Finite Group Databasebeta

← #34111655

Subgroups — #34111655

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×#19097790C211yesmaximal, characteristic, complemented
2#23757922C211yesmaximal, characteristic, complemented
3#23757917C211yesmaximal, characteristic, complemented
4#23757664C211yesmaximal, characteristic, complemented
5#23757531C211yesmaximal, characteristic, complemented
6#23757499C211yesmaximal, characteristic, complemented
7#23757486C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757424C211yesmaximal, characteristic, complemented
12#23757415C211yesmaximal, characteristic, complemented
13#23757385C211yesmaximal, characteristic, complemented
14#23757342C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578953C2211yescharacteristic, complemented
18#19097790—24no—
19C2×Dic628C2211yescharacteristic, complemented
20C2×#14578944C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic
22C2×#14578942C2211yescharacteristic
23C2×#14578942C2211yescharacteristic, complemented
24C2×C8×D157C2211yescharacteristic, complemented
25#19097769C2211yescharacteristic, complemented
26#19097758C2211yescharacteristic
27#19097754—21no—
28#19097752—21no—
29#19097752—21no—
30#19097750C2211yescharacteristic
31#19097750C2211yescharacteristic
32#19097747C2211yescharacteristic, complemented
33#19097743—21no—
34#19097743—21no—
35#19097742C2211yescharacteristic
36#19097740—21no—
37D4×Dic157C2211yescharacteristic
38D4×Dic157C2211yescharacteristic, complemented
39C4×C22⋊D157C2211yescharacteristic
40C2×#14578933—21no—
41C2×C22⋊Dic157—21no—
42#19097703C2211yescharacteristic, complemented
43#19097700C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097696C2211yescharacteristic, complemented
46#19097695C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic, complemented
50C22⋊#11929434C2211yescharacteristic
51#19097683C2211yescharacteristic, complemented
52#19097681C2211yescharacteristic, complemented
53C2×#41×C157C2211yescharacteristic
54C22×D4×C157C2211yescharacteristic
55AGL(1,ℤ/8)×C157—24no—
56C2×SD16×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic
58C2×D8×C157C2211yescharacteristic
59C2×D8×C157C2211yescharacteristic
60C2×M16×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—22no—
63#14578953—41no—
64#14578953—41no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157D411yescharacteristic
71C22×Dic157—21no—
72C22×Dic157—21no—
73C22×Dic157—21no—
74C22×Dic157—21no—
75C2×C4×D157C2311yescharacteristic
76Dic628—22no—
77#14578944—41no—
78#14578942—22no—
79#14578942—22no—
80#14578942—41no—
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C8×D157—22no—
84C4⋊Dic157C2311yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86#14578933—21no—
87#14578933—21no—
88#14578933—21no—
89#14578933—21no—
90#14578933—21no—
91#14578933D411yescharacteristic
92C22⋊Dic157—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96C22⋊Dic157—21no—
97C22⋊Dic157—21no—
98C22⋊Dic157—21no—
99C22⋊Dic157D411yescharacteristic
100#14578931—21no—
101#14578931D411yescharacteristic
102C4×Dic157C2311yescharacteristic
103C24×C157—21no—
104#41×C157—22no—
105#41×C157—41no—
106C2×Q8×C157C2311yescharacteristic
107C2×D4×C157—22no—
108C2×D4×C157—21no—
109C2×D4×C157—21no—
110C2×D4×C157—41no—
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116SD16×C157—22no—
117SD16×C157—22no—
118D8×C157—22no—
119D8×C157D414yescomplemented
120M16×C157—22no—
121C2×C8×C157C2311yescharacteristic
122C2×C8×C157C2311yescharacteristic
123C22×D157C2×D411yescharacteristic
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128C22⋊D157—41no—
129C22⋊D157—42no—
130C22⋊D157—42no—
131C22⋊D157—81no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—42no—
137C2×Dic157—42no—
138C2×Dic157—42no—
139C2×Dic157—81no—
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C4×D157—21no—
144C4×D157—21no—
145#11929434—21no—
146#11929434—21no—
147C23×C157—21no—
148C23×C157—21no—
149C23×C157—21no—
150C23×C157—42no—
151C23×C157—42no—
152C23×C157—81no—
153C23×C157C2×D411yescharacteristic
154Q8×C157—21no—
155Q8×C157—21no—
156D4×C157—21no—
157D4×C157—21no—
158D4×C157—21no—
159D4×C157—21no—
160D4×C157—42no—
161D4×C157—42no—
162D4×C157C2×D412yes—
163D4×C157C2×D412yes—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—42no—
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—21no—
173C22×C157—22no—
174C22×C157—22no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—41no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—84no—
185C22×C157C22×D411yescharacteristic
186C2×D157—21no—
187C2×D157—21no—
188C2×D157—21no—
189C4×C157—21no—
190C4×C157—41no—
191C4×C157C22×D411yescharacteristic
192C4×C157H(5,𝔽2)11yescharacteristic
193Dic157—21no—
194Dic157—21no—
195Dic157—21no—
196Dic157—41no—
197Dic157—41no—
198Dic157—41no—
199Dic157—41no—
200C2×C157—41no—
201C2×C157—42no—
202C2×C157—42no—
203C2×C157—81no—
204C2×C157D4211yescharacteristic
205C2×C157#57511yescharacteristic
206C2×C157#57711yescharacteristic
207D157—41no—
208C157#326711yescharacteristic, complemented, Sylow
209#3267—1571nomaximal, Sylow
210C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
211C2×#135—1571no—
212#547—1571no—
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×#41—1571no—
227C2×#41C2×D15711yescharacteristic
228C22×D4C2×D15711yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—3144no—
231C2×Q16—1571no—
232C2×Q16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16C2×D15711yescharacteristic
237C2×SD16C2×D15711yescharacteristic
238C2×D8C2×D15711yescharacteristic
239C2×D8C2×D15711yescharacteristic
240C2×M16C2×D15711yescharacteristic
241C22×C8—1571no—
242C4⋊Q8—1571no—
243#124—1571no—
244#123—3141no—
245#123—3141no—
246#123—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8—1571no—
249C22⋊Q8—1571no—
250#121—3141no—
251#121—3141no—
252#121—1571no—
253#120—3141no—
254C4×D4—1571no—
255C4×D4—1571no—
256C4×D4—1571no—
257C2×C4⋊C4—3141no—
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—21no—
271#41—22no—
272#41—41no—
273#41—3142no—
274#41—3142no—
275#41—6281no—
276#41—6281no—
277C2×Q8—1571no—
278C2×Q8—1571no—
279C2×Q8C22×D15711yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—1571no—
288C2×D4C22×D15711yescharacteristic
289C2×D4C22×D15711yescharacteristic
290C2×D4C22×D15711yescharacteristic
291C22×C4—21no—
292C22×C4—3141no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4C22×D15711yescharacteristic
298Q16—3142no—
299Q16—6281no—
300SD16—22no—
301SD16—22no—
302SD16—3142no—
303SD16—3142no—
304SD16—6281no—
305D8—22no—
306D8C22⋊D15714yescomplemented
307M16—22no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—3141no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—3141no—
318C4⋊C4—1571no—
319C4⋊C4—1571no—
320C4⋊C4—1571no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—1571no—
330C22⋊C4—1571no—
331C42—1571no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1571no—
339C23D4×D15711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3141no—
343Q8—3141no—
344Q8—3141no—
345Q8—3141no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—6281no—
353D4—6282no—
354D4—6282no—
355D4—12561no—
356D4C2×C22⋊D15712yes—
357D4C2×C22⋊D15712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4—1571no—
374C2×C4—12561no—
375C2×C4D4×D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—3141no—
379C8—3141no—
380C8C2×C22⋊D15712yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3141no—
395C22—3141no—
396C22—3141no—
397C22C2×D4×D15711yescharacteristic
398C4—21no—
399C4—41no—
400C4—3141no—
401C4—3141no—
402C4—3141no—
403C4—6281no—
404C4—6281no—
405C4—6281no—
406C4—6281no—
407C4C22×C22⋊D15711yescharacteristic
408C4#1909781611yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—6281no—
414C2D4×C22⋊D15711yescharacteristic
415C2#2375806911yescharacteristic
416C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3267
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097790C2semidirect
2. #23757922C2semidirect
3. #23757917C2semidirect
4. #23757664C2semidirect
5. #23757531C2semidirect
6. #23757499C2semidirect
7. #23757486C2semidirect
8. #23757476C2semidirect
9. #23757435C2semidirect
10. #23757432C2semidirect
11. #23757424C2semidirect
12. #23757415C2semidirect
13. #23757385C2semidirect
14. #23757342C2semidirect
15. C2×AGL(1,ℤ/8)×C157C2semidirect
17. C2×#14578953C22semidirect
19. C2×Dic628C22semidirect
20. C2×#14578944C22semidirect
23. C2×#14578942C22semidirect
24. C2×C8×D157C22semidirect
25. #19097769C22semidirect
32. #19097747C22semidirect
38. D4×Dic157C22semidirect
42. #19097703C22semidirect
43. #19097700C22semidirect
44. #19097696C22semidirect
45. #19097696C22semidirect
46. #19097695C22semidirect
49. #19097692C22semidirect
51. #19097683C22semidirect
52. #19097681C22semidirect
119. D8×C157D4semidirect
208. C157#3267semidirect
210. C2×AGL(1,ℤ/8)D157semidirect
306. D8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977902
#237579222
#237579172
#237576642
#237575312
#237574992
#237574862
#237574762
#237574352
#237574322
#237574242
#237574152
#237573852
#237573422
C2×AGL(1,ℤ/8)×C1572
#3267157