Finite Group Databasebeta

← #34111911

Subgroups — #34111911

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#23758086C212yesmaximal, complemented
2#23758079C212yesmaximal, complemented
3#23757545C212yesmaximal, complemented
4#23757533C212yesmaximal, complemented
5#23757455C212yesmaximal, complemented
6#23757450C211yesmaximal, characteristic, complemented
7#23757445C211yesmaximal, characteristic, complemented
8#23757396C212yesmaximal, complemented
9#497×C157C211yesmaximal, characteristic, complemented
10#19097819C2212yes—
11#19097819C2211yescharacteristic, complemented
12#19097816C2211yescharacteristic
13#19097803—22no—
14#19097803—22no—
15#19097802C2212yes—
16#19097802C2212yes—
17#19097802C2212yescomplemented
18#19097800—22no—
19#19097798—22no—
20#19097797C2212yes—
21#19097794—22no—
22#19097793C2212yes—
23#19097792—22no—
24Q16×D157—22no—
25SD16×D157—22no—
26#19097777C2212yes—
27#19097777C2212yescomplemented
28#19097707C2212yescomplemented
29#19097699C2212yes—
30#19097699C2212yescomplemented
31#19097690C2211yescharacteristic, complemented
32#19097690C2212yescomplemented
33#19097688C2211yescharacteristic
34C2×Q16×C157C2212yescomplemented
35C2×SD16×C157C2212yes—
36C4⋊Q8×C157C2211yescharacteristic, complemented
37#124×C157C2211yescharacteristic
38#97×C157C2211yescharacteristic, complemented
39#14578955—22no—
40#14578955—22no—
41#14578955—22no—
42#14578954—42no—
43#14578954—41no—
44#14578954C2312yes—
45#14578954C2312yes—
46#14578953—22no—
47Q8×D157—22no—
48Q8×D157—22no—
49Q8×D157—22no—
50C2×C22⋊D157—41no—
51D4×D157—22no—
52Dic628D412yes—
53#14578944—22no—
54#14578944—22no—
55#14578944—22no—
56#14578943D412yes—
57#14578943D412yes—
58#14578942—22no—
59#14578941—22no—
60#14578941—22no—
61#14578941—22no—
62D1256D412yes—
63#14578939—22no—
64#14578938C2312yes—
65#14578938C2312yes—
66#14578937—22no—
67#14578937—22no—
68C8×D157—22no—
69C8×D157—22no—
70C2×Q8×C157C2311yescharacteristic
71C2×Q8×C157C2312yes—
72C2×D4×C157C2311yescharacteristic
73Q16×C157—22no—
74Q16×C157—22no—
75SD16×C157—22no—
76SD16×C157—22no—
77C2×C8×C157C2312yes—
78C4⋊C4×C157—41no—
79C22⋊C4×C157—41no—
80C42×C157C2311yescharacteristic
81C22×D157—41no—
82Dic314—41no—
83Dic314—42no—
84Dic314C2×D412yes—
85Dic314C2×D412yes—
86D628—41no—
87D628—42no—
88D628C2×D412yes—
89D628C2×D412yes—
90C22⋊D157—22no—
91C22⋊D157—22no—
92C22⋊D157—42no—
93C2×Dic157—41no—
94C4×D157—22no—
95C4×D157—22no—
96C4×D157—42no—
97C4×D157—42no—
98C4×D157—42no—
99#11929434—22no—
100#11929434—22no—
101C23×C157—21no—
102Q8×C157—22no—
103Q8×C157—22no—
104Q8×C157—22no—
105D4×C157—22no—
106C2×C4×C157—21no—
107C2×C4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157C2411yescharacteristic
110C8×C157C2×D412yes—
111C8×C157C2×D412yes—
112C22×C157—41no—
113C22×C157—81no—
114C22×C157H(5,𝔽2)11yescharacteristic
115C2×D157—22no—
116C2×D157—22no—
117C2×D157—161no—
118C4×C157—41no—
119C4×C157—41no—
120C4×C157—42no—
121C4×C157C22×D412yes—
122Dic157—22no—
123Dic157—22no—
124C2×C157—21no—
125C2×C157—81no—
126C2×C157D4211yescharacteristic
127D157—42no—
128D157—42no—
129C157#328911yescharacteristic, complemented, Sylow
130#3289—1571nomaximal, Sylow
131#578—1572no—
132#577—1572no—
133#497D15711yescharacteristic, complemented
134#473—1572no—
135#471—1572no—
136#456—1572no—
137#455—1571no—
138#454—1571no—
139#444—1572no—
1402−1+4—1572no—
1412−1+4—1571no—
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)—1572no—
151#135—3142no—
152#135—3142no—
153#135—1572no—
154C2×Q16—3142no—
155C2×Q16C2×D15712yescomplemented
156C2×SD16—3142no—
157C2×SD16C2×D15712yes—
158#131—1572no—
159#131—1572no—
160C4⋊Q8C2×D15711yescharacteristic, complemented
161#124C2×D15711yescharacteristic
162#108—1572no—
163#104—1572no—
164#104—1572no—
165#101—1571no—
166#101—1572no—
167#100—1571no—
168#97C2×D15711yescharacteristic, complemented
169#41—3142no—
170#41—3142no—
171#41—3142no—
172#41—3142no—
173#41—6281no—
174#41—6282no—
175#41—1572no—
176#41—1572no—
177C2×Q8—3142no—
178C2×Q8—3142no—
179C2×Q8—3142no—
180C2×Q8C22×D15711yescharacteristic
181C2×Q8C22×D15712yes—
182C2×D4—3142no—
183C2×D4—6281no—
184C2×D4C22×D15711yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—3142no—
188Q16—3142no—
189Q16—3142no—
190Q16—1572no—
191SD16—22no—
192SD16—22no—
193SD16—3142no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—1572no—
198SD16—1572no—
199D8—3142no—
200D8—1572no—
201M16—3142no—
202M16—3142no—
203M16—1572no—
204M16—1572no—
205C2×C8—3142no—
206C2×C8—3142no—
207C2×C8C22×D15712yes—
208C4⋊C4—41no—
209C22⋊C4—41no—
210C42C22×D15711yescharacteristic
211C23—21no—
212C23—6281no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—6282no—
217Q8—6281no—
218Q8—1572no—
219Q8—1572no—
220D4—22no—
221D4—3142no—
222D4—3142no—
223D4—6281no—
224D4—6282no—
225D4—6282no—
226D4—1572no—
227D4—1572no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—3142no—
232C2×C4—3142no—
233C2×C4—6282no—
234C2×C4—6282no—
235C2×C4—6282no—
236C2×C4—6281no—
237C2×C4C23×D15711yescharacteristic
238C8—3142no—
239C8—3142no—
240C8D4×D15712yes—
241C8D4×D15712yes—
242C22—41no—
243C22—81no—
244C22—3142no—
245C22—3142no—
246C22—25121no—
247C22#1909781611yescharacteristic
248C4—41no—
249C4—41no—
250C4—42no—
251C4—3142no—
252C4—3142no—
253C4C2×D4×D15712yes—
254C2—21no—
255C2—81no—
256C2—6282no—
257C2—6282no—
258C2#2375790811yescharacteristic
subgroupG/H
centerC2#23757908
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#497×C157C2
socleC2×C157D42
hypercenter#497D157
hypocenterC157#3289
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #23758086C2semidirect
2. #23758079C2semidirect
3. #23757545C2semidirect
4. #23757533C2semidirect
5. #23757455C2semidirect
6. #23757450C2semidirect
7. #23757445C2semidirect
8. #23757396C2semidirect
9. #497×C157C2semidirect
11. #19097819C22semidirect
17. #19097802C22semidirect
27. #19097777C22semidirect
28. #19097707C22semidirect
30. #19097699C22semidirect
31. #19097690C22semidirect
32. #19097690C22semidirect
34. C2×Q16×C157C22semidirect
36. C4⋊Q8×C157C22semidirect
38. #97×C157C22semidirect
129. C157#3289semidirect
133. #497D157semidirect
155. C2×Q16C2×D157semidirect
160. C4⋊Q8C2×D157semidirect
168. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580862
#237580792
#237575452
#237575332
#237574552
#237574502
#237574452
#237573962
#497×C1572
#3289157