Finite Group Databasebeta

← #34109484

Subgroups — #34109484

Order 20096.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#23758109C211yesmaximal, characteristic, complemented
2#23758071C212yesmaximal, complemented
3#23757455C212yesmaximal, complemented
4#23757450C212yesmaximal, complemented
5#23757445C212yesmaximal, complemented
6#23757440C212yesmaximal, complemented
7C2×#19097699C212yesmaximal, complemented
8#23757257C211yesmaximal, characteristic, complemented
9#532×C157C211yesmaximal, characteristic, complemented
10#19097819C2212yes—
11#19097818—22no—
12#19097817—22no—
13#19097816—22no—
14#19097816C2212yes—
15C2×#14578954—22no—
16C2×#14578954C2212yes—
17#41×D157—24no—
18#19097802C2214yescomplemented
19#19097801—22no—
20#19097797C2214yescomplemented
21#19097796—22no—
22#19097791—24no—
23C2×#14578938C2212yescomplemented
24#19097699C2214yescomplemented
25#19097699C2214yescomplemented
26#19097690C2212yescomplemented
27#19097688C2212yescomplemented
28C2×#41×C157C2211yescharacteristic
29C2×#41×C157C2211yescharacteristic, complemented
30C4⋊Q8×C157C2211yescharacteristic, complemented
31#127×C157C2211yescharacteristic, complemented
32#124×C157C2211yescharacteristic, complemented
33#124×C157C2211yescharacteristic, complemented
34#121×C157—22no—
35C4×D4×C157—22no—
36C2×C42×C157C2211yescharacteristic, complemented
37#14578955—24no—
38#14578954—24no—
39#14578954—24no—
40#14578954—22no—
41#14578954—24no—
42#14578954—22no—
43#14578954D414yescomplemented
44#14578954C2314yes—
45#14578953—28no—
46#14578953—24no—
47C2×Dic314—22no—
48C2×Dic314D412yescomplemented
49Q8×D157—24no—
50C2×D628—22no—
51C2×D628D412yescomplemented
52C2×C22⋊D157—24no—
53C2×C22⋊D157—22no—
54D4×D157—28no—
55D4×D157—24no—
56C2×C4×D157—24no—
57C2×C4×D157—22no—
58#14578944—24no—
59#14578942—24no—
60#14578941—24no—
61#14578939—24no—
62#14578938—22no—
63#14578938C2314yes—
64C2×#11929434—22no—
65#41×C157—22no—
66#41×C157—22no—
67#41×C157—22no—
68#41×C157—22no—
69C2×Q8×C157C2311yescharacteristic
70C2×Q8×C157C2311yescharacteristic
71C2×D4×C157—22no—
72C2×D4×C157—21no—
73C2×D4×C157D412yes—
74C2×D4×C157C2311yescharacteristic
75C2×D4×C157C2311yescharacteristic
76C22×C4×C157—21no—
77C22×C4×C157—21no—
78C22×C4×C157D412yes—
79C22×C4×C157C2311yescharacteristic
80C4⋊C4×C157—22no—
81C22⋊C4×C157—22no—
82C22⋊C4×C157—22no—
83C42×C157—21no—
84C42×C157C2311yescharacteristic
85C42×C157C2311yescharacteristic
86C22×D157—24no—
87C22×D157—22no—
88Dic314—24no—
89Dic314—22no—
90Dic314C2×D414yes—
91D628—24no—
92D628—22no—
93D628C2×D414yes—
94C22⋊D157—28no—
95C22⋊D157—24no—
96C22⋊D157—24no—
97C2×Dic157—24no—
98C2×Dic157—22no—
99C4×D157—28no—
100C4×D157—24no—
101C4×D157—24no—
102#11929434—24no—
103C23×C157—21no—
104C23×C157C2×D412yes—
105C23×C157C2×D411yescharacteristic
106Q8×C157—22no—
107Q8×C157—22no—
108D4×C157—24no—
109D4×C157—22no—
110D4×C157—22no—
111D4×C157—44no—
112D4×C157—42no—
113C2×C4×C157—21no—
114C2×C4×C157—22no—
115C2×C4×C157—22no—
116C2×C4×C157—21no—
117C2×C4×C157—21no—
118C2×C4×C157—42no—
119C2×C4×C157—41no—
120C2×C4×C157—41no—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2×D412yes—
124C2×C4×C157C2×D411yescharacteristic
125C2×C4×C157C2411yescharacteristic
126C22×C157—22no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157—42no—
130C22×C157—41no—
131C22×C157#12012yes—
132C22×C157C22×D411yescharacteristic
133C2×D157—24no—
134C2×D157—82no—
135C2×D157—84no—
136C4×C157—22no—
137C4×C157—22no—
138C4×C157—41no—
139C4×C157—41no—
140C4×C157#12011yescharacteristic
141C4×C157#12011yescharacteristic
142C4×C157C22×D412yes—
143Dic157—24no—
144C2×C157—21no—
145C2×C157—22no—
146C2×C157—42no—
147C2×C157—81no—
148C2×C157C2×#12011yescharacteristic
149D157—44no—
150C157#301811yescharacteristic, complemented, Sylow
151#3018—1571nomaximal, Sylow
152#585—1571no—
153#575—1572no—
154#532D15711yescharacteristic, complemented
155#456—1572no—
156#455—1572no—
157#454—1572no—
158#453—1572no—
159C2×#104—1572no—
160#413—1571no—
1612−1+4—3142no—
1622−1+4—1572no—
163H(5,𝔽2)—3142no—
164H(5,𝔽2)—3142no—
165H(5,𝔽2)—1572no—
166C2×#41—3144no—
167C2×#41—3142no—
168C2×#41—1572no—
169C2×#41C2×D15711yescharacteristic
170C2×#41C2×D15711yescharacteristic, complemented
171#137—3142no—
172#137—1574no—
173AGL(1,ℤ/8)—3142no—
174AGL(1,ℤ/8)—1574no—
175#135—3144no—
176C2×M16—1572no—
177C4⋊Q8C2×D15711yescharacteristic, complemented
178#127C2×D15711yescharacteristic, complemented
179#124C2×D15711yescharacteristic, complemented
180#124C2×D15711yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D15711yescharacteristic, complemented
184#104—1574no—
185#104—1574no—
186#101—1572no—
187#100—1572no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—3144no—
193#41—3144no—
194#41—3148no—
195#41—3144no—
196#41—3144no—
197#41—3142no—
198#41—3142no—
199#41—3144no—
200#41—1574no—
201#41—1574no—
202C2×Q8—3142no—
203C2×Q8—3144no—
204C2×Q8—1572no—
205C2×Q8C22×D15711yescharacteristic
206C2×Q8C22×D15711yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—3142no—
210C2×D4—3148no—
211C2×D4—3144no—
212C2×D4—3144no—
213C2×D4—3142no—
214C2×D4—1572no—
215C2×D4C22⋊D15712yes—
216C2×D4C22×D15711yescharacteristic
217C2×D4C22×D15711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—3144no—
221C22×C4—3142no—
222C22×C4C22⋊D15712yes—
223C22×C4C22×D15711yescharacteristic
224Q16—3144no—
225SD16—3144no—
226SD16—3144no—
227D8—3144no—
228M16—3142no—
229M16—1574no—
230C2×C8—3142no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D15711yescharacteristic
236C42C22×D15711yescharacteristic
237C23—21no—
238C23—3144no—
239C23—3142no—
240C23C2×C22⋊D15712yes—
241C23C2×C22⋊D15711yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—3144no—
245Q8—3142no—
246Q8—1574no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—3144no—
253D4—3148no—
254D4—3144no—
255D4—3144no—
256D4—3142no—
257D4—1574no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3148no—
267C2×C4—3144no—
268C2×C4—3142no—
269C2×C4—3144no—
270C2×C4—3144no—
271C2×C4D4×D15712yes—
272C2×C4D4×D15712yes—
273C2×C4C2×C22⋊D15712yes—
274C2×C4C2×C22⋊D15711yescharacteristic
275C2×C4C23×D15711yescharacteristic
276C8—3144no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—3144no—
283C22—12562no—
284C22—12564no—
285C22#1909773912yes—
286C22C22×C22⋊D15711yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3144no—
292C4#1909773911yescharacteristic
293C4#1909773911yescharacteristic
294C4C2×D4×D15712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—6284no—
300C2C2×#1909773911yescharacteristic
subgroupG/H
centerC4#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#532×C157C2
socleC2×C157C2×#120
hypercenter#532D157
hypocenterC157#3018
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23758071C2semidirect
3. #23757455C2semidirect
4. #23757450C2semidirect
5. #23757445C2semidirect
6. #23757440C2semidirect
7. C2×#19097699C2semidirect
8. #23757257C2semidirect
9. #532×C157C2semidirect
18. #19097802C22semidirect
20. #19097797C22semidirect
23. C2×#14578938C22semidirect
24. #19097699C22semidirect
25. #19097699C22semidirect
26. #19097690C22semidirect
27. #19097688C22semidirect
29. C2×#41×C157C22semidirect
30. C4⋊Q8×C157C22semidirect
31. #127×C157C22semidirect
32. #124×C157C22semidirect
33. #124×C157C22semidirect
36. C2×C42×C157C22semidirect
43. #14578954D4semidirect
48. C2×Dic314D4semidirect
51. C2×D628D4semidirect
150. C157#3018semidirect
154. #532D157semidirect
170. C2×#41C2×D157semidirect
177. C4⋊Q8C2×D157semidirect
178. #127C2×D157semidirect
179. #124C2×D157semidirect
180. #124C2×D157semidirect
183. C2×C42C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237580712
#237574552
#237574502
#237574452
#237574402
C2×#190976992
#237572572
#532×C1572
#3018157