Finite Group Databasebeta

← #34109512

Subgroups — #34109512

Order 20096.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#23758109C211yesmaximal, characteristic
2C2×#19097802C211yesmaximal, characteristic, complemented
3#23757847C211yesmaximal, characteristic, complemented
4#23757454C214yesmaximal, complemented
5#23757449C214yesmaximal, complemented
6#23757290C211yesmaximal, characteristic, complemented
7#23757289C211yesmaximal, characteristic, complemented
8#23757255C211yesmaximal, characteristic, complemented
9C2×#137×C157C211yesmaximal, characteristic, complemented
10#19097819—21no—
11#19097818—21no—
12#19097818C2212yes—
13#19097817—21no—
14#19097817C2212yes—
15#19097816—21no—
16#19097816—22no—
17C2×#14578954—22no—
18C2×#14578954C2211yescharacteristic
19C2×#14578954C2211yescharacteristic
20#41×D157—22no—
21#41×D157—22no—
22#19097802—22no—
23#19097802C2214yescomplemented
24C2×#14578944—21no—
25C2×#14578941—21no—
26C2×#14578938C2211yescharacteristic, complemented
27#19097769C2212yescomplemented
28#19097760C2212yescomplemented
29#19097754—22no—
30#19097747—22no—
31#19097730C2211yescharacteristic, complemented
32#19097698C2214yescomplemented
33#19097697C2214yescomplemented
34#19097689C2212yescomplemented
35#19097687C2212yescomplemented
36C2×#41×C157C2211yescharacteristic
37C22×Q8×C157C2211yescharacteristic, complemented
38#137×C157—22no—
39#137×C157C2214yescomplemented
40C2×Q16×C157—21no—
41C2×SD16×C157—21no—
42C2×M16×C157C2211yescharacteristic, complemented
43#14578955—22no—
44#14578955—22no—
45#14578954—24no—
46#14578954—24no—
47#14578954—22no—
48#14578954—22no—
49#14578954—21no—
50#14578954—21no—
51#14578954—21no—
52#14578954—21no—
53#14578954—21no—
54#14578954—21no—
55#14578954D412yes—
56#14578954C2312yes—
57#14578953—24no—
58#14578953—22no—
59#14578953—24no—
60#14578953—22no—
61C2×Dic314—22no—
62C2×Dic314D411yescharacteristic
63C2×Dic314C2311yescharacteristic
64Q8×D157—22no—
65Q8×D157—22no—
66C2×D628—22no—
67C2×D628D411yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—24no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157D412yes—
72D4×D157—24no—
73D4×D157—22no—
74D4×D157—24no—
75D4×D157—22no—
76C2×C4×D157—24no—
77C2×C4×D157—21no—
78C2×C4×D157D412yes—
79#14578944—24no—
80#14578941—24no—
81#14578938—21no—
82#14578938C2312yes—
83C2×#11929434—21no—
84C4⋊Dic157—21no—
85#14578933—24no—
86C22⋊Dic157—21no—
87#14578931—24no—
88C4×Dic157C2312yes—
89#41×C157—21no—
90#41×C157—21no—
91#41×C157D412yes—
92#41×C157C2312yes—
93C2×Q8×C157—21no—
94C2×Q8×C157—21no—
95C2×Q8×C157—21no—
96C2×Q8×C157—21no—
97C2×Q8×C157—21no—
98C2×Q8×C157D411yescharacteristic
99C2×Q8×C157C2312yes—
100C2×D4×C157—21no—
101C2×D4×C157D411yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105Q16×C157—24no—
106SD16×C157—24no—
107M16×C157—21no—
108M16×C157C2312yes—
109C2×C8×C157—21no—
110C22×D157—24no—
111C22×D157—21no—
112C22×D157C2×D412yes—
113Dic314—24no—
114Dic314—21no—
115Dic314—21no—
116Dic314—21no—
117Dic314C2×D412yes—
118D628—24no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628C2×D412yes—
123C22⋊D157—24no—
124C22⋊D157—24no—
125C22⋊D157—22no—
126C22⋊D157—22no—
127C22⋊D157—22no—
128C22⋊D157—22no—
129C2×Dic157—24no—
130C2×Dic157—21no—
131C2×Dic157—22no—
132C2×Dic157C2×D412yes—
133C4×D157—24no—
134C4×D157—24no—
135C4×D157—22no—
136C4×D157—22no—
137C4×D157—22no—
138C4×D157—22no—
139#11929434—22no—
140C23×C157—21no—
141C23×C157C2×D411yescharacteristic
142Q8×C157—21no—
143Q8×C157—22no—
144Q8×C157—22no—
145Q8×C157—41no—
146Q8×C157—41no—
147Q8×C157C2×D412yes—
148D4×C157—22no—
149D4×C157—22no—
150D4×C157—21no—
151D4×C157C2×D412yes—
152C2×C4×C157—21no—
153C2×C4×C157—22no—
154C2×C4×C157—22no—
155C2×C4×C157—22no—
156C2×C4×C157—41no—
157C2×C4×C157—41no—
158C2×C4×C157C2×D411yescharacteristic
159C2×C4×C157C2×D411yescharacteristic
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2×D411yescharacteristic
162C2×C4×C157C2×D411yescharacteristic
163C2×C4×C157C2411yescharacteristic
164C8×C157—22no—
165C22×C157—22no—
166C22×C157—41no—
167C22×C157—81no—
168C22×C157#12011yescharacteristic
169C22×C157#12011yescharacteristic
170C22×C157C22×D411yescharacteristic
171C2×D157—22no—
172C2×D157—22no—
173C2×D157—42no—
174C2×D157—81no—
175C2×D157—84no—
176C4×C157—22no—
177C4×C157—42no—
178C4×C157#12011yescharacteristic
179C4×C157#12011yescharacteristic
180C4×C157C22×D411yescharacteristic
181C4×C157C22×D411yescharacteristic
182Dic157—22no—
183Dic157—22no—
184Dic157—42no—
185C2×C157—21no—
186C2×C157—21no—
187C2×C157—21no—
188C2×C157—42no—
189C2×C157C2×#12011yescharacteristic
190D157—42no—
191D157—42no—
192C157#302211yescharacteristic, complemented, Sylow
193#3022—1571nomaximal, Sylow
194#585—1571no—
195C2×#137—1571no—
196C2×#137D15711yescharacteristic, complemented
197#536—1571no—
198#456—1574no—
199#455—1574no—
200#421—1571no—
201#421—1571no—
202#413—1571no—
2032−1+4—3141no—
2042−1+4—3141no—
2052−1+4—1572no—
206H(5,𝔽2)—3142no—
207H(5,𝔽2)—3141no—
208H(5,𝔽2)—3141no—
209H(5,𝔽2)—1572no—
210C2×#41—3142no—
211C2×#41—3142no—
212C2×#41—3142no—
213C2×#41—1571no—
214C2×#41—1571no—
215C2×#41C2×D15711yescharacteristic
216C22×Q8C2×D15711yescharacteristic, complemented
217#137—22no—
218#137—3142no—
219#137—1574no—
220#137C2×D15714yescomplemented
221C2×Q16—21no—
222C2×Q16—3141no—
223C2×SD16—21no—
224C2×SD16—3141no—
225C2×M16—1571no—
226C2×M16C2×D15711yescharacteristic, complemented
227C4⋊Q8—1572no—
228#124—1572no—
229#123—3142no—
230C22⋊Q8—3142no—
231#117—1571no—
232#104—1574no—
233#104—1574no—
234#101—1572no—
235#100—1572no—
236#41—21no—
237#41—21no—
238#41—3144no—
239#41—3144no—
240#41—3142no—
241#41—3142no—
242#41—3144no—
243#41—3142no—
244#41—3142no—
245#41—3142no—
246#41—3142no—
247#41—3144no—
248#41—3141no—
249#41—3141no—
250#41—3141no—
251#41—3141no—
252#41—3141no—
253#41—3141no—
254#41—1572no—
255#41—1572no—
256#41C22⋊D15712yes—
257#41C22×D15712yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—3142no—
264C2×Q8—3142no—
265C2×Q8—3142no—
266C2×Q8—1571no—
267C2×Q8—1571no—
268C2×Q8C22⋊D15711yescharacteristic
269C2×Q8C22×D15712yes—
270C2×D4—21no—
271C2×D4—3142no—
272C2×D4—3144no—
273C2×D4—3142no—
274C2×D4—3144no—
275C2×D4—3144no—
276C2×D4—3141no—
277C2×D4—3142no—
278C2×D4—1572no—
279C2×D4—1571no—
280C2×D4—1571no—
281C2×D4C22⋊D15711yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—3144no—
285C22×C4—3141no—
286C22×C4—1572no—
287C22×C4C22×D15711yescharacteristic
288Q16—24no—
289Q16—3144no—
290SD16—24no—
291SD16—3144no—
292M16—21no—
293M16—3141no—
294M16—1572no—
295M16C22×D15712yes—
296C2×C8—21no—
297C2×C8—3141no—
298C4⋊C4—3144no—
299C4⋊C4—3141no—
300C22⋊C4—3144no—
301C22⋊C4—3141no—
302C42—1572no—
303C23—21no—
304C23—3144no—
305C23—3141no—
306C23—1572no—
307C23D4×D15711yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—3144no—
314Q8—3141no—
315Q8—3141no—
316Q8—3141no—
317Q8—1572no—
318Q8C2×C22⋊D15712yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—3144no—
323D4—3144no—
324D4—3144no—
325D4—3142no—
326D4—3142no—
327D4—3142no—
328D4—3142no—
329D4—3141no—
330D4—3141no—
331D4—3141no—
332D4—1572no—
333D4C2×C22⋊D15712yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—3144no—
341C2×C4—3144no—
342C2×C4—3144no—
343C2×C4—3141no—
344C2×C4—3142no—
345C2×C4—3142no—
346C2×C4—3142no—
347C2×C4—3142no—
348C2×C4—3142no—
349C2×C4—1572no—
350C2×C4D4×D15711yescharacteristic
351C2×C4D4×D15711yescharacteristic
352C2×C4D4×D15711yescharacteristic
353C2×C4C2×C22⋊D15711yescharacteristic
354C2×C4C2×C22⋊D15711yescharacteristic
355C2×C4C23×D15711yescharacteristic
356C8—22no—
357C8—3142no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—3142no—
362C22—3142no—
363C22—6282no—
364C22—12561no—
365C22—12564no—
366C22#1909773911yescharacteristic
367C22#1909773911yescharacteristic
368C22C2×D4×D15711yescharacteristic
369C4—22no—
370C4—42no—
371C4—3142no—
372C4—3142no—
373C4—6282no—
374C4#1909773911yescharacteristic
375C4#1909773911yescharacteristic
376C4C2×D4×D15711yescharacteristic
377C4C22×C22⋊D15711yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—6282no—
383C2—6282no—
384C2C2×#1909773911yescharacteristic
subgroupG/H
centerC2C2×#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#137×C157C2
socleC2×C157C2×#120
prime radical#23758109C2
hypercenterC2×#137D157
hypocenterC157#3022
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19097802C2semidirect
3. #23757847C2semidirect
4. #23757454C2semidirect
5. #23757449C2semidirect
6. #23757290C2semidirect
7. #23757289C2semidirect
8. #23757255C2semidirect
9. C2×#137×C157C2semidirect
23. #19097802C22semidirect
26. C2×#14578938C22semidirect
27. #19097769C22semidirect
28. #19097760C22semidirect
31. #19097730C22semidirect
32. #19097698C22semidirect
33. #19097697C22semidirect
34. #19097689C22semidirect
35. #19097687C22semidirect
37. C22×Q8×C157C22semidirect
39. #137×C157C22semidirect
42. C2×M16×C157C22semidirect
192. C157#3022semidirect
196. C2×#137D157semidirect
216. C22×Q8C2×D157semidirect
220. #137C2×D157semidirect
226. C2×M16C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
C2×#190978022
#237578472
#237574542
#237574492
#237572902
#237572892
#237572552
C2×#137×C1572
#3022157