Finite Group Databasebeta

← #34110276

Subgroups — #34110276

Order 20096.

450 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#23757807C211yesmaximal, characteristic, complemented
2C2×C4×D628C211yesmaximal, characteristic, complemented
3#23757604C211yesmaximal, characteristic, complemented
4#23757587C211yesmaximal, characteristic, complemented
5#23757574C212yesmaximal, complemented
6#23757489C212yesmaximal, complemented
7#23757426C211yesmaximal, characteristic, complemented
8#23757409C211yesmaximal, characteristic
9#23757386C212yesmaximal, complemented
10#23757375C212yesmaximal, complemented
11#432×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578943C2212yes—
15C2×D1256C2212yes—
16#19097765—21no—
17#19097763C2211yescharacteristic, complemented
18#19097759C2211yescharacteristic, complemented
19#19097754—21no—
20#19097750C2211yescharacteristic
21#19097745C2211yescharacteristic
22C4×Dic314C2211yescharacteristic, complemented
23C4×D628—21no—
24C4×D628—21no—
25C4×D628C2212yes—
26C4×D628C2212yes—
27C4×D628C2211yescharacteristic, complemented
28C4×C22⋊D157—21no—
29#19097728—21no—
30C2×C4⋊Dic157C2211yescharacteristic
31C2×#14578931—21no—
32#19097705C2212yescomplemented
33#19097703C2212yescomplemented
34#19097696C2212yescomplemented
35#19097693C2212yes—
36#19097693C2212yes—
37#19097693C2212yescomplemented
38C2×C42×C157C2211yescharacteristic
39C4⋊C8×C157C2212yescomplemented
40C22⋊C8×C157C2211yescharacteristic
41C22⋊C8×C157C2211yescharacteristic
42#97×C157C2212yescomplemented
43C23×D157—21no—
44C2×Dic314C2311yescharacteristic
45C2×D628—24no—
46C2×D628—21no—
47C2×D628C2311yescharacteristic
48C2×D628C2312yes—
49C2×C22⋊D157—21no—
50C22×Dic157—21no—
51C2×C4×D157—21no—
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157—44no—
55#14578943—24no—
56D1256—24no—
57C4⋊Dic157—21no—
58C4⋊Dic157C2311yescharacteristic
59C4⋊Dic157C2311yescharacteristic
60C4⋊Dic157C2312yes—
61#14578933—21no—
62#14578933—21no—
63#14578933—21no—
64#14578933—21no—
65C22⋊Dic157—21no—
66#14578931—22no—
67#14578931—22no—
68#14578931—21no—
69#14578931—21no—
70#14578931—21no—
71#14578931—21no—
72C4×Dic157—21no—
73C22×C4×C157D411yescharacteristic
74C22×C4×C157D411yescharacteristic
75C22×C4×C157C2311yescharacteristic
76C2×C8×C157C2312yes—
77C2×C8×C157C2312yes—
78C42×C157D411yescharacteristic
79C42×C157D411yescharacteristic
80C42×C157C2311yescharacteristic
81C42×C157C2311yescharacteristic
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—21no—
85C22×D157—42no—
86C22×D157—42no—
87Dic314—22no—
88D628—24no—
89D628—22no—
90D628—22no—
91D628#4114yes—
92C22⋊D157—42no—
93C2×Dic157—22no—
94C2×Dic157—21no—
95C2×Dic157—21no—
96C2×Dic157—21no—
97C2×Dic157—42no—
98C4×D157—42no—
99C4×D157—44no—
100C4×D157—44no—
101C23×C157C2×D411yescharacteristic
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—41no—
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2411yescharacteristic
111C8×C157—22no—
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157AGL(1,ℤ/8)12yes—
116C22×C157C22×D411yescharacteristic
117C2×D157—24no—
118C2×D157—48no—
119C2×D157—42no—
120C2×D157—42no—
121C2×D157—41no—
122C2×D157—41no—
123C2×D157—41no—
124C4×C157—21no—
125C4×C157—21no—
126C4×C157—21no—
127C4×C157—21no—
128C4×C157—21no—
129C4×C157C2×#4112yes—
130Dic157—41no—
131Dic157—42no—
132Dic157—41no—
133Dic157—41no—
134C2×C157—22no—
135C2×C157#52511yescharacteristic
136C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C157#57511yescharacteristic
138D157—44no—
139D157—81no—
140C157#311411yescharacteristic, complemented, Sylow
141#3114—1571nomaximal, Sylow
142#529—1571no—
143C2×C4×D4—1571no—
144#483—1571no—
145#480—1571no—
146#478—1572no—
147#463—1572no—
148#450—1571no—
149#447—1571no—
150#442—1572no—
151#440—1572no—
152#432D15711yescharacteristic, complemented
153C22×D4—1571no—
154C23×C4—3141no—
155C2×SD16—1572no—
156C2×D8—1572no—
157#126—3141no—
158#125—1571no—
159#124—1571no—
160#123—3141no—
161C22⋊Q8—1571no—
162#121—1571no—
163C4×Q8—1571no—
164C4×D4—3141no—
165C4×D4—3141no—
166C4×D4—3141no—
167C4×D4—1572no—
168C4×D4—1572no—
169C4×D4—1571no—
170#117—3141no—
171C2×C4⋊C4—1571no—
172C2×C22⋊C4—3141no—
173C2×C42C2×D15711yescharacteristic
174#107—1572no—
175#106—1572no—
176C4⋊C8C2×D15712yescomplemented
177#103—1572no—
178#102—1572no—
179#102—1572no—
180#102—1572no—
181C22⋊C8C2×D15711yescharacteristic
182C22⋊C8C2×D15711yescharacteristic
183#97C2×D15712yescomplemented
184C24—3141no—
185C2×Q8—1571no—
186C2×D4—3144no—
187C2×D4—3141no—
188C2×D4—3141no—
189C2×D4—1571no—
190C2×D4—1572no—
191C22×C4—3141no—
192C22×C4—3141no—
193C22×C4—3142no—
194C22×C4—3142no—
195C22×C4—6284no—
196C22×C4D62811yescharacteristic
197C22×C4D62811yescharacteristic
198C22×C4C22×D15711yescharacteristic
199SD16—3144no—
200D8—3144no—
201C2×C8C22×D15712yes—
202C2×C8C22×D15712yes—
203C4⋊C4—3141no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—1571no—
209C4⋊C4—1571no—
210C4⋊C4—1572no—
211C22⋊C4—3141no—
212C22⋊C4—3142no—
213C22⋊C4—3142no—
214C22⋊C4—3141no—
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C42—3141no—
219C42D62811yescharacteristic
220C42D62811yescharacteristic
221C42C22×D15711yescharacteristic
222C42C22×D15711yescharacteristic
223C23—3144no—
224C23—3142no—
225C23—3141no—
226C23—6282no—
227C23—6282no—
228C23C2×D62811yescharacteristic
229Q8—3142no—
230D4—3144no—
231D4—3142no—
232D4—3142no—
233D4—6282no—
234D4—1574no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—3141no—
239C2×C4—3141no—
240C2×C4—3141no—
241C2×C4—3142no—
242C2×C4—6284no—
243C2×C4—6284no—
244C2×C4—6282no—
245C2×C4—6282no—
246C2×C4C2×D62811yescharacteristic
247C2×C4C2×D62811yescharacteristic
248C2×C4C2×D62811yescharacteristic
249C2×C4C2×D62811yescharacteristic
250C2×C4C2×D62811yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—22no—
253C8—22no—
254C22—21no—
255C22—21no—
256C22—3144no—
257C22—6288no—
258C22—6282no—
259C22—6282no—
260C22—6281no—
261C22—6281no—
262C22—6281no—
263C22#1909779912yes—
264C22C22×D62811yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—6281no—
271C4—6282no—
272C4—6281no—
273C4—6281no—
274C4#41×D15712yes—
275C2—22no—
276C2—6284no—
277C2—12561no—
278C2#2375778511yescharacteristic
279C2C2×#1909779911yescharacteristic
280C2#2375807311yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#432×C157C2
socleC22×C157C22×D4
prime radical#23757409C2
hypercenter#432D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #23757807C2semidirect
2. C2×C4×D628C2semidirect
3. #23757604C2semidirect
4. #23757587C2semidirect
5. #23757574C2semidirect
6. #23757489C2semidirect
7. #23757426C2semidirect
9. #23757386C2semidirect
10. #23757375C2semidirect
11. #432×C157C2semidirect
17. #19097763C22semidirect
18. #19097759C22semidirect
22. C4×Dic314C22semidirect
27. C4×D628C22semidirect
32. #19097705C22semidirect
33. #19097703C22semidirect
34. #19097696C22semidirect
37. #19097693C22semidirect
39. C4⋊C8×C157C22semidirect
42. #97×C157C22semidirect
140. C157#3114semidirect
152. #432D157semidirect
176. C4⋊C8C2×D157semidirect
183. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578072
C2×C4×D6282
#237576042
#237575872
#237575742
#237574892
#237574262
#237574092
#237573862
#237573752
#432×C1572
#3114157