Finite Group Databasebeta

← #34101138

Subgroups — #34101138

Order 20096.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#14578941C211yesmaximal, characteristic, complemented
2C2×#19097747C211yesmaximal, characteristic, complemented
3C2×#19097746C211yesmaximal, characteristic
4C2×#19097693C211yesmaximal, characteristic, complemented
5C2×#19097683C211yesmaximal, characteristic, complemented
6#23757035C211yesmaximal, characteristic, complemented
7C2×#103×C157C211yesmaximal, characteristic, complemented
8C22×D628—21no—
9C22×D628C2211yescharacteristic
10C22×C4×D157C2211yescharacteristic
11C2×#14578941—22no—
12C2×#14578941—22no—
13C2×#14578941—22no—
14C22×#11929434C2211yescharacteristic
15#19097747—22no—
16#19097747—22no—
17#19097746—22no—
18#19097746—22no—
19C2×C4⋊Dic157C2211yescharacteristic
20C2×#14578933—21no—
21C2×#14578931—21no—
22C2×#14578931—21no—
23#19097693—22no—
24#19097683—22no—
25C22×Q8×C157C2211yescharacteristic
26C22×C8×C157C2211yescharacteristic
27C2×C4⋊C4×C157C2211yescharacteristic
28#103×C157—22no—
29C23×D157—21no—
30C23×D157D411yescharacteristic
31C2×D628—24no—
32C2×D628—21no—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×D628—21no—
37C2×D628—42no—
38C2×D628—42no—
39C2×D628D412yes—
40C22×Dic157—21no—
41C22×Dic157D411yescharacteristic
42C2×C4×D157—21no—
43C2×C4×D157—21no—
44C2×C4×D157—21no—
45C2×C4×D157—21no—
46C2×C4×D157—21no—
47C2×C4×D157—21no—
48#14578941—28no—
49C2×#11929434—21no—
50C2×#11929434—21no—
51C2×#11929434D412yescomplemented
52C4⋊Dic157—21no—
53C4⋊Dic157D412yescomplemented
54#14578933—24no—
55#14578931—24no—
56#14578931—24no—
57C2×Q8×C157—21no—
58C2×Q8×C157—21no—
59C2×Q8×C157—21no—
60C2×Q8×C157—21no—
61C2×Q8×C157—21no—
62C2×Q8×C157D412yescomplemented
63C22×C4×C157—21no—
64C22×C4×C157—21no—
65C22×C4×C157C2311yescharacteristic
66C2×C8×C157—21no—
67C2×C8×C157—21no—
68C2×C8×C157—21no—
69C4⋊C4×C157—21no—
70C4⋊C4×C157D412yes—
71C22×D157—21no—
72C22×D157—21no—
73C22×D157—21no—
74C22×D157—21no—
75C22×D157—21no—
76C22×D157—21no—
77C22×D157—22no—
78C22×D157—41no—
79C22×D157—41no—
80C22×D157—41no—
81C22×D157—41no—
82C22×D157—41no—
83C22×D157—41no—
84C22×D157SD1612yescomplemented
85D628—22no—
86D628—22no—
87D628—48no—
88D628—41no—
89D628—41no—
90C2×Dic157—21no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—41no—
95C2×Dic157—41no—
96C4×D157—41no—
97C4×D157—41no—
98C4×D157—41no—
99C4×D157—41no—
100#11929434—22no—
101C23×C157C2×D411yescharacteristic
102Q8×C157—22no—
103Q8×C157—22no—
104Q8×C157—41no—
105Q8×C157—41no—
106C2×C4×C157—22no—
107C2×C4×C157—22no—
108C2×C4×C157—41no—
109C2×C4×C157—41no—
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—41no—
119C22×C157#12011yescharacteristic
120C22×C157#12011yescharacteristic
121C22×C157#12111yescharacteristic
122C22×C157C2×SD1611yescharacteristic
123C22×C157#13511yescharacteristic
124C22×C157AGL(1,ℤ/8)11yescharacteristic
125C22×C157#13711yescharacteristic
126C2×D157—22no—
127C2×D157—22no—
128C2×D157—22no—
129C2×D157—41no—
130C2×D157—41no—
131C2×D157—41no—
132C2×D157—41no—
133C2×D157—42no—
134C2×D157—42no—
135C2×D157—42no—
136C2×D157—81no—
137C2×D157—81no—
138C2×D157—81no—
139C2×D157—81no—
140C4×C157—42no—
141C4×C157—42no—
142C4×C157#12011yescharacteristic
143C4×C157#12111yescharacteristic
144C4×C157#12111yescharacteristic
145C4×C157#12711yescharacteristic
146Dic157—41no—
147Dic157—42no—
148C2×C157#39211yescharacteristic
149C2×C157#44811yescharacteristic
150C2×C157#44911yescharacteristic
151C2×C157#45011yescharacteristic
152C2×C157#45211yescharacteristic
153C2×C157#46511yescharacteristic
154C2×C157#46811yescharacteristic
155D157—42no—
156D157—82no—
157C157#200011yescharacteristic, complemented, Sylow
158#2000—1571nomaximal, Sylow
159C22×SD16—1571no—
160C2×C22⋊Q8—1571no—
161C2×#121—1571no—
162C2×#103D15711yescharacteristic, complemented
163C2×#102—1571no—
164C2×C22⋊C8—1571no—
165#340—1571no—
166C22×Q8C2×D15711yescharacteristic
167C22×D4—3141no—
168C22×D4—1571no—
169C23×C4—1571no—
170C2×SD16—3142no—
171C2×SD16—3142no—
172C2×SD16—3142no—
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⋊C4C2×D15711yescharacteristic
182C2×C22⋊C4—3141no—
183C2×C22⋊C4—3141no—
184#103—22no—
185#102—3142no—
186C22⋊C8—3142no—
187C24—3141no—
188C24—1571no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8D62812yescomplemented
195C2×D4—3144no—
196C2×D4—3141no—
197C2×D4—3141no—
198C2×D4—3141no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—6282no—
202C2×D4—6282no—
203C2×D4—1572no—
204C22×C4—21no—
205C22×C4—21no—
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—3148no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—3141no—
220C2×C8—3141no—
221C2×C8—1572no—
222C4⋊C4—21no—
223C4⋊C4—3144no—
224C4⋊C4—3141no—
225C4⋊C4—1572no—
226C4⋊C4C22⋊D15712yes—
227C22⋊C4—3144no—
228C22⋊C4—3144no—
229C23—3141no—
230C23—3141no—
231C23—3141no—
232C23—3141no—
233C23—3141no—
234C23—3141no—
235C23—3142no—
236C23—6281no—
237C23—6281no—
238C23—6281no—
239C23—6281no—
240C23—6281no—
241C23—6281no—
242C23—1572no—
243C23D4×D15711yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—3142no—
249D4—3142no—
250D4—6288no—
251D4—6281no—
252D4—6281no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3141no—
260C2×C4—3142no—
261C2×C4—3141no—
262C2×C4—3141no—
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×C4D4×D15711yescharacteristic
272C2×C4C2×C22⋊D15711yescharacteristic
273C2×C4C2×D62811yescharacteristic
274C2×C4#1457895411yescharacteristic
275C8—41no—
276C8—3142no—
277C22—3142no—
278C22—3142no—
279C22—3142no—
280C22—6281no—
281C22—6281no—
282C22—6281no—
283C22—6281no—
284C22—6282no—
285C22—6282no—
286C22—6282no—
287C22—12561no—
288C22—12561no—
289C22—12561no—
290C22—12561no—
291C22#1909773911yescharacteristic
292C22#1909774111yescharacteristic
293C22#1909774411yescharacteristic
294C22SD16×D15711yescharacteristic
295C22#1909779411yescharacteristic
296C22#1909779811yescharacteristic
297C22#1909780311yescharacteristic
298C4—42no—
299C4—42no—
300C4—6282no—
301C4—6281no—
302C4#1909774111yescharacteristic
303C4#1909774411yescharacteristic
304C4#1909774511yescharacteristic
305C4#1909776711yescharacteristic
306C2—6282no—
307C2—12562no—
308C2#2375718211yescharacteristic
309C2#2375741311yescharacteristic
310C2#2375742111yescharacteristic
311C2#2375742211yescharacteristic
312C2#2375743411yescharacteristic
313C2#2375750511yescharacteristic
314C2#2375752211yescharacteristic
subgroupG/H
centerC23D4×D157
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#103×C157C2
socleC23×C157C2×D4
prime radicalC2×#19097746C2
hypercenterC2×#103D157
hypocenterC157#2000
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#14578941C2semidirect
2. C2×#19097747C2semidirect
4. C2×#19097693C2semidirect
5. C2×#19097683C2semidirect
6. #23757035C2semidirect
7. C2×#103×C157C2semidirect
51. C2×#11929434D4semidirect
53. C4⋊Dic157D4semidirect
62. C2×Q8×C157D4semidirect
84. C22×D157SD16semidirect
157. C157#2000semidirect
162. C2×#103D157semidirect
194. C2×Q8D628semidirect

Maximal subgroups

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

subgroupindex
C22×#145789412
C2×#190977472
C2×#190977462
C2×#190976932
C2×#190976832
#237570352
C2×#103×C1572
#2000157