Finite Group Databasebeta

← #34111827

Subgroups — #34111827

Order 20096.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D157C212yesmaximal, complemented
2Q8×D628C211yesmaximal, characteristic, complemented
3#23757908C211yesmaximal, characteristic
4#23757645C211yesmaximal, characteristic, complemented
5#23757501C212yesmaximal, complemented
6#23757479C211yesmaximal, characteristic, complemented
7#23757425C212yesmaximal, complemented
8#23757410C212yesmaximal, complemented
9C4×#14578941C211yesmaximal, characteristic, complemented
10#23757338C211yesmaximal, characteristic, complemented
11#460×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2212yescomplemented
13C2×D4×D157—21no—
14C2×D4×D157C2212yes—
15C2×#14578943C2212yes—
16C2×#14578942C2212yescomplemented
17C2×#14578942C2212yescomplemented
18C2×#14578941C2211yescharacteristic
19SD16×D157—28no—
20C2×C8×D157C2212yescomplemented
21#19097771—21no—
22#19097771C2211yescharacteristic, complemented
23#19097751—22no—
24#19097751C2212yescomplemented
25#19097742—21no—
26#19097742C2212yes—
27#19097739—22no—
28C4×D628—21no—
29C4×D628C2211yescharacteristic
30C4×D628C2211yescharacteristic
31#19097702C2211yescharacteristic, complemented
32#19097696C2212yescomplemented
33#19097695C2211yescharacteristic, complemented
34#19097691C2211yescharacteristic
35#19097683C2212yescomplemented
36C4×#11929434C2211yescharacteristic, complemented
37C2×SD16×C157C2212yescomplemented
38#127×C157C2211yescharacteristic
39C4×Q8×C157C2211yescharacteristic, complemented
40C4⋊C8×C157C2211yescharacteristic, complemented
41#102×C157C2212yescomplemented
42C23×D157—22no—
43C2×Dic314—22no—
44C2×Dic314C2312yes—
45Q8×D157—24no—
46Q8×D157—42no—
47C2×D628D411yescharacteristic
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—22no—
50C2×C22⋊D157—22no—
51D4×D157—24no—
52D4×D157—42no—
53D4×D157—44no—
54C2×C4×D157—22no—
55C2×C4×D157—21no—
56C2×C4×D157C2312yes—
57#14578943—24no—
58#14578942—24no—
59#14578942—42no—
60#14578941D414yescomplemented
61C2×#11929434C2311yescharacteristic
62C2×#11929434C2311yescharacteristic
63C8×D157—24no—
64C4⋊Dic157—22no—
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67C4⋊Dic157D411yescharacteristic
68C4⋊Dic157C2311yescharacteristic
69C22⋊Dic157—22no—
70#14578931—22no—
71#14578931—21no—
72#14578931D412yes—
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157—21no—
76C2×D4×C157C2312yes—
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C42×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—44no—
85C22×D157—44no—
86C22×D157—82no—
87C22×D157C2×D412yes—
88Dic314—22no—
89Dic314—22no—
90Dic314—44no—
91D628—21no—
92D628SD1614yescomplemented
93D628C2×D412yes—
94C22⋊D157—44no—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C2×Dic157—22no—
98C2×Dic157C2×D412yes—
99C4×D157—22no—
100C4×D157—22no—
101C4×D157—44no—
102C4×D157—42no—
103#11929434—21no—
104#11929434C2×D412yescomplemented
105C23×C157—22no—
106Q8×C157—21no—
107Q8×C157C2×D412yescomplemented
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—42no—
111D4×C157—81no—
112C2×C4×C157—21no—
113C2×C4×C157—21no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2411yescharacteristic
117C8×C157—22no—
118C22×C157—42no—
119C22×C157—42no—
120C22×C157—42no—
121C22×C157C22×D411yescharacteristic
122C2×D157—22no—
123C2×D157—22no—
124C2×D157—22no—
125C2×D157—41no—
126C2×D157—41no—
127C2×D157—88no—
128C2×D157C2×SD1614yescomplemented
129C4×C157—22no—
130C4×C157—21no—
131C4×C157—41no—
132C4×C157C2×SD1612yescomplemented
133C4×C157C22×D411yescharacteristic
134C4×C157H(5,𝔽2)11yescharacteristic
135Dic157—22no—
136Dic157—42no—
137Dic157—41no—
138C2×C157—82no—
139C2×C157D4211yescharacteristic
140C2×C157C22×SD1611yescharacteristic
141C2×C157#57711yescharacteristic
142D157—24no—
143D157—42no—
144C157D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1571nomaximal, Sylow
146C22×SD16—1572no—
147D4×Q8—1571no—
148D42—1571no—
149#492—1571no—
150#465—1572no—
151#461—1571no—
152#460D15711yescharacteristic, complemented
153#450—1572no—
154#448—1572no—
155C4×SD16—1571no—
156C8×D4—1571no—
157C22×Q8—1572no—
158C22×D4—3141no—
159C22×D4—1572no—
160C2×SD16—3148no—
161C2×SD16—1572no—
162C2×SD16—1572no—
163C2×SD16—1571no—
164C2×SD16—1572no—
165C2×SD16C2×D15712yescomplemented
166C22×C8—1572no—
167C4⋊Q8—3141no—
168C4⋊Q8—1571no—
169#127C2×D15711yescharacteristic
170C22⋊Q8—3142no—
171C22⋊Q8—1572no—
172#121—3141no—
173#121—1572no—
174#120—3142no—
175C4×Q8C2×D15711yescharacteristic, complemented
176C4×D4—3141no—
177C4×D4—1571no—
178C4×D4—1571no—
179#106—1571no—
180C4⋊C8C2×D15711yescharacteristic, complemented
181#103—1571no—
182#103—1572no—
183#102—1571no—
184#102C2×D15712yescomplemented
185C22⋊C8—1572no—
186C4×C8—1571no—
187C24—3142no—
188C2×Q8—3142no—
189C2×Q8—3144no—
190C2×Q8—6282no—
191C2×Q8—1572no—
192C2×Q8C22×D15711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—3144no—
196C2×D4—3142no—
197C2×D4—3142no—
198C2×D4—6282no—
199C2×D4—6284no—
200C2×D4—1571no—
201C2×D4—1571no—
202C2×D4C22×D15712yes—
203C22×C4—3142no—
204C22×C4—3141no—
205C22×C4—1572no—
206SD16—24no—
207SD16—3144no—
208SD16—3144no—
209SD16—6282no—
210SD16—1574no—
211C2×C8—3144no—
212C2×C8—1571no—
213C2×C8—1571no—
214C2×C8C22×D15712yes—
215C4⋊C4—21no—
216C4⋊C4—3142no—
217C4⋊C4—3141no—
218C4⋊C4—3141no—
219C4⋊C4—1571no—
220C4⋊C4—1571no—
221C4⋊C4C22×D15711yescharacteristic
222C22⋊C4—3142no—
223C22⋊C4—3142no—
224C22⋊C4—3141no—
225C22⋊C4—1572no—
226C42—21no—
227C42C22×D15711yescharacteristic
228C23—22no—
229C23—3141no—
230C23—6284no—
231C23—6284no—
232C23—1572no—
233C23—12562no—
234Q8—21no—
235Q8—3142no—
236Q8—3142no—
237Q8—6284no—
238Q8D4×D15712yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3141no—
244D4—6284no—
245D4—6284no—
246D4—1574no—
247D4—1572no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—3142no—
251C2×C4—3142no—
252C2×C4—3141no—
253C2×C4—3142no—
254C2×C4—6284no—
255C2×C4—6282no—
256C2×C4—1572no—
257C2×C4D4×D15711yescharacteristic
258C2×C4D4×D15711yescharacteristic
259C2×C4C23×D15711yescharacteristic
260C8—22no—
261C8—3141no—
262C8—1572no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—3142no—
267C22—3142no—
268C22—3142no—
269C22—6281no—
270C22—6281no—
271C22—1574no—
272C22—12568no—
273C22C2×D4×D15711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—3142no—
278C4—6282no—
279C4—6281no—
280C4SD16×D15712yescomplemented
281C4C2×D4×D15711yescharacteristic
282C4#1909781711yescharacteristic
283C2—82no—
284C2—3144no—
285C2—6282no—
286C2#2375790711yescharacteristic
287C2C2×SD16×D15711yescharacteristic
288C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#460×C157C2
socleC22×C157C22×D4
prime radical#23757908C2
hypercenter#460D157
hypocenterC157D4×SD16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D157C2semidirect
2. Q8×D628C2semidirect
4. #23757645C2semidirect
5. #23757501C2semidirect
6. #23757479C2semidirect
7. #23757425C2semidirect
8. #23757410C2semidirect
9. C4×#14578941C2semidirect
10. #23757338C2semidirect
11. #460×C157C2semidirect
12. C2×Q8×D157C22semidirect
16. C2×#14578942C22semidirect
17. C2×#14578942C22semidirect
20. C2×C8×D157C22semidirect
22. #19097771C22semidirect
24. #19097751C22semidirect
31. #19097702C22semidirect
32. #19097696C22semidirect
33. #19097695C22semidirect
35. #19097683C22semidirect
36. C4×#11929434C22semidirect
37. C2×SD16×C157C22semidirect
39. C4×Q8×C157C22semidirect
40. C4⋊C8×C157C22semidirect
41. #102×C157C22semidirect
60. #14578941D4semidirect
92. D628SD16semidirect
104. #11929434C2×D4semidirect
107. Q8×C157C2×D4semidirect
128. C2×D157C2×SD16semidirect
132. C4×C157C2×SD16semidirect
144. C157D4×SD16semidirect
152. #460D157semidirect
165. C2×SD16C2×D157semidirect
175. C4×Q8C2×D157semidirect
180. C4⋊C8C2×D157semidirect
184. #102C2×D157semidirect
238. Q8D4×D157semidirect
280. C4SD16×D157semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1572
Q8×D6282
#237579082
#237576452
#237575012
#237574792
#237574252
#237574102
C4×#145789412
#237573382
#460×C1572
D4×SD16157