Finite Group Databasebeta

← #34113268

Subgroups — #34113268

Order 20096.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19097790C211yesmaximal, characteristic, complemented
2#135×D157C211yesmaximal, characteristic, complemented
3#23757716C211yesmaximal, characteristic, complemented
4#23757713C211yesmaximal, characteristic, complemented
5#23757712C212yesmaximal, complemented
6#23757709C211yesmaximal, characteristic, complemented
7#23757707C212yesmaximal, complemented
8#23757704C212yesmaximal, complemented
9#23757702C212yesmaximal, complemented
10M32×D157C211yesmaximal, characteristic, complemented
11#509×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—21no—
13#41×D157—21no—
14#19097794C2211yescharacteristic, complemented
15#19097793C2211yescharacteristic
16#19097792—21no—
17#19097791—21no—
18#19097790—21no—
19#19097790—21no—
20#19097790C2212yes—
21#19097790C2211yescharacteristic, complemented
22#19097790C2212yescomplemented
23C2×Dic628C2211yescharacteristic, complemented
24Q16×D157C2211yescharacteristic, complemented
25C2×#14578942—21no—
26SD16×D157—21no—
27D8×D157C2211yescharacteristic, complemented
28C2×C8×D157C2211yescharacteristic
29Dic1256C2212yescomplemented
30#19097717C2212yescomplemented
31#19097716C2212yescomplemented
32#19097715C2212yescomplemented
33#19097715C2212yescomplemented
34#19097712C2212yescomplemented
35#19097711C2211yescharacteristic, complemented
36#19097710C2212yescomplemented
37C16×D157C2212yescomplemented
38#135×C157C2211yescharacteristic
39C2×D8×C157C2211yescharacteristic, complemented
40SD32×C157C2212yescomplemented
41D16×C157C2212yescomplemented
42M32×C157C2211yescharacteristic, complemented
43#14578955—21no—
44#14578954—21no—
45#14578954—41no—
46#14578953—22no—
47#14578953—22no—
48#14578953—21no—
49#14578953—41no—
50#14578953—41no—
51#14578953—41no—
52C2×Dic314—21no—
53Q8×D157—21no—
54C2×C22⋊D157—41no—
55D4×D157—21no—
56D4×D157—41no—
57C22×Dic157—41no—
58C2×C4×D157—41no—
59C2×C4×D157D411yescharacteristic
60Dic628—21no—
61Dic628C2311yescharacteristic
62Dic628C2312yes—
63#14578944—21no—
64#14578943—21no—
65#14578942—21no—
66#14578942—22no—
67#14578942—22no—
68#14578941—21no—
69D1256C2311yescharacteristic
70#14578939—21no—
71C2×#11929434D411yescharacteristic
72C8×D157D411yescharacteristic
73C8×D157D411yescharacteristic
74C8×D157C2311yescharacteristic
75C8×D157C2311yescharacteristic
76#14578929C2312yes—
77#41×C157—21no—
78C2×D4×C157—21no—
79Q16×C157C2311yescharacteristic
80SD16×C157—21no—
81D8×C157—21no—
82D8×C157C2311yescharacteristic
83D8×C157C2312yes—
84C2×C8×C157C2311yescharacteristic
85C16×C157C2312yes—
86C22×D157—41no—
87C22×D157D811yescharacteristic
88Dic314—21no—
89Dic314—22no—
90Dic314—41no—
91Dic314—41no—
92D628—21no—
93D628—41no—
94C22⋊D157—41no—
95C22⋊D157—41no—
96C22⋊D157—42no—
97C22⋊D157—42no—
98C22⋊D157—41no—
99C2×Dic157—41no—
100C2×Dic157—41no—
101C2×Dic157—42no—
102C2×Dic157—42no—
103C2×Dic157D811yescharacteristic
104C4×D157—41no—
105C4×D157—41no—
106C4×D157—41no—
107C4×D157D811yescharacteristic
108C4×D157D811yescharacteristic
109C4×D157C2×D411yescharacteristic
110C4×D157C2×D411yescharacteristic
111#11929434C2×D411yescharacteristic
112#11929434C2×D411yescharacteristic
113C23×C157—41no—
114Q8×C157—21no—
115D4×C157—21no—
116D4×C157—22no—
117D4×C157—41no—
118D4×C157—41no—
119C2×C4×C157—41no—
120C2×C4×C157C2×D411yescharacteristic
121C8×C157C2×D411yescharacteristic
122C8×C157C2411yescharacteristic
123C22×C157—41no—
124C22×C157—42no—
125C22×C157—82no—
126C22×C157C2×D811yescharacteristic
127C2×D157—41no—
128C2×D157—41no—
129C2×D157—82no—
130C2×D157C2×D811yescharacteristic
131C2×D157C2×D811yescharacteristic
132C4×C157—41no—
133C4×C157C2×D811yescharacteristic
134C4×C157C22×D411yescharacteristic
135Dic157—41no—
136Dic157—42no—
137Dic157C2×D811yescharacteristic
138Dic157C2×D811yescharacteristic
139C2×C157—21no—
140C2×C157—81no—
141C2×C157—82no—
142C2×C157C22×D811yescharacteristic
143D157—21no—
144D157—21no—
145D157—81no—
146C157#341411yescharacteristic, complemented, Sylow
147#3414—1571nomaximal, Sylow
148C2×#135—1571no—
149C2×#135—1571no—
150#510—1571no—
151#510—1572no—
152#510—1571no—
153#509—1571no—
154#509—1572no—
155#509D15711yescharacteristic, complemented
156#508—1572no—
157#508—1572no—
158C2×M32—1571no—
159C2×#41—3141no—
160C2×#41—3141no—
161#135—3141no—
162#135—3141no—
163#135—3141no—
164#135—3141no—
165#135—1571no—
166#135—1572no—
167#135—1572no—
168#135—1571no—
169#135—1571no—
170#135C2×D15711yescharacteristic
171C2×Q16—1571no—
172C2×Q16—1571no—
173C2×SD16—3141no—
174C2×SD16—3141no—
175C2×D8—1571no—
176C2×D8C2×D15711yescharacteristic, complemented
177C22×C8—1571no—
178Q32—1572no—
179Q32—1572no—
180SD32—1572no—
181SD32—1572no—
182SD32—1572no—
183SD32C2×D15712yescomplemented
184D16—1572no—
185D16C2×D15712yescomplemented
186M32—1571no—
187M32—1572no—
188M32C2×D15711yescharacteristic, complemented
189C2×C16—1572no—
190#41—21no—
191#41—3142no—
192#41—3142no—
193#41—3141no—
194#41—3141no—
195#41—3141no—
196#41—6281no—
197#41—6281no—
198#41—6281no—
199#41—6281no—
200C2×Q8—3141no—
201C2×Q8—3141no—
202C2×D4—21no—
203C2×D4—3141no—
204C2×D4—6281no—
205C2×D4—6281no—
206C22×C4—6281no—
207C22×C4—6281no—
208C22×C4—1571no—
209Q16—3141no—
210Q16—3141no—
211Q16—1571no—
212Q16—1572no—
213Q16C22×D15711yescharacteristic
214SD16—21no—
215SD16—3142no—
216SD16—3142no—
217SD16—3141no—
218SD16—3141no—
219SD16—3141no—
220D8—21no—
221D8—3141no—
222D8—1571no—
223D8C22×D15712yes—
224D8C22×D15711yescharacteristic
225C2×C8—1571no—
226C2×C8—1571no—
227C2×C8—1571no—
228C2×C8—1571no—
229C2×C8—1571no—
230C2×C8C22×D15711yescharacteristic
231C16—1572no—
232C16C22×D15712yes—
233C23—41no—
234C23—6281no—
235C23—1571no—
236Q8—21no—
237Q8—3142no—
238Q8—3141no—
239Q8—6281no—
240Q8—6281no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—3141no—
246D4—6281no—
247D4—6281no—
248D4—6282no—
249D4—6282no—
250D4—6281no—
251D4—6281no—
252C2×C4—41no—
253C2×C4—6281no—
254C2×C4—6281no—
255C2×C4—6281no—
256C2×C4—6281no—
257C2×C4—6282no—
258C2×C4—6282no—
259C2×C4—6281no—
260C2×C4—1571no—
261C2×C4—1571no—
262C2×C4—1571no—
263C2×C4—1571no—
264C2×C4—1571no—
265C2×C4D4×D15711yescharacteristic
266C8—1571no—
267C8—1571no—
268C8D4×D15711yescharacteristic
269C8C23×D15711yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—6281no—
274C22—6281no—
275C22—1571no—
276C22—1571no—
277C22—12562no—
278C22D8×D15711yescharacteristic
279C4—41no—
280C4—6281no—
281C4—6282no—
282C4—1571no—
283C4—1571no—
284C4D8×D15711yescharacteristic
285C4C2×D4×D15711yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—3141no—
290C2—3141no—
291C2—12561no—
292C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
Fitting#509×C157C2
socleC2×C157C22×D8
hypercenter#509D157
hypocenterC157#3414
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#19097790C2semidirect
2. #135×D157C2semidirect
3. #23757716C2semidirect
4. #23757713C2semidirect
5. #23757712C2semidirect
6. #23757709C2semidirect
7. #23757707C2semidirect
8. #23757704C2semidirect
9. #23757702C2semidirect
10. M32×D157C2semidirect
11. #509×C157C2semidirect
14. #19097794C22semidirect
21. #19097790C22semidirect
22. #19097790C22semidirect
23. C2×Dic628C22semidirect
24. Q16×D157C22semidirect
27. D8×D157C22semidirect
29. Dic1256C22semidirect
30. #19097717C22semidirect
31. #19097716C22semidirect
32. #19097715C22semidirect
33. #19097715C22semidirect
34. #19097712C22semidirect
35. #19097711C22semidirect
36. #19097710C22semidirect
37. C16×D157C22semidirect
39. C2×D8×C157C22semidirect
40. SD32×C157C22semidirect
41. D16×C157C22semidirect
42. M32×C157C22semidirect
146. C157#3414semidirect
155. #509D157semidirect
176. C2×D8C2×D157semidirect
183. SD32C2×D157semidirect
185. D16C2×D157semidirect
188. M32C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977902
#135×D1572
#237577162
#237577132
#237577122
#237577092
#237577072
#237577042
#237577022
M32×D1572
#509×C1572
#3414157