Finite Group Databasebeta

← #23775005

Subgroups — #23775005

Order 10112.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14579185C211yesmaximal, characteristic, complemented
2#19099207C211yesmaximal, characteristic, complemented
3#19099201C211yesmaximal, characteristic, complemented
4#19098944C211yesmaximal, characteristic, complemented
5#19098809C211yesmaximal, characteristic, complemented
6#19098806C211yesmaximal, characteristic, complemented
7#19098780C211yesmaximal, characteristic, complemented
8#19098776C211yesmaximal, characteristic, complemented
9#19098725C211yesmaximal, characteristic, complemented
10#19098722C211yesmaximal, characteristic, complemented
11#19098697C211yesmaximal, characteristic, complemented
12#19098695C211yesmaximal, characteristic, complemented
13D8×Dic79C211yesmaximal, characteristic, complemented
14#19098639C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C79C211yesmaximal, characteristic, complemented
16C2×#11929623C2211yescharacteristic
17C2×D4×D79C2211yescharacteristic, complemented
18#14579185—24no—
19C2×#11929614C2211yescharacteristic, complemented
20C2×#11929613C2211yescharacteristic, complemented
21C2×#11929612C2211yescharacteristic, complemented
22C2×#11929609C2211yescharacteristic
23C2×#11929609C2211yescharacteristic
24C2×#11929607C2211yescharacteristic, complemented
25#14579150C2211yescharacteristic, complemented
26#14579148C2211yescharacteristic, complemented
27#14579144—21no—
28#14579142—21no—
29#14579140C2211yescharacteristic, complemented
30#14579137C2211yescharacteristic, complemented
31#14579135C2211yescharacteristic
32#14579133—21no—
33#14579133—21no—
34#14579132C2211yescharacteristic
35#14579130—21no—
36#14579129—21no—
37D4×Dic79C2211yescharacteristic, complemented
38D4×Dic79C2211yescharacteristic, complemented
39C4×C22⋊D79C2211yescharacteristic
40C2×C22⋊Dic79—21no—
41C2×(C2×D79)⋊C4—21no—
42#14579095C2211yescharacteristic, complemented
43Dic79⋊C8C2211yescharacteristic, complemented
44#14579086C2211yescharacteristic, complemented
45#14579085C2211yescharacteristic, complemented
46#14579083C2211yescharacteristic, complemented
47#14579082C2211yescharacteristic, complemented
48#14579082C2211yescharacteristic, complemented
49#14579082C2211yescharacteristic, complemented
50C22⋊(C79⋊C8)C2211yescharacteristic, complemented
51(C2×D79)⋊C8C2211yescharacteristic, complemented
52C8×Dic79C2211yescharacteristic, complemented
53C2×#41×C79C2211yescharacteristic
54C22×D4×C79C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C79—24no—
56C2×SD16×C79C2211yescharacteristic, complemented
57C2×SD16×C79C2211yescharacteristic, complemented
58C2×D8×C79C2211yescharacteristic
59C2×D8×C79C2211yescharacteristic
60C2×M16×C79C2211yescharacteristic, complemented
61C23×D79—21no—
62#11929623—22no—
63#11929623—41no—
64C2×Dic158C2311yescharacteristic
65C2×D316C2311yescharacteristic
66C2×C22⋊D79—21no—
67C2×C22⋊D79—21no—
68C2×C22⋊D79—21no—
69C2×C22⋊D79—21no—
70C2×C22⋊D79D411yescharacteristic
71D4×D79—22no—
72D4×D79—41no—
73C22×Dic79—21no—
74C22×Dic79—21no—
75C22×Dic79—21no—
76C2×C4×D79C2311yescharacteristic
77#11929614—41no—
78#11929613—22no—
79#11929612—22no—
80#11929609—22no—
81#11929609—41no—
82#11929607—22no—
83C2×C79⋊C8C2311yescharacteristic
84C2×C79⋊C8C2311yescharacteristic
85C4⋊Dic79C2311yescharacteristic
86C4⋊Dic79C2311yescharacteristic
87Dic79⋊C4—21no—
88Dic79⋊C4—21no—
89Dic79⋊C4D411yescharacteristic
90C22⋊Dic79—21no—
91C22⋊Dic79—21no—
92C22⋊Dic79—21no—
93C22⋊Dic79—21no—
94C22⋊Dic79—21no—
95C22⋊Dic79—21no—
96C22⋊Dic79D411yescharacteristic
97(C2×D79)⋊C4—21no—
98(C2×D79)⋊C4—21no—
99(C2×D79)⋊C4—21no—
100(C2×D79)⋊C4—21no—
101(C2×D79)⋊C4D411yescharacteristic
102C4×Dic79C2311yescharacteristic
103C24×C79—21no—
104#41×C79—22no—
105#41×C79—41no—
106C2×Q8×C79C2311yescharacteristic
107C2×D4×C79—22no—
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—41no—
111C2×D4×C79C2311yescharacteristic
112C2×D4×C79C2311yescharacteristic
113C2×D4×C79C2311yescharacteristic
114C22×C4×C79—21no—
115C22×C4×C79C2311yescharacteristic
116SD16×C79—22no—
117SD16×C79—22no—
118D8×C79—22no—
119D8×C79D414yescomplemented
120M16×C79—22no—
121C2×C8×C79C2311yescharacteristic
122C2×C8×C79C2311yescharacteristic
123C23×C79—21no—
124C23×C79—21no—
125C23×C79—21no—
126C23×C79—42no—
127C23×C79—42no—
128C23×C79—81no—
129C23×C79C2×D411yescharacteristic
130C22×D79—21no—
131C22×D79—42no—
132C22×D79—42no—
133C22×D79—81no—
134C22×D79C2×D411yescharacteristic
135Q8×C79—21no—
136Q8×C79—21no—
137D4×C79—21no—
138D4×C79—21no—
139D4×C79—21no—
140D4×C79—21no—
141D4×C79—42no—
142D4×C79—42no—
143D4×C79C2×D412yes—
144D4×C79C2×D412yes—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—42no—
149C2×C4×C79C2×D411yescharacteristic
150C2×C4×C79C2411yescharacteristic
151C22⋊D79—41no—
152C22⋊D79—42no—
153C22⋊D79—42no—
154C22⋊D79—81no—
155C22⋊D79—81no—
156C2×Dic79—21no—
157C2×Dic79—21no—
158C2×Dic79—21no—
159C2×Dic79—42no—
160C2×Dic79—81no—
161C2×Dic79—81no—
162C2×Dic79C2×D411yescharacteristic
163C2×Dic79C2×D411yescharacteristic
164C2×Dic79C2×D411yescharacteristic
165D316—21no—
166D316—21no—
167C4×D79—21no—
168C4×D79—21no—
169Dic158—21no—
170Dic158—21no—
171C8×C79—21no—
172C8×C79C2×D412yes—
173C79⋊C8—21no—
174C79⋊C8—21no—
175C22×C79—22no—
176C22×C79—22no—
177C22×C79—21no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—41no—
181C22×C79—41no—
182C22×C79—41no—
183C22×C79—41no—
184C22×C79—41no—
185C22×C79—41no—
186C22×C79—41no—
187C22×C79—84no—
188C22×C79C22×D411yescharacteristic
189C2×D79—21no—
190C2×D79—21no—
191C2×D79—21no—
192C2×D79—41no—
193C2×D79—41no—
194C2×D79—41no—
195C2×D79—84no—
196C4×C79—21no—
197C4×C79—41no—
198C4×C79C22×D411yescharacteristic
199C4×C79H(5,𝔽2)11yescharacteristic
200Dic79—21no—
201Dic79—21no—
202Dic79—21no—
203Dic79—41no—
204Dic79—41no—
205Dic79—41no—
206C2×C79—42no—
207C2×C79—41no—
208C2×C79—42no—
209C2×C79—81no—
210C2×C79D4211yescharacteristic
211C2×C79#57511yescharacteristic
212C2×C79#57611yescharacteristic
213D79—41no—
214D79—81no—
215#3273—791nomaximal, Sylow
216C79#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—791no—
218C2×AGL(1,ℤ/8)D7911yescharacteristic, complemented
219#546—791no—
220#546—791no—
221#495—791no—
222#468—791no—
223#468—791no—
224#463—791no—
225#463—791no—
226#452—791no—
227#452—791no—
228#447—791no—
229#447—791no—
230C4×D8—791no—
231#436—791no—
232C2×#41—791no—
233C2×#41C2×D7911yescharacteristic
234C22×D4—791no—
235C22×D4C2×D7911yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—1584no—
238C2×Q16—791no—
239C2×SD16—791no—
240C2×SD16—791no—
241C2×SD16C2×D7911yescharacteristic, complemented
242C2×SD16C2×D7911yescharacteristic, complemented
243C2×D8—791no—
244C2×D8—791no—
245C2×D8C2×D7911yescharacteristic
246C2×D8C2×D7911yescharacteristic
247C2×M16—791no—
248C2×M16C2×D7911yescharacteristic, complemented
249#124—791no—
250#124—791no—
251#123—1581no—
252#123—1581no—
253C22⋊Q8—791no—
254C22⋊Q8—791no—
255#121—791no—
256#121—791no—
257#121—1581no—
258#121—1581no—
259#120—1581no—
260#120—1581no—
261C4×D4—791no—
262C4×D4—791no—
263C4×D4—791no—
264C2×C22⋊C4—1581no—
265C2×C22⋊C4—1581no—
266#107—791no—
267C4⋊C8—791no—
268#103—791no—
269#103—791no—
270#102—791no—
271#102—791no—
272#102—791no—
273#102—791no—
274C22⋊C8—791no—
275C22⋊C8—791no—
276C4×C8—791no—
277C24—21no—
278C24—1581no—
279#41—22no—
280#41—41no—
281#41—3161no—
282#41—1582no—
283C2×Q8—791no—
284C2×Q8C22×D7911yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3161no—
290C2×D4—791no—
291C2×D4—791no—
292C2×D4—1581no—
293C2×D4—1581no—
294C2×D4—1582no—
295C2×D4—1581no—
296C2×D4—1581no—
297C2×D4C22×D7911yescharacteristic
298C2×D4C22×D7911yescharacteristic
299C2×D4C22×D7911yescharacteristic
300C22×C4—21no—
301C22×C4—791no—
302C22×C4—1581no—
303C22×C4—1581no—
304C22×C4—1581no—
305C22×C4C22×D7911yescharacteristic
306Q16—3161no—
307SD16—22no—
308SD16—22no—
309SD16—1582no—
310SD16—1582no—
311D8—22no—
312D8—3161no—
313D8—1582no—
314D8C22⋊D7914yescomplemented
315M16—22no—
316M16—1582no—
317C2×C8—791no—
318C2×C8—791no—
319C2×C8C22×D7911yescharacteristic
320C2×C8C22×D7911yescharacteristic
321C4⋊C4—791no—
322C4⋊C4—791no—
323C4⋊C4—791no—
324C4⋊C4—1581no—
325C4⋊C4—1581no—
326C22⋊C4—791no—
327C22⋊C4—791no—
328C22⋊C4—1581no—
329C22⋊C4—1581no—
330C22⋊C4—1581no—
331C22⋊C4—1581no—
332C22⋊C4—1581no—
333C22⋊C4—1581no—
334C22⋊C4—1581no—
335C22⋊C4—1581no—
336C22⋊C4—1581no—
337C22⋊C4—1581no—
338C42—791no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—3162no—
346C23—3162no—
347C23—791no—
348C23—6321no—
349C23—1581no—
350C23D4×D7911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1581no—
354Q8—1581no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—3162no—
362D4—3162no—
363D4—3161no—
364D4—6321no—
365D4—6321no—
366D4—1581no—
367D4—1581no—
368D4C2×C22⋊D7912yes—
369D4C2×C22⋊D7912yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—3162no—
375C2×C4—791no—
376C2×C4—791no—
377C2×C4—791no—
378C2×C4—6321no—
379C2×C4—6321no—
380C2×C4—1581no—
381C2×C4—1581no—
382C2×C4—1581no—
383C2×C4—1581no—
384C2×C4—1581no—
385C2×C4D4×D7911yescharacteristic
386C2×C4C23×D7911yescharacteristic
387C8—21no—
388C8—1581no—
389C8—1581no—
390C8C2×C22⋊D7912yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3161no—
405C22—3161no—
406C22—3161no—
407C22—6324no—
408C22—1581no—
409C22—1581no—
410C22—1581no—
411C22C2×D4×D7911yescharacteristic
412C4—21no—
413C4—41no—
414C4—3161no—
415C4—3161no—
416C4—3161no—
417C4—1581no—
418C4—1581no—
419C4—1581no—
420C4C22×C22⋊D7911yescharacteristic
421C4#1457920611yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—3161no—
427C2—6321no—
428C2D4×C22⋊D7911yescharacteristic
429C2#1909935911yescharacteristic
430C2#1909936711yescharacteristic
subgroupG/H
centerC22C2×D4×D79
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
FittingC2×AGL(1,ℤ/8)×C79C2
socleC22×C79C22×D4
hypercenterC2×AGL(1,ℤ/8)D79
hypocenterC79#3273
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
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×#14579185C2semidirect
2. #19099207C2semidirect
3. #19099201C2semidirect
4. #19098944C2semidirect
5. #19098809C2semidirect
6. #19098806C2semidirect
7. #19098780C2semidirect
8. #19098776C2semidirect
9. #19098725C2semidirect
10. #19098722C2semidirect
11. #19098697C2semidirect
12. #19098695C2semidirect
13. D8×Dic79C2semidirect
14. #19098639C2semidirect
15. C2×AGL(1,ℤ/8)×C79C2semidirect
17. C2×D4×D79C22semidirect
19. C2×#11929614C22semidirect
20. C2×#11929613C22semidirect
21. C2×#11929612C22semidirect
24. C2×#11929607C22semidirect
25. #14579150C22semidirect
26. #14579148C22semidirect
29. #14579140C22semidirect
30. #14579137C22semidirect
37. D4×Dic79C22semidirect
38. D4×Dic79C22semidirect
42. #14579095C22semidirect
43. Dic79⋊C8C22semidirect
44. #14579086C22semidirect
45. #14579085C22semidirect
46. #14579083C22semidirect
47. #14579082C22semidirect
48. #14579082C22semidirect
49. #14579082C22semidirect
50. C22⋊(C79⋊C8)C22semidirect
51. (C2×D79)⋊C8C22semidirect
52. C8×Dic79C22semidirect
54. C22×D4×C79C22semidirect
56. C2×SD16×C79C22semidirect
57. C2×SD16×C79C22semidirect
60. C2×M16×C79C22semidirect
119. D8×C79D4semidirect
216. C79#3273semidirect
218. C2×AGL(1,ℤ/8)D79semidirect
235. C22×D4C2×D79semidirect
241. C2×SD16C2×D79semidirect
242. C2×SD16C2×D79semidirect
248. C2×M16C2×D79semidirect
314. D8C22⋊D79semidirect