Finite Group Databasebeta

← #34100543

Subgroups — #34100543

Order 20096.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19097741C211yesmaximal, characteristic
2#23757746C211yesmaximal, characteristic, complemented
3C2×#19097685C211yesmaximal, characteristic, complemented
4#23757128C211yesmaximal, characteristic, complemented
5#23757128C211yesmaximal, characteristic, complemented
6#23757039C211yesmaximal, characteristic, complemented
7#342×C157C211yesmaximal, characteristic, complemented
8C24×D157C2211yescharacteristic
9C22×D628—21no—
10C22×C22⋊D157C2211yescharacteristic
11#19097741—22no—
12#19097741C414yescomplemented
13C4×C22⋊D157—22no—
14#19097731—22no—
15#19097727—21no—
16C2×C22⋊Dic157C2211yescharacteristic
17C2×C22⋊Dic157C2211yescharacteristic
18C2×#14578931—22no—
19C2×#14578931—21no—
20C2×#14578931—22no—
21C22⋊C4×D157—21no—
22#19097685—22no—
23C2×C22⋊C4×C157C2211yescharacteristic
24C2×C22⋊C4×C157C2211yescharacteristic
25C2×C22⋊C4×C157C2211yescharacteristic
26C23×D157—21no—
27C23×D157—21no—
28C23×D157—21no—
29C23×D157—21no—
30C23×D157—21no—
31C23×D157—21no—
32C23×D157—44no—
33C23×D157C2×C412yescomplemented
34C23×D157D411yescharacteristic
35C2×D628—24no—
36C2×D628—44no—
37C2×C22⋊D157—21no—
38C2×C22⋊D157—21no—
39C2×C22⋊D157—21no—
40C2×C22⋊D157—21no—
41C2×C22⋊D157C2×C412yes—
42C2×C22⋊D157D412yes—
43C22×Dic157—21no—
44C22×Dic157D411yescharacteristic
45C22×Dic157D412yescomplemented
46C2×C4×D157—22no—
47#14578933—22no—
48C22⋊Dic157—21no—
49C22⋊Dic157—21no—
50C22⋊Dic157—21no—
51C22⋊Dic157—21no—
52#14578931—22no—
53#14578931—24no—
54#14578931—44no—
55#14578931—44no—
56C4×Dic157—22no—
57C24×C157C2311yescharacteristic
58C22×C4×C157—21no—
59C22×C4×C157—21no—
60C22×C4×C157D412yescomplemented
61C22⋊C4×C157—21no—
62C22⋊C4×C157—21no—
63C22⋊C4×C157—21no—
64C22⋊C4×C157—21no—
65C22⋊C4×C157—21no—
66C22⋊C4×C157C2×C412yescomplemented
67C22×D157—24no—
68C22×D157—24no—
69C22×D157—24no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157—22no—
73C22×D157—21no—
74C22×D157—21no—
75C22×D157—44no—
76C22×D157—44no—
77C22×D157—44no—
78C22×D157—44no—
79C22×D157—42no—
80C22×D157—42no—
81C22×D157—42no—
82C22×D157—41no—
83C22×D157—41no—
84C22×D157—41no—
85C22×D157—41no—
86C22×D157—42no—
87C22×D157C22⋊C412yescomplemented
88D628—48no—
89C22⋊D157—22no—
90C22⋊D157—22no—
91C22⋊D157—22no—
92C22⋊D157—22no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—22no—
96C2×Dic157—42no—
97C2×Dic157—42no—
98C2×Dic157—41no—
99C2×Dic157C22⋊C412yescomplemented
100C4×D157—82no—
101C23×C157—42no—
102C23×C157C22×C411yescharacteristic
103C23×C157C2×D411yescharacteristic
104C23×C157C2×D411yescharacteristic
105C23×C157C2×D411yescharacteristic
106C23×C157C2×D411yescharacteristic
107C23×C157#4111yescharacteristic
108C23×C157#4111yescharacteristic
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—42no—
112C2×C4×C157—42no—
113C2×C4×C157—42no—
114C2×C4×C157—41no—
115C22×C157—22no—
116C22×C157—22no—
117C22×C157—22no—
118C22×C157—41no—
119C22×C157—41no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157C2×C22⋊C411yescharacteristic
123C22×C157C4×D411yescharacteristic
124C22×C157C4×D411yescharacteristic
125C22×C157#12011yescharacteristic
126C22×C157#12111yescharacteristic
127C22×C157#12111yescharacteristic
128C22×C157#12311yescharacteristic
129C2×D157—24no—
130C2×D157—24no—
131C2×D157—24no—
132C2×D157—22no—
133C2×D157—44no—
134C2×D157—42no—
135C2×D157—44no—
136C2×D157—42no—
137C2×D157—42no—
138C2×D157—44no—
139C2×D157—42no—
140C2×D157—42no—
141C2×D157—44no—
142C2×D157—42no—
143C2×D157—81no—
144C4×C157—42no—
145C4×C157—42no—
146C4×C157—81no—
147Dic157—22no—
148Dic157—42no—
149Dic157—81no—
150C2×C157—21no—
151C2×C157—21no—
152C2×C157—21no—
153C2×C157—21no—
154C2×C157—21no—
155C2×C157—21no—
156C2×C157#38611yescharacteristic
157C2×C157B(4,𝔽2)12yes—
158D157—44no—
159D157—42no—
160C157#188911yescharacteristic, complemented, Sylow
161#1889—1571nomaximal, Sylow
162C2×#120—1571no—
163#518—1571no—
164C2×#99—1571no—
165#379—1571no—
166#379—1571no—
167#342—1571no—
168#342D15711yescharacteristic, complemented
169C25—1571no—
170C22×D4—3141no—
171C22×D4—1571no—
172#120—3142no—
173#120—1574no—
174C4×D4—3142no—
175C4×D4—3142no—
176#117—3141no—
177C2×C22⋊C4—3141no—
178C2×C22⋊C4—3141no—
179C2×C22⋊C4—3142no—
180C2×C22⋊C4—3142no—
181C2×C22⋊C4—1571no—
182C2×C22⋊C4—1571no—
183C2×C22⋊C4C2×D15711yescharacteristic
184C2×C22⋊C4C2×D15711yescharacteristic
185C2×C22⋊C4C2×D15711yescharacteristic
186#99—3142no—
187C24—3141no—
188C24—3141no—
189C24—3141no—
190C24—3141no—
191C24—3141no—
192C24—3141no—
193C24—6284no—
194C24—1572no—
195C24—1571no—
196C24C22×D15711yescharacteristic
197C2×D4—3144no—
198C2×D4—3141no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—3141no—
202C2×D4—6284no—
203C2×D4—1572no—
204C2×D4—1572no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3142no—
208C22×C4—3141no—
209C22×C4—1571no—
210C22×C4—1572no—
211C22×C4D62812yescomplemented
212C4⋊C4—3142no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—3142no—
219C22⋊C4—3144no—
220C22⋊C4—3141no—
221C22⋊C4—3141no—
222C22⋊C4—3141no—
223C22⋊C4—3141no—
224C22⋊C4—6284no—
225C22⋊C4—6284no—
226C22⋊C4C4×D15712yescomplemented
227C42—3142no—
228C23—42no—
229C23—3144no—
230C23—3144no—
231C23—3144no—
232C23—3142no—
233C23—3142no—
234C23—3142no—
235C23—3141no—
236C23—3141no—
237C23—6282no—
238C23—6282no—
239C23—6282no—
240C23—6282no—
241C23—6284no—
242C23—6284no—
243C23—6284no—
244C23—6284no—
245C23—6281no—
246C23—6281no—
247C23—6281no—
248C23—6281no—
249C23—1572no—
250C23C2×C4×D15711yescharacteristic
251C23D4×D15711yescharacteristic
252C23D4×D15711yescharacteristic
253C23D4×D15711yescharacteristic
254C23C2×D62811yescharacteristic
255C23#1457895411yescharacteristic
256C23#1457895511yescharacteristic
257D4—3142no—
258D4—3142no—
259D4—3142no—
260D4—3142no—
261D4—6288no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—3141no—
269C2×C4—3141no—
270C2×C4—3142no—
271C2×C4—6282no—
272C2×C4—6282no—
273C2×C4—6281no—
274C2×C4—1572no—
275C2×C4—12562no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—3144no—
284C22—3144no—
285C22—3144no—
286C22—3142no—
287C22—6282no—
288C22—6282no—
289C22—6284no—
290C22—6284no—
291C22—6282no—
292C22—6282no—
293C22—6284no—
294C22—6282no—
295C22—6282no—
296C22—6284no—
297C22—12561no—
298C22C22⋊C4×D15711yescharacteristic
299C22#1909773311yescharacteristic
300C22C4×D62811yescharacteristic
301C22#1909774111yescharacteristic
302C22#1909774411yescharacteristic
303C22#1909774611yescharacteristic
304C22#1909775611yescharacteristic
305C4—42no—
306C4—42no—
307C4—81no—
308C4—3142no—
309C4—6282no—
310C4—12561no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—6284no—
318C2—6282no—
319C2#2375715111yescharacteristic
320C2#2375746112yes—
subgroupG/H
centerC22#19097741
commutatorC23×C157C22×C4
FrattiniC24C22×D157
Fitting#342×C157C2
socleC22×C157#120
prime radicalC2×#19097741C2
cosocleC24×C157C23
hypercenter#342D157
hypocenterC157#1889
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #23757746C2semidirect
3. C2×#19097685C2semidirect
4. #23757128C2semidirect
5. #23757128C2semidirect
6. #23757039C2semidirect
7. #342×C157C2semidirect
12. #19097741C4semidirect
33. C23×D157C2×C4semidirect
45. C22×Dic157D4semidirect
60. C22×C4×C157D4semidirect
66. C22⋊C4×C157C2×C4semidirect
87. C22×D157C22⋊C4semidirect
99. C2×Dic157C22⋊C4semidirect
160. C157#1889semidirect
168. #342D157semidirect
211. C22×C4D628semidirect
226. C22⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977412
#237577462
C2×#190976852
#237571282
#237570392
#342×C1572
#1889157