Finite Group Databasebeta

← #34100075

Subgroups — #34100075

Order 20096.

612 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14578955C211yesmaximal, characteristic
2C2×#19097695C211yesmaximal, characteristic, complemented
3C2×#19097693C211yesmaximal, characteristic, complemented
4C2×#19097691C211yesmaximal, characteristic, complemented
5C2×#19097683C211yesmaximal, characteristic, complemented
6#23757130C211yesmaximal, characteristic, complemented
7C2×#103×C157C211yesmaximal, characteristic, complemented
8C2×#14578955—22no—
9C2×#14578955—22no—
10C2×#14578955—22no—
11C2×#14578955—22no—
12C2×#14578955—22no—
13C2×#14578955C414yescomplemented
14C22×D628—21no—
15C22×D628C2211yescharacteristic
16C22×C4×D157—21no—
17C22×C4×D157C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19C2×C4⋊Dic157C2211yescharacteristic
20C2×#14578931—21no—
21#19097695—22no—
22#19097693—22no—
23#19097691—22no—
24#19097683—22no—
25#19097678—21no—
26C22×Q8×C157C2211yescharacteristic
27C22×C8×C157C2211yescharacteristic
28C2×C4⋊C4×C157C2211yescharacteristic
29#103×C157—22no—
30C23×D157—21no—
31C23×D157D411yescharacteristic
32#14578955—28no—
33#14578955—28no—
34#14578955—28no—
35#14578955—28no—
36C2×D628—24no—
37C2×D628—24no—
38C2×D628—24no—
39C2×D628—21no—
40C2×D628—21no—
41C2×D628—21no—
42C2×D628C2×C412yes—
43C2×D628D412yes—
44C2×D628D412yes—
45C22×Dic157—21no—
46C22×Dic157D411yescharacteristic
47C2×C4×D157—24no—
48C2×C4×D157—24no—
49C2×C4×D157—24no—
50C2×C4×D157—21no—
51C2×C4×D157—21no—
52C2×C4×D157—21no—
53C2×C4×D157—21no—
54C2×C4×D157C2×C412yes—
55C2×C4×D157D412yes—
56C2×#11929434—21no—
57C2×#11929434—21no—
58C2×#11929434—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157—21no—
61#14578931—42no—
62C2×Q8×C157—21no—
63C2×Q8×C157—21no—
64C2×Q8×C157—21no—
65C2×Q8×C157C2×C412yes—
66C2×Q8×C157D412yes—
67C2×Q8×C157D412yes—
68C22×C4×C157—21no—
69C22×C4×C157—21no—
70C22×C4×C157C2311yescharacteristic
71C2×C8×C157—21no—
72C2×C8×C157—21no—
73C2×C8×C157—21no—
74C4⋊C4×C157—21no—
75C4⋊C4×C157—21no—
76C22×D157—22no—
77C22×D157—21no—
78C22×D157—21no—
79C22×D157—22no—
80C22×D157—22no—
81C22×D157—21no—
82C22×D157—21no—
83C22×D157—21no—
84C22×D157—21no—
85C22×D157—41no—
86C22×D157—41no—
87C22×D157—41no—
88C22×D157—41no—
89C22×D157C22⋊C412yes—
90D628—216no—
91D628—22no—
92D628—22no—
93D628—22no—
94D628C22⋊C414yescomplemented
95C2×Dic157—21no—
96C2×Dic157—21no—
97C2×Dic157—41no—
98C2×Dic157—41no—
99C2×Dic157—41no—
100C2×Dic157C22⋊C412yes—
101C4×D157—216no—
102C4×D157—22no—
103C4×D157—22no—
104C4×D157—22no—
105C4×D157—22no—
106#11929434—41no—
107C23×C157C2×D411yescharacteristic
108Q8×C157—22no—
109Q8×C157—22no—
110Q8×C157—22no—
111Q8×C157C22⋊C414yescomplemented
112C2×C4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157—41no—
116C2×C4×C157—41no—
117C2×C4×C157—41no—
118C2×C4×C157C22×C411yescharacteristic
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D411yescharacteristic
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157C2×D411yescharacteristic
124C8×C157—41no—
125C22×C157C2×C22⋊C411yescharacteristic
126C22×C157#12011yescharacteristic
127C22×C157#12011yescharacteristic
128C22×C157#13511yescharacteristic
129C22×C157#13511yescharacteristic
130C22×C157AGL(1,ℤ/8)11yescharacteristic
131C22×C157#13711yescharacteristic
132C2×D157—22no—
133C2×D157—24no—
134C2×D157—22no—
135C2×D157—22no—
136C2×D157—42no—
137C2×D157—42no—
138C2×D157—41no—
139C2×D157—41no—
140C2×D157—42no—
141C2×D157—42no—
142C2×D157—41no—
143C2×D157—41no—
144C2×D157—42no—
145C2×D157—42no—
146C4×C157—24no—
147C4×C157—81no—
148C4×C157C2×C22⋊C411yescharacteristic
149C4×C157C2×C22⋊C411yescharacteristic
150C4×C157#12011yescharacteristic
151C4×C157#12011yescharacteristic
152Dic157—22no—
153Dic157—81no—
154C2×C157#37911yescharacteristic
155C2×C157#41611yescharacteristic
156C2×C157#41711yescharacteristic
157C2×C157#44911yescharacteristic
158C2×C157#44911yescharacteristic
159C2×C157#45211yescharacteristic
160C2×C157#45211yescharacteristic
161D157—42no—
162D157—44no—
163C157#178811yescharacteristic, complemented, Sylow
164#1788—1571nomaximal, Sylow
165C22×#41—1571no—
166C2×#103—1571no—
167C2×#103D15711yescharacteristic, complemented
168C2×#102—1571no—
169C2×#102—1571no—
170C2×C22⋊C8—1571no—
171#380—1571no—
172C2×#41—3142no—
173C2×#41—3142no—
174C2×#41—3142no—
175C2×#41—3142no—
176C2×#41—3142no—
177C2×#41—1574no—
178C22×Q8C2×D15711yescharacteristic
179C22×D4—3141no—
180C22×D4—1571no—
181C23×C4—3141no—
182C23×C4—1571no—
183C22×C8—1571no—
184C22×C8C2×D15711yescharacteristic
185C2×C4⋊C4—1571no—
186C2×C4⋊C4C2×D15711yescharacteristic
187C2×C22⋊C4—3141no—
188#103—22no—
189#103—3142no—
190#102—3142no—
191#102—3142no—
192C22⋊C8—3142no—
193#95—3141no—
194C24—3141no—
195C24—1571no—
196#41—3148no—
197#41—3148no—
198#41—3148no—
199#41—3148no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8C4×D15712yes—
204C2×Q8C22⋊D15712yes—
205C2×Q8D62812yes—
206C2×D4—3144no—
207C2×D4—3144no—
208C2×D4—3144no—
209C2×D4—3141no—
210C2×D4—3141no—
211C2×D4—3141no—
212C2×D4—1572no—
213C2×D4—1572no—
214C2×D4—1572no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—3144no—
218C22×C4—3144no—
219C22×C4—3144no—
220C22×C4—3141no—
221C22×C4—3141no—
222C22×C4—3141no—
223C22×C4—3141no—
224C22×C4—3141no—
225C22×C4—1571no—
226C22×C4—1572no—
227C22×C4—1572no—
228C22×C4C22×D15711yescharacteristic
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—3141no—
233C2×C8—3141no—
234C2×C8—3141no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—3141no—
238C4⋊C4—3141no—
239C22⋊C4—6282no—
240C23—3142no—
241C23—3141no—
242C23—3142no—
243C23—3141no—
244C23—3142no—
245C23—3141no—
246C23—3141no—
247C23—3141no—
248C23—3141no—
249C23—6281no—
250C23—6281no—
251C23—6281no—
252C23—6281no—
253C23—1572no—
254C23D4×D15711yescharacteristic
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8#1457893114yescomplemented
259D4—31416no—
260D4—3142no—
261D4—3142no—
262D4—3142no—
263D4—1574no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—31416no—
271C2×C4—3141no—
272C2×C4—3141no—
273C2×C4—3142no—
274C2×C4—3142no—
275C2×C4—3142no—
276C2×C4—3142no—
277C2×C4—6281no—
278C2×C4—6281no—
279C2×C4—6281no—
280C2×C4—1572no—
281C2×C4C2×C4×D15711yescharacteristic
282C2×C4D4×D15711yescharacteristic
283C2×C4D4×D15711yescharacteristic
284C2×C4D4×D15711yescharacteristic
285C2×C4C2×C22⋊D15711yescharacteristic
286C2×C4C2×D62811yescharacteristic
287C8—41no—
288C8—6281no—
289C22—3142no—
290C22—3144no—
291C22—3142no—
292C22—3142no—
293C22—6281no—
294C22—6282no—
295C22—6281no—
296C22—6282no—
297C22—6281no—
298C22—6282no—
299C22—6281no—
300C22—6282no—
301C22—6282no—
302C22—6282no—
303C22C22⋊C4×D15711yescharacteristic
304C22#1909773911yescharacteristic
305C22#1909774111yescharacteristic
306C22#1909779211yescharacteristic
307C22#1909779411yescharacteristic
308C22#1909779811yescharacteristic
309C22#1909780311yescharacteristic
310C4—24no—
311C4—81no—
312C4—3142no—
313C4—12561no—
314C4C22⋊C4×D15711yescharacteristic
315C4C2×#1457893111yescharacteristic
316C4#1909773911yescharacteristic
317C4#1909774111yescharacteristic
318C2—6282no—
319C2—6284no—
320C2#2375712811yescharacteristic
321C2#2375726811yescharacteristic
322C2#2375727311yescharacteristic
323C2#2375741711yescharacteristic
324C2#2375742111yescharacteristic
325C2#2375743411yescharacteristic
326C2#2375743811yescharacteristic
subgroupG/H
centerC23D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#103×C157C2
socleC23×C157C2×D4
prime radicalC22×#14578955C2
cosocleC22×C4×C157C23
hypercenterC2×#103D157
hypocenterC157#1788
LCSG ▷ C2×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
2. C2×#19097695C2semidirect
3. C2×#19097693C2semidirect
4. C2×#19097691C2semidirect
5. C2×#19097683C2semidirect
6. #23757130C2semidirect
7. C2×#103×C157C2semidirect
13. C2×#14578955C4semidirect
94. D628C22⋊C4semidirect
111. Q8×C157C22⋊C4semidirect
163. C157#1788semidirect
167. C2×#103D157semidirect
258. Q8#14578931semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#190976952
C2×#190976932
C2×#190976912
C2×#190976832
#237571302
C2×#103×C1572
#1788157