Finite Group Databasebeta

← #34109408

Subgroups — #34109408

Order 20096.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097816C211yesmaximal, characteristic
2C2×#19097800C211yesmaximal, characteristic, complemented
3C2×SD16×D157C211yesmaximal, characteristic, complemented
4#23757846C211yesmaximal, characteristic, complemented
5#23757437C211yesmaximal, characteristic, complemented
6#23757436C211yesmaximal, characteristic, complemented
7#23757433C211yesmaximal, characteristic, complemented
8#23757432C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic, complemented
10#23757424C211yesmaximal, characteristic, complemented
11#23757422C211yesmaximal, characteristic, complemented
12#23757276C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14#23757238C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16#19097816—24no—
17#19097816—24no—
18C2×#14578954C2211yescharacteristic
19C2×#14578954C2211yescharacteristic
20C2×#14578953—21no—
21C2×#14578953C2211yescharacteristic
22C2×#14578953C2211yescharacteristic
23C2×Q8×D157C2211yescharacteristic, complemented
24C22×C22⋊D157—21no—
25C22×C22⋊D157—21no—
26C2×D4×D157—21no—
27C2×D4×D157C2211yescharacteristic
28C2×D4×D157C2211yescharacteristic
29#19097800—24no—
30C2×Dic628C2211yescharacteristic, complemented
31C2×#14578944C2211yescharacteristic, complemented
32C2×#14578943C2211yescharacteristic, complemented
33C2×#14578942C2211yescharacteristic, complemented
34C2×#14578942C2211yescharacteristic, complemented
35C2×#14578941C2211yescharacteristic, complemented
36SD16×D157—24no—
37C2×#14578937C2211yescharacteristic, complemented
38C2×C8×D157C2211yescharacteristic, complemented
39#19097770—21no—
40#19097757—21no—
41#19097756—21no—
42#19097753—21no—
43#19097751C2211yescharacteristic, complemented
44#19097750C2211yescharacteristic, complemented
45#19097747C2211yescharacteristic, complemented
46#19097729C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097691C2211yescharacteristic, complemented
53C22⋊#11929434C2211yescharacteristic, complemented
54#19097683C2211yescharacteristic, complemented
55#19097683C2211yescharacteristic, complemented
56C22×D4×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic, complemented
58C2×SD16×C157C2211yescharacteristic, complemented
59C22⋊Q8×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62C22⋊C8×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64C23×D157—21no—
65#14578954—22no—
66#14578954—22no—
67#14578954—22no—
68#14578954—22no—
69#14578953—28no—
70#14578953—22no—
71#14578953—22no—
72#14578953—22no—
73#14578953D414yes—
74C2×Dic314—21no—
75C2×Dic314D411yescharacteristic
76C2×Dic314C2311yescharacteristic
77C2×Dic314C2311yescharacteristic
78Q8×D157—22no—
79Q8×D157—22no—
80C2×D628D411yescharacteristic
81C2×D628C2311yescharacteristic
82C2×C22⋊D157—28no—
83C2×C22⋊D157—28no—
84C2×C22⋊D157—21no—
85C2×C22⋊D157—21no—
86C2×C22⋊D157—21no—
87C2×C22⋊D157—21no—
88C2×C22⋊D157—21no—
89C2×C22⋊D157D411yescharacteristic
90C2×C22⋊D157D411yescharacteristic
91D4×D157—28no—
92D4×D157—22no—
93D4×D157—22no—
94D4×D157—22no—
95D4×D157D414yes—
96C22×Dic157—21no—
97C22×Dic157—21no—
98C2×C4×D157—21no—
99C2×C4×D157—21no—
100C2×C4×D157C2311yescharacteristic
101C2×C4×D157C2311yescharacteristic
102Dic628—22no—
103#14578944—22no—
104#14578943—22no—
105#14578942—22no—
106#14578942—22no—
107#14578941—22no—
108#14578937—22no—
109C2×#11929434C2311yescharacteristic
110C2×#11929434C2311yescharacteristic
111C8×D157—22no—
112C4⋊Dic157—21no—
113C4⋊Dic157C2311yescharacteristic
114#14578933—21no—
115#14578933—21no—
116C22⋊Dic157—21no—
117#14578931—21no—
118#14578931—21no—
119#14578931—21no—
120C4×Dic157—21no—
121C24×C157—21no—
122C2×Q8×C157C2311yescharacteristic
123C2×D4×C157—22no—
124C2×D4×C157—22no—
125C2×D4×C157—21no—
126C2×D4×C157C2311yescharacteristic
127C2×D4×C157C2311yescharacteristic
128C22×C4×C157C2311yescharacteristic
129SD16×C157—22no—
130SD16×C157—22no—
131C2×C8×C157C2311yescharacteristic
132C2×C8×C157C2311yescharacteristic
133C4⋊C4×C157—21no—
134C4⋊C4×C157C2311yescharacteristic
135C22⋊C4×C157—21no—
136C22×D157—21no—
137C22×D157—24no—
138C22×D157—24no—
139C22×D157—81no—
140C22×D157—81no—
141C22×D157—81no—
142C22×D157—81no—
143C22×D157C2×D411yescharacteristic
144C22×D157C2×D411yescharacteristic
145Dic314—21no—
146Dic314—22no—
147Dic314—21no—
148Dic314—21no—
149Dic314—42no—
150Dic314C2×D412yes—
151D628—22no—
152D628—21no—
153D628C2×D412yes—
154C22⋊D157—24no—
155C22⋊D157—24no—
156C22⋊D157—24no—
157C22⋊D157—24no—
158C22⋊D157—22no—
159C22⋊D157—24no—
160C22⋊D157—22no—
161C2×Dic157—24no—
162C2×Dic157—21no—
163C2×Dic157—24no—
164C2×Dic157—21no—
165C2×Dic157—21no—
166C2×Dic157C2×D411yescharacteristic
167C2×Dic157C2×D411yescharacteristic
168C4×D157—24no—
169C4×D157—21no—
170C4×D157—21no—
171C4×D157—42no—
172C4×D157C2×D412yes—
173C4×D157C2×D412yes—
174#11929434—21no—
175#11929434—21no—
176C23×C157—24no—
177C23×C157—21no—
178C23×C157—21no—
179C23×C157—81no—
180C23×C157—81no—
181C23×C157C2×D411yescharacteristic
182Q8×C157—21no—
183Q8×C157—21no—
184D4×C157—24no—
185D4×C157—21no—
186D4×C157—21no—
187D4×C157C2×D412yes—
188D4×C157C2×D412yes—
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157C2×D411yescharacteristic
194C2×C4×C157C2411yescharacteristic
195C8×C157—21no—
196C8×C157—21no—
197C22×C157—21no—
198C22×C157—22no—
199C22×C157—22no—
200C22×C157—21no—
201C22×C157—21no—
202C22×C157—41no—
203C22×C157—41no—
204C22×C157—41no—
205C22×C157—41no—
206C22×C157—84no—
207C22×C157C22×D411yescharacteristic
208C2×D157—21no—
209C2×D157—21no—
210C2×D157—21no—
211C2×D157—22no—
212C2×D157—21no—
213C2×D157—21no—
214C2×D157—41no—
215C2×D157—41no—
216C2×D157—84no—
217C2×D157—84no—
218C2×D157#12012yes—
219C4×C157—21no—
220C4×C157—41no—
221C4×C157—41no—
222C4×C157C22×D411yescharacteristic
223C4×C157C22×D411yescharacteristic
224Dic157—22no—
225Dic157—21no—
226Dic157—41no—
227Dic157—41no—
228Dic157#12012yes—
229C2×C157—42no—
230C2×C157—42no—
231C2×C157—41no—
232C2×C157C2×#12011yescharacteristic
233C2×C157#57711yescharacteristic
234C2×C157#57711yescharacteristic
235D157—22no—
236D157—41no—
237D157—42no—
238C157#300911yescharacteristic, complemented, Sylow
239#3009—1571nomaximal, Sylow
240C2×H(5,𝔽2)—1571no—
241C2×#137—1571no—
242C22×SD16—1571no—
243#536—1571no—
244#452—1571no—
245#452—1571no—
246#452—1571no—
247#452—1571no—
248#450—1571no—
249#450—1571no—
250#450—1571no—
251#450D15711yescharacteristic, complemented
252#418—1571no—
253#416—1571no—
254#408—1571no—
255H(5,𝔽2)—3144no—
256H(5,𝔽2)—3144no—
257C2×#41—3141no—
258C2×#41—1571no—
259C2×#41—1571no—
260C2×#41—1571no—
261C2×#41—1571no—
262C22×Q8—1571no—
263C22×D4—3141no—
264C22×D4—3141no—
265C22×D4—3141no—
266C22×D4—1571no—
267C22×D4—1571no—
268C22×D4C2×D15711yescharacteristic
269#137—3144no—
270C2×Q16—1571no—
271C2×Q16—1571no—
272C2×SD16—3144no—
273C2×SD16—1571no—
274C2×SD16—1571no—
275C2×SD16—1571no—
276C2×SD16—1571no—
277C2×SD16C2×D15711yescharacteristic, complemented
278C2×SD16C2×D15711yescharacteristic, complemented
279C2×M16—1571no—
280C22×C8—1571no—
281C4⋊Q8—3141no—
282#124—3141no—
283#123—3141no—
284#123—3141no—
285C22⋊Q8—1571no—
286C22⋊Q8—1571no—
287C22⋊Q8—1571no—
288C22⋊Q8C2×D15711yescharacteristic, complemented
289#117—1571no—
290#103—1571no—
291#103—1571no—
292#102—1571no—
293#102—1571no—
294#102—1571no—
295#102—1571no—
296#102C2×D15711yescharacteristic, complemented
297#102C2×D15711yescharacteristic, complemented
298C22⋊C8—1571no—
299C22⋊C8—1571no—
300C22⋊C8—1571no—
301C22⋊C8C2×D15711yescharacteristic, complemented
302C24—21no—
303C24—3141no—
304C24—3141no—
305#41—3142no—
306#41—3142no—
307#41—3142no—
308#41—3142no—
309#41—3148no—
310#41—3142no—
311#41—3142no—
312#41—3142no—
313#41—1574no—
314C2×Q8—3141no—
315C2×Q8—3142no—
316C2×Q8—3142no—
317C2×Q8—1571no—
318C2×Q8—1571no—
319C2×Q8—1571no—
320C2×Q8C22×D15711yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—3148no—
325C2×D4—3142no—
326C2×D4—3148no—
327C2×D4—3148no—
328C2×D4—3141no—
329C2×D4—3141no—
330C2×D4—3141no—
331C2×D4—3142no—
332C2×D4—3142no—
333C2×D4—3141no—
334C2×D4—3141no—
335C2×D4—1571no—
336C2×D4—1571no—
337C2×D4—1574no—
338C2×D4—1571no—
339C2×D4—1571no—
340C2×D4C22×D15711yescharacteristic
341C2×D4C22×D15711yescharacteristic
342C22×C4—3141no—
343C22×C4—3141no—
344C22×C4—3141no—
345C22×C4—3141no—
346C22×C4—1571no—
347C22×C4—1571no—
348C22×C4C22×D15711yescharacteristic
349Q16—3142no—
350Q16—3142no—
351SD16—22no—
352SD16—22no—
353SD16—3142no—
354SD16—3142no—
355SD16—3142no—
356SD16—3142no—
357M16—3142no—
358C2×C8—3142no—
359C2×C8—1571no—
360C2×C8—1571no—
361C2×C8C22×D15711yescharacteristic
362C2×C8C22×D15711yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—3141no—
365C4⋊C4—3141no—
366C4⋊C4—3141no—
367C4⋊C4—1571no—
368C4⋊C4C22×D15711yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—3141no—
371C22⋊C4—3141no—
372C22⋊C4—3141no—
373C22⋊C4—3141no—
374C42—3141no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—3144no—
381C23—3144no—
382C23—3141no—
383C23—1571no—
384C23—1571no—
385C23—12561no—
386C23—12561no—
387C23—12561no—
388C23—12561no—
389C23D4×D15711yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—3142no—
393Q8—3141no—
394Q8—3141no—
395Q8—3141no—
396Q8—6282no—
397Q8—1572no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—3142no—
402D4—3144no—
403D4—3144no—
404D4—3144no—
405D4—3144no—
406D4—3142no—
407D4—3142no—
408D4—3144no—
409D4—3141no—
410D4—1572no—
411D4D4×D15712yes—
412D4D4×D15712yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—3144no—
418C2×C4—3144no—
419C2×C4—3141no—
420C2×C4—3144no—
421C2×C4—3141no—
422C2×C4—3141no—
423C2×C4—3141no—
424C2×C4—3141no—
425C2×C4—6282no—
426C2×C4—1571no—
427C2×C4—1571no—
428C2×C4—1572no—
429C2×C4—1572no—
430C2×C4D4×D15711yescharacteristic
431C2×C4C23×D15711yescharacteristic
432C8—21no—
433C8—21no—
434C8—3141no—
435C8—3141no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—3141no—
447C22—3141no—
448C22—3141no—
449C22—3141no—
450C22—3142no—
451C22—3141no—
452C22—6281no—
453C22—6281no—
454C22—1572no—
455C22—12564no—
456C22—12564no—
457C22C2×D4×D15711yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—3142no—
462C4—3141no—
463C4—6281no—
464C4—6281no—
465C4—1572no—
466C4C2×D4×D15711yescharacteristic
467C4C2×D4×D15711yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—3142no—
472C2—6281no—
473C2—6282no—
474C2#120×D15711yescharacteristic
475C2#2375807911yescharacteristic
476C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097816C2
hypercenter#450D157
hypocenterC157#3009
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#19097800C2semidirect
3. C2×SD16×D157C2semidirect
4. #23757846C2semidirect
5. #23757437C2semidirect
6. #23757436C2semidirect
7. #23757433C2semidirect
8. #23757432C2semidirect
9. #23757425C2semidirect
10. #23757424C2semidirect
11. #23757422C2semidirect
12. #23757276C2semidirect
13. #23757266C2semidirect
14. #23757238C2semidirect
15. #450×C157C2semidirect
23. C2×Q8×D157C22semidirect
30. C2×Dic628C22semidirect
31. C2×#14578944C22semidirect
32. C2×#14578943C22semidirect
33. C2×#14578942C22semidirect
34. C2×#14578942C22semidirect
35. C2×#14578941C22semidirect
37. C2×#14578937C22semidirect
38. C2×C8×D157C22semidirect
43. #19097751C22semidirect
44. #19097750C22semidirect
45. #19097747C22semidirect
46. #19097729C22semidirect
47. #19097696C22semidirect
48. #19097694C22semidirect
49. #19097693C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
52. #19097691C22semidirect
53. C22⋊#11929434C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
57. C2×SD16×C157C22semidirect
58. C2×SD16×C157C22semidirect
59. C22⋊Q8×C157C22semidirect
60. #102×C157C22semidirect
61. #102×C157C22semidirect
62. C22⋊C8×C157C22semidirect
238. C157#3009semidirect
251. #450D157semidirect
277. C2×SD16C2×D157semidirect
278. C2×SD16C2×D157semidirect
288. C22⋊Q8C2×D157semidirect
296. #102C2×D157semidirect
297. #102C2×D157semidirect
301. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190978002
C2×SD16×D1572
#237578462
#237574372
#237574362
#237574332
#237574322
#237574252
#237574242
#237574222
#237572762
#237572662
#237572382
#450×C1572
#3009157