Finite Group Databasebeta

← #34111746

Subgroups — #34111746

Order 20096.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#19097795C212yesmaximal, complemented
2#23757918C212yesmaximal, complemented
3#23757657C211yesmaximal, characteristic, complemented
4#23757520C212yesmaximal, complemented
5#23757488C212yesmaximal, complemented
6#23757436C212yesmaximal, complemented
7#23757408C212yesmaximal, complemented
8#23757352C211yesmaximal, characteristic, complemented
9C4×D8×C157C211yesmaximal, characteristic, complemented
10C2×#14578953C2212yes—
11C2×D4×D157C2212yescomplemented
12#19097795—28no—
13C2×Dic628C2211yescharacteristic, complemented
14C2×#14578943C2212yescomplemented
15C2×#14578942C2212yescomplemented
16C2×D1256C2211yescharacteristic
17C2×#14578939C2212yes—
18C2×#14578937C2212yescomplemented
19#19097759C2212yescomplemented
20#19097755—22no—
21#19097751C2212yescomplemented
22#19097746C2212yes—
23#19097745—22no—
24#19097741—22no—
25C4×D628C2211yescharacteristic
26C2×#14578931—22no—
27C4⋊#11929434C2211yescharacteristic, complemented
28#19097694C2212yescomplemented
29#19097691C2212yescomplemented
30#19097683C2212yescomplemented
31C2×D8×C157C2211yescharacteristic
32C4×D4×C157C2212yescomplemented
33#107×C157C2211yescharacteristic, complemented
34#102×C157C2212yescomplemented
35C4×C8×C157C2211yescharacteristic, complemented
36C23×D157—22no—
37#14578953—24no—
38#14578953—42no—
39C2×Dic314C2312yes—
40C2×D628—22no—
41C2×D628D411yescharacteristic
42C2×D628C2312yes—
43C2×C22⋊D157—22no—
44C2×C22⋊D157—22no—
45D4×D157—24no—
46D4×D157—42no—
47C22×Dic157—22no—
48C2×C4×D157C2312yes—
49Dic628—41no—
50#14578943—24no—
51#14578942—24no—
52D1256—41no—
53#14578939—24no—
54#14578937—24no—
55C2×#11929434C2312yes—
56C4⋊Dic157—22no—
57C4⋊Dic157D411yescharacteristic
58#14578931—22no—
59#14578931—22no—
60#14578931—22no—
61#14578931—22no—
62#14578931D412yes—
63C2×D4×C157C2312yes—
64C22×C4×C157—22no—
65D8×C157D414yescomplemented
66C2×C8×C157C2311yescharacteristic
67C2×C8×C157C2311yescharacteristic
68C4⋊C4×C157C2312yes—
69C22⋊C4×C157—22no—
70C42×C157C2311yescharacteristic
71C22×D157—22no—
72C22×D157—44no—
73C22×D157—44no—
74C22×D157—82no—
75C22×D157C2×D412yes—
76Dic314—22no—
77Dic314—22no—
78D628—22no—
79D628—22no—
80D628—41no—
81D628—82no—
82C22⋊D157—44no—
83C22⋊D157—44no—
84C2×Dic157—22no—
85C2×Dic157—44no—
86C2×Dic157C2×D412yes—
87C4×D157—22no—
88C4×D157—22no—
89#11929434—22no—
90C23×C157—22no—
91D4×C157—22no—
92D4×C157C2×D414yes—
93C2×C4×C157—22no—
94C2×C4×C157—82no—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2411yescharacteristic
98C8×C157—21no—
99C8×C157C2×D412yes—
100C22×C157—24no—
101C22×C157—42no—
102C22×C157—42no—
103C22×C157C22×D411yescharacteristic
104C2×D157—22no—
105C2×D157—22no—
106C2×D157—22no—
107C2×D157—42no—
108C2×D157—42no—
109C2×D157—42no—
110C2×D157—88no—
111C4×C157—21no—
112C4×C157—21no—
113C4×C157—42no—
114C4×C157C22×D411yescharacteristic
115C4×C157H(5,𝔽2)11yescharacteristic
116Dic157—22no—
117Dic157—42no—
118C2×C157—44no—
119C2×C157D4211yescharacteristic
120C2×C157#57511yescharacteristic
121C2×C157#57611yescharacteristic
122D157—42no—
123D157—82no—
124C157#327311yescharacteristic, complemented, Sylow
125#3273—1571nomaximal, Sylow
126C2×AGL(1,ℤ/8)—1572no—
127#546—1572no—
128#495—1571no—
129#468—1572no—
130#463—1572no—
131#452—1572no—
132#447—1572no—
133C4×D8D15711yescharacteristic, complemented
134#436—1571no—
135C2×#41—1572no—
136C22×D4—1572no—
137AGL(1,ℤ/8)—3148no—
138C2×Q16—1571no—
139C2×SD16—1572no—
140C2×SD16—1572no—
141C2×D8—1572no—
142C2×D8—1571no—
143C2×D8C2×D15711yescharacteristic
144C2×M16—1572no—
145#124—1572no—
146#123—3142no—
147C22⋊Q8—1572no—
148#121—3142no—
149#121—1572no—
150#120—3142no—
151C4×D4—1571no—
152C4×D4C2×D15712yescomplemented
153C2×C22⋊C4—3142no—
154#107C2×D15711yescharacteristic, complemented
155C4⋊C8—1571no—
156#103—1572no—
157#102—1572no—
158#102C2×D15712yescomplemented
159C22⋊C8—1572no—
160C4×C8C2×D15711yescharacteristic, complemented
161C24—3142no—
162#41—3144no—
163#41—6282no—
164C2×Q8—1572no—
165C2×D4—3142no—
166C2×D4—3144no—
167C2×D4—3142no—
168C2×D4—3142no—
169C2×D4—6282no—
170C2×D4—1571no—
171C2×D4—1572no—
172C2×D4C22×D15712yes—
173C22×C4—22no—
174C22×C4—3142no—
175C22×C4—1572no—
176Q16—6281no—
177SD16—3144no—
178SD16—3144no—
179D8—3144no—
180D8—6281no—
181D8D62814yescomplemented
182M16—3144no—
183C2×C8—1572no—
184C2×C8C22×D15711yescharacteristic
185C2×C8C22×D15711yescharacteristic
186C4⋊C4—3142no—
187C4⋊C4—1571no—
188C4⋊C4C22×D15712yes—
189C22⋊C4—22no—
190C22⋊C4—3142no—
191C22⋊C4—3142no—
192C22⋊C4—3142no—
193C22⋊C4—3142no—
194C22⋊C4—1572no—
195C42C22×D15711yescharacteristic
196C23—22no—
197C23—3142no—
198C23—6284no—
199C23—6284no—
200C23—1572no—
201C23—12562no—
202Q8—3142no—
203Q8—3142no—
204D4—22no—
205D4—3142no—
206D4—3142no—
207D4—6281no—
208D4—6284no—
209D4—6284no—
210D4—12562no—
211D4C2×D62814yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—3142no—
215C2×C4—3142no—
216C2×C4—3142no—
217C2×C4—6284no—
218C2×C4—1572no—
219C2×C4D4×D15711yescharacteristic
220C2×C4D4×D15711yescharacteristic
221C2×C4C23×D15711yescharacteristic
222C8—21no—
223C8—3142no—
224C8C2×D62812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—3142no—
229C22—3142no—
230C22—3142no—
231C22—6282no—
232C22—6282no—
233C22—6282no—
234C22—12568no—
235C22C2×D4×D15711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—3142no—
240C4—6282no—
241C4C22×D62811yescharacteristic
242C4#1909781711yescharacteristic
243C2—44no—
244C2—6282no—
245C2—12562no—
246C2D4×D62811yescharacteristic
247C2#2375807011yescharacteristic
248C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×D8×C157C2
socleC22×C157C22×D4
hypercenterC4×D8D157
hypocenterC157#3273
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. #23757918C2semidirect
3. #23757657C2semidirect
4. #23757520C2semidirect
5. #23757488C2semidirect
6. #23757436C2semidirect
7. #23757408C2semidirect
8. #23757352C2semidirect
9. C4×D8×C157C2semidirect
11. C2×D4×D157C22semidirect
13. C2×Dic628C22semidirect
14. C2×#14578943C22semidirect
15. C2×#14578942C22semidirect
18. C2×#14578937C22semidirect
19. #19097759C22semidirect
21. #19097751C22semidirect
27. C4⋊#11929434C22semidirect
28. #19097694C22semidirect
29. #19097691C22semidirect
30. #19097683C22semidirect
32. C4×D4×C157C22semidirect
33. #107×C157C22semidirect
34. #102×C157C22semidirect
35. C4×C8×C157C22semidirect
65. D8×C157D4semidirect
124. C157#3273semidirect
133. C4×D8D157semidirect
152. C4×D4C2×D157semidirect
154. #107C2×D157semidirect
158. #102C2×D157semidirect
160. C4×C8C2×D157semidirect
181. D8D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
#237579182
#237576572
#237575202
#237574882
#237574362
#237574082
#237573522
C4×D8×C1572
#3273157