Finite Group Databasebeta

← #34103461

Subgroups — #34103461

Order 20096.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D157C211yesmaximal, characteristic
2C2×#19097753C212yesmaximal, complemented
3C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
4#23757189C212yesmaximal, complemented
5#23757188C212yesmaximal, complemented
6#23757149C212yesmaximal, complemented
7#23757148C211yesmaximal, characteristic, complemented
8#23757145C211yesmaximal, characteristic, complemented
9#23757129C211yesmaximal, characteristic, complemented
10#23757128C211yesmaximal, characteristic, complemented
11#379×C157C211yesmaximal, characteristic, complemented
12C24×D157C2211yescharacteristic
13C22×C22⋊D157—24no—
14C22×C22⋊D157—28no—
15C22×C22⋊D157C2212yes—
16C22×C22⋊D157C2212yes—
17C23×Dic157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic, complemented
19C22×C4×D157C2211yescharacteristic, complemented
20#19097753—28no—
21C2×C4⋊Dic157C2212yescomplemented
22C2×#14578933C2212yescomplemented
23C2×C22⋊Dic157C2211yescharacteristic
24C2×C22⋊Dic157C2211yescharacteristic
25C2×C22⋊Dic157C2212yescomplemented
26C2×C22⋊Dic157C2212yescomplemented
27C2×#14578931C2212yescomplemented
28C2×#14578931C2211yescharacteristic, complemented
29C2×#14578931C2211yescharacteristic, complemented
30C2×#14578931C2212yescomplemented
31C22⋊C4×D157—24no—
32#19097678C2212yescomplemented
33#19097678C2212yescomplemented
34C25×C157C2211yescharacteristic
35C2×C22⋊C4×C157C2211yescharacteristic
36C2×C22⋊C4×C157C2211yescharacteristic
37C2×C22⋊C4×C157C2212yescomplemented
38C2×C22⋊C4×C157C2212yescomplemented
39C23×D157—24no—
40C23×D157—22no—
41C23×D157—22no—
42C23×D157—42no—
43C23×D157D414yes—
44C23×D157C2311yescharacteristic
45C23×D157C2311yescharacteristic
46C2×C22⋊D157—232no—
47C2×C22⋊D157—24no—
48C2×C22⋊D157—24no—
49C2×C22⋊D157—24no—
50C2×C22⋊D157—24no—
51C2×C22⋊D157—24no—
52C2×C22⋊D157D418yes—
53C22×Dic157—24no—
54C22×Dic157—22no—
55C22×Dic157C2311yescharacteristic
56C22×Dic157C2312yes—
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2312yes—
59C2×C4×D157—22no—
60C2×C4×D157—22no—
61C2×C4×D157—24no—
62C2×C4×D157—42no—
63C4⋊Dic157—24no—
64#14578933—24no—
65C22⋊Dic157—22no—
66C22⋊Dic157—24no—
67C22⋊Dic157—22no—
68C22⋊Dic157—42no—
69#14578931—22no—
70#14578931—22no—
71#14578931—24no—
72#14578931—42no—
73C24×C157—24no—
74C24×C157—22no—
75C24×C157—22no—
76C24×C157—22no—
77C24×C157—42no—
78C24×C157C2312yes—
79C24×C157C2311yescharacteristic
80C22×C4×C157C2311yescharacteristic
81C22×C4×C157C2312yes—
82C22×C4×C157C2311yescharacteristic
83C22⋊C4×C157—22no—
84C22⋊C4×C157—24no—
85C22⋊C4×C157—22no—
86C22⋊C4×C157—42no—
87C22×D157—21no—
88C22×D157—21no—
89C22×D157—22no—
90C22×D157—28no—
91C22×D157—22no—
92C22×D157—21no—
93C22×D157—21no—
94C22×D157—22no—
95C22×D157—28no—
96C22×D157—44no—
97C22×D157—48no—
98C22×D157—48no—
99C22×D157C2×D412yes—
100C22×D157C2×D412yes—
101C22×D157C2×D414yes—
102C22⋊D157—216no—
103C22⋊D157—216no—
104C2×Dic157—28no—
105C2×Dic157—22no—
106C2×Dic157—22no—
107C2×Dic157—21no—
108C2×Dic157—21no—
109C2×Dic157—22no—
110C2×Dic157—22no—
111C2×Dic157—22no—
112C2×Dic157—22no—
113C2×Dic157—22no—
114C2×Dic157C2×D414yes—
115C4×D157—44no—
116C4×D157—44no—
117C23×C157—22no—
118C23×C157—22no—
119C23×C157—28no—
120C23×C157—22no—
121C23×C157—22no—
122C23×C157—22no—
123C23×C157—22no—
124C23×C157—22no—
125C23×C157—22no—
126C23×C157—48no—
127C23×C157—48no—
128C23×C157—44no—
129C23×C157—44no—
130C23×C157C2×D414yes—
131C23×C157C2×D412yes—
132C23×C157#4112yes—
133C23×C157#4112yes—
134C23×C157C2411yescharacteristic
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—22no—
138C2×C4×C157—22no—
139C2×C4×C157—22no—
140C2×C4×C157—22no—
141C2×C4×C157—22no—
142C22×C157—22no—
143C22×C157—22no—
144C22×C157—24no—
145C22×C157—24no—
146C22×C157—24no—
147C22×C157—24no—
148C22×C157—24no—
149C22×C157—416no—
150C22×C157—42no—
151C22×C157—42no—
152C22×C157—42no—
153C22×C157—42no—
154C22×C157#12314yescomplemented
155C22×C157C22×D411yescharacteristic
156C22×C157C22×D412yes—
157C22×C157C2×#4111yescharacteristic
158C22×C157C2×#4111yescharacteristic
159C22×C157H(5,𝔽2)12yes—
160C2×D157—22no—
161C2×D157—22no—
162C2×D157—24no—
163C2×D157—24no—
164C2×D157—22no—
165C2×D157—22no—
166C2×D157—22no—
167C2×D157—416no—
168C2×D157—42no—
169C2×D157—42no—
170C2×D157#12014yes—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157—42no—
174Dic157—24no—
175Dic157—41no—
176Dic157—41no—
177Dic157—42no—
178C2×C157—24no—
179C2×C157—44no—
180C2×C157C2×#12011yescharacteristic
181C2×C157C2×#12311yescharacteristic
182C2×C157#53411yescharacteristic
183C2×C157#54612yes—
184C2×C157#54612yes—
185D157—24no—
186D157—42no—
187C157#240511yescharacteristic, complemented, Sylow
188#2405—1571nomaximal, Sylow
189C23×D4—1571no—
190C2×#123—1572no—
191C22×C22⋊C4—1571no—
192#394—1572no—
193#394—1572no—
194#386—1572no—
195#385—1571no—
196#385—1571no—
197#379—1571no—
198#379—1571no—
199#379D15711yescharacteristic, complemented
200C25—1571no—
201C25C2×D15711yescharacteristic
202C22×D4—3144no—
203C22×D4—3148no—
204C22×D4—1572no—
205C22×D4—1572no—
206C23×C4—1571no—
207C23×C4—1571no—
208C23×C4—1571no—
209#123—3148no—
210C2×C4⋊C4—1572no—
211C2×C4⋊C4—1572no—
212C2×C22⋊C4—3144no—
213C2×C22⋊C4—1571no—
214C2×C22⋊C4—1572no—
215C2×C22⋊C4—1571no—
216C2×C22⋊C4—1571no—
217C2×C22⋊C4—1571no—
218C2×C22⋊C4—1572no—
219C2×C22⋊C4—1572no—
220C2×C22⋊C4—1572no—
221C2×C22⋊C4C2×D15711yescharacteristic
222C2×C22⋊C4C2×D15711yescharacteristic
223C2×C22⋊C4C2×D15712yescomplemented
224C2×C22⋊C4C2×D15712yescomplemented
225#95—1572no—
226#95—1572no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—3144no—
233C24—3142no—
234C24—3142no—
235C24—6282no—
236C24—1571no—
237C24—1574no—
238C24—1571no—
239C24C22×D15712yes—
240C24C22×D15711yescharacteristic
241C2×D4—31432no—
242C2×D4—3144no—
243C2×D4—3144no—
244C2×D4—3144no—
245C2×D4—3144no—
246C2×D4—3144no—
247C2×D4—1578no—
248C22×C4—3142no—
249C22×C4—3142no—
250C22×C4—3144no—
251C22×C4—3142no—
252C22×C4—3144no—
253C22×C4—6282no—
254C22×C4—1571no—
255C22×C4—1572no—
256C22×C4—1571no—
257C22×C4—1572no—
258C22×C4C22×D15711yescharacteristic
259C22×C4C22×D15712yes—
260C22×C4C22×D15711yescharacteristic
261C4⋊C4—3144no—
262C4⋊C4—3144no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—3142no—
268C22⋊C4—3142no—
269C22⋊C4—3144no—
270C22⋊C4—3142no—
271C22⋊C4—3144no—
272C22⋊C4—3142no—
273C22⋊C4—6282no—
274C22⋊C4—6282no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—3148no—
289C23—3141no—
290C23—3141no—
291C23—3142no—
292C23—3148no—
293C23—3142no—
294C23—3141no—
295C23—3141no—
296C23—3142no—
297C23—6284no—
298C23—6288no—
299C23—6288no—
300C23—1572no—
301C23—1574no—
302C23—1572no—
303C23D4×D15714yes—
304C23D4×D15712yes—
305C23#1457895312yes—
306C23#1457895412yes—
307C23C23×D15711yescharacteristic
308D4—31416no—
309D4—31416no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—3148no—
318C2×C4—3142no—
319C2×C4—3142no—
320C2×C4—3142no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3142no—
324C2×C4—3142no—
325C2×C4—3142no—
326C2×C4—3142no—
327C2×C4—6284no—
328C2×C4—6284no—
329C2×C4—1574no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—3142no—
343C22—3144no—
344C22—3144no—
345C22—3142no—
346C22—3142no—
347C22—3142no—
348C22—3142no—
349C22—62816no—
350C22—6282no—
351C22—6282no—
352C22—1574no—
353C22#1909775314yescomplemented
354C22C2×D4×D15711yescharacteristic
355C22C2×D4×D15712yes—
356C22C2×#1457895311yescharacteristic
357C22C2×#1457895411yescharacteristic
358C22#1909781612yes—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—3144no—
363C4—6281no—
364C4—6282no—
365C4—6281no—
366C2—24no—
367C2—44no—
368C2—3144no—
369C2—6282no—
370C2#120×D15711yescharacteristic
371C2C2×#1909775311yescharacteristic
372C2#2375783311yescharacteristic
373C2#2375791012yes—
374C2#2375791312yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#379×C157C2
prime radicalC23×C22⋊D157C2
hypercenter#379D157
hypocenterC157#2405
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#19097753C2semidirect
3. C2×C22⋊C4×D157C2semidirect
4. #23757189C2semidirect
5. #23757188C2semidirect
6. #23757149C2semidirect
7. #23757148C2semidirect
8. #23757145C2semidirect
9. #23757129C2semidirect
10. #23757128C2semidirect
11. #379×C157C2semidirect
18. C22×C4×D157C22semidirect
19. C22×C4×D157C22semidirect
21. C2×C4⋊Dic157C22semidirect
22. C2×#14578933C22semidirect
25. C2×C22⋊Dic157C22semidirect
26. C2×C22⋊Dic157C22semidirect
27. C2×#14578931C22semidirect
28. C2×#14578931C22semidirect
29. C2×#14578931C22semidirect
30. C2×#14578931C22semidirect
32. #19097678C22semidirect
33. #19097678C22semidirect
37. C2×C22⋊C4×C157C22semidirect
38. C2×C22⋊C4×C157C22semidirect
154. C22×C157#123semidirect
187. C157#2405semidirect
199. #379D157semidirect
223. C2×C22⋊C4C2×D157semidirect
224. C2×C22⋊C4C2×D157semidirect
353. C22#19097753semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1572
C2×#190977532
C2×C22⋊C4×D1572
#237571892
#237571882
#237571492
#237571482
#237571452
#237571292
#237571282
#379×C1572
#2405157