Finite Group Databasebeta

← #34109535

Subgroups — #34109535

Order 20096.

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

normal subgroups only

This group has 800 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
1C2×#19097816C211yesmaximal, characteristic, complemented
2#23757835C212yesmaximal, complemented
3#23757832C211yesmaximal, characteristic, complemented
4#23757464C214yesmaximal, complemented
5#23757461C214yesmaximal, complemented
6C2×#19097685C212yesmaximal, complemented
7C2×#99×C157C211yesmaximal, characteristic, complemented
8#19097816—22no—
9#19097816—24no—
10#19097816C2214yes—
11C2×#14578954—21no—
12C2×#14578953—22no—
13C22×C22⋊D157—21no—
14C22×C22⋊D157C2212yes—
15C22×C22⋊D157C2211yescharacteristic, complemented
16C2×D4×D157—22no—
17#19097755—22no—
18#19097755C2214yescomplemented
19#19097754—21no—
20#19097752C2212yescomplemented
21#19097745—24no—
22#19097743—22no—
23#19097741—22no—
24#19097741C2214yescomplemented
25#19097740—21no—
26#19097739C2212yescomplemented
27C2×C22⋊Dic157C2211yescharacteristic, complemented
28#19097685C2218yescomplemented
29C22×D4×C157C2211yescharacteristic
30C2×C22⋊C4×C157C2212yescomplemented
31#99×C157C2214yescomplemented
32C23×D157—21no—
33C23×D157D412yescomplemented
34C23×D157C2311yescharacteristic, complemented
35#14578954—24no—
36#14578954—24no—
37#14578953—28no—
38#14578953—28no—
39C2×Dic314—21no—
40C2×D628—21no—
41C2×D628—24no—
42C2×C22⋊D157—22no—
43C2×C22⋊D157—24no—
44C2×C22⋊D157—22no—
45C2×C22⋊D157—22no—
46C2×C22⋊D157—24no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—21no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157—21no—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54C2×C22⋊D157D414yescomplemented
55C2×C22⋊D157C2314yes—
56D4×D157—28no—
57D4×D157—28no—
58C22×Dic157—21no—
59C22×Dic157—21no—
60C22×Dic157D412yescomplemented
61C22×Dic157C2311yescharacteristic, complemented
62C2×C4×D157—22no—
63C4⋊Dic157—24no—
64#14578933—22no—
65C22⋊Dic157—21no—
66C22⋊Dic157—22no—
67C22⋊Dic157C2312yescomplemented
68#14578931—24no—
69#14578931—24no—
70#14578931—22no—
71C24×C157D411yescharacteristic, complemented
72C24×C157C2311yescharacteristic
73C2×D4×C157—22no—
74C2×D4×C157—22no—
75C2×D4×C157D412yescomplemented
76C2×D4×C157C2312yes—
77C22×C4×C157—22no—
78C22×C4×C157D411yescharacteristic, complemented
79C22⋊C4×C157—22no—
80C22⋊C4×C157C2314yescomplemented
81C22×D157—22no—
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—21no—
85C22×D157—22no—
86C22×D157—81no—
87C22×D157—82no—
88C22×D157—82no—
89C22×D157C2×D414yes—
90Dic314—24no—
91D628—24no—
92D628—84no—
93C22⋊D157—28no—
94C22⋊D157—24no—
95C22⋊D157—24no—
96C22⋊D157—24no—
97C22⋊D157—24no—
98C22⋊D157—41no—
99C22⋊D157—41no—
100C22⋊D157—42no—
101C22⋊D157—82no—
102C2×Dic157—24no—
103C2×Dic157—22no—
104C2×Dic157—22no—
105C2×Dic157—22no—
106C2×Dic157—22no—
107C2×Dic157—21no—
108C2×Dic157—81no—
109C2×Dic157C2×D414yes—
110C4×D157—28no—
111C23×C157—21no—
112C23×C157—21no—
113C23×C157—81no—
114C23×C157—81no—
115C23×C157C2×D412yes—
116C23×C157C2×D412yes—
117C23×C157C2×D412yes—
118C23×C157C2×D411yescharacteristic
119C23×C157C2×D411yescharacteristic
120C23×C157C2411yescharacteristic
121D4×C157—24no—
122D4×C157—24no—
123C2×C4×C157—21no—
124C2×C4×C157—21no—
125C2×C4×C157—24no—
126C2×C4×C157—82no—
127C2×C4×C157C2×D412yes—
128C22×C157—22no—
129C22×C157—42no—
130C22×C157—41no—
131C22×C157—41no—
132C22×C157—42no—
133C22×C157—41no—
134C22×C157—42no—
135C22×C157—42no—
136C22×C157—81no—
137C22×C157#12011yescharacteristic
138C22×C157#12011yescharacteristic
139C22×C157#12012yes—
140C22×C157C22×D412yes—
141C22×C157C22×D411yescharacteristic
142C2×D157—24no—
143C2×D157—44no—
144C2×D157—44no—
145C2×D157—42no—
146C2×D157—44no—
147C2×D157—41no—
148C2×D157—82no—
149C2×D157—84no—
150C4×C157—22no—
151C4×C157—44no—
152Dic157—24no—
153Dic157—42no—
154Dic157—41no—
155C2×C157—21no—
156C2×C157—22no—
157C2×C157—21no—
158C2×C157—22no—
159C2×C157—21no—
160C2×C157—42no—
161C2×C157C2×#12011yescharacteristic
162D157—44no—
163D157—81no—
164C157#302611yescharacteristic, complemented, Sylow
165#3026—1571nomaximal, Sylow
166C2×H(5,𝔽2)—1571no—
167#534—1571no—
168#534—1572no—
169#458—1574no—
170B(4,𝔽2)—1574no—
171C2×#99—1572no—
172C2×#99D15711yescharacteristic, complemented
173H(5,𝔽2)—3144no—
174H(5,𝔽2)—3142no—
175H(5,𝔽2)—1574no—
176C2×#41—3142no—
177C2×#41—3141no—
178C22×D4—3142no—
179C22×D4—3141no—
180C22×D4—1571no—
181C22×D4—1572no—
182C22×D4C2×D15711yescharacteristic
183#123—3142no—
184#123—3141no—
185#123—1572no—
186#123—1574no—
187#121—3144no—
188#121—3142no—
189#120—3142no—
190#120—3141no—
191#120—1572no—
192#120—1574no—
193C2×C22⋊C4—1571no—
194C2×C22⋊C4C2×D15712yescomplemented
195#99—1578no—
196#99C2×D15714yescomplemented
197C24—3141no—
198C24—1571no—
199C24—1572no—
200C24D62811yescharacteristic, complemented
201C24C22×D15711yescharacteristic
202#41—3144no—
203#41—3148no—
204#41—3148no—
205#41—3144no—
206C2×Q8—3141no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—3141no—
210C2×D4—3144no—
211C2×D4—3148no—
212C2×D4—3148no—
213C2×D4—3142no—
214C2×D4—3144no—
215C2×D4—3142no—
216C2×D4—3142no—
217C2×D4—3142no—
218C2×D4—3144no—
219C2×D4—3142no—
220C2×D4—3142no—
221C2×D4—3142no—
222C2×D4—3141no—
223C2×D4—3141no—
224C2×D4—3142no—
225C2×D4—1574no—
226C2×D4—1574no—
227C2×D4D62812yescomplemented
228C2×D4C22×D15712yes—
229C22×C4—22no—
230C22×C4—3142no—
231C22×C4—3141no—
232C22×C4—3141no—
233C22×C4—1571no—
234C22×C4—1572no—
235C22×C4D62811yescharacteristic, complemented
236C4⋊C4—3144no—
237C4⋊C4—3142no—
238C22⋊C4—22no—
239C22⋊C4—3144no—
240C22⋊C4—3144no—
241C22⋊C4—3141no—
242C22⋊C4—3142no—
243C22⋊C4—3142no—
244C22⋊C4—1572no—
245C22⋊C4C22×D15714yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—3144no—
251C23—3142no—
252C23—3142no—
253C23—3142no—
254C23—3141no—
255C23—1574no—
256C23—12561no—
257C23—12562no—
258C23—12562no—
259C23D4×D15712yes—
260C23D4×D15712yes—
261C23C2×D62812yes—
262C23C2×D62811yescharacteristic
263C23C2×D62811yescharacteristic
264C23C23×D15711yescharacteristic
265Q8—3144no—
266D4—24no—
267D4—24no—
268D4—3144no—
269D4—3148no—
270D4—3144no—
271D4—3144no—
272D4—3144no—
273D4—3144no—
274D4—6281no—
275D4—6281no—
276D4—6282no—
277D4—12564no—
278D4—12562no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—3148no—
284C2×C4—3144no—
285C2×C4—3142no—
286C2×C4—3142no—
287C2×C4—3142no—
288C2×C4—3142no—
289C2×C4—3141no—
290C2×C4—1574no—
291C2×C4—12561no—
292C2×C4C2×D62812yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—3144no—
303C22—6284no—
304C22—6284no—
305C22—6284no—
306C22—6282no—
307C22—6281no—
308C22—12562no—
309C22—12564no—
310C22#1909774111yescharacteristic
311C22#1909774111yescharacteristic
312C22#1909774112yes—
313C22C2×D4×D15712yes—
314C22C22×D62811yescharacteristic
315C4—22no—
316C4—44no—
317C4—3144no—
318C4—6282no—
319C4—6281no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—6284no—
327C2—12561no—
328C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
FittingC2×#99×C157C2
socleC2×C157C2×#120
hypercenterC2×#99D157
hypocenterC157#3026
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. #23757835C2semidirect
3. #23757832C2semidirect
4. #23757464C2semidirect
5. #23757461C2semidirect
6. C2×#19097685C2semidirect
7. C2×#99×C157C2semidirect
15. C22×C22⋊D157C22semidirect
18. #19097755C22semidirect
20. #19097752C22semidirect
24. #19097741C22semidirect
26. #19097739C22semidirect
27. C2×C22⋊Dic157C22semidirect
28. #19097685C22semidirect
30. C2×C22⋊C4×C157C22semidirect
31. #99×C157C22semidirect
33. C23×D157D4semidirect
34. C23×D157C23semidirect
54. C2×C22⋊D157D4semidirect
60. C22×Dic157D4semidirect
61. C22×Dic157C23semidirect
67. C22⋊Dic157C23semidirect
71. C24×C157D4semidirect
75. C2×D4×C157D4semidirect
78. C22×C4×C157D4semidirect
80. C22⋊C4×C157C23semidirect
164. C157#3026semidirect
172. C2×#99D157semidirect
194. C2×C22⋊C4C2×D157semidirect
196. #99C2×D157semidirect
200. C24D628semidirect
227. C2×D4D628semidirect
235. C22×C4D628semidirect
245. C22⋊C4C22×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
#237578352
#237578322
#237574642
#237574612
C2×#190976852
C2×#99×C1572
#3026157