Finite Group Databasebeta

← #34111662

Subgroups — #34111662

Order 20096.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097796C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757919C211yesmaximal, characteristic, complemented
4#23757665C211yesmaximal, characteristic, complemented
5#23757529C211yesmaximal, characteristic, complemented
6#23757504C211yesmaximal, characteristic, complemented
7#23757488C211yesmaximal, characteristic, complemented
8#23757475C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757433C211yesmaximal, characteristic, complemented
11#23757423C211yesmaximal, characteristic, complemented
12#23757418C211yesmaximal, characteristic, complemented
13#23757383C211yesmaximal, characteristic, complemented
14#23757345C211yesmaximal, characteristic, complemented
15#547×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18#19097796—24no—
19#19097791—24no—
20C2×#14578944C2211yescharacteristic, complemented
21C2×#14578944C2211yescharacteristic, complemented
22C2×#14578942C2211yescharacteristic, complemented
23C2×#14578941C2211yescharacteristic
24C2×#14578941C2211yescharacteristic
25C2×#14578941C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578938C2211yescharacteristic
30C22×#11929434C2211yescharacteristic, complemented
31#19097759C2211yescharacteristic
32#19097756—21no—
33#19097750C2211yescharacteristic
34#19097745C2211yescharacteristic
35#19097744—21no—
36#19097739—21no—
37C4×D628C2211yescharacteristic
38C22⋊C4×D157—21no—
39#19097702C2211yescharacteristic, complemented
40C4⋊#11929434C2211yescharacteristic, complemented
41#19097695C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097694C2211yescharacteristic, complemented
44#19097692C2211yescharacteristic, complemented
45#19097691C2211yescharacteristic
46#19097691C2211yescharacteristic
47C22⋊#11929434C2211yescharacteristic
48C22⋊#11929434C2211yescharacteristic, complemented
49#19097682C2211yescharacteristic, complemented
50C2×#41×C157C2211yescharacteristic
51C2×#41×C157C2211yescharacteristic, complemented
52C4⋊Q8×C157C2211yescharacteristic, complemented
53#123×C157—21no—
54#123×C157—21no—
55C22⋊Q8×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157—21no—
58C4×D4×C157C2211yescharacteristic
59C4×D4×C157C2211yescharacteristic, complemented
60C2×C4⋊C4×C157—21no—
61C23×D157—21no—
62#14578954—22no—
63#14578954—41no—
64C2×Dic314C2311yescharacteristic
65C2×D628—22no—
66C2×D628—21no—
67C2×D628—41no—
68C2×D628C2311yescharacteristic
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74#14578944—22no—
75#14578944—41no—
76#14578942—22no—
77#14578941—22no—
78#14578941—22no—
79#14578941—22no—
80#14578941—41no—
81#14578939—22no—
82#14578939D414yescomplemented
83#14578938—22no—
84C2×#11929434—22no—
85C2×#11929434C2311yescharacteristic
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C2×#11929434C2311yescharacteristic
89C4⋊Dic157C2311yescharacteristic
90#14578933—21no—
91C22⋊Dic157—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#41×C157—22no—
97#41×C157—22no—
98#41×C157—41no—
99#41×C157—41no—
100C2×Q8×C157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C2×D4×C157—21no—
103C2×D4×C157—21no—
104C2×D4×C157D411yescharacteristic
105C2×D4×C157C2311yescharacteristic
106C22×C4×C157—21no—
107C22×C4×C157—21no—
108C22×C4×C157—21no—
109C22×C4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111C4⋊C4×C157—21no—
112C4⋊C4×C157—21no—
113C4⋊C4×C157—21no—
114C4⋊C4×C157—21no—
115C4⋊C4×C157D411yescharacteristic
116C4⋊C4×C157C2311yescharacteristic
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C22⋊C4×C157D411yescharacteristic
121C22⋊C4×C157D411yescharacteristic
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—81no—
128Dic314—21no—
129Dic314—21no—
130D628—21no—
131D628—21no—
132D628—21no—
133D628—42no—
134D628C2×D412yes—
135C22⋊D157—42no—
136C22⋊D157—81no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C4×D157—42no—
140C4×D157—81no—
141#11929434—21no—
142#11929434—21no—
143#11929434—21no—
144#11929434C2×D412yes—
145C23×C157—21no—
146C23×C157C2×D411yescharacteristic
147C23×C157C2×D411yescharacteristic
148Q8×C157—21no—
149Q8×C157—21no—
150Q8×C157—21no—
151Q8×C157—21no—
152D4×C157—21no—
153D4×C157—41no—
154D4×C157—42no—
155D4×C157—42no—
156D4×C157C2×D412yes—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—42no—
165C2×C4×C157—42no—
166C2×C4×C157—42no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2×D411yescharacteristic
170C2×C4×C157C2×D411yescharacteristic
171C2×C4×C157C2411yescharacteristic
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—22no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—21no—
178C22×C157—21no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157C22×D411yescharacteristic
182C2×D157—22no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—21no—
192C4×C157—21no—
193C4×C157—41no—
194C4×C157—41no—
195C4×C157—41no—
196C4×C157C22×D411yescharacteristic
197C4×C157H(5,𝔽2)11yescharacteristic
198Dic157—41no—
199Dic157—41no—
200C2×C157—41no—
201C2×C157—41no—
202C2×C157—42no—
203C2×C157D4211yescharacteristic
204C2×C157#57511yescharacteristic
205C2×C157#57711yescharacteristic
206D157—42no—
207D157—81no—
208C157#326711yescharacteristic, complemented, Sylow
209#3267—1571nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1571no—
211C2×#135—1571no—
212#547D15711yescharacteristic, complemented
213#546—1571no—
214#497—1571no—
215#470—1571no—
216#465—1571no—
217#463—1571no—
218#461—1571no—
219#452—1571no—
220#452—1571no—
221#450—1571no—
222#449—1571no—
223#442—1571no—
224#435—1571no—
225C2×#41—1571no—
226C2×#41C2×D15711yescharacteristic
227C2×#41C2×D15711yescharacteristic, complemented
228C22×D4—1571no—
229AGL(1,ℤ/8)—3144no—
230#135—3144no—
231C2×Q16—1571no—
232C2×Q16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16—1571no—
238C2×D8—1571no—
239C2×D8—1571no—
240C2×M16—1571no—
241C22×C8—1571no—
242C4⋊Q8C2×D15711yescharacteristic, complemented
243#124—1571no—
244#123—21no—
245#123—21no—
246#123—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8C2×D15711yescharacteristic
249C22⋊Q8C2×D15711yescharacteristic, complemented
250#121—21no—
251#121—3141no—
252#121—1571no—
253#120—3141no—
254C4×D4—1571no—
255C4×D4C2×D15711yescharacteristic
256C4×D4C2×D15711yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—3141no—
259#106—1571no—
260C4⋊C8—1571no—
261#103—1571no—
262#103—1571no—
263#103—1571no—
264#102—1571no—
265#102—1571no—
266#102—1571no—
267C22⋊C8—1571no—
268C22⋊C8—1571no—
269#97—1571no—
270C24—3141no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—3142no—
276#41—6281no—
277C2×Q8—1571no—
278C2×Q8C22×D15711yescharacteristic
279C2×Q8C22×D15711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—3142no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—6281no—
287C2×D4—1571no—
288C2×D4—1571no—
289C2×D4C22⋊D15711yescharacteristic
290C2×D4C22×D15711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4C22×D15711yescharacteristic
297C22×C4C22×D15711yescharacteristic
298Q16—3142no—
299Q16—6281no—
300SD16—3142no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—6281no—
305D8—3142no—
306D8—1574no—
307M16—3142no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8—1571no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—3141no—
318C4⋊C4—1571no—
319C4⋊C4C22⋊D15711yescharacteristic
320C4⋊C4C22×D15711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4C22⋊D15711yescharacteristic
330C22⋊C4C22⋊D15711yescharacteristic
331C42C22×D15711yescharacteristic
332C23—21no—
333C23—3141no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—12561no—
338C23C2×C22⋊D15711yescharacteristic
339C23C2×C22⋊D15711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3141no—
345Q8—3141no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—3141no—
351D4—3141no—
352D4—3141no—
353D4—6282no—
354D4—6282no—
355D4—1572no—
356D4—12561no—
357D4D4×D15712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—3141no—
369C2×C4—3141no—
370C2×C4—6282no—
371C2×C4—12561no—
372C2×C4C2×C22⋊D15711yescharacteristic
373C2×C4C2×C22⋊D15711yescharacteristic
374C2×C4C2×C22⋊D15711yescharacteristic
375C2×C4C2×C22⋊D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—3141no—
378C8—3141no—
379C8—3141no—
380C8—1572no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—3142no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22—6281no—
396C22—12564no—
397C22C22×C22⋊D15711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—6281no—
406C4—6281no—
407C4C2×D4×D15711yescharacteristic
408C4#1909781611yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—6282no—
413C2—12561no—
414C2D4×C22⋊D15711yescharacteristic
415C2#2375807111yescharacteristic
416C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#547×C157C2
socleC22×C157C22×D4
hypercenter#547D157
hypocenterC157#3267
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19097796C2semidirect
2. C2×#19097791C2semidirect
3. #23757919C2semidirect
4. #23757665C2semidirect
5. #23757529C2semidirect
6. #23757504C2semidirect
7. #23757488C2semidirect
8. #23757475C2semidirect
9. #23757435C2semidirect
10. #23757433C2semidirect
11. #23757423C2semidirect
12. #23757418C2semidirect
13. #23757383C2semidirect
14. #23757345C2semidirect
15. #547×C157C2semidirect
20. C2×#14578944C22semidirect
21. C2×#14578944C22semidirect
22. C2×#14578942C22semidirect
30. C22×#11929434C22semidirect
39. #19097702C22semidirect
40. C4⋊#11929434C22semidirect
41. #19097695C22semidirect
42. #19097695C22semidirect
43. #19097694C22semidirect
44. #19097692C22semidirect
48. C22⋊#11929434C22semidirect
49. #19097682C22semidirect
51. C2×#41×C157C22semidirect
52. C4⋊Q8×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
59. C4×D4×C157C22semidirect
82. #14578939D4semidirect
208. C157#3267semidirect
212. #547D157semidirect
227. C2×#41C2×D157semidirect
242. C4⋊Q8C2×D157semidirect
249. C22⋊Q8C2×D157semidirect
256. C4×D4C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977962
C2×#190977912
#237579192
#237576652
#237575292
#237575042
#237574882
#237574752
#237574352
#237574332
#237574232
#237574182
#237573832
#237573452
#547×C1572
#3267157