristic
169C2×C157C2×#13511yescharacteristic
170C2×C157#57511yescharacteristic
171D157—22no—
172D157—41no—
173D157—44no—
174D157—81no—
175C157#300211yescharacteristic, complemented, Sylow
176#3002—1571nomaximal, Sylow
177C22×#41—1571no—
178C2×#135—1572no—
179#525—1571no—
180#452—1572no—
181#451D15711yescharacteristic, complemented
182#450—1571no—
183#449—1572no—
184#448—1571no—
185#447—1571no—
186#416—1572no—
187C2×C22⋊C8—1571no—
188C2×#41—3148no—
189C2×#41—3141no—
190C2×#41—3141no—
191C2×#41—1571no—
192C2×#41—1572no—
193C2×#41—1572no—
194C22×Q8C2×D15711yescharacteristic
195C22×D4—3141no—
196C22×D4—1571no—
197C23×C4—3141no—
198C23×C4—1571no—
199#135—3148no—
200C2×Q16C2×D15712yescomplemented
201C2×SD16—1572no—
202C2×SD16—1572no—
203C2×D8—1572no—
204C22×C8—1572no—
205#123—3141no—
206C22⋊Q8—1571no—
207C22⋊Q8C2×D15711yescharacteristic
208#121—1571no—
209#121—1571no—
210#120—3141no—
211C4×D4—3141no—
212C4×D4—3141no—
213#117—1571no—
214#103—1572no—
215#103C2×D15712yescomplemented
216#102—1572no—
217#102—1572no—
218C22⋊C8—1571no—
219C22⋊C8—1572no—
220C22⋊C8C2×D15711yescharacteristic
221C24—3141no—
222C24—1571no—
223#41—3148no—
224#41—3148no—
225#41—3144no—
226#41—3144no—
227#41—3144no—
228#41—3144no—
229#41—1578no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D15711yescharacteristic
233C2×Q8C22×D15712yes—
234C2×D4—3148no—
235C2×D4—3142no—
236C2×D4—3142no—
237C2×D4—3144no—
238C2×D4—3141no—
239C2×D4—3141no—
240C2×D4—3141no—
241C2×D4—3141no—
242C2×D4—1571no—
243C2×D4—1572no—
244C22×C4—21no—
245C22×C4—3148no—
246C22×C4—3142no—
247C22×C4—3142no—
248C22×C4—3144no—
249C22×C4—3141no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4—1571no—
253C22×C4—1571no—
254C22×C4—1571no—
255C22×C4C22×D15711yescharacteristic
256Q16—24no—
257SD16—3144no—
258SD16—3144no—
259D8—3144no—
260C2×C8—3144no—
261C2×C8—1572no—
262C2×C8C22×D15712yes—
263C4⋊C4—21no—
264C4⋊C4—3141no—
265C4⋊C4—1571no—
266C4⋊C4C22×D15711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3141no—
269C22⋊C4—3141no—
270C22⋊C4—3141no—
271C22⋊C4—3141no—
272C42—3141no—
273C23—3144no—
274C23—3142no—
275C23—3142no—
276C23—3142no—
277C23—3141no—
278C23—6281no—
279C23—6282no—
280C23—6282no—
281C23—1571no—
282C23—1571no—
283C23D4×D15711yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D15714yes—
288D4—3148no—
289D4—3144no—
290D4—3148no—
291D4—3142no—
292D4—3142no—
293D4—6282no—
294D4—6282no—
295D4—6282no—
296D4—1574no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—3148no—
303C2×C4—3142no—
304C2×C4—3148no—
305C2×C4—3142no—
306C2×C4—3142no—
307C2×C4—3142no—
308C2×C4—3141no—
309C2×C4—6282no—
310C2×C4—1571no—
311C2×C4—1571no—
312C2×C4—1574no—
313C2×C4D4×D15711yescharacteristic
314C2×C4C23×D15711yescharacteristic
315C8—22no—
316C8—3142no—
317C22—21no—
318C22—21no—
319C22—3141no—
320C22—3141no—
321C22—3141no—
322C22—3144no—
323C22—3141no—
324C22—3141no—
325C22—6284no—
326C22—6288no—
327C22—6282no—
328C22—6282no—
329C22—6281no—
330C22—6281no—
331C22—6281no—
332C22—1572no—
333C22#1909779412yescomplemented
334C22C2×D4×D15711yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—3141no—
340C4—6281no—
341C4—1572no—
342C4C2×D4×D15712yes—
343C2—22no—
344C2—3142no—
345C2—6281no—
346C2—6284no—
347C2—12561no—
348C2#120×D15711yescharacteristic
349C2C2×#1909779411yescharacteristic
350C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
hypercenter#451D157
hypocenterC157#3002
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. C2×#19097794C2semidirect
3. #23757786C2semidirect
4. #23757434C2semidirect
5. #23757423C2semidirect
6. #23757421C2semidirect
7. #23757412C2semidirect
8. #23757409C2semidirect
9. #23757268C2semidirect
10. C22⋊C8×D157C2semidirect
11. #451×C157C2semidirect
17. C2×#14578955C22semidirect
23. C2×#14578941C22semidirect
24. C2×#14578941C22semidirect
25. C2×D1256C22semidirect
26. C2×C8×D157C22semidirect
28. #19097747C22semidirect
29. #19097746C22semidirect
34. #19097729C22semidirect
35. #19097695C22semidirect
36. #19097693C22semidirect
37. #19097691C22semidirect
39. #19097683C22semidirect
41. C2×Q16×C157C22semidirect
43. #103×C157C22semidirect
53. #14578955D4semidirect
143. C22×C157#135semidirect
175. C157#3002semidirect
181. #451D157semidirect
200. C2×Q16C2×D157semidirect
215. #103C2×D157semidirect
333. C22#19097794semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#190977942
#237577862
#237574342
#237574232
#237574212
#237574122
#237574092
#237572682
C22⋊C8×D1572
#451×C1572
#3002157