Finite Group Databasebeta

← #34103446

Subgroups — #34103446

Order 20096.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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, complemented
2C2×#19097740C212yesmaximal, complemented
3C2×#19097739C212yesmaximal, complemented
4#23757181C214yesmaximal, complemented
5#23757153C212yesmaximal, complemented
6#23757152C212yesmaximal, complemented
7#23757129C211yesmaximal, characteristic, complemented
8C23×D4×C157C211yesmaximal, characteristic, complemented
9C24×D157C2211yescharacteristic, complemented
10C22×C22⋊D157—24no—
11C22×C22⋊D157—28no—
12C22×C22⋊D157C2212yes—
13C22×C22⋊D157C2212yescomplemented
14C22×C22⋊D157C2214yescomplemented
15C23×Dic157C2211yescharacteristic, complemented
16#19097740—28no—
17#19097739—28no—
18C2×C22⋊Dic157C2212yescomplemented
19C2×C22⋊Dic157C2212yescomplemented
20C2×C22⋊Dic157C2214yescomplemented
21C2×C22⋊Dic157C2212yescomplemented
22C2×#14578931C2214yescomplemented
23#19097678C2214yescomplemented
24C25×C157C2211yescharacteristic
25C25×C157C2211yescharacteristic, complemented
26C22×D4×C157—24no—
27C22×D4×C157—28no—
28C22×D4×C157C2212yes—
29C22×D4×C157C2212yescomplemented
30C23×C4×C157C2211yescharacteristic, complemented
31C23×D157—24no—
32C23×D157—22no—
33C23×D157—24no—
34C23×D157—42no—
35C23×D157C2312yes—
36C2×C22⋊D157—232no—
37C2×C22⋊D157—24no—
38C2×C22⋊D157—28no—
39C2×C22⋊D157—24no—
40C2×C22⋊D157—24no—
41C2×C22⋊D157—24no—
42C2×C22⋊D157—24no—
43C2×C22⋊D157—28no—
44C2×C22⋊D157—44no—
45C2×C22⋊D157D418yescomplemented
46C22×Dic157—24no—
47C22×Dic157—22no—
48C22×Dic157C2314yes—
49C22×Dic157C2312yes—
50C22⋊Dic157—28no—
51C22⋊Dic157—24no—
52C22⋊Dic157—24no—
53C22⋊Dic157—42no—
54#14578931—28no—
55C24×C157—24no—
56C24×C157—22no—
57C24×C157—24no—
58C24×C157—24no—
59C24×C157—24no—
60C24×C157—42no—
61C24×C157—42no—
62C24×C157D414yescomplemented
63C24×C157C2311yescharacteristic
64C24×C157C2312yes—
65C24×C157C2311yescharacteristic
66C24×C157C2311yescharacteristic
67C2×D4×C157—232no—
68C2×D4×C157—24no—
69C2×D4×C157—24no—
70C2×D4×C157—24no—
71C2×D4×C157—24no—
72C2×D4×C157—24no—
73C2×D4×C157D418yescomplemented
74C22×C4×C157—22no—
75C22×C4×C157—24no—
76C22×C4×C157C2312yes—
77C22×D157—28no—
78C22×D157—22no—
79C22×D157—22no—
80C22×D157—24no—
81C22×D157—24no—
82C22×D157—416no—
83C22×D157—44no—
84C22×D157—44no—
85C22×D157C2×D414yescomplemented
86C22⋊D157—216no—
87C22⋊D157—216no—
88C22⋊D157—416no—
89C2×Dic157—28no—
90C2×Dic157—22no—
91C2×Dic157—22no—
92C2×Dic157—24no—
93C2×Dic157—24no—
94C2×Dic157—24no—
95C2×Dic157C2×D414yescomplemented
96C23×C157—22no—
97C23×C157—22no—
98C23×C157—22no—
99C23×C157—22no—
100C23×C157—28no—
101C23×C157—28no—
102C23×C157—28no—
103C23×C157—22no—
104C23×C157—24no—
105C23×C157—22no—
106C23×C157—22no—
107C23×C157—22no—
108C23×C157—24no—
109C23×C157—22no—
110C23×C157—44no—
111C23×C157—416no—
112C23×C157—416no—
113C23×C157—44no—
114C23×C157—44no—
115C23×C157C2×D414yes—
116C23×C157C2×D412yes—
117C23×C157C2×D412yes—
118C23×C157C2×D412yes—
119C23×C157C2×D414yes—
120C23×C157C2×D414yescomplemented
121C23×C157C2411yescharacteristic
122D4×C157—216no—
123D4×C157—216no—
124C2×C4×C157—28no—
125C2×C4×C157—22no—
126C2×C4×C157—22no—
127C2×C4×C157C2×D414yescomplemented
128C22×C157—22no—
129C22×C157—22no—
130C22×C157—22no—
131C22×C157—22no—
132C22×C157—28no—
133C22×C157—28no—
134C22×C157—22no—
135C22×C157—24no—
136C22×C157—22no—
137C22×C157—22no—
138C22×C157—28no—
139C22×C157—416no—
140C22×C157—416no—
141C22×C157—42no—
142C22×C157—42no—
143C22×C157—44no—
144C22×C157—44no—
145C22×C157#12014yescomplemented
146C22×C157#12014yescomplemented
147C22×C157C22×D412yes—
148C22×C157C22×D411yescharacteristic
149C22×C157C22×D412yes—
150C22×C157H(5,𝔽2)12yes—
151C2×D157—24no—
152C2×D157—28no—
153C2×D157—416no—
154C2×D157—42no—
155C2×D157—42no—
156C2×D157—44no—
157C4×C157—24no—
158Dic157—24no—
159Dic157—44no—
160C2×C157—24no—
161C2×C157—24no—
162C2×C157—42no—
163C2×C157—44no—
164C2×C157C2×#12011yescharacteristic
165C2×C157C2×#12011yescharacteristic
166C2×C157#53411yescharacteristic
167C2×C157D4214yes—
168D157—44no—
169C157#240311yescharacteristic, complemented, Sylow
170#2403—1571nomaximal, Sylow
171C23×D4—1571no—
172C23×D4D15711yescharacteristic, complemented
173C2×#120—1572no—
174C2×#120—1572no—
175#392—1574no—
176#386—1572no—
177#386—1572no—
178#379—1571no—
179C25—1571no—
180C25C2×D15711yescharacteristic
181C25C2×D15711yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—3144no—
185C22×D4—3148no—
186C22×D4—1572no—
187C22×D4—1574no—
188C22×D4—1572no—
189C22×D4C2×D15712yes—
190C22×D4C2×D15712yescomplemented
191C23×C4—1571no—
192C23×C4C2×D15711yescharacteristic, complemented
193#120—3148no—
194#120—3148no—
195C2×C22⋊C4—1572no—
196C2×C22⋊C4—1572no—
197C2×C22⋊C4—1574no—
198C2×C22⋊C4—1574no—
199C2×C22⋊C4—1572no—
200#95—1574no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—3144no—
209C24—3142no—
210C24—3144no—
211C24—6282no—
212C24—1572no—
213C24C22⋊D15714yescomplemented
214C24C22×D15711yescharacteristic
215C24C22×D15712yes—
216C24C22×D15711yescharacteristic
217C24C22×D15711yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—31432no—
225C2×D4—3144no—
226C2×D4—3144no—
227C2×D4—3144no—
228C2×D4—3144no—
229C2×D4—3148no—
230C2×D4—3148no—
231C2×D4—3144no—
232C2×D4—6284no—
233C2×D4—1578no—
234C2×D4C22⋊D15718yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—3142no—
238C22×C4—3144no—
239C22×C4—1574no—
240C22×C4—1572no—
241C22×C4C22×D15712yes—
242C22⋊C4—3144no—
243C22⋊C4—3144no—
244C22⋊C4—3148no—
245C22⋊C4—3148no—
246C22⋊C4—6282no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—3148no—
267C23—3142no—
268C23—3144no—
269C23—3142no—
270C23—3144no—
271C23—62816no—
272C23—6284no—
273C23—6284no—
274C23—1574no—
275C23D4×D15714yes—
276C23C2×C22⋊D15712yes—
277C23C2×C22⋊D15712yes—
278C23C2×C22⋊D15714yes—
279C23C2×C22⋊D15712yes—
280C23C2×C22⋊D15714yescomplemented
281C23C23×D15711yescharacteristic
282D4—216no—
283D4—216no—
284D4—31416no—
285D4—31416no—
286D4—62816no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—3148no—
291C2×C4—3142no—
292C2×C4—3142no—
293C2×C4—3144no—
294C2×C4—3144no—
295C2×C4—3144no—
296C2×C4—1574no—
297C2×C4C2×C22⋊D15714yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—3148no—
316C22—3144no—
317C22—62816no—
318C22—6282no—
319C22—6284no—
320C22—6282no—
321C22#1909773914yescomplemented
322C22#1909774014yescomplemented
323C22C2×D4×D15712yes—
324C22C22×C22⋊D15711yescharacteristic
325C22C22×C22⋊D15712yes—
326C22#1909781612yes—
327C4—24no—
328C4—3144no—
329C4—6284no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—6284no—
335C2C2×#1909773911yescharacteristic
336C2C2×#1909774011yescharacteristic
337C2#2375783211yescharacteristic
338C2D4×C22⋊D15714yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
FittingC23×D4×C157C2
hypercenterC23×D4D157
hypocenterC157#2403
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D157C2semidirect
2. C2×#19097740C2semidirect
3. C2×#19097739C2semidirect
4. #23757181C2semidirect
5. #23757153C2semidirect
6. #23757152C2semidirect
7. #23757129C2semidirect
8. C23×D4×C157C2semidirect
9. C24×D157C22semidirect
13. C22×C22⋊D157C22semidirect
14. C22×C22⋊D157C22semidirect
15. C23×Dic157C22semidirect
18. C2×C22⋊Dic157C22semidirect
19. C2×C22⋊Dic157C22semidirect
20. C2×C22⋊Dic157C22semidirect
21. C2×C22⋊Dic157C22semidirect
22. C2×#14578931C22semidirect
23. #19097678C22semidirect
25. C25×C157C22semidirect
29. C22×D4×C157C22semidirect
30. C23×C4×C157C22semidirect
45. C2×C22⋊D157D4semidirect
62. C24×C157D4semidirect
73. C2×D4×C157D4semidirect
85. C22×D157C2×D4semidirect
95. C2×Dic157C2×D4semidirect
120. C23×C157C2×D4semidirect
127. C2×C4×C157C2×D4semidirect
145. C22×C157#120semidirect
146. C22×C157#120semidirect
169. C157#2403semidirect
172. C23×D4D157semidirect
181. C25C2×D157semidirect
190. C22×D4C2×D157semidirect
192. C23×C4C2×D157semidirect
213. C24C22⋊D157semidirect
234. C2×D4C22⋊D157semidirect
280. C23C2×C22⋊D157semidirect
297. C2×C4C2×C22⋊D157semidirect
321. C22#19097739semidirect
322. C22#19097740semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1572
C2×#190977402
C2×#190977392
#237571812
#237571532
#237571522
#237571292
C23×D4×C1572
#2403157