Finite Group Databasebeta

← #28347955

Subgroups — #28347955

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293285C211yesmaximal, characteristic, complemented
4#20293262C211yesmaximal, characteristic, complemented
5#20292870C211yesmaximal, characteristic, complemented
6#20292860C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292824C211yesmaximal, characteristic, complemented
9#20292776C211yesmaximal, characteristic, complemented
10#20292769C211yesmaximal, characteristic, complemented
11#20292764C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292729C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#496×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic, complemented
17C2×Q8×D107C2211yescharacteristic, complemented
18C2×D4×D107—21no—
19C2×D4×D107C2211yescharacteristic
20C2×D4×D107C2211yescharacteristic
21#16812133—24no—
22C2×#13133807C2211yescharacteristic
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic
26SD16×D107—24no—
27C2×#13133803C2211yescharacteristic
28C2×#13133801C2211yescharacteristic, complemented
29C2×C8×D107C2211yescharacteristic, complemented
30#16812095—21no—
31#16812086—21no—
32#16812085C2211yescharacteristic, complemented
33#16812082—21no—
34#16812080—21no—
35#16812080C2211yescharacteristic
36#16812080C2211yescharacteristic
37#16812080C2211yescharacteristic, complemented
38#16812077—21no—
39#16812077—21no—
40C4×Dic214C2211yescharacteristic, complemented
41C4×D428C2211yescharacteristic
42C4×D428C2211yescharacteristic
43(C22⋊D107)⋊C4—21no—
44#16812043C2211yescharacteristic, complemented
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46#16812034C2211yescharacteristic, complemented
47#16812033C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic, complemented
57C2×D8×C107C2211yescharacteristic, complemented
58#127×C107C2211yescharacteristic
59#124×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133817—22no—
64#13133817—41no—
65C2×Dic214—21no—
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—41no—
69C2×D428D411yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—41no—
81D4×D107—44no—
82C22×Dic107—21no—
83C2×C4×D107—21no—
84C2×C4×D107—21no—
85C2×C4×D107C2311yescharacteristic
86C2×C4×D107C2311yescharacteristic
87#13133807D414yescomplemented
88#13133806—22no—
89#13133806—22no—
90#13133805—22no—
91#13133803—22no—
92#13133801—22no—
93C2×C107⋊C8C2311yescharacteristic
94C2×C107⋊C8C2311yescharacteristic
95C8×D107—22no—
96C4⋊Dic107D411yescharacteristic
97C4⋊Dic107C2311yescharacteristic
98C4⋊Dic107C2311yescharacteristic
99Dic107⋊C4—21no—
100Dic107⋊C4—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4D411yescharacteristic
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107—21no—
109C2×Q8×C107C2311yescharacteristic
110C2×D4×C107—21no—
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115SD16×C107—22no—
116SD16×C107—41no—
117D8×C107—22no—
118D8×C107—41no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—81no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—42no—
145D4×C107—81no—
146C2×C4×C107—21no—
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2411yescharacteristic
150C22⋊D107—42no—
151C22⋊D107—42no—
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107—21no—
159C2×Dic107—42no—
160C2×Dic107C2×D411yescharacteristic
161C2×Dic107C2×D411yescharacteristic
162D428—22no—
163D428—21no—
164D428C2×D412yes—
165C4×D107—21no—
166C4×D107—21no—
167C4×D107—21no—
168C4×D107—21no—
169C4×D107—42no—
170C4×D107—42no—
171Dic214—21no—
172Dic214—42no—
173Dic214C2×D412yescomplemented
174C8×C107—21no—
175C8×C107C2×D412yescomplemented
176C107⋊C8—21no—
177C107⋊C8—21no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107C22×D411yescharacteristic
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—22no—
192C2×D107—21no—
193C2×D107—21no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—84no—
197C2×D107—84no—
198C2×D107AGL(1,ℤ/8)12yescomplemented
199C4×C107—21no—
200C4×C107—21no—
201C4×C107—41no—
202C4×C107C22×D411yescharacteristic
203C4×C107H(5,𝔽2)11yescharacteristic
204Dic107—22no—
205Dic107—21no—
206Dic107—21no—
207Dic107—41no—
208Dic107—41no—
209C2×C107—81no—
210C2×C107—81no—
211C2×C107—81no—
212C2×C107D4211yescharacteristic
213C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C107#57711yescharacteristic
215D107—22no—
216D107—41no—
217D107—42no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42—1071no—
224#496D10711yescharacteristic, complemented
225#468—1071no—
226#466—1071no—
227#463—1071no—
228#460—1071no—
229#450—1071no—
230#449—1071no—
231#448—1071no—
232#447—1071no—
233#440—1071no—
234#435—1071no—
235C2×#41—1071no—
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—2144no—
246C2×SD16C2×D10711yescharacteristic, complemented
247C2×SD16C2×D10711yescharacteristic, complemented
248C2×D8—1071no—
249C2×D8C2×D10711yescharacteristic, complemented
250C2×D8C2×D10711yescharacteristic, complemented
251C2×M16—1071no—
252C22×C8—1071no—
253#127C2×D10711yescharacteristic
254#124—2141no—
255#124C2×D10711yescharacteristic, complemented
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—1071no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121—2141no—
263#120—2141no—
264#120—2141no—
265C4×Q8—1071no—
266C4×D4—1071no—
267C4×D4—1071no—
268C4×D4—2141no—
269#107—1071no—
270C4⋊C8—1071no—
271#103—1071no—
272#103—1071no—
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102—1071no—
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97C2×D10711yescharacteristic, complemented
280C24—2141no—
281C24—2141no—
282#41—4281no—
283#41—2142no—
284C2×Q8—1071no—
285C2×Q8—4281no—
286C2×Q8—2141no—
287C2×Q8—2142no—
288C2×Q8C22×D10711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—4284no—
294C2×D4—4281no—
295C2×D4—4281no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2142no—
299C2×D4—2142no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4C22×D10711yescharacteristic
305C2×D4C22×D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—1071no—
308C22×C4—1071no—
309C22×C4—2141no—
310C22×C4—2141no—
311C22×C4—2141no—
312SD16—22no—
313SD16—41no—
314SD16—1074no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—22no—
319D8—41no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—2142no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—1071no—
330C4⋊C4—2141no—
331C4⋊C4—2141no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—1071no—
335C22⋊C4—1071no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C42—2141no—
342C42C22×D10711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—8561no—
347C23—8561no—
348C23—1071no—
349C23—1071no—
350C23—4282no—
351C23—4282no—
352C23—4282no—
353C23—4282no—
354C23—2141no—
355Q8—21no—
356Q8—21no—
357Q8—1072no—
358Q8—4282no—
359Q8—2141no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—1072no—
369D4—4282no—
370D4—4282no—
371D4—4282no—
372D4—4282no—
373D4—4282no—
374D4—4282no—
375D4—2142no—
376D4—2141no—
377C2×C4—21no—
378C2×C4—1071no—
379C2×C4—1071no—
380C2×C4—4282no—
381C2×C4—4282no—
382C2×C4—4282no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4D4×D10711yescharacteristic
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—2141no—
395C8—2141no—
396C8D4×D10712yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—8564no—
407C22—8564no—
408C22—1072no—
409C22—4281no—
410C22—4281no—
411C22—2141no—
412C22—2141no—
413C22—2141no—
414C22—2142no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—4281no—
422C4—4281no—
423C4—2141no—
424C4—2141no—
425C4—2142no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—4282no—
432C2—4281no—
433C2—2142no—
434C2#2029326211yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293285C2semidirect
4. #20293262C2semidirect
5. #20292870C2semidirect
6. #20292860C2semidirect
7. #20292839C2semidirect
8. #20292824C2semidirect
9. #20292776C2semidirect
10. #20292769C2semidirect
11. #20292764C2semidirect
12. #20292759C2semidirect
13. #20292729C2semidirect
14. #20292700C2semidirect
15. #496×C107C2semidirect
16. C2×#13133817C22semidirect
17. C2×Q8×D107C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133806C22semidirect
28. C2×#13133801C22semidirect
29. C2×C8×D107C22semidirect
32. #16812085C22semidirect
37. #16812080C22semidirect
40. C4×Dic214C22semidirect
44. #16812043C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. #16812034C22semidirect
47. #16812033C22semidirect
49. #16812030C22semidirect
50. #16812030C22semidirect
51. #16812030C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. (C2×D107)⋊C8C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C2×D8×C107C22semidirect
57. C2×D8×C107C22semidirect
59. #124×C107C22semidirect
60. #97×C107C22semidirect
87. #13133807D4semidirect
173. Dic214C2×D4semidirect
175. C8×C107C2×D4semidirect
198. C2×D107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
224. #496D107semidirect
246. C2×SD16C2×D107semidirect
247. C2×SD16C2×D107semidirect
249. C2×D8C2×D107semidirect
250. C2×D8C2×D107semidirect
255. #124C2×D107semidirect
279. #97C2×D107semidirect
396. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×SD16×D1072
#202932852
#202932622
#202928702
#202928602
#202928392
#202928242
#202927762
#202927692
#202927642
#202927592
#202927292
#202927002
#496×C1072
#3265107