Finite Group Databasebeta

← #34109758

Subgroups — #34109758

Order 20096.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19097796C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757780C211yesmaximal, characteristic, complemented
4C2×#19097746C211yesmaximal, characteristic, complemented
5#23757525C212yesmaximal, complemented
6#23757522C212yesmaximal, complemented
7#23757508C212yesmaximal, complemented
8#23757505C212yesmaximal, complemented
9#23757313C211yesmaximal, characteristic, complemented
10#23757279C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12C22×D628—21no—
13C22×D628C2211yescharacteristic
14C22×C4×D157C2211yescharacteristic
15#19097796—24no—
16C2×#14578941C2212yescomplemented
17C2×#14578939C2212yescomplemented
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2212yescomplemented
20M16×D157—24no—
21C2×C8×D157C2212yescomplemented
22#19097752—21no—
23#19097747C2211yescharacteristic, complemented
24#19097747C2211yescharacteristic, complemented
25#19097746—21no—
26#19097746—21no—
27#19097746C2212yes—
28#19097746C2212yes—
29#19097742C2211yescharacteristic, complemented
30#19097742C2211yescharacteristic, complemented
31#19097739—21no—
32C4×C22⋊D157—21no—
33C4×C22⋊D157—21no—
34#19097730C2211yescharacteristic
35C2×#14578931—21no—
36#19097704C2212yescomplemented
37#19097702C2212yescomplemented
38#19097693C2212yescomplemented
39#19097693C2212yescomplemented
40C2×#41×C157C2211yescharacteristic
41C2×M16×C157C2211yescharacteristic
42C2×C4⋊C4×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44#102×C157C2212yescomplemented
45C23×D157—21no—
46C23×D157D411yescharacteristic
47C2×D628—22no—
48C2×D628—22no—
49C2×D628—22no—
50C2×D628—21no—
51C2×D628—22no—
52C2×D628—44no—
53C2×D628C2312yes—
54C2×C22⋊D157—21no—
55C2×C22⋊D157—21no—
56C22×Dic157D411yescharacteristic
57C2×C4×D157—22no—
58C2×C4×D157—22no—
59C2×C4×D157D411yescharacteristic
60C2×C4×D157D411yescharacteristic
61C2×C4×D157C2311yescharacteristic
62C2×C4×D157C2311yescharacteristic
63#14578941—24no—
64#14578939—24no—
65#14578938D414yescomplemented
66#14578937—24no—
67C2×#11929434C2312yes—
68C8×D157—24no—
69C4⋊Dic157C2311yescharacteristic
70C4⋊Dic157C2311yescharacteristic
71#14578933—21no—
72#14578933—21no—
73C22⋊Dic157—21no—
74C22⋊Dic157—21no—
75C22⋊Dic157—21no—
76#14578931—21no—
77#14578931—21no—
78#14578931—22no—
79#14578931—22no—
80C4×Dic157—21no—
81#41×C157—22no—
82#41×C157—22no—
83C2×Q8×C157C2311yescharacteristic
84C2×D4×C157—21no—
85C2×D4×C157C2311yescharacteristic
86C22×C4×C157—21no—
87C22×C4×C157—21no—
88C22×C4×C157C2311yescharacteristic
89M16×C157—22no—
90C2×C8×C157C2312yes—
91C4⋊C4×C157—21no—
92C4⋊C4×C157C2312yes—
93C22×D157—22no—
94C22×D157—22no—
95C22×D157—22no—
96C22×D157—41no—
97C22×D157—42no—
98C22×D157—42no—
99C22×D157—42no—
100C22×D157C2×D411yescharacteristic
101C22×D157C2×D411yescharacteristic
102D628—22no—
103D628—22no—
104D628—44no—
105D628—44no—
106D628—42no—
107C22⋊D157—42no—
108C22⋊D157—42no—
109C2×Dic157—22no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157C2×D411yescharacteristic
113C2×Dic157C2×D411yescharacteristic
114C4×D157—22no—
115C4×D157—22no—
116C4×D157—21no—
117C4×D157—21no—
118C4×D157—21no—
119C4×D157—21no—
120#11929434C2×D414yescomplemented
121C23×C157—21no—
122C23×C157C2×D411yescharacteristic
123Q8×C157—21no—
124Q8×C157—21no—
125D4×C157—21no—
126D4×C157—21no—
127D4×C157—42no—
128C2×C4×C157—22no—
129C2×C4×C157—21no—
130C2×C4×C157—42no—
131C2×C4×C157—42no—
132C2×C4×C157C2×D411yescharacteristic
133C2×C4×C157C2×D412yes—
134C2×C4×C157#4112yes—
135C2×C4×C157C2411yescharacteristic
136C8×C157—22no—
137C22×C157—21no—
138C22×C157—21no—
139C22×C157—41no—
140C22×C157—41no—
141C22×C157—41no—
142C22×C157#12112yes—
143C22×C157C22×D411yescharacteristic
144C2×D157—21no—
145C2×D157—21no—
146C2×D157—21no—
147C2×D157—21no—
148C2×D157—21no—
149C2×D157—44no—
150C2×D157—42no—
151C2×D157—42no—
152C2×D157—42no—
153C2×D157—84no—
154C2×D157AGL(1,ℤ/8)12yescomplemented
155C4×C157—41no—
156C4×C157—42no—
157C4×C157#12112yes—
158C4×C157C22×D411yescharacteristic
159C4×C157C2×#4111yescharacteristic
160Dic157—21no—
161Dic157—21no—
162Dic157—41no—
163Dic157—41no—
164C2×C157—22no—
165C2×C157—81no—
166C2×C157C2×#12111yescharacteristic
167C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C157#57511yescharacteristic
169D157—22no—
170D157—41no—
171D157—82no—
172C157#305511yescharacteristic, complemented, Sylow
173#3055—1571nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1571no—
175C22×M16—1571no—
176#525—1571no—
177C2×#121—1571no—
178#469—1572no—
179#468—1572no—
180#466—1572no—
181#465—1572no—
182#428—1571no—
183#418—1571no—
184#417D15711yescharacteristic, complemented
185C2×#41C2×D15711yescharacteristic
186C22×D4—3141no—
187C22×D4—1571no—
188C23×C4—1571no—
189AGL(1,ℤ/8)—3144no—
190C2×SD16—1572no—
191C2×D8—1572no—
192C2×M16—3144no—
193C2×M16—1571no—
194C2×M16—1572no—
195C2×M16C2×D15711yescharacteristic
196C22×C8—1572no—
197#123—3141no—
198C22⋊Q8—1571no—
199C22⋊Q8—1571no—
200#121—3141no—
201#121—3141no—
202#121—1571no—
203#121—1571no—
204#121—1572no—
205#121—1572no—
206#120—3141no—
207C4×D4—3141no—
208C4×D4—3141no—
209#117—1571no—
210C2×C4⋊C4C2×D15711yescharacteristic
211C2×C22⋊C4—3141no—
212#107—1572no—
213#106—1572no—
214#103C2×D15712yescomplemented
215#102—1572no—
216#102—1572no—
217#102C2×D15712yescomplemented
218C24—3141no—
219C24—1571no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D15711yescharacteristic
223C2×D4—21no—
224C2×D4—3142no—
225C2×D4—3142no—
226C2×D4—3141no—
227C2×D4—3141no—
228C2×D4—3142no—
229C2×D4—3141no—
230C2×D4—3142no—
231C2×D4—6284no—
232C2×D4—1572no—
233C2×D4C22×D15711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—3142no—
237C22×C4—3142no—
238C22×C4—1571no—
239C22×C4—1571no—
240C22×C4—1571no—
241C22×C4—1571no—
242C22×C4—1571no—
243C22×C4C22×D15711yescharacteristic
244SD16—3144no—
245D8—3144no—
246M16—22no—
247M16—3144no—
248M16—1574no—
249C2×C8—3144no—
250C2×C8—1572no—
251C2×C8C22×D15712yes—
252C4⋊C4—21no—
253C4⋊C4—3141no—
254C4⋊C4—3141no—
255C4⋊C4—1571no—
256C4⋊C4—1571no—
257C4⋊C4C22×D15712yes—
258C22⋊C4—3141no—
259C22⋊C4—3141no—
260C22⋊C4—3142no—
261C22⋊C4—3142no—
262C22⋊C4—3141no—
263C22⋊C4—3141no—
264C22⋊C4—3141no—
265C42—3141no—
266C23—21no—
267C23—3142no—
268C23—3142no—
269C23—3142no—
270C23—6281no—
271C23—6282no—
272C23—6282no—
273C23—6282no—
274C23—1571no—
275C23—1571no—
276C23D4×D15711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—3142no—
283D4—3142no—
284D4—6284no—
285D4—6284no—
286D4—6282no—
287D4—6282no—
288D4—6282no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—3142no—
294C2×C4—3142no—
295C2×C4—3142no—
296C2×C4—3141no—
297C2×C4—3141no—
298C2×C4—3141no—
299C2×C4—3141no—
300C2×C4—3141no—
301C2×C4—3141no—
302C2×C4—1571no—
303C2×C4—1571no—
304C2×C4D4×D15711yescharacteristic
305C2×C4D4×D15712yes—
306C2×C4#1457895412yes—
307C2×C4C23×D15711yescharacteristic
308C8—22no—
309C8—1574no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3141no—
316C22—3141no—
317C22—3141no—
318C22—3141no—
319C22—3141no—
320C22—6284no—
321C22—6282no—
322C22—6282no—
323C22—6282no—
324C22—1572no—
325C22—12564no—
326C22#1909774412yes—
327C22C2×D4×D15711yescharacteristic
328C4—41no—
329C4—42no—
330C4—3141no—
331C4—3141no—
332C4—6281no—
333C4—6281no—
334C4#1909774412yes—
335C4C2×D4×D15711yescharacteristic
336C4C2×#1457895411yescharacteristic
337C2—22no—
338C2—81no—
339C2—3142no—
340C2—6281no—
341C2—12562no—
342C2C2×#1909774411yescharacteristic
343C2AGL(1,ℤ/8)×D15711yescharacteristic
344C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
hypercenter#417D157
hypocenterC157#3055
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19097796C2semidirect
2. C2×M16×D157C2semidirect
3. #23757780C2semidirect
4. C2×#19097746C2semidirect
5. #23757525C2semidirect
6. #23757522C2semidirect
7. #23757508C2semidirect
8. #23757505C2semidirect
9. #23757313C2semidirect
10. #23757279C2semidirect
11. #417×C157C2semidirect
16. C2×#14578941C22semidirect
17. C2×#14578939C22semidirect
19. C2×#14578937C22semidirect
21. C2×C8×D157C22semidirect
23. #19097747C22semidirect
24. #19097747C22semidirect
29. #19097742C22semidirect
30. #19097742C22semidirect
36. #19097704C22semidirect
37. #19097702C22semidirect
38. #19097693C22semidirect
39. #19097693C22semidirect
43. #103×C157C22semidirect
44. #102×C157C22semidirect
65. #14578938D4semidirect
120. #11929434C2×D4semidirect
154. C2×D157AGL(1,ℤ/8)semidirect
172. C157#3055semidirect
184. #417D157semidirect
214. #103C2×D157semidirect
217. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977962
C2×M16×D1572
#237577802
C2×#190977462
#237575252
#237575222
#237575082
#237575052
#237573132
#237572792
#417×C1572
#3055157