Finite Group Databasebeta

← #34103456

Subgroups — #34103456

Order 20096.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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, complemented
2C2×#19097741C212yesmaximal, complemented
3C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
4#23757184C211yesmaximal, characteristic, complemented
5#23757183C211yesmaximal, characteristic, complemented
6#23757182C211yesmaximal, characteristic, complemented
7#23757151C212yesmaximal, complemented
8#23757151C212yesmaximal, complemented
9#23757149C212yesmaximal, complemented
10#23757135C211yesmaximal, characteristic, complemented
11#393×C157C211yesmaximal, characteristic, complemented
12C24×D157C2211yescharacteristic
13C2×#14578955—28no—
14C2×#14578955—24no—
15C22×D628C2212yes—
16C22×D628C2212yes—
17C22×D628C2211yescharacteristic, complemented
18C22×C22⋊D157C2212yes—
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21C22×C4×D157C2212yescomplemented
22#19097741—28no—
23C2×C22⋊Dic157C2211yescharacteristic
24C2×#14578931C2212yes—
25C2×#14578931C2211yescharacteristic
26C2×#14578931C2212yes—
27C2×#14578931C2211yescharacteristic
28C2×#14578931C2212yescomplemented
29C2×#14578931C2212yescomplemented
30C22⋊C4×D157—24no—
31#19097678C2212yescomplemented
32#19097678C2211yescharacteristic, complemented
33#19097678C2212yescomplemented
34C22×Q8×C157C2211yescharacteristic, complemented
35C2×C22⋊C4×C157C2212yes—
36C2×C22⋊C4×C157C2211yescharacteristic
37#95×C157C2211yescharacteristic, complemented
38#95×C157C2212yescomplemented
39C23×D157—22no—
40C23×D157—22no—
41C23×D157—24no—
42C23×D157—22no—
43C23×D157—42no—
44C23×D157C2312yes—
45C23×D157C2311yescharacteristic
46C23×D157C2311yescharacteristic
47#14578955—232no—
48C2×D628—24no—
49C2×D628—24no—
50C2×D628—24no—
51C2×D628—24no—
52C2×D628—24no—
53C2×D628—22no—
54C2×D628—42no—
55C2×D628D418yescomplemented
56C2×C22⋊D157—24no—
57C2×C22⋊D157—24no—
58C2×C22⋊D157—42no—
59C22×Dic157C2312yes—
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C2×C4×D157—24no—
63C2×C4×D157—24no—
64C2×C4×D157—24no—
65C2×C4×D157—22no—
66C2×C4×D157—24no—
67C2×C4×D157—42no—
68C2×C4×D157D414yescomplemented
69C22⋊Dic157—22no—
70#14578931—24no—
71#14578931—22no—
72#14578931—22no—
73#14578931—24no—
74#14578931—42no—
75#14578931—42no—
76C24×C157C2311yescharacteristic
77C2×Q8×C157—24no—
78C2×Q8×C157—22no—
79C22×C4×C157C2312yes—
80C22×C4×C157C2311yescharacteristic
81C22×C4×C157C2311yescharacteristic
82C22×C4×C157C2312yes—
83C22⋊C4×C157—22no—
84C22⋊C4×C157—24no—
85C22×D157—21no—
86C22×D157—21no—
87C22×D157—22no—
88C22×D157—28no—
89C22×D157—22no—
90C22×D157—22no—
91C22×D157—22no—
92C22×D157—21no—
93C22×D157—22no—
94C22×D157—22no—
95C22×D157—21no—
96C22×D157—22no—
97C22×D157—22no—
98C22×D157—22no—
99C22×D157—48no—
100C22×D157—48no—
101C22×D157—44no—
102C22×D157—44no—
103C22×D157C2×D414yes—
104C22×D157C2×D412yes—
105C22×D157C2×D414yes—
106C22×D157#4112yescomplemented
107D628—28no—
108D628—216no—
109D628—48no—
110C22⋊D157—48no—
111C2×Dic157—22no—
112C2×Dic157—21no—
113C2×Dic157—22no—
114C2×Dic157—22no—
115C2×Dic157—22no—
116C2×Dic157—22no—
117C2×Dic157C2×D412yes—
118C4×D157—216no—
119C4×D157—28no—
120C4×D157—44no—
121C23×C157—22no—
122C23×C157—21no—
123C23×C157—21no—
124C23×C157—22no—
125C23×C157#4112yescomplemented
126C23×C157C2411yescharacteristic
127Q8×C157—28no—
128C2×C4×C157—22no—
129C2×C4×C157—22no—
130C2×C4×C157—22no—
131C2×C4×C157—21no—
132C2×C4×C157—22no—
133C2×C4×C157—22no—
134C2×C4×C157—22no—
135C2×C4×C157—22no—
136C2×C4×C157C2×D414yes—
137C2×C4×C157C2×D412yes—
138C22×C157—22no—
139C22×C157—24no—
140C22×C157—41no—
141C22×C157—41no—
142C22×C157—42no—
143C22×C157C22×D411yescharacteristic
144C22×C157C22×D412yes—
145C22×C157C2×#4111yescharacteristic
146C22×C157C2×#4111yescharacteristic
147C22×C157H(5,𝔽2)12yes—
148C2×D157—22no—
149C2×D157—22no—
150C2×D157—24no—
151C2×D157—24no—
152C2×D157—24no—
153C2×D157—22no—
154C2×D157—24no—
155C2×D157—24no—
156C2×D157—24no—
157C2×D157—416no—
158C2×D157—41no—
159C2×D157—42no—
160C2×D157—42no—
161C2×D157—41no—
162C2×D157—42no—
163C2×D157—42no—
164C2×D157—42no—
165C2×D157#12014yes—
166C4×C157—24no—
167C4×C157—22no—
168C4×C157—41no—
169C4×C157—42no—
170Dic157—22no—
171Dic157—41no—
172Dic157—42no—
173C2×C157—42no—
174C2×C157C2×#12011yescharacteristic
175C2×C157#52511yescharacteristic
176C2×C157#53511yescharacteristic
177C2×C157#54612yes—
178C2×C157#54612yes—
179D157—24no—
180D157—44no—
181D157—42no—
182C157#240411yescharacteristic, complemented, Sylow
183#2404—1571nomaximal, Sylow
184C22×#41—1571no—
185C2×#120—1572no—
186C22×C22⋊C4—1571no—
187#393—1571no—
188#393D15711yescharacteristic, complemented
189#392—1571no—
190#392—1571no—
191#386—1572no—
192#386—1572no—
193#386—1572no—
194#381—1571no—
195C25—1571no—
196C2×#41—3148no—
197C2×#41—3144no—
198C22×Q8C2×D15711yescharacteristic, complemented
199C22×D4—1572no—
200C22×D4—1571no—
201C22×D4—1572no—
202C22×D4—1572no—
203C23×C4—1572no—
204C23×C4—1571no—
205C23×C4—1571no—
206#120—3148no—
207C2×C22⋊C4—3144no—
208C2×C22⋊C4—1571no—
209C2×C22⋊C4—1571no—
210C2×C22⋊C4—1571no—
211C2×C22⋊C4—1572no—
212C2×C22⋊C4—1572no—
213C2×C22⋊C4—1572no—
214C2×C22⋊C4—1572no—
215C2×C22⋊C4C2×D15711yescharacteristic
216C2×C22⋊C4C2×D15712yes—
217#95—1572no—
218#95—1571no—
219#95—1572no—
220#95C2×D15711yescharacteristic, complemented
221#95C2×D15712yescomplemented
222C24—3142no—
223C24—3142no—
224C24—3144no—
225C24—3142no—
226C24—6282no—
227C24—1571no—
228C24—1572no—
229C24—1571no—
230C24C22×D15711yescharacteristic
231#41—31432no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×D4—3144no—
235C2×D4—3144no—
236C2×D4—3144no—
237C2×D4—3144no—
238C2×D4—3144no—
239C2×D4—3144no—
240C2×D4—3144no—
241C2×D4—3142no—
242C2×D4—6282no—
243C2×D4—6282no—
244C2×D4—1578no—
245C22×C4—3144no—
246C22×C4—3144no—
247C22×C4—3144no—
248C22×C4—3142no—
249C22×C4—3144no—
250C22×C4—6282no—
251C22×C4—1572no—
252C22×C4—1571no—
253C22×C4—1571no—
254C22×C4—1574no—
255C22×C4C22×D15712yes—
256C22×C4C22×D15711yescharacteristic
257C22×C4C22×D15712yes—
258C22×C4C22×D15711yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—3144no—
262C22⋊C4—3142no—
263C22⋊C4—3142no—
264C22⋊C4—3142no—
265C22⋊C4—3144no—
266C22⋊C4—6282no—
267C22⋊C4—6282no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—3148no—
273C23—3141no—
274C23—3141no—
275C23—3142no—
276C23—3141no—
277C23—3142no—
278C23—3142no—
279C23—3142no—
280C23—3142no—
281C23—3142no—
282C23—3141no—
283C23—3142no—
284C23—3142no—
285C23—3142no—
286C23—6288no—
287C23—6288no—
288C23—6284no—
289C23—6284no—
290C23—1574no—
291C23—1572no—
292C23—1574no—
293C23—1572no—
294C23#1457895512yescomplemented
295C23C23×D15711yescharacteristic
296Q8—28no—
297D4—3148no—
298D4—31416no—
299D4—6288no—
300D4—6288no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—31416no—
310C2×C4—3148no—
311C2×C4—3142no—
312C2×C4—3141no—
313C2×C4—3142no—
314C2×C4—3142no—
315C2×C4—3142no—
316C2×C4—3142no—
317C2×C4—6284no—
318C2×C4—1572no—
319C2×C4D4×D15714yes—
320C2×C4D4×D15712yes—
321C22—24no—
322C22—22no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—3142no—
327C22—3144no—
328C22—3144no—
329C22—3142no—
330C22—3144no—
331C22—3144no—
332C22—3144no—
333C22—3144no—
334C22—3142no—
335C22—62816no—
336C22—6281no—
337C22—6282no—
338C22—6282no—
339C22—6282no—
340C22—6282no—
341C22—6281no—
342C22—6282no—
343C22—1574no—
344C22C2×D4×D15711yescharacteristic
345C22C2×D4×D15712yes—
346C22#41×D15711yescharacteristic
347C22C2×#1457895511yescharacteristic
348C22#1909781712yes—
349C4—24no—
350C4—22no—
351C4—41no—
352C4—42no—
353C4—3142no—
354C4—6281no—
355C4—6282no—
356C2—42no—
357C2—3144no—
358C2—6282no—
359C2—6284no—
360C2#120×D15711yescharacteristic
361C2#2375778611yescharacteristic
362C2#2375784111yescharacteristic
363C2#2375791412yes—
364C2#2375791612yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#393×C157C2
hypercenter#393D157
hypocenterC157#2404
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. C2×#19097741C2semidirect
3. C2×C22⋊C4×D157C2semidirect
4. #23757184C2semidirect
5. #23757183C2semidirect
6. #23757182C2semidirect
7. #23757151C2semidirect
8. #23757151C2semidirect
9. #23757149C2semidirect
10. #23757135C2semidirect
11. #393×C157C2semidirect
17. C22×D628C22semidirect
21. C22×C4×D157C22semidirect
28. C2×#14578931C22semidirect
29. C2×#14578931C22semidirect
31. #19097678C22semidirect
32. #19097678C22semidirect
33. #19097678C22semidirect
34. C22×Q8×C157C22semidirect
37. #95×C157C22semidirect
38. #95×C157C22semidirect
55. C2×D628D4semidirect
68. C2×C4×D157D4semidirect
106. C22×D157#41semidirect
125. C23×C157#41semidirect
182. C157#2404semidirect
188. #393D157semidirect
198. C22×Q8C2×D157semidirect
220. #95C2×D157semidirect
221. #95C2×D157semidirect
294. C23#14578955semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#190977412
C2×C22⋊C4×D1572
#237571842
#237571832
#237571822
#237571512
#237571492
#237571352
#393×C1572
#2404157