Finite Group Databasebeta

← #34109511

Subgroups — #34109511

Order 20096.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 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#23758108C211yesmaximal, characteristic
2#137×D157C212yesmaximal, complemented
3#23757843C211yesmaximal, characteristic, complemented
4#23757457C212yesmaximal, complemented
5#23757454C212yesmaximal, complemented
6#23757453C211yesmaximal, characteristic, complemented
7#23757451C211yesmaximal, characteristic, complemented
8#23757450C211yesmaximal, characteristic, complemented
9#23757288C212yesmaximal, complemented
10#23757254C211yesmaximal, characteristic, complemented
11#455×C157C211yesmaximal, characteristic, complemented
12#19097818—22no—
13#19097818C2212yes—
14#19097816—22no—
15#19097816—22no—
16#19097816C2211yescharacteristic
17C2×#14578953—22no—
18C2×#14578953—22no—
19C2×#14578953C2211yescharacteristic
20#41×D157—22no—
21#41×D157C2212yes—
22C2×Q8×D157C2211yescharacteristic, complemented
23#19097804C2212yescomplemented
24#19097803—22no—
25#19097802C2212yescomplemented
26#19097801C2212yescomplemented
27#19097800—22no—
28Q16×D157—22no—
29SD16×D157—22no—
30M16×D157C2212yescomplemented
31#19097771C2211yescharacteristic, complemented
32#19097769C2211yescharacteristic, complemented
33#19097758C2211yescharacteristic, complemented
34#19097753—22no—
35#19097748—22no—
36#19097728C2211yescharacteristic, complemented
37#19097699C2212yescomplemented
38#19097698C2212yescomplemented
39#19097697C2212yescomplemented
40#19097689C2212yescomplemented
41#19097688C2211yescharacteristic, complemented
42H(5,𝔽2)×C157C2211yescharacteristic
43#137×C157C2212yescomplemented
44#124×C157C2211yescharacteristic, complemented
45#104×C157C2212yescomplemented
46#100×C157C2211yescharacteristic, complemented
47#14578955D412yes—
48#14578954—24no—
49#14578954—22no—
50#14578954—22no—
51#14578954C2312yes—
52#14578953—24no—
53#14578953—24no—
54#14578953—24no—
55#14578953—22no—
56#14578953—22no—
57#14578953—24no—
58#14578953—22no—
59#14578953—22no—
60#14578953—22no—
61#14578953D412yes—
62C2×Dic314—22no—
63C2×Dic314—22no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—22no—
69Q8×D157—22no—
70Q8×D157D412yes—
71C2×C22⋊D157—24no—
72C2×C22⋊D157—22no—
73C2×C22⋊D157—22no—
74C2×C22⋊D157D412yes—
75D4×D157—24no—
76D4×D157—22no—
77D4×D157—22no—
78D4×D157D412yes—
79C22×Dic157—22no—
80C22×Dic157D412yes—
81C2×C4×D157—22no—
82C2×C4×D157—22no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85Dic628—22no—
86#14578944—22no—
87#14578944—22no—
88#14578943—22no—
89#14578942—22no—
90#14578941—22no—
91#14578938C2312yes—
92#14578937—22no—
93C8×D157—22no—
94C4⋊Dic157—22no—
95#14578933—21no—
96#14578933—22no—
97C22⋊Dic157—22no—
98#14578931—21no—
99C4×Dic157C2311yescharacteristic
100#41×C157—22no—
101#41×C157C2312yes—
102C2×Q8×C157C2311yescharacteristic
103C2×D4×C157—22no—
104C2×D4×C157—22no—
105C2×D4×C157C2311yescharacteristic
106Q16×C157—22no—
107SD16×C157—22no—
108M16×C157C2312yes—
109C22⋊C4×C157—22no—
110C42×C157C2311yescharacteristic
111C22×D157—22no—
112C22×D157—22no—
113C22×D157C2×D411yescharacteristic
114Dic314—24no—
115Dic314—22no—
116Dic314—22no—
117Dic314—22no—
118Dic314—41no—
119Dic314—41no—
120Dic314C2×D412yes—
121D628—22no—
122D628C2×D412yes—
123C22⋊D157—24no—
124C22⋊D157—24no—
125C22⋊D157—22no—
126C22⋊D157—22no—
127C22⋊D157—24no—
128C22⋊D157—22no—
129C2×Dic157—22no—
130C2×Dic157—24no—
131C2×Dic157—22no—
132C2×Dic157—22no—
133C2×Dic157—22no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157C2×D412yes—
137C2×Dic157C2×D411yescharacteristic
138C4×D157—24no—
139C4×D157—22no—
140C4×D157—22no—
141C4×D157—22no—
142C4×D157—42no—
143C4×D157C2×D412yes—
144C4×D157C2×D412yes—
145#11929434—22no—
146C23×C157—22no—
147C23×C157C2×D412yes—
148Q8×C157—22no—
149Q8×C157C2×D412yes—
150D4×C157—24no—
151D4×C157—22no—
152D4×C157—22no—
153D4×C157C2×D412yes—
154C2×C4×C157—22no—
155C2×C4×C157—22no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157C2411yescharacteristic
159C8×C157—22no—
160C22×C157—22no—
161C22×C157—22no—
162C22×C157—42no—
163C22×C157—82no—
164C22×C157C22×D411yescharacteristic
165C2×D157—22no—
166C2×D157—41no—
167C2×D157—82no—
168C2×D157—82no—
169C2×D157#12012yes—
170C4×C157—22no—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157C22×D412yes—
174Dic157—22no—
175Dic157—22no—
176Dic157—41no—
177Dic157—41no—
178Dic157#12011yescharacteristic
179Dic157#12011yescharacteristic
180C2×C157—21no—
181C2×C157—42no—
182C2×C157—42no—
183C2×C157C2×#12011yescharacteristic
184D157—22no—
185D157—42no—
186C157#302211yescharacteristic, complemented, Sylow
187#3022—1571nomaximal, Sylow
188#585—1571no—
189C2×#137—1572no—
190#536—1571no—
191#456—1572no—
192#456—1572no—
193#455—1571no—
194#455—1571no—
195#455—1571no—
196#455D15711yescharacteristic, complemented
197#421—1572no—
198#413—1571no—
1992−1+4—3142no—
2002−1+4—1572no—
201H(5,𝔽2)—3142no—
202H(5,𝔽2)—3142no—
203H(5,𝔽2)—1571no—
204H(5,𝔽2)C2×D15711yescharacteristic
205C2×#41—3142no—
206C2×#41—3142no—
207C2×#41—3142no—
208C2×#41—1572no—
209C2×#41—1571no—
210C22×Q8—1571no—
211#137—3142no—
212#137—3142no—
213#137—1572no—
214#137—1572no—
215#137—1572no—
216#137C2×D15712yescomplemented
217C2×Q16—3142no—
218C2×SD16—3142no—
219C2×M16—1572no—
220C4⋊Q8—1571no—
221C4⋊Q8—1571no—
222#124—1571no—
223#124C2×D15711yescharacteristic, complemented
224#123—3142no—
225C22⋊Q8—3142no—
226#117—1571no—
227#104—1572no—
228#104—1572no—
229#104—1572no—
230#104C2×D15712yescomplemented
231#101—1572no—
232#100—1571no—
233#100C2×D15711yescharacteristic, complemented
234#41—22no—
235#41—3144no—
236#41—3142no—
237#41—3142no—
238#41—3144no—
239#41—3144no—
240#41—3144no—
241#41—3142no—
242#41—3142no—
243#41—3142no—
244#41—3142no—
245#41—3144no—
246#41—3142no—
247#41—1572no—
248#41—1572no—
249#41—1572no—
250#41C22×D15712yes—
251C2×Q8—3142no—
252C2×Q8—3142no—
253C2×Q8—3141no—
254C2×Q8—3142no—
255C2×Q8—3142no—
256C2×Q8—3142no—
257C2×Q8—1572no—
258C2×Q8—1571no—
259C2×Q8—1571no—
260C2×Q8C22×D15711yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—3144no—
264C2×D4—3142no—
265C2×D4—3142no—
266C2×D4—3144no—
267C2×D4—3142no—
268C2×D4—3142no—
269C2×D4—1572no—
270C2×D4—1572no—
271C2×D4C22×D15711yescharacteristic
272C22×C4—3142no—
273C22×C4—3142no—
274C22×C4—3141no—
275C22×C4—3142no—
276C22×C4—1572no—
277C22×C4—1571no—
278Q16—22no—
279Q16—3142no—
280Q16—3142no—
281Q16—3142no—
282SD16—22no—
283SD16—3142no—
284SD16—3142no—
285SD16—3142no—
286M16—3142no—
287M16—1572no—
288M16C22×D15712yes—
289C2×C8—3142no—
290C4⋊C4—3141no—
291C4⋊C4—3142no—
292C4⋊C4—3142no—
293C22⋊C4—22no—
294C22⋊C4—3141no—
295C22⋊C4—3142no—
296C42—1571no—
297C42C22×D15711yescharacteristic
298C23—22no—
299C23—3142no—
300C23—3142no—
301C23—1571no—
302C23D4×D15712yes—
303Q8—22no—
304Q8—3144no—
305Q8—3142no—
306Q8—3142no—
307Q8—3142no—
308Q8—6281no—
309Q8—6281no—
310Q8—1572no—
311Q8D4×D15712yes—
312D4—24no—
313D4—22no—
314D4—22no—
315D4—3142no—
316D4—3144no—
317D4—3144no—
318D4—3142no—
319D4—3144no—
320D4—3142no—
321D4—3142no—
322D4—1572no—
323D4D4×D15712yes—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3144no—
329C2×C4—3142no—
330C2×C4—3142no—
331C2×C4—3142no—
332C2×C4—3144no—
333C2×C4—3142no—
334C2×C4—3142no—
335C2×C4—3142no—
336C2×C4—3142no—
337C2×C4—3141no—
338C2×C4—3141no—
339C2×C4—6282no—
340C2×C4—1572no—
341C2×C4—1571no—
342C2×C4—1572no—
343C2×C4—1572no—
344C2×C4C23×D15711yescharacteristic
345C8—22no—
346C8—3142no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—82no—
351C22—3142no—
352C22—6281no—
353C22—1572no—
354C22—12562no—
355C22—12562no—
356C22C2×D4×D15711yescharacteristic
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3142no—
361C4—3142no—
362C4—6281no—
363C4—6281no—
364C4—1571no—
365C4—1571no—
366C4C2×D4×D15712yes—
367C2—21no—
368C2—42no—
369C2—42no—
370C2—3142no—
371C2—6282no—
372C2#120×D15711yescharacteristic
subgroupG/H
centerC2#120×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#455×C157C2
socleC2×C157C2×#120
prime radical#23758108C2
hypercenter#455D157
hypocenterC157#3022
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #137×D157C2semidirect
3. #23757843C2semidirect
4. #23757457C2semidirect
5. #23757454C2semidirect
6. #23757453C2semidirect
7. #23757451C2semidirect
8. #23757450C2semidirect
9. #23757288C2semidirect
10. #23757254C2semidirect
11. #455×C157C2semidirect
22. C2×Q8×D157C22semidirect
23. #19097804C22semidirect
25. #19097802C22semidirect
26. #19097801C22semidirect
30. M16×D157C22semidirect
31. #19097771C22semidirect
32. #19097769C22semidirect
33. #19097758C22semidirect
36. #19097728C22semidirect
37. #19097699C22semidirect
38. #19097698C22semidirect
39. #19097697C22semidirect
40. #19097689C22semidirect
41. #19097688C22semidirect
43. #137×C157C22semidirect
44. #124×C157C22semidirect
45. #104×C157C22semidirect
46. #100×C157C22semidirect
186. C157#3022semidirect
196. #455D157semidirect
216. #137C2×D157semidirect
223. #124C2×D157semidirect
230. #104C2×D157semidirect
233. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581082
#137×D1572
#237578432
#237574572
#237574542
#237574532
#237574512
#237574502
#237572882
#237572542
#455×C1572
#3022157