Finite Group Databasebeta

← #34109932

Subgroups — #34109932

Order 20096.

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

normal subgroups only

This group has 440 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
1#23757826C211yesmaximal, characteristic, complemented
2C2×#19097733C211yesmaximal, characteristic
3#23757576C212yesmaximal, complemented
4#23757572C212yesmaximal, complemented
5#23757562C212yesmaximal, complemented
6#23757552C212yesmaximal, complemented
7#23757313C211yesmaximal, characteristic, complemented
8#23757296C211yesmaximal, characteristic, complemented
9#23757279C211yesmaximal, characteristic
10#23757277C211yesmaximal, characteristic
11#428×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578938C2211yescharacteristic
15#19097770C2211yescharacteristic, complemented
16#19097770C2211yescharacteristic, complemented
17#19097762C2211yescharacteristic, complemented
18#19097762C2211yescharacteristic, complemented
19#19097748—21no—
20#19097748—21no—
21C4×Dic314—21no—
22C4×Dic314—21no—
23#19097733—21no—
24#19097733—21no—
25#19097733C2212yes—
26#19097733C2212yes—
27#19097730C2211yescharacteristic
28C2×#14578931—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097704C2212yescomplemented
31#19097702C2212yescomplemented
32#19097700C2212yescomplemented
33#19097700C2212yescomplemented
34#19097693C2212yes—
35#19097693C2212yes—
36#19097691C2212yes—
37#19097691C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157—21no—
44C2×Dic314—21no—
45C2×Dic314—21no—
46C2×D628—21no—
47C2×D628—22no—
48C2×D628Q814yescomplemented
49C2×D628C2312yes—
50C22×Dic157D411yescharacteristic
51C22×Dic157D411yescharacteristic
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157—44no—
55#14578938—22no—
56C2×#11929434C2312yes—
57C4⋊Dic157—21no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2311yescharacteristic
60C4⋊Dic157C2311yescharacteristic
61#14578933—21no—
62#14578933—21no—
63#14578933—21no—
64#14578933—21no—
65C22⋊Dic157—21no—
66#14578931—22no—
67#14578931—22no—
68C4×Dic157—21no—
69C4×Dic157D411yescharacteristic
70C4×Dic157D411yescharacteristic
71C4×Dic157C2311yescharacteristic
72C4×Dic157C2311yescharacteristic
73C22×C4×C157—21no—
74C22×C4×C157C2311yescharacteristic
75M16×C157—22no—
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2311yescharacteristic
79C4⋊C4×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C42×C157—21no—
83C22×D157—24no—
84C22×D157—22no—
85C22×D157—42no—
86C22×D157—42no—
87Dic314—42no—
88Dic314—42no—
89D628—24no—
90D628—22no—
91D628C2×Q814yes—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—41no—
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C4×D157—44no—
101C4×D157—44no—
102#11929434—22no—
103C23×C157C2×D411yescharacteristic
104C2×C4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—21no—
107C2×C4×C157—42no—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×Q812yes—
110C2×C4×C157#4112yes—
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157C22⋊Q812yes—
116C22×C157C22×D411yescharacteristic
117C2×D157—24no—
118C2×D157—48no—
119C2×D157—42no—
120C2×D157—42no—
121C4×C157—42no—
122C4×C157—41no—
123C4×C157—41no—
124C4×C157C22⋊Q812yes—
125C4×C157C22×Q811yescharacteristic
126C4×C157C2×#4111yescharacteristic
127Dic157—21no—
128Dic157—21no—
129Dic157—21no—
130Dic157—41no—
131Dic157—41no—
132Dic157AGL(1,ℤ/8)12yes—
133C2×C157—22no—
134C2×C157C2×C22⋊Q811yescharacteristic
135C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C157#57511yescharacteristic
137D157—44no—
138C157#307711yescharacteristic, complemented, Sylow
139#3077—1571nomaximal, Sylow
140#533—1571no—
141C2×C4×D4—1571no—
142#478—1572no—
143#478—1572no—
144#476—1572no—
145#474—1572no—
146#428—1571no—
147#428D15711yescharacteristic, complemented
148#423—1571no—
149#418—1571no—
150#418—1571no—
151C22×D4—1571no—
152C23×C4—3141no—
153C2×M16—1571no—
154C2×M16C2×D15711yescharacteristic
155C4⋊Q8—1571no—
156C4⋊Q8—1571no—
157#125—1571no—
158#125—1571no—
159C22⋊Q8—3141no—
160C22⋊Q8—3141no—
161C4×Q8—3141no—
162C4×Q8—3141no—
163C4×D4—3141no—
164C4×D4—3141no—
165C4×D4—1572no—
166C4×D4—1572no—
167#117—1571no—
168#117C2×D15711yescharacteristic
169C2×C4⋊C4C2×D15711yescharacteristic
170C2×C22⋊C4—3141no—
171C2×C42—1571no—
172#107—1572no—
173#107C2×D15712yescomplemented
174#106—1572no—
175#106C2×D15712yescomplemented
176C4⋊C8—1572no—
177C4⋊C8—1572no—
178#102—1572no—
179#102—1572no—
180#102—1572no—
181#102—1572no—
182C24—3141no—
183C2×Q8—3141no—
184C2×Q8—3141no—
185C2×D4—3141no—
186C2×D4—3142no—
187C2×D4—1574no—
188C2×D4—1572no—
189C22×C4—21no—
190C22×C4—3142no—
191C22×C4—3142no—
192C22×C4—6284no—
193C22×C4—1571no—
194C22×C4—1571no—
195C22×C4C22×D15711yescharacteristic
196M16—22no—
197M16—3142no—
198C2×C8—1572no—
199C2×C8C22×D15712yes—
200C4⋊C4—21no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3141no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—1571no—
208C4⋊C4—1571no—
209C4⋊C4C22×D15711yescharacteristic
210C4⋊C4C22×D15711yescharacteristic
211C4⋊C4C22×D15712yes—
212C22⋊C4—21no—
213C22⋊C4—3142no—
214C22⋊C4—3142no—
215C22⋊C4—3141no—
216C42—21no—
217C42—3141no—
218C42—1571no—
219C42—1571no—
220C42—1571no—
221C42—1571no—
222C23—3144no—
223C23—3142no—
224C23—6282no—
225C23—6282no—
226C23D4×D15711yescharacteristic
227Q8—6282no—
228Q8—6282no—
229D4—3144no—
230D4—3142no—
231D4—1574no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—3142no—
237C2×C4—3141no—
238C2×C4—3141no—
239C2×C4—6284no—
240C2×C4—6284no—
241C2×C4—6281no—
242C2×C4—1571no—
243C2×C4—1571no—
244C2×C4—1571no—
245C2×C4—1571no—
246C2×C4D4×D15711yescharacteristic
247C2×C4Q8×D15712yes—
248C2×C4#1457895412yes—
249C2×C4C23×D15711yescharacteristic
250C8—22no—
251C8—3142no—
252C22—21no—
253C22—21no—
254C22—3144no—
255C22—6288no—
256C22—6282no—
257C22—6282no—
258C22#1909774912yes—
259C22C2×D4×D15711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—3141no—
264C4—3141no—
265C4—3141no—
266C4—6281no—
267C4—6281no—
268C4—1572no—
269C4#1909774912yes—
270C4C2×Q8×D15711yescharacteristic
271C4C2×#1457895411yescharacteristic
272C2—22no—
273C2—6284no—
274C2C2×#1909774911yescharacteristic
275C2AGL(1,ℤ/8)×D15711yescharacteristic
276C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC22×D628C22
hypercenter#428D157
hypocenterC157#3077
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #23757826C2semidirect
3. #23757576C2semidirect
4. #23757572C2semidirect
5. #23757562C2semidirect
6. #23757552C2semidirect
7. #23757313C2semidirect
8. #23757296C2semidirect
11. #428×C157C2semidirect
15. #19097770C22semidirect
16. #19097770C22semidirect
17. #19097762C22semidirect
18. #19097762C22semidirect
30. #19097704C22semidirect
31. #19097702C22semidirect
32. #19097700C22semidirect
33. #19097700C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
48. C2×D628Q8semidirect
138. C157#3077semidirect
147. #428D157semidirect
173. #107C2×D157semidirect
175. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578262
C2×#190977332
#237575762
#237575722
#237575622
#237575522
#237573132
#237572962
#237572792
#237572772
#428×C1572
#3077157