Finite Group Databasebeta

← #34100049

Subgroups — #34100049

Order 20096.

880 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 105 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×D4×D157C211yesmaximal, characteristic
2C2×#19097686C211yesmaximal, characteristic, complemented
3C2×#19097685C212yesmaximal, complemented
4#23757128C212yesmaximal, complemented
5C2×#99×C157C211yesmaximal, characteristic, complemented
6C24×D157C411yescharacteristic, complemented
7C24×D157C2211yescharacteristic
8C22×D628C411yescharacteristic, complemented
9C22×C22⋊D157C411yescharacteristic, complemented
10C22×C22⋊D157C2211yescharacteristic
11C2×D4×D157—22no—
12C2×D4×D157—22no—
13C2×D4×D157—24no—
14C2×D4×D157—24no—
15C22×C4×D157C411yescharacteristic, complemented
16C2×C22⋊Dic157C2212yescomplemented
17C2×#14578931—24no—
18#19097686—22no—
19#19097685—24no—
20C22×D4×C157C2211yescharacteristic
21C2×C22⋊C4×C157C2212yescomplemented
22#99×C157—22no—
23C23×D157—22no—
24C23×D157—22no—
25C23×D157—22no—
26C23×D157—22no—
27C23×D157—21no—
28C23×D157—21no—
29C23×D157—44no—
30C23×D157—44no—
31C23×D157C2×C411yescharacteristic
32C23×D157C2×C411yescharacteristic
33C23×D157C2×C411yescharacteristic
34C23×D157D414yes—
35C2×D628—22no—
36C2×D628—22no—
37C2×D628—22no—
38C2×C22⋊D157—22no—
39C2×C22⋊D157—22no—
40C2×C22⋊D157—22no—
41C2×C22⋊D157—22no—
42C2×C22⋊D157—21no—
43C2×C22⋊D157—21no—
44C2×C22⋊D157D414yes—
45D4×D157—216no—
46D4×D157—216no—
47C22×Dic157—22no—
48C22×Dic157C2×C411yescharacteristic
49C2×C4×D157—22no—
50C2×C4×D157—22no—
51C2×C4×D157—22no—
52C22⋊Dic157—22no—
53C22⋊Dic157—22no—
54#14578931—48no—
55C24×C157C2×C411yescharacteristic
56C24×C157C2311yescharacteristic
57C2×D4×C157—22no—
58C2×D4×C157—22no—
59C2×D4×C157D412yes—
60C2×D4×C157D412yes—
61C22×C4×C157—22no—
62C22×C4×C157C2×C411yescharacteristic
63C22⋊C4×C157—22no—
64C22⋊C4×C157—22no—
65C22×D157—21no—
66C22×D157—21no—
67C22×D157—24no—
68C22×D157—28no—
69C22×D157—21no—
70C22×D157—21no—
71C22×D157—21no—
72C22×D157—21no—
73C22×D157—24no—
74C22×D157—24no—
75C22×D157—28no—
76C22×D157—42no—
77C22×D157—44no—
78C22×D157—48no—
79C22×D157—44no—
80C22×D157—44no—
81C22×D157—48no—
82C22×D157—48no—
83C22×D157—44no—
84C22×D157—44no—
85C22×D157—42no—
86C22×D157C22⋊C412yes—
87C22×D157C22⋊C412yes—
88C22×D157C22⋊C412yescomplemented
89D628—28no—
90C22⋊D157—28no—
91C22⋊D157—24no—
92C22⋊D157—24no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—44no—
96C2×Dic157—42no—
97C2×Dic157C22⋊C412yescomplemented
98C4×D157—28no—
99C23×C157—21no—
100C23×C157—21no—
101C23×C157—42no—
102C23×C157—42no—
103C23×C157C22⋊C412yescomplemented
104C23×C157C22×C411yescharacteristic
105C23×C157C2×D412yes—
106C23×C157C2×D412yes—
107C23×C157C2×D411yescharacteristic
108C23×C157C2×D411yescharacteristic
109D4×C157—24no—
110D4×C157—24no—
111C2×C4×C157—21no—
112C2×C4×C157—21no—
113C2×C4×C157—44no—
114C2×C4×C157—42no—
115C2×C4×C157C22⋊C412yescomplemented
116C22×C157—24no—
117C22×C157—22no—
118C22×C157—22no—
119C22×C157—42no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157—42no—
123C22×C157—42no—
124C22×C157—42no—
125C22×C157C2×C22⋊C411yescharacteristic
126C22×C157C2×C22⋊C411yescharacteristic
127C22×C157C2×C22⋊C411yescharacteristic
128C22×C157#12012yes—
129C22×C157#12012yes—
130C2×D157—24no—
131C2×D157—24no—
132C2×D157—22no—
133C2×D157—22no—
134C2×D157—22no—
135C2×D157—22no—
136C2×D157—24no—
137C2×D157—48no—
138C2×D157—48no—
139C2×D157—416no—
140C2×D157—42no—
141C2×D157—42no—
142C2×D157—42no—
143C2×D157—42no—
144C2×D157—42no—
145C2×D157#9914yescomplemented
146C4×C157—22no—
147C4×C157—82no—
148Dic157—22no—
149Dic157—82no—
150C2×C157—22no—
151C2×C157—22no—
152C2×C157—21no—
153C2×C157—21no—
154C2×C157—42no—
155C2×C157#37911yescharacteristic
156C2×C157C2×#9912yes—
157D157—24no—
158D157—42no—
159D157—42no—
160C157#178111yescharacteristic, complemented, Sylow
161#1781—1571nomaximal, Sylow
162C23×D4—1571no—
163C2×#99—1571no—
164C2×#99—1572no—
165C2×#99D15711yescharacteristic, complemented
166#379—1572no—
167C25—1571no—
168C25—1571no—
169C22×D4—3144no—
170C22×D4—3144no—
171C22×D4—3142no—
172C22×D4—3142no—
173C22×D4—1571no—
174C22×D4—1571no—
175C22×D4—1571no—
176C22×D4C2×D15711yescharacteristic
177C23×C4—1571no—
178C2×C22⋊C4—3144no—
179C2×C22⋊C4—1572no—
180C2×C22⋊C4C2×D15712yescomplemented
181#99—22no—
182#99—3144no—
183#99—3142no—
184C24—3142no—
185C24—3142no—
186C24—3142no—
187C24—3141no—
188C24—3141no—
189C24—3142no—
190C24—6284no—
191C24—6284no—
192C24—1571no—
193C24—1574no—
194C24—1571no—
195C24—1571no—
196C24C4×D15711yescharacteristic
197C24C22×D15711yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—3142no—
201C2×D4—3142no—
202C2×D4—31416no—
203C2×D4—31416no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3141no—
208C2×D4—3142no—
209C2×D4—3142no—
210C2×D4—3142no—
211C2×D4—1574no—
212C2×D4C22⋊D15712yes—
213C2×D4D62812yes—
214C22×C4—22no—
215C22×C4—3142no—
216C22×C4—3142no—
217C22×C4—3142no—
218C22×C4—3142no—
219C22×C4—1571no—
220C22×C4C4×D15711yescharacteristic
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—3142no—
224C22⋊C4—3142no—
225C22⋊C4—6288no—
226C23—21no—
227C23—21no—
228C23—42no—
229C23—42no—
230C23—3144no—
231C23—3148no—
232C23—3148no—
233C23—3141no—
234C23—3141no—
235C23—3141no—
236C23—3141no—
237C23—3141no—
238C23—3141no—
239C23—3144no—
240C23—3144no—
241C23—6284no—
242C23—6282no—
243C23—6282no—
244C23—6288no—
245C23—6284no—
246C23—6284no—
247C23—6288no—
248C23—6288no—
249C23—6284no—
250C23—6284no—
251C23—1572no—
252C23—1572no—
253C23—1572no—
254C23#1457893112yescomplemented
255C23C2×C4×D15711yescharacteristic
256C23D4×D15712yes—
257C23D4×D15712yes—
258C23C2×C22⋊D15711yescharacteristic
259C23C2×D62811yescharacteristic
260D4—24no—
261D4—24no—
262D4—3148no—
263D4—3148no—
264D4—3144no—
265D4—3144no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—3148no—
271C2×C4—3141no—
272C2×C4—3141no—
273C2×C4—6284no—
274C2×C4—6282no—
275C2×C4—1572no—
276C2×C4#1457893112yescomplemented
277C22—24no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—3144no—
287C22—3144no—
288C22—3142no—
289C22—3142no—
290C22—3142no—
291C22—3142no—
292C22—3144no—
293C22—6288no—
294C22—6288no—
295C22—6282no—
296C22—6282no—
297C22—6282no—
298C22—6282no—
299C22—62816no—
300C22—6282no—
301C22—1574no—
302C22C22⋊C4×D15711yescharacteristic
303C22C22⋊C4×D15711yescharacteristic
304C22C2×#1457893111yescharacteristic
305C22#1909773912yes—
306C22#1909774112yes—
307C4—22no—
308C4—82no—
309C4—3142no—
310C4—12562no—
311C2—22no—
312C2—22no—
313C2—21no—
314C2—21no—
315C2—42no—
316C2—3144no—
317C2—6282no—
318C2—6282no—
319C2#2375712811yescharacteristic
320C2#99×D15712yes—
subgroupG/H
centerC22C22⋊C4×D157
commutatorC23×C157C22×C4
FrattiniC24C22×D157
FittingC2×#99×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC22×D4×D157C2
cosocleC24×C157C23
hypercenterC2×#99D157
hypocenterC157#1781
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#19097686C2semidirect
3. C2×#19097685C2semidirect
4. #23757128C2semidirect
5. C2×#99×C157C2semidirect
6. C24×D157C4semidirect
8. C22×D628C4semidirect
9. C22×C22⋊D157C4semidirect
15. C22×C4×D157C4semidirect
16. C2×C22⋊Dic157C22semidirect
21. C2×C22⋊C4×C157C22semidirect
88. C22×D157C22⋊C4semidirect
97. C2×Dic157C22⋊C4semidirect
103. C23×C157C22⋊C4semidirect
115. C2×C4×C157C22⋊C4semidirect
145. C2×D157#99semidirect
160. C157#1781semidirect
165. C2×#99D157semidirect
180. C2×C22⋊C4C2×D157semidirect
254. C23#14578931semidirect
276. C2×C4#14578931semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×#190976862
C2×#190976852
#237571282
C2×#99×C1572
#1781157