Finite Group Databasebeta

← #34111913

Subgroups — #34111913

Order 20096.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#23758088C212yesmaximal, complemented
2#23758083C212yesmaximal, complemented
3#23757668C211yesmaximal, characteristic, complemented
4#23757547C212yesmaximal, complemented
5#23757534C212yesmaximal, complemented
6#23757457C212yesmaximal, complemented
7#23757450C211yesmaximal, characteristic, complemented
8#23757392C212yesmaximal, complemented
9#454×C157C211yesmaximal, characteristic, complemented
10#19097818C2212yes—
11#19097816C2211yescharacteristic
12#19097804C2212yes—
13#19097802—22no—
14#19097802C2212yes—
15#19097801—22no—
16#19097801C2212yescomplemented
17#19097800—22no—
18#19097795—22no—
19#19097791—22no—
20#19097791C2212yes—
21#19097790—22no—
22C2×#14578944—22no—
23C2×#14578944C2212yescomplemented
24C2×#14578942C2212yes—
25SD16×D157—22no—
26#19097779C2212yes—
27#19097778C2212yescomplemented
28#19097771C2211yescharacteristic, complemented
29#19097706C2212yescomplemented
30#19097699C2212yes—
31#19097689C2212yescomplemented
32#19097688C2211yescharacteristic
33#19097682C2211yescharacteristic, complemented
342−1+4×C157C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C157C2212yes—
36#124×C157C2211yescharacteristic
37#104×C157C2212yescomplemented
38#101×C157C2211yescharacteristic, complemented
39#14578954—22no—
40#14578954C2312yes—
41#14578953—22no—
42#14578953—22no—
43#14578953—42no—
44C2×Dic314—22no—
45C2×Dic314C2312yes—
46Q8×D157—22no—
47C2×C22⋊D157—41no—
48D4×D157—22no—
49Dic628—22no—
50#14578944—22no—
51#14578944—22no—
52#14578944—22no—
53#14578944—22no—
54#14578944D412yes—
55#14578943—22no—
56#14578942—22no—
57#14578942—22no—
58#14578942—22no—
59#14578942D412yes—
60#14578941—22no—
61#14578941D412yes—
62#14578939D412yes—
63#14578938—22no—
64#14578938C2312yes—
65#14578937—22no—
66C2×#11929434—22no—
67C2×#11929434C2312yes—
68C8×D157—22no—
69C4⋊Dic157—41no—
70#41×C157—22no—
71#41×C157—41no—
72#41×C157C2312yes—
73C2×Q8×C157—22no—
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157C2311yescharacteristic
76SD16×C157—22no—
77D8×C157—22no—
78M16×C157C2312yes—
79C22⋊C4×C157—41no—
80C42×C157C2311yescharacteristic
81C22×D157—41no—
82Dic314—22no—
83Dic314—22no—
84Dic314—42no—
85Dic314C2×D412yes—
86D628C2×D412yes—
87C22⋊D157—22no—
88C22⋊D157—42no—
89C22⋊D157—42no—
90C2×Dic157—22no—
91C2×Dic157—41no—
92C2×Dic157—42no—
93C4×D157—22no—
94C4×D157—42no—
95#11929434—22no—
96#11929434C2×D412yes—
97#11929434C2×D412yes—
98C23×C157—21no—
99Q8×C157—22no—
100Q8×C157—41no—
101Q8×C157C2×D412yes—
102D4×C157—22no—
103D4×C157—22no—
104D4×C157—41no—
105D4×C157C2×D412yes—
106C2×C4×C157—22no—
107C2×C4×C157—21no—
108C2×C4×C157—21no—
109C2×C4×C157—42no—
110C2×C4×C157C2411yescharacteristic
111C8×C157—22no—
112C22×C157—22no—
113C22×C157—41no—
114C22×C157—81no—
115C22×C157H(5,𝔽2)11yescharacteristic
116C2×D157—22no—
117C2×D157—161no—
118C4×C157—22no—
119C4×C157—41no—
120C4×C157—41no—
121C4×C157C22×D412yes—
122Dic157—22no—
123Dic157—42no—
124C2×C157—21no—
125C2×C157—42no—
126C2×C157—81no—
127C2×C157D4211yescharacteristic
128D157—42no—
129C157#328911yescharacteristic, complemented, Sylow
130#3289—1571nomaximal, Sylow
131#578—1572no—
132#577—1572no—
133#497—1571no—
134#473—1572no—
135#471—1572no—
136#456—1572no—
137#455—1571no—
138#454D15711yescharacteristic, complemented
139#444—1572no—
1402−1+4—1572no—
1412−1+4C2×D15711yescharacteristic, complemented
142H(5,𝔽2)—1571no—
143#137—3142no—
144#137—3142no—
145#137—3142no—
146#137—1572no—
147#137—1572no—
148#137—1572no—
149AGL(1,ℤ/8)—3142no—
150AGL(1,ℤ/8)C2×D15712yes—
151#135—3142no—
152#135—3142no—
153#135—1572no—
154C2×Q16—3142no—
155C2×Q16—1572no—
156C2×SD16—3142no—
157C2×SD16—1572no—
158#131—1572no—
159#131—1572no—
160C4⋊Q8—1571no—
161#124C2×D15711yescharacteristic
162#108—1572no—
163#104—1572no—
164#104C2×D15712yescomplemented
165#101—1572no—
166#101C2×D15711yescharacteristic, complemented
167#100—1571no—
168#97—1571no—
169#41—22no—
170#41—41no—
171#41—3142no—
172#41—3142no—
173#41—3142no—
174#41—6282no—
175#41—1572no—
176#41C22×D15712yes—
177C2×Q8—22no—
178C2×Q8—3142no—
179C2×Q8—3142no—
180C2×Q8—1572no—
181C2×Q8C22×D15711yescharacteristic
182C2×D4—3142no—
183C2×D4—6281no—
184C2×D4C22×D15711yescharacteristic
185Q16—3142no—
186Q16—3142no—
187Q16—3142no—
188Q16—3142no—
189Q16—3142no—
190Q16—1572no—
191SD16—22no—
192SD16—3142no—
193SD16—3142no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—1572no—
198SD16—1572no—
199D8—22no—
200D8—1572no—
201M16—3142no—
202M16—3142no—
203M16—1572no—
204M16C22×D15712yes—
205C2×C8—3142no—
206C2×C8—3142no—
207C2×C8—1572no—
208C4⋊C4—6281no—
209C22⋊C4—41no—
210C42C22×D15711yescharacteristic
211C23—21no—
212C23—6281no—
213Q8—22no—
214Q8—41no—
215Q8—3142no—
216Q8—3142no—
217Q8—6282no—
218Q8—1572no—
219Q8D4×D15712yes—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—3142no—
224D4—6282no—
225D4—6282no—
226D4—1572no—
227D4D4×D15712yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—3142no—
233C2×C4—3142no—
234C2×C4—6282no—
235C2×C4—6281no—
236C2×C4—6282no—
237C2×C4C23×D15711yescharacteristic
238C8—22no—
239C8—3142no—
240C8—1572no—
241C8—1572no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—3142no—
246C22—25121no—
247C22#1909781611yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—3142no—
252C4—6282no—
253C4C2×D4×D15712yes—
254C2—21no—
255C2—42no—
256C2—81no—
257C2—6282no—
258C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#454×C157C2
socleC2×C157D42
hypercenter#454D157
hypocenterC157#3289
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23758088C2semidirect
2. #23758083C2semidirect
3. #23757668C2semidirect
4. #23757547C2semidirect
5. #23757534C2semidirect
6. #23757457C2semidirect
7. #23757450C2semidirect
8. #23757392C2semidirect
9. #454×C157C2semidirect
16. #19097801C22semidirect
23. C2×#14578944C22semidirect
27. #19097778C22semidirect
28. #19097771C22semidirect
29. #19097706C22semidirect
31. #19097689C22semidirect
33. #19097682C22semidirect
34. 2−1+4×C157C22semidirect
37. #104×C157C22semidirect
38. #101×C157C22semidirect
129. C157#3289semidirect
138. #454D157semidirect
141. 2−1+4C2×D157semidirect
164. #104C2×D157semidirect
166. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580882
#237580832
#237576682
#237575472
#237575342
#237574572
#237574502
#237573922
#454×C1572
#3289157