Finite Group Databasebeta

← #34110519

Subgroups — #34110519

Order 20096.

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

normal subgroups only

This group has 400 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#23757829C211yesmaximal, characteristic, complemented
2#124×D157C211yesmaximal, characteristic
3#23757634C212yesmaximal, complemented
4#23757631C212yesmaximal, complemented
5#23757630C211yesmaximal, characteristic, complemented
6#23757628C211yesmaximal, characteristic, complemented
7#23757627C211yesmaximal, characteristic, complemented
8#23757280C212yesmaximal, complemented
9#23757276C212yesmaximal
10C4×#14578937C211yesmaximal, characteristic, complemented
11#488×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C2×D4×D157C2211yescharacteristic
14C2×#14578937C2212yes—
15#19097770C2212yescomplemented
16#19097763C2211yescharacteristic, complemented
17#19097760C2211yescharacteristic
18#19097759C2211yescharacteristic
19#19097758C2211yescharacteristic
20#19097757—22no—
21#19097749—22no—
22#19097736—22no—
23#19097729C2212yes—
24C22⋊C4×D157—22no—
25C42×D157C2211yescharacteristic
26#19097696C2212yescomplemented
27#19097695C2212yescomplemented
28#19097694C2212yescomplemented
29#19097693C2212yes—
30#19097692C2212yes—
31#19097691C2212yes—
32#19097681C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
34#125×C157C2211yescharacteristic, complemented
35#124×C157C2211yescharacteristic
36#103×C157C2212yescomplemented
37#102×C157C2212yes—
38C4×C8×C157C2211yescharacteristic, complemented
39C23×D157—21no—
40C2×Dic314—22no—
41C2×Dic314—21no—
42C2×Dic314C2311yescharacteristic
43Q8×D157—24no—
44C2×D628C2311yescharacteristic
45C2×C22⋊D157—21no—
46D4×D157—24no—
47C2×C4×D157—21no—
48C2×C4×D157D412yes—
49C2×C4×D157C2311yescharacteristic
50#14578937—24no—
51C2×#11929434C2312yes—
52C4⋊Dic157C2312yes—
53#14578933—22no—
54#14578933—22no—
55C22⋊Dic157—22no—
56#14578931—22no—
57#14578931—22no—
58#14578931—22no—
59C4×Dic157—22no—
60C4×Dic157D412yes—
61C4×Dic157C2311yescharacteristic
62C2×Q8×C157C2311yescharacteristic
63C2×D4×C157C2311yescharacteristic
64C2×C8×C157C2312yes—
65C4⋊C4×C157—22no—
66C4⋊C4×C157C2312yes—
67C22⋊C4×C157—22no—
68C42×C157C2311yescharacteristic
69C22×D157—21no—
70C22×D157—44no—
71C22×D157—42no—
72C22×D157C2×D411yescharacteristic
73Dic314—22no—
74Dic314—44no—
75Dic314—42no—
76D628—22no—
77C22⋊D157—42no—
78C2×Dic157—21no—
79C2×Dic157—22no—
80C2×Dic157C2×D412yes—
81C2×Dic157C2×D411yescharacteristic
82C4×D157—42no—
83C4×D157—42no—
84C4×D157#4114yes—
85#11929434—22no—
86C23×C157—21no—
87Q8×C157—22no—
88D4×C157—22no—
89C2×C4×C157—21no—
90C2×C4×C157—22no—
91C2×C4×C157C2×D412yes—
92C2×C4×C157C2411yescharacteristic
93C8×C157—22no—
94C22×C157—41no—
95C22×C157—42no—
96C22×C157C22×D411yescharacteristic
97C2×D157—22no—
98C2×D157—41no—
99C2×D157—42no—
100C2×D157—84no—
101C2×D157#12412yes—
102C4×C157—22no—
103C4×C157—41no—
104C4×C157—42no—
105C4×C157C2×#4112yes—
106Dic157—22no—
107Dic157—41no—
108Dic157—42no—
109Dic157#12412yes—
110C2×C157—81no—
111C2×C157C2×#12411yescharacteristic
112C2×C157#57512yes—
113D157—22no—
114D157—81no—
115C157#314011yescharacteristic, complemented, Sylow
116#3140—1571nomaximal, Sylow
117#533—1571no—
118C2×#124—1571no—
119#489—1572no—
120#489—1572no—
121#488—1571no—
122#488—1571no—
123#488—1571no—
124#488D15711yescharacteristic, complemented
125#419—1572no—
126#418—1572no—
127C4×M16—1571no—
128C22×Q8—1571no—
129C22×D4—1571no—
130C2×M16—1572no—
131C4⋊Q8—1572no—
132#125—1571no—
133#125C2×D15711yescharacteristic, complemented
134#124—3142no—
135#124—1571no—
136#124—1571no—
137#124—1571no—
138#124C2×D15711yescharacteristic
139C22⋊Q8—3142no—
140C4×Q8—3142no—
141#117—1572no—
142C2×C22⋊C4—3142no—
143C2×C42—1571no—
144#103—1572no—
145#103—1572no—
146#103—1572no—
147#103C2×D15712yescomplemented
148#102—1572no—
149#102—1572no—
150#102—1572no—
151#102C2×D15712yes—
152#97—1572no—
153C4×C8—1571no—
154C4×C8C2×D15711yescharacteristic, complemented
155C24—3141no—
156C2×Q8—3142no—
157C2×Q8—3141no—
158C2×Q8—3144no—
159C2×Q8—1571no—
160C2×Q8C22×D15711yescharacteristic
161C2×D4—3144no—
162C2×D4—3141no—
163C2×D4—1571no—
164C2×D4C22×D15711yescharacteristic
165C22×C4—3141no—
166C22×C4—1572no—
167C22×C4—1571no—
168M16—3144no—
169C2×C8—1572no—
170C2×C8C22×D15712yes—
171C4⋊C4—22no—
172C4⋊C4—3142no—
173C4⋊C4—3142no—
174C4⋊C4—1572no—
175C4⋊C4C22×D15712yes—
176C22⋊C4—22no—
177C22⋊C4—3142no—
178C22⋊C4—3142no—
179C22⋊C4—3142no—
180C22⋊C4—3142no—
181C42—3142no—
182C42—1572no—
183C42—1571no—
184C42C22×D15711yescharacteristic
185C23—21no—
186C23—3141no—
187C23—6284no—
188C23—6282no—
189C23—1571no—
190Q8—22no—
191Q8—3142no—
192Q8—6284no—
193Q8—6282no—
194D4—22no—
195D4—3142no—
196D4—6282no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3142no—
200C2×C4—3141no—
201C2×C4—6282no—
202C2×C4—6282no—
203C2×C4—1572no—
204C2×C4—1571no—
205C2×C4—1574no—
206C2×C4D4×D15712yes—
207C2×C4C23×D15711yescharacteristic
208C8—22no—
209C8—3142no—
210C22—41no—
211C22—42no—
212C22—3142no—
213C22—6281no—
214C22—6282no—
215C22—1572no—
216C22—12564no—
217C22C2×D4×D15711yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—3142no—
222C4—6281no—
223C4—6282no—
224C4—1572no—
225C4#41×D15712yes—
226C2—81no—
227C2—3142no—
228C2—12561no—
229C2#124×D15711yescharacteristic
230C2#2375807012yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#488×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#488D157
hypocenterC157#3140
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #23757829C2semidirect
3. #23757634C2semidirect
4. #23757631C2semidirect
5. #23757630C2semidirect
6. #23757628C2semidirect
7. #23757627C2semidirect
8. #23757280C2semidirect
10. C4×#14578937C2semidirect
11. #488×C157C2semidirect
15. #19097770C22semidirect
16. #19097763C22semidirect
26. #19097696C22semidirect
27. #19097695C22semidirect
28. #19097694C22semidirect
32. #19097681C22semidirect
33. C4×#11929434C22semidirect
34. #125×C157C22semidirect
36. #103×C157C22semidirect
38. C4×C8×C157C22semidirect
115. C157#3140semidirect
124. #488D157semidirect
133. #125C2×D157semidirect
147. #103C2×D157semidirect
154. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578292
#124×D1572
#237576342
#237576312
#237576302
#237576282
#237576272
#237572802
#237572762
C4×#145789372
#488×C1572
#3140157