Finite Group Databasebeta

← #28348078

Subgroups — #28348078

Order 13696.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3#20293295C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20293002C211yesmaximal, characteristic, complemented
6#20292881C211yesmaximal, characteristic, complemented
7#20292826C211yesmaximal, characteristic, complemented
8#20292821C211yesmaximal, characteristic, complemented
9#20292773C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292714C211yesmaximal, characteristic, complemented
14#20292703C211yesmaximal, characteristic, complemented
15#469×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C2×#13133817C2211yescharacteristic, complemented
18C22×D428C2211yescharacteristic
19C2×D4×D107—21no—
20C2×D4×D107C2211yescharacteristic
21#16812137—24no—
22#16812133—24no—
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133806C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic
27C2×D856C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic
29C2×#13133803C2211yescharacteristic, complemented
30C2×#13133803C2211yescharacteristic, complemented
31C2×#13133801C2211yescharacteristic, complemented
32#16812105—21no—
33#16812105C2211yescharacteristic
34#16812105C2211yescharacteristic, complemented
35#16812084C2211yescharacteristic
36#16812084C2211yescharacteristic, complemented
37#16812082—21no—
38#16812082—21no—
39#16812081—21no—
40#16812079—21no—
41#16812077—21no—
42#16812074C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic
44C4×C22⋊D107C2211yescharacteristic
45(C22⋊D107)⋊C4—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic, complemented
56#121×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic, complemented
58#106×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133818—22no—
64#13133818—41no—
65#13133817—22no—
66#13133817—41no—
67C2×Dic214C2311yescharacteristic
68C2×D428—22no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—41no—
73C2×D428C2311yescharacteristic
74C2×D428C2311yescharacteristic
75C2×D428C2311yescharacteristic
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81C2×C22⋊D107—21no—
82C2×C22⋊D107—21no—
83C2×C22⋊D107D411yescharacteristic
84D4×D107—22no—
85D4×D107—41no—
86D4×D107—44no—
87C22×Dic107—21no—
88C2×C4×D107—21no—
89C2×C4×D107—21no—
90C2×C4×D107C2311yescharacteristic
91#13133807—22no—
92#13133807D414yescomplemented
93#13133806—22no—
94D856—22no—
95D856—22no—
96#13133803—22no—
97#13133803—41no—
98#13133803—41no—
99#13133801—22no—
100C2×C107⋊C8C2311yescharacteristic
101C2×C107⋊C8C2311yescharacteristic
102Dic107⋊C4—21no—
103Dic107⋊C4D411yescharacteristic
104C22⋊Dic107D411yescharacteristic
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107—21no—
110C4×Dic107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116D8×C107—22no—
117M16×C107—22no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107C2×D411yescharacteristic
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135C22×D107—81no—
136C22×D107C2×D411yescharacteristic
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—42no—
142D4×C107—81no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2411yescharacteristic
149C22⋊D107—22no—
150C22⋊D107—42no—
151C22⋊D107—42no—
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—81no—
156C22⋊D107—81no—
157C2×Dic107—21no—
158C2×Dic107—42no—
159C2×Dic107C2×D411yescharacteristic
160C2×Dic107C2×D411yescharacteristic
161C2×Dic107C2×D411yescharacteristic
162D428—21no—
163D428—21no—
164D428—21no—
165D428—21no—
166D428—21no—
167D428—42no—
168D428—42no—
169D428—81no—
170D428C2×D412yes—
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—42no—
174C4×D107—42no—
175Dic214—21no—
176Dic214C2×D412yescomplemented
177C8×C107—21no—
178C8×C107C2×D412yescomplemented
179C107⋊C8—21no—
180C107⋊C8—21no—
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107C22×D411yescharacteristic
191C2×D107—21no—
192C2×D107—22no—
193C2×D107—21no—
194C2×D107—21no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—84no—
204C2×D107—84no—
205C4×C107—21no—
206C4×C107—41no—
207C4×C107—41no—
208C4×C107C22×D411yescharacteristic
209C4×C107H(5,𝔽2)11yescharacteristic
210Dic107—22no—
211Dic107—21no—
212Dic107—21no—
213Dic107AGL(1,ℤ/8)12yescomplemented
214C2×C107—41no—
215C2×C107—81no—
216C2×C107—81no—
217C2×C107D4211yescharacteristic
218C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C107#57611yescharacteristic
220D107—41no—
221D107—42no—
222D107—81no—
223D107—81no—
224#3274—1071nomaximal, Sylow
225C107#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1071no—
227C2×AGL(1,ℤ/8)—1071no—
228#550—1071no—
229D42—1071no—
230#493—1071no—
231#469—1071no—
232#469D10711yescharacteristic, complemented
233#460—1071no—
234#459—1071no—
235#449—1071no—
236#449—1071no—
237#447—1071no—
238#447—1071no—
239C4×SD16—1071no—
240#436—1071no—
241C2×#41—1071no—
242C2×#41—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—2141no—
246AGL(1,ℤ/8)—2144no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16—1071no—
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8C2×D10711yescharacteristic, complemented
257C2×M16—1071no—
258C2×M16C2×D10711yescharacteristic, complemented
259#127—1071no—
260#127—1071no—
261#127—2141no—
262#121—1071no—
263#121—1071no—
264#121—2141no—
265#121—2141no—
266#121—2141no—
267#121C2×D10711yescharacteristic
268#121C2×D10711yescharacteristic, complemented
269#120—2141no—
270#120—2141no—
271C4×Q8—1071no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4—2141no—
275#106C2×D10711yescharacteristic, complemented
276C4⋊C8—1071no—
277#103—1071no—
278#102—1071no—
279#102—1071no—
280#102—1071no—
281#102C2×D10711yescharacteristic, complemented
282#102C2×D10711yescharacteristic, complemented
283C22⋊C8—1071no—
284C22⋊C8—1071no—
285C4×C8—1071no—
286C24—2141no—
287C24—2141no—
288#41—4281no—
289#41—4281no—
290#41—2142no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—1071no—
298C2×D4—1071no—
299C2×D4—4281no—
300C2×D4—4284no—
301C2×D4—4281no—
302C2×D4—2141no—
303C2×D4—2142no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2142no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4—2141no—
314C2×D4C22×D10711yescharacteristic
315C2×D4C22×D10711yescharacteristic
316C22×C4—1071no—
317C22×C4—2141no—
318C22×C4—2141no—
319C22×C4—2141no—
320C22×C4C22×D10711yescharacteristic
321SD16—1074no—
322SD16—2142no—
323SD16—2142no—
324D8—22no—
325D8—4281no—
326D8—4281no—
327D8—2142no—
328D8—2142no—
329D8—2142no—
330M16—22no—
331M16—2142no—
332C2×C8—1071no—
333C2×C8—1071no—
334C2×C8C22×D10711yescharacteristic
335C2×C8C22×D10711yescharacteristic
336C4⋊C4—1071no—
337C4⋊C4—2141no—
338C4⋊C4C22×D10711yescharacteristic
339C4⋊C4C22×D10711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—1071no—
343C22⋊C4—1071no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C42—1071no—
348C42—2141no—
349C23—21no—
350C23—21no—
351C23—8561no—
352C23—8561no—
353C23—1071no—
354C23—4282no—
355C23—4282no—
356C23—4282no—
357C23—4282no—
358C23—2141no—
359C23—2141no—
360C23—2141no—
361C23D4×D10711yescharacteristic
362Q8—1072no—
363Q8—2141no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—81no—
370D4—8561no—
371D4—8561no—
372D4—8561no—
373D4—1072no—
374D4—4282no—
375D4—4282no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—2142no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—1071no—
392C2×C4—1071no—
393C2×C4—1071no—
394C2×C4—4282no—
395C2×C4—4282no—
396C2×C4—4282no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4D4×D10711yescharacteristic
401C2×C4C23×D10711yescharacteristic
402C8—21no—
403C8—2141no—
404C8—2141no—
405C8D4×D10712yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—8564no—
416C22—8564no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—2141no—
426C22—2141no—
427C22—2141no—
428C22—2142no—
429C22C2×D4×D10711yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—1072no—
434C4—2141no—
435C4—2141no—
436C4—2142no—
437C4C2×D4×D10711yescharacteristic
438C4#1681215511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—8561no—
443C2—8561no—
444C2—4282no—
445C2—4281no—
446C2#2029326111yescharacteristic
447C2AGL(1,ℤ/8)×D10711yescharacteristic
448C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107#3274
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812133C2semidirect
3. #20293295C2semidirect
4. #20293261C2semidirect
5. #20293002C2semidirect
6. #20292881C2semidirect
7. #20292826C2semidirect
8. #20292821C2semidirect
9. #20292773C2semidirect
10. #20292770C2semidirect
11. #20292762C2semidirect
12. #20292760C2semidirect
13. #20292714C2semidirect
14. #20292703C2semidirect
15. #469×C107C2semidirect
16. C2×#13133818C22semidirect
17. C2×#13133817C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133806C22semidirect
27. C2×D856C22semidirect
29. C2×#13133803C22semidirect
30. C2×#13133803C22semidirect
31. C2×#13133801C22semidirect
34. #16812105C22semidirect
36. #16812084C22semidirect
42. #16812074C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
49. #16812029C22semidirect
50. #16812029C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C2×D8×C107C22semidirect
55. C2×M16×C107C22semidirect
57. #121×C107C22semidirect
58. #106×C107C22semidirect
59. #102×C107C22semidirect
60. #102×C107C22semidirect
92. #13133807D4semidirect
176. Dic214C2×D4semidirect
178. C8×C107C2×D4semidirect
213. Dic107AGL(1,ℤ/8)semidirect
225. C107#3274semidirect
232. #469D107semidirect
256. C2×D8C2×D107semidirect
258. C2×M16C2×D107semidirect
268. #121C2×D107semidirect
275. #106C2×D107semidirect
281. #102C2×D107semidirect
282. #102C2×D107semidirect
405. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121332
#202932952
#202932612
#202930022
#202928812
#202928262
#202928212
#202927732
#202927702
#202927622
#202927602
#202927142
#202927032
#469×C1072
#3274107