Finite Group Databasebeta

← #34111831

Subgroups — #34111831

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
2#23757938C211yesmaximal, characteristic, complemented
3#23757908C211yesmaximal, characteristic
4#23757499C212yesmaximal, complemented
5#23757475C211yesmaximal, characteristic, complemented
6#23757470C211yesmaximal, characteristic, complemented
7#23757422C212yesmaximal, complemented
8#23757410C212yesmaximal, complemented
9C4×#14578943C211yesmaximal, characteristic, complemented
10C8×D628C211yesmaximal, characteristic, complemented
11#492×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2212yescomplemented
13C2×D4×D157—21no—
14C2×D4×D157C2212yes—
15C2×#14578943C2211yescharacteristic
16C2×#14578942C2212yescomplemented
17C2×#14578941C2212yes—
18SD16×D157—28no—
19C2×C8×D157C2212yescomplemented
20#19097770—21no—
21#19097749—22no—
22#19097747C2212yescomplemented
23#19097742—21no—
24#19097742C2212yes—
25#19097739—22no—
26C4×Dic314C2211yescharacteristic, complemented
27C4×D628C2211yescharacteristic
28C4×D628C2211yescharacteristic
29#19097733—21no—
30#19097703C2211yescharacteristic, complemented
31C4⋊#11929434C2211yescharacteristic, complemented
32#19097696C2211yescharacteristic, complemented
33#19097695C2212yescomplemented
34#19097693C2211yescharacteristic
35#19097692C2212yescomplemented
36#19097683C2212yescomplemented
37C2×SD16×C157C2212yescomplemented
38C2×SD16×C157C2212yescomplemented
39C4⋊Q8×C157C2211yescharacteristic, complemented
40#127×C157C2211yescharacteristic
41C4×C8×C157C2211yescharacteristic, complemented
42C23×D157—22no—
43C2×Dic314—22no—
44C2×Dic314C2311yescharacteristic
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#14578943D414yescomplemented
58#14578942—24no—
59#14578941—24no—
60C2×#11929434C2312yes—
61C8×D157—24no—
62C4⋊Dic157D411yescharacteristic
63C4⋊Dic157C2311yescharacteristic
64C4⋊Dic157C2311yescharacteristic
65#14578933—22no—
66#14578933—21no—
67C22⋊Dic157—22no—
68#14578931—22no—
69#14578931—21no—
70#14578931D412yes—
71C4×Dic157—21no—
72C2×Q8×C157C2312yes—
73C2×D4×C157—21no—
74C2×D4×C157—21no—
75C2×D4×C157C2312yes—
76SD16×C157—24no—
77SD16×C157—42no—
78C2×C8×C157C2311yescharacteristic
79C2×C8×C157C2311yescharacteristic
80C4⋊C4×C157—21no—
81C4⋊C4×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—44no—
85C22×D157—44no—
86C22×D157—82no—
87C22×D157C2×D412yes—
88Dic314—21no—
89Dic314—44no—
90Dic314C2×D412yescomplemented
91D628—21no—
92D628SD1614yescomplemented
93D628C2×D412yes—
94C22⋊D157—44no—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C2×Dic157—21no—
98C2×Dic157—21no—
99C2×Dic157C2×D412yes—
100C4×D157—22no—
101C4×D157—22no—
102C4×D157—44no—
103C4×D157—42no—
104#11929434—22no—
105C23×C157—22no—
106Q8×C157—22no—
107Q8×C157—22no—
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—42no—
111D4×C157—81no—
112C2×C4×C157—22no—
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2411yescharacteristic
116C8×C157—21no—
117C8×C157C2×D412yescomplemented
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—21no—
130C4×C157—42no—
131C4×C157C2×SD1612yescomplemented
132C4×C157C22×D411yescharacteristic
133C4×C157H(5,𝔽2)11yescharacteristic
134Dic157—22no—
135Dic157—22no—
136Dic157—41no—
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#492D15711yescharacteristic, complemented
150#465—1572no—
151#461—1571no—
152#460—1571no—
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×SD16C2×D15712yescomplemented
165C2×SD16C2×D15712yescomplemented
166C22×C8—1572no—
167C4⋊Q8—3141no—
168C4⋊Q8C2×D15711yescharacteristic, complemented
169#127C2×D15711yescharacteristic
170C22⋊Q8—3142no—
171C22⋊Q8—1572no—
172#121—3141no—
173#121—1572no—
174#120—3142no—
175C4×Q8—1571no—
176C4×D4—3141no—
177C4×D4—1571no—
178C4×D4—1571no—
179#106—1571no—
180C4⋊C8—1571no—
181#103—1572no—
182#103—1571no—
183#102—1572no—
184#102—1571no—
185C22⋊C8—1572no—
186C4×C8C2×D15711yescharacteristic, complemented
187C24—3142no—
188C2×Q8—3142no—
189C2×Q8—3144no—
190C2×Q8—6282no—
191C2×Q8—1571no—
192C2×Q8C22×D15712yes—
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—42no—
208SD16—3144no—
209SD16—3144no—
210SD16—1574no—
211C2×C8—3144no—
212C2×C8—1572no—
213C2×C8C22×D15711yescharacteristic
214C2×C8C22×D15711yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—3142no—
218C4⋊C4—3141no—
219C4⋊C4—1571no—
220C4⋊C4—1571no—
221C4⋊C4—1571no—
222C22⋊C4—3142no—
223C22⋊C4—3142no—
224C22⋊C4—3141no—
225C22⋊C4—1572no—
226C42—3141no—
227C42C22×D15711yescharacteristic
228C23—22no—
229C23—3141no—
230C23—6284no—
231C23—6284no—
232C23—1572no—
233C23—12562no—
234Q8—22no—
235Q8—22no—
236Q8—3141no—
237Q8—6284no—
238Q8—1572no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3141no—
244D4—6284no—
245D4—6284no—
246D4—1574no—
247D4—1572no—
248C2×C4—22no—
249C2×C4—3141no—
250C2×C4—3142no—
251C2×C4—3142no—
252C2×C4—3141no—
253C2×C4—3141no—
254C2×C4—6284no—
255C2×C4—6282no—
256C2×C4—1572no—
257C2×C4D4×D15711yescharacteristic
258C2×C4D4×D15711yescharacteristic
259C2×C4C23×D15711yescharacteristic
260C8—21no—
261C8—3142no—
262C8D4×D15712yescomplemented
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—42no—
276C4—3142no—
277C4—3142no—
278C4—6281no—
279C4—6281no—
280C4SD16×D15712yescomplemented
281C4C2×D4×D15711yescharacteristic
282C4#1909781611yescharacteristic
283C2—82no—
284C2—3144no—
285C2—6282no—
286C2#2375790811yescharacteristic
287C2C2×SD16×D15711yescharacteristic
288C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#492×C157C2
socleC22×C157C22×D4
prime radical#23757908C2
hypercenter#492D157
hypocenterC157D4×SD16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D157C2semidirect
2. #23757938C2semidirect
4. #23757499C2semidirect
5. #23757475C2semidirect
6. #23757470C2semidirect
7. #23757422C2semidirect
8. #23757410C2semidirect
9. C4×#14578943C2semidirect
10. C8×D628C2semidirect
11. #492×C157C2semidirect
12. C2×Q8×D157C22semidirect
16. C2×#14578942C22semidirect
19. C2×C8×D157C22semidirect
22. #19097747C22semidirect
26. C4×Dic314C22semidirect
30. #19097703C22semidirect
31. C4⋊#11929434C22semidirect
32. #19097696C22semidirect
33. #19097695C22semidirect
35. #19097692C22semidirect
36. #19097683C22semidirect
37. C2×SD16×C157C22semidirect
38. C2×SD16×C157C22semidirect
39. C4⋊Q8×C157C22semidirect
41. C4×C8×C157C22semidirect
57. #14578943D4semidirect
90. Dic314C2×D4semidirect
92. D628SD16semidirect
117. C8×C157C2×D4semidirect
128. C2×D157C2×SD16semidirect
131. C4×C157C2×SD16semidirect
144. C157D4×SD16semidirect
149. #492D157semidirect
164. C2×SD16C2×D157semidirect
165. C2×SD16C2×D157semidirect
168. C4⋊Q8C2×D157semidirect
186. C4×C8C2×D157semidirect
262. C8D4×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
#237579382
#237579082
#237574992
#237574752
#237574702
#237574222
#237574102
C4×#145789432
C8×D6282
#492×C1572
D4×SD16157