Finite Group Databasebeta

← #34111954

Subgroups — #34111954

Order 20096.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D628C211yesmaximal, characteristic
2#23757621C211yesmaximal, characteristic, complemented
3#23757591C212yesmaximal, complemented
4#23757560C211yesmaximal, characteristic, complemented
5#23757504C212yesmaximal, complemented
6#23757473C211yesmaximal, characteristic, complemented
7#23757423C212yesmaximal, complemented
8C4×#14578941C211yesmaximal, characteristic, complemented
9D4×#11929434C211yesmaximal, characteristic, complemented
10C2×#19097691C212yesmaximal, complemented
11#554×C157C211yesmaximal, characteristic, complemented
12C22×D628C2212yes—
13C2×D4×D157—21no—
14C2×#14578941C2211yescharacteristic
15C2×#14578941C2212yescomplemented
16C22×#11929434C2212yescomplemented
17#19097768C2211yescharacteristic
18#19097746—21no—
19#19097745C2212yes—
20#19097741—22no—
21C4×D628C2211yescharacteristic
22#19097702C2211yescharacteristic, complemented
23#19097702C2212yescomplemented
24C4⋊#11929434C2211yescharacteristic, complemented
25#19097695C2211yescharacteristic, complemented
26#19097691C2211yescharacteristic
27#19097691C2212yes—
28#19097691C2212yescomplemented
29#19097691C2212yescomplemented
30#19097691C2212yescomplemented
31C22⋊#11929434C2212yescomplemented
32C4×#11929434C2211yescharacteristic, complemented
33C4⋊Q8×C157C2211yescharacteristic, complemented
34#125×C157—21no—
35C22⋊Q8×C157—22no—
36C22⋊Q8×C157C2212yescomplemented
37C4×Q8×C157C2211yescharacteristic, complemented
38C4×D4×C157—21no—
39C4×D4×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2212yescomplemented
41C23×D157—22no—
42C2×D628—24no—
43C2×D628—22no—
44C2×D628—21no—
45C2×D628—21no—
46C2×D628—42no—
47C2×D628C2312yes—
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—22no—
50D4×D157—44no—
51C2×C4×D157—21no—
52#14578941—22no—
53#14578941—42no—
54C2×#11929434—42no—
55C2×#11929434C2311yescharacteristic
56C2×#11929434C2311yescharacteristic
57C2×#11929434C2312yes—
58C4⋊Dic157C2311yescharacteristic
59#14578931—21no—
60#14578931—22no—
61C2×Q8×C157—21no—
62C2×Q8×C157C2311yescharacteristic
63C2×D4×C157D411yescharacteristic
64C22×C4×C157—22no—
65C22×C4×C157C2312yes—
66C4⋊C4×C157—22no—
67C4⋊C4×C157—21no—
68C4⋊C4×C157—21no—
69C4⋊C4×C157—22no—
70C4⋊C4×C157D411yescharacteristic
71C4⋊C4×C157C2311yescharacteristic
72C4⋊C4×C157C2312yes—
73C22⋊C4×C157—22no—
74C22⋊C4×C157D412yes—
75C42×C157—21no—
76C42×C157C2311yescharacteristic
77C22×D157—21no—
78C22×D157—22no—
79C22×D157—44no—
80C22×D157—44no—
81C22×D157—82no—
82D628—22no—
83D628—21no—
84D628—22no—
85D628—44no—
86D628—42no—
87D628—81no—
88D628#4112yes—
89C22⋊D157—44no—
90C2×Dic157—21no—
91C4×D157—42no—
92#11929434—21no—
93#11929434—21no—
94#11929434—22no—
95C23×C157C2×D412yes—
96Q8×C157—21no—
97Q8×C157—21no—
98Q8×C157—81no—
99D4×C157SD1614yescomplemented
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—21no—
103C2×C4×C157—21no—
104C2×C4×C157—22no—
105C2×C4×C157—82no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2411yescharacteristic
110C22×C157—22no—
111C22×C157—22no—
112C22×C157C2×SD1614yes—
113C22×C157C22×D411yescharacteristic
114C2×D157—22no—
115C2×D157—41no—
116C2×D157—41no—
117C2×D157—42no—
118C2×D157—42no—
119C2×D157—42no—
120C2×D157—88no—
121C4×C157—22no—
122C4×C157—21no—
123C4×C157—41no—
124C4×C157—41no—
125C4×C157—42no—
126C4×C157C2×SD1612yes—
127C4×C157C2×#4111yescharacteristic
128C4×C157H(5,𝔽2)11yescharacteristic
129Dic157—41no—
130C2×C157—24no—
131C2×C157#54611yescharacteristic
132C2×C157C22×SD1611yescharacteristic
133C2×C157#57611yescharacteristic
134D157—42no—
135D157—82no—
136C157#329511yescharacteristic, complemented, Sylow
137#3295—1571nomaximal, Sylow
138#554D15711yescharacteristic, complemented
139D42—1571no—
140#486—1571no—
141#481—1572no—
142#476—1571no—
143#465—1572no—
144#460—1571no—
145#450—1572no—
146C4×SD16—1571no—
147C8×D4—1571no—
148C2×#102—1572no—
149C22×D4—3141no—
150C22×D4—1572no—
151C2×SD16—1571no—
152C2×SD16—1572no—
153C22×C8—1572no—
154C4⋊Q8C2×D15711yescharacteristic, complemented
155#127—1571no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D15712yescomplemented
159#121—3141no—
160#121—1572no—
161#120—3142no—
162C4×Q8C2×D15711yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—1571no—
165C4×D4C2×D15711yescharacteristic
166C2×C4⋊C4C2×D15712yescomplemented
167#106—1571no—
168#106—1572no—
169C4⋊C8—1571no—
170#103—1571no—
171#102—1571no—
172#102—1572no—
173#102—1572no—
174#102—1572no—
175#102—1572no—
176C22⋊C8—1572no—
177C4×C8—1571no—
178C24—3142no—
179C2×Q8—21no—
180C2×Q8C22×D15711yescharacteristic
181C2×D4—3144no—
182C2×D4—3142no—
183C2×D4—3142no—
184C2×D4—3141no—
185C2×D4—3141no—
186C2×D4—6282no—
187C2×D4—6284no—
188C2×D4—1572no—
189C2×D4—1571no—
190C2×D4C22⋊D15711yescharacteristic
191C22×C4—22no—
192C22×C4—3141no—
193C22×C4C22×D15712yes—
194SD16—3142no—
195SD16—6282no—
196C2×C8—6282no—
197C2×C8—1571no—
198C2×C8—1571no—
199C2×C8—1572no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—1571no—
205C4⋊C4C22⋊D15711yescharacteristic
206C4⋊C4C22×D15711yescharacteristic
207C4⋊C4C22×D15712yes—
208C22⋊C4—22no—
209C22⋊C4—3141no—
210C22⋊C4—3142no—
211C22⋊C4C22⋊D15712yes—
212C42—21no—
213C42C22×D15711yescharacteristic
214C23—3141no—
215C23—3142no—
216C23—6284no—
217C23—6284no—
218C23—12562no—
219C23C2×C22⋊D15712yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—3142no—
224D4—3141no—
225D4—3142no—
226D4—6284no—
227D4—6284no—
228D4—6282no—
229D4—1572no—
230D4—12561no—
231D4#1457894114yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—3141no—
239C2×C4—6282no—
240C2×C4C2×C22⋊D15711yescharacteristic
241C2×C4C2×C22⋊D15712yes—
242C2×C4C2×C22⋊D15711yescharacteristic
243C2×C4C23×D15711yescharacteristic
244C8—3141no—
245C8—3141no—
246C8—3142no—
247C22—22no—
248C22—22no—
249C22—3142no—
250C22—6281no—
251C22—6281no—
252C22—6282no—
253C22—6282no—
254C22—6282no—
255C22—12568no—
256C22C2×#1457894114yes—
257C22C22×C22⋊D15711yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—6281no—
264C4C2×#1457894112yes—
265C4#41×D15711yescharacteristic
266C4#1909781711yescharacteristic
267C2—24no—
268C2—6282no—
269C2—12562no—
270C2#2375791111yescharacteristic
271C2C22×#1457894111yescharacteristic
272C2#2375807611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#554×C157C2
socleC22×C157C22×D4
prime radicalD4×D628C2
hypercenter#554D157
hypocenterC157#3295
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #23757621C2semidirect
3. #23757591C2semidirect
4. #23757560C2semidirect
5. #23757504C2semidirect
6. #23757473C2semidirect
7. #23757423C2semidirect
8. C4×#14578941C2semidirect
9. D4×#11929434C2semidirect
10. C2×#19097691C2semidirect
11. #554×C157C2semidirect
15. C2×#14578941C22semidirect
16. C22×#11929434C22semidirect
22. #19097702C22semidirect
23. #19097702C22semidirect
24. C4⋊#11929434C22semidirect
25. #19097695C22semidirect
28. #19097691C22semidirect
29. #19097691C22semidirect
30. #19097691C22semidirect
31. C22⋊#11929434C22semidirect
32. C4×#11929434C22semidirect
33. C4⋊Q8×C157C22semidirect
36. C22⋊Q8×C157C22semidirect
37. C4×Q8×C157C22semidirect
40. C2×C4⋊C4×C157C22semidirect
99. D4×C157SD16semidirect
136. C157#3295semidirect
138. #554D157semidirect
154. C4⋊Q8C2×D157semidirect
158. C22⋊Q8C2×D157semidirect
162. C4×Q8C2×D157semidirect
166. C2×C4⋊C4C2×D157semidirect
231. D4#14578941semidirect

Maximal subgroups

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

subgroupindex
D4×D6282
#237576212
#237575912
#237575602
#237575042
#237574732
#237574232
C4×#145789412
D4×#119294342
C2×#190976912
#554×C1572
#3295157