Finite Group Databasebeta

← #34101136

Subgroups — #34101136

Order 20096.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19097747C211yesmaximal, characteristic, complemented
2C2×#19097742C211yesmaximal, characteristic
3C2×#19097695C211yesmaximal, characteristic, complemented
4C2×#19097692C211yesmaximal, characteristic, complemented
5C2×#19097683C211yesmaximal, characteristic, complemented
6#23757036C211yesmaximal, characteristic, complemented
7C22×SD16×C157C211yesmaximal, characteristic, complemented
8C22×C22⋊D157—21no—
9C22×C4×D157C2211yescharacteristic
10C22×#11929434C2211yescharacteristic
11#19097747—22no—
12#19097747—22no—
13#19097742—22no—
14#19097742—22no—
15C2×C4⋊Dic157C2211yescharacteristic
16C2×C4⋊Dic157C2211yescharacteristic
17C2×#14578933—21no—
18C2×C22⋊Dic157—21no—
19C2×#14578931—21no—
20#19097695—22no—
21#19097692—22no—
22#19097683—22no—
23C22×Q8×C157C2211yescharacteristic
24C22×D4×C157C2211yescharacteristic
25C2×SD16×C157—22no—
26C2×SD16×C157—22no—
27C2×SD16×C157—22no—
28C22×C8×C157C2211yescharacteristic
29C23×D157D411yescharacteristic
30C2×C22⋊D157—24no—
31C2×C22⋊D157—42no—
32C2×C22⋊D157—42no—
33C22×Dic157—21no—
34C22×Dic157—21no—
35C22×Dic157D411yescharacteristic
36C2×C4×D157—21no—
37C2×C4×D157—21no—
38C2×C4×D157—21no—
39C2×C4×D157—21no—
40C2×C4×D157—21no—
41C2×C4×D157—21no—
42C2×#11929434—21no—
43C2×#11929434—21no—
44C2×#11929434—21no—
45C4⋊Dic157—21no—
46C4⋊Dic157—21no—
47C4⋊Dic157D412yes—
48C4⋊Dic157D412yescomplemented
49#14578933—24no—
50C22⋊Dic157—24no—
51#14578931—24no—
52C24×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157—21no—
57C2×Q8×C157—21no—
58C2×Q8×C157D412yescomplemented
59C2×D4×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157—21no—
63C2×D4×C157—21no—
64C2×D4×C157D412yes—
65C22×C4×C157—21no—
66C22×C4×C157C2311yescharacteristic
67SD16×C157—28no—
68C2×C8×C157—21no—
69C2×C8×C157—21no—
70C2×C8×C157D412yescomplemented
71C22×D157—21no—
72C22×D157—21no—
73C22×D157—21no—
74C22×D157—21no—
75C22×D157—21no—
76C22×D157—21no—
77C22×D157SD1612yescomplemented
78C22⋊D157—48no—
79C2×Dic157—21no—
80C2×Dic157—21no—
81C2×Dic157—22no—
82C2×Dic157—22no—
83C2×Dic157—21no—
84C2×Dic157—41no—
85C2×Dic157—41no—
86C2×Dic157—41no—
87C2×Dic157—41no—
88C4×D157—41no—
89C4×D157—41no—
90C4×D157—41no—
91C4×D157—41no—
92#11929434—41no—
93C23×C157—22no—
94C23×C157—41no—
95C23×C157—41no—
96C23×C157—41no—
97C23×C157—41no—
98C23×C157—41no—
99C23×C157—41no—
100C23×C157C2×D411yescharacteristic
101Q8×C157—22no—
102Q8×C157—22no—
103Q8×C157—41no—
104Q8×C157—41no—
105D4×C157—22no—
106D4×C157—22no—
107D4×C157—41no—
108D4×C157—41no—
109C2×C4×C157—22no—
110C2×C4×C157—41no—
111C2×C4×C157—41no—
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157#4111yescharacteristic
118C8×C157—22no—
119C22×C157—42no—
120C22×C157—42no—
121C22×C157—42no—
122C22×C157—81no—
123C22×C157—81no—
124C22×C157—81no—
125C22×C157—81no—
126C22×C157#12011yescharacteristic
127C22×C157#12011yescharacteristic
128C22×C157#12111yescharacteristic
129C22×C157C2×SD1611yescharacteristic
130C22×C157#13511yescharacteristic
131C22×C157AGL(1,ℤ/8)11yescharacteristic
132C22×C157#13711yescharacteristic
133C2×D157—22no—
134C2×D157—22no—
135C2×D157—22no—
136C2×D157—41no—
137C2×D157—41no—
138C2×D157—41no—
139C2×D157—41no—
140C4×C157—42no—
141C4×C157#12011yescharacteristic
142C4×C157#12111yescharacteristic
143C4×C157#12111yescharacteristic
144C4×C157#12711yescharacteristic
145Dic157—41no—
146Dic157—42no—
147Dic157—42no—
148C2×C157—82no—
149C2×C157#39211yescharacteristic
150C2×C157#44811yescharacteristic
151C2×C157#44911yescharacteristic
152C2×C157#45011yescharacteristic
153C2×C157#45211yescharacteristic
154C2×C157#46511yescharacteristic
155C2×C157#46811yescharacteristic
156D157—42no—
157C157#200011yescharacteristic, complemented, Sylow
158#2000—1571nomaximal, Sylow
159C22×SD16D15711yescharacteristic, complemented
160C2×C22⋊Q8—1571no—
161C2×#121—1571no—
162C2×#103—1571no—
163C2×#102—1571no—
164C2×C22⋊C8—1571no—
165#340—1571no—
166C22×Q8C2×D15711yescharacteristic
167C22×D4—3141no—
168C22×D4C2×D15711yescharacteristic
169C23×C4—1571no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—22no—
173C22×C8—1571no—
174C22×C8C2×D15711yescharacteristic
175C22⋊Q8—3142no—
176C22⋊Q8—3142no—
177#121—3142no—
178#121—3142no—
179C2×C4⋊C4—3141no—
180C2×C4⋊C4—1571no—
181C2×C4⋊C4—1571no—
182C2×C22⋊C4—3141no—
183C2×C22⋊C4—3141no—
184#103—3142no—
185#102—3142no—
186C22⋊C8—3142no—
187C24—21no—
188C24—1571no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8C22⋊D15712yescomplemented
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—3144no—
201C2×D4—6282no—
202C2×D4—6282no—
203C2×D4C22⋊D15712yes—
204C22×C4—21no—
205C22×C4—3141no—
206C22×C4—3141no—
207C22×C4—3141no—
208C22×C4—3141no—
209C22×C4—3141no—
210C22×C4—3141no—
211C22×C4—3141no—
212C22×C4—3141no—
213C22×C4—1571no—
214C22×C4C22×D15711yescharacteristic
215SD16—28no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—3141no—
219C2×C8—3141no—
220C2×C8—3141no—
221C2×C8C22⋊D15712yescomplemented
222C4⋊C4—3144no—
223C4⋊C4—3141no—
224C4⋊C4—3141no—
225C4⋊C4—1572no—
226C4⋊C4—1572no—
227C22⋊C4—3144no—
228C22⋊C4—3144no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—3141no—
237C23—3141no—
238C23—3141no—
239C23—3141no—
240C23—3141no—
241C23—3141no—
242C23—1572no—
243C23D4×D15711yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—6288no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—3141no—
257C2×C4—3142no—
258C2×C4—3142no—
259C2×C4—3141no—
260C2×C4—3141no—
261C2×C4—6281no—
262C2×C4—6281no—
263C2×C4—6281no—
264C2×C4—6281no—
265C2×C4—6281no—
266C2×C4—6281no—
267C2×C4—6281no—
268C2×C4—6281no—
269C2×C4D4×D15711yescharacteristic
270C2×C4D4×D15711yescharacteristic
271C2×C4C2×C22⋊D15711yescharacteristic
272C2×C4C2×C22⋊D15711yescharacteristic
273C2×C4C2×C22⋊D15711yescharacteristic
274C2×C4#1457895311yescharacteristic
275C8—22no—
276C8—6281no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—3142no—
285C22—3142no—
286C22—3142no—
287C22—6281no—
288C22—6281no—
289C22—6281no—
290C22—6281no—
291C22#1909773911yescharacteristic
292C22#1909773911yescharacteristic
293C22#1909774211yescharacteristic
294C22SD16×D15711yescharacteristic
295C22#1909779211yescharacteristic
296C22#1909779811yescharacteristic
297C22#1909780011yescharacteristic
298C4—42no—
299C4—6282no—
300C4—6282no—
301C4—6281no—
302C4#1909774011yescharacteristic
303C4#1909774311yescharacteristic
304C4#1909774311yescharacteristic
305C4#1909776711yescharacteristic
306C2—82no—
307C2—6282no—
308C2#2375718111yescharacteristic
309C2#2375741011yescharacteristic
310C2#2375741711yescharacteristic
311C2#2375742211yescharacteristic
312C2#2375743211yescharacteristic
313C2#2375749911yescharacteristic
314C2#2375751611yescharacteristic
subgroupG/H
centerC23D4×D157
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC22×SD16×C157C2
socleC23×C157C2×D4
prime radicalC2×#19097742C2
hypercenterC22×SD16D157
hypocenterC157#2000
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19097747C2semidirect
3. C2×#19097695C2semidirect
4. C2×#19097692C2semidirect
5. C2×#19097683C2semidirect
6. #23757036C2semidirect
7. C22×SD16×C157C2semidirect
48. C4⋊Dic157D4semidirect
58. C2×Q8×C157D4semidirect
70. C2×C8×C157D4semidirect
77. C22×D157SD16semidirect
157. C157#2000semidirect
159. C22×SD16D157semidirect
194. C2×Q8C22⋊D157semidirect
221. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977472
C2×#190977422
C2×#190976952
C2×#190976922
C2×#190976832
#237570362
C22×SD16×C1572
#2000157