Finite Group Databasebeta

← #34109406

Subgroups — #34109406

Order 20096.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19097817C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C22×D1256C211yesmaximal, characteristic, complemented
4#23757837C211yesmaximal, characteristic, complemented
5#23757420C214yesmaximal, complemented
6#23757409C214yesmaximal, complemented
7#23757279C211yesmaximal, characteristic, complemented
8#23757270C211yesmaximal, characteristic, complemented
9#408×C157C211yesmaximal, characteristic, complemented
10#19097817—24no—
11#19097817—24no—
12C2×#14578955—21no—
13C2×#14578954C2211yescharacteristic
14C2×#14578954C2212yescomplemented
15C22×D628C2211yescharacteristic
16C22×D628C2211yescharacteristic, complemented
17C22×D628C2212yescomplemented
18C2×D4×D157—21no—
19C2×D4×D157—22no—
20#19097799—24no—
21C2×#14578943C2212yescomplemented
22C2×D1256—24no—
23C2×D1256C2212yescomplemented
24C2×D1256C2214yescomplemented
25#19097767—21no—
26#19097760—21no—
27#19097745C2214yescomplemented
28#19097739—22no—
29#19097730C2211yescharacteristic, complemented
30#19097696C2212yescomplemented
31#19097693C2212yescomplemented
32#19097693C2214yescomplemented
33C2×#41×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic, complemented
35C22×C8×C157C2211yescharacteristic, complemented
36C22⋊C8×C157C2214yescomplemented
37C23×D157—21no—
38C23×D157—22no—
39C23×D157—21no—
40#14578955—28no—
41#14578954—24no—
42#14578954—24no—
43#14578954—22no—
44#14578954D414yescomplemented
45C2×Dic314C2311yescharacteristic
46C2×D628—24no—
47C2×D628—24no—
48C2×D628—21no—
49C2×D628—22no—
50C2×D628—22no—
51C2×D628—22no—
52C2×D628—22no—
53C2×D628—21no—
54C2×D628D414yescomplemented
55C2×D628C2312yes—
56C2×D628C2312yes—
57C2×D628C2311yescharacteristic
58C2×C22⋊D157—21no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—24no—
61D4×D157—216no—
62D4×D157—28no—
63C2×C4×D157—21no—
64C2×C4×D157—22no—
65#14578943—24no—
66D1256—28no—
67D1256—24no—
68C4⋊Dic157C2312yes—
69C22⋊Dic157—21no—
70#14578931—24no—
71C4×Dic157—21no—
72#41×C157—22no—
73#41×C157—22no—
74C2×Q8×C157D411yescharacteristic
75C2×D4×C157D412yes—
76C2×D4×C157D411yescharacteristic
77C22×C4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79M16×C157—22no—
80C2×C8×C157—22no—
81C2×C8×C157C2312yes—
82C2×C8×C157C2312yes—
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—28no—
86C22×D157—24no—
87C22×D157—22no—
88C22×D157—42no—
89C22×D157—82no—
90C22×D157—84no—
91C22×D157—82no—
92Dic314—21no—
93Dic314C2×D412yes—
94D628—28no—
95D628—24no—
96D628—22no—
97D628—21no—
98D628—22no—
99D628—22no—
100D628—42no—
101D628C2×D412yes—
102D628C2×D414yes—
103C22⋊D157—28no—
104C22⋊D157—24no—
105C22⋊D157—84no—
106C2×Dic157—21no—
107C2×Dic157—22no—
108C4×D157—28no—
109C4×D157—24no—
110C23×C157C2×D412yes—
111C23×C157C2×D411yescharacteristic
112Q8×C157—22no—
113D4×C157—24no—
114D4×C157—22no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2411yescharacteristic
122C8×C157—22no—
123C8×C157—22no—
124C22×C157—22no—
125C22×C157—22no—
126C22×C157—24no—
127C22×C157#12012yes—
128C22×C157C22×D411yescharacteristic
129C2×D157—24no—
130C2×D157—22no—
131C2×D157—44no—
132C2×D157—42no—
133C2×D157—42no—
134C2×D157—44no—
135C2×D157—84no—
136C2×D157—88no—
137C2×D157—84no—
138C4×C157—22no—
139C4×C157#12012yes—
140C4×C157C22×D411yescharacteristic
141C4×C157C22×D411yescharacteristic
142Dic157—22no—
143Dic157—42no—
144C2×C157—22no—
145C2×C157—42no—
146C2×C157C2×#12011yescharacteristic
147C2×C157#57612yes—
148D157—44no—
149D157—42no—
150D157—82no—
151C157#300811yescharacteristic, complemented, Sylow
152#3008—1571nomaximal, Sylow
153C2×H(5,𝔽2)—1571no—
154C2×AGL(1,ℤ/8)—1571no—
155C22×D8—1571no—
156#535—1571no—
157#449—1574no—
158#447—1574no—
159#418—1571no—
160#416—1571no—
161#408D15711yescharacteristic, complemented
162H(5,𝔽2)—3144no—
163H(5,𝔽2)—3144no—
164C2×#41—3141no—
165C2×#41—1572no—
166C2×#41—1571no—
167C2×#41C2×D15711yescharacteristic
168C22×D4—3142no—
169C22×D4—3141no—
170C22×D4—1571no—
171C22×D4—1572no—
172C22×D4—1571no—
173AGL(1,ℤ/8)—3144no—
174C2×SD16—1572no—
175C2×D8—3144no—
176C2×D8—1574no—
177C2×D8—1572no—
178C2×M16C2×D15711yescharacteristic, complemented
179C22×C8C2×D15711yescharacteristic, complemented
180#127—3141no—
181#124—3141no—
182#121—1574no—
183#120—3142no—
184#117—1571no—
185#103—1572no—
186#102—1572no—
187#102—1574no—
188C22⋊C8C2×D15714yescomplemented
189C24—3142no—
190C24—3141no—
191C24—3141no—
192#41—22no—
193#41—22no—
194#41—3144no—
195#41—3144no—
196#41—3148no—
197#41—3142no—
198#41—1574no—
199C2×Q8—1571no—
200C2×Q8D62811yescharacteristic
201C2×D4—3144no—
202C2×D4—3144no—
203C2×D4—31416no—
204C2×D4—3148no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3144no—
208C2×D4—3142no—
209C2×D4—3142no—
210C2×D4—3142no—
211C2×D4—3142no—
212C2×D4—3141no—
213C2×D4—3141no—
214C2×D4—1574no—
215C2×D4—1572no—
216C2×D4—1571no—
217C2×D4—1572no—
218C2×D4D62812yes—
219C2×D4D62811yescharacteristic
220C22×C4—3142no—
221C22×C4—3141no—
222C22×C4C22×D15712yes—
223C22×C4C22×D15711yescharacteristic
224SD16—3144no—
225D8—3148no—
226D8—3144no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D15712yes—
230C2×C8C22×D15712yes—
231C4⋊C4—1572no—
232C22⋊C4—3141no—
233C22⋊C4—3144no—
234C42—3141no—
235C23—3148no—
236C23—3142no—
237C23—3141no—
238C23—3144no—
239C23—3142no—
240C23—6282no—
241C23—12562no—
242C23—12562no—
243C23—12564no—
244C23C2×D62812yes—
245C23C2×D62811yescharacteristic
246Q8—22no—
247Q8—3141no—
248Q8—1572no—
249D4—24no—
250D4—22no—
251D4—3148no—
252D4—3148no—
253D4—3144no—
254D4—3144no—
255D4—3142no—
256D4—3141no—
257D4—3142no—
258D4—3142no—
259D4—6282no—
260D4—1572no—
261D4—1574no—
262D4—12564no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—3148no—
266C2×C4—3141no—
267C2×C4—3144no—
268C2×C4—3142no—
269C2×C4D4×D15712yes—
270C2×C4D4×D15712yes—
271C2×C4C2×D62812yes—
272C2×C4C2×D62811yescharacteristic
273C2×C4C23×D15711yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—3144no—
280C22—3142no—
281C22—6284no—
282C22—6282no—
283C22—6282no—
284C22—6284no—
285C22—12564no—
286C22—12564no—
287C22—12568no—
288C22#1909774112yes—
289C22C22×D62811yescharacteristic
290C4—22no—
291C4—3142no—
292C4—6282no—
293C4#1909774112yes—
294C4C2×D4×D15711yescharacteristic
295C4C2×D4×D15711yescharacteristic
296C2—22no—
297C2—42no—
298C2—6284no—
299C2—6282no—
300C2—12562no—
301C2C2×#1909774111yescharacteristic
302C2#2375807812yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#408×C157C2
socleC22×C157C22×D4
hypercenter#408D157
hypocenterC157#3008
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#19097817C2semidirect
2. C2×#19097799C2semidirect
3. C22×D1256C2semidirect
4. #23757837C2semidirect
5. #23757420C2semidirect
6. #23757409C2semidirect
7. #23757279C2semidirect
8. #23757270C2semidirect
9. #408×C157C2semidirect
14. C2×#14578954C22semidirect
16. C22×D628C22semidirect
17. C22×D628C22semidirect
21. C2×#14578943C22semidirect
23. C2×D1256C22semidirect
24. C2×D1256C22semidirect
27. #19097745C22semidirect
29. #19097730C22semidirect
30. #19097696C22semidirect
31. #19097693C22semidirect
32. #19097693C22semidirect
34. C2×M16×C157C22semidirect
35. C22×C8×C157C22semidirect
36. C22⋊C8×C157C22semidirect
44. #14578954D4semidirect
54. C2×D628D4semidirect
151. C157#3008semidirect
161. #408D157semidirect
178. C2×M16C2×D157semidirect
179. C22×C8C2×D157semidirect
188. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190977992
C22×D12562
#237578372
#237574202
#237574092
#237572792
#237572702
#408×C1572
#3008157