Finite Group Databasebeta

← #28347995

Subgroups — #28347995

Order 13696.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812130C211yesmaximal, characteristic, complemented
2#20293296C211yesmaximal, characteristic, complemented
3#20293271C211yesmaximal, characteristic, complemented
4#20293015C211yesmaximal, characteristic, complemented
5#20292873C211yesmaximal, characteristic, complemented
6#20292860C211yesmaximal, characteristic, complemented
7#20292847C211yesmaximal, characteristic, complemented
8#20292820C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292771C211yesmaximal, characteristic, complemented
11#20292769C211yesmaximal, characteristic, complemented
12#20292761C211yesmaximal, characteristic, complemented
13#20292728C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C2×#13133817C2211yescharacteristic
18#16812130—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133807C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133803C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×C8×D107C2211yescharacteristic, complemented
25#16812105—21no—
26#16812105C2211yescharacteristic, complemented
27#16812096C2211yescharacteristic
28#16812090—21no—
29#16812088C2211yescharacteristic
30#16812083—21no—
31#16812083C2211yescharacteristic
32#16812081—21no—
33#16812080—21no—
34#16812080C2211yescharacteristic
35#16812080C2211yescharacteristic, complemented
36#16812078—21no—
37Q8×Dic107C2211yescharacteristic, complemented
38D4×Dic107C2211yescharacteristic
39C4×C22⋊D107—21no—
40C4×C22⋊D107C2211yescharacteristic
41C2×C22⋊Dic107—21no—
42#16812043C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic, complemented
50C22⋊(C107⋊C8)C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52#16812019C2211yescharacteristic, complemented
53C2×#41×C107C2211yescharacteristic
54C22×D4×C107C2211yescharacteristic
55AGL(1,ℤ/8)×C107—24no—
56C2×SD16×C107C2211yescharacteristic
57C2×SD16×C107C2211yescharacteristic
58C2×D8×C107C2211yescharacteristic
59C2×D8×C107C2211yescharacteristic
60C2×M16×C107C2211yescharacteristic
61#13133819—22no—
62#13133819—41no—
63#13133817—22no—
64#13133817—41no—
65C2×Dic214C2311yescharacteristic
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107D411yescharacteristic
75C22×Dic107—21no—
76C22×Dic107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79#13133808—22no—
80#13133807—22no—
81#13133806—41no—
82#13133803—22no—
83#13133803—41no—
84C2×C107⋊C8C2311yescharacteristic
85C2×C107⋊C8C2311yescharacteristic
86C8×D107—22no—
87C4⋊Dic107—21no—
88C4⋊Dic107C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107—21no—
98C22⋊Dic107D411yescharacteristic
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4D411yescharacteristic
101C4×Dic107—21no—
102C4×Dic107C2311yescharacteristic
103C24×C107—21no—
104#41×C107—22no—
105#41×C107—41no—
106C2×Q8×C107C2311yescharacteristic
107C2×D4×C107—22no—
108C2×D4×C107—21no—
109C2×D4×C107—21no—
110C2×D4×C107—41no—
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116SD16×C107—22no—
117SD16×C107D414yescomplemented
118D8×C107—22no—
119D8×C107—22no—
120M16×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—42no—
127C23×C107—42no—
128C23×C107—81no—
129C23×C107C2×D411yescharacteristic
130C22×D107—21no—
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107C2×D412yes—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—42no—
140D4×C107—42no—
141D4×C107C2×D412yes—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—42no—
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2411yescharacteristic
148C22⋊D107—41no—
149C22⋊D107—42no—
150C22⋊D107—42no—
151C22⋊D107—81no—
152C22⋊D107—81no—
153C22⋊D107—81no—
154C22⋊D107—81no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107—42no—
159C2×Dic107—81no—
160C2×Dic107C2×D411yescharacteristic
161C2×Dic107C2×D411yescharacteristic
162C2×Dic107C2×D411yescharacteristic
163D428—21no—
164D428—21no—
165D428—42no—
166C4×D107—21no—
167C4×D107—21no—
168C4×D107—42no—
169Dic214—21no—
170Dic214—21no—
171C8×C107—21no—
172C8×C107C2×D412yes—
173C107⋊C8—21no—
174C107⋊C8—21no—
175C22×C107—22no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—21no—
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×C107—84no—
188C22×C107C22×D411yescharacteristic
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—21no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C4×C107—22no—
196C4×C107—21no—
197C4×C107C22×D411yescharacteristic
198C4×C107H(5,𝔽2)11yescharacteristic
199Dic107—21no—
200Dic107—21no—
201Dic107—21no—
202Dic107—41no—
203Dic107—41no—
204Dic107—41no—
205C2×C107—41no—
206C2×C107—42no—
207C2×C107—81no—
208C2×C107—81no—
209C2×C107D4211yescharacteristic
210C2×C107#57511yescharacteristic
211C2×C107#57611yescharacteristic
212D107—41no—
213D107—81no—
214#3268—1071nomaximal, Sylow
215C107#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
217C2×#135—1071no—
218#550—1071no—
219#546—1071no—
220#496—1071no—
221#468—1071no—
222#466—1071no—
223#464—1071no—
224#459—1071no—
225#452—1071no—
226#449—1071no—
227#449—1071no—
228#447—1071no—
229#440—1071no—
230#435—1071no—
231C2×#41—1071no—
232C2×#41—1071no—
233C2×#41C2×D10711yescharacteristic
234C22×D4C2×D10711yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16C2×D10711yescharacteristic
241C2×SD16C2×D10711yescharacteristic
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8C2×D10711yescharacteristic
245C2×D8C2×D10711yescharacteristic
246C2×M16C2×D10711yescharacteristic
247C22×C8—1071no—
248#127—1071no—
249#127—2141no—
250#124—1071no—
251#123—2141no—
252C22⋊Q8—1071no—
253#121—1071no—
254#121—1071no—
255#121—1071no—
256#121—2141no—
257#121—2141no—
258#121—2141no—
259#120—2141no—
260C4×Q8—1071no—
261C4×D4—1071no—
262C4×D4—1071no—
263C4×D4—2141no—
264C2×C22⋊C4—2141no—
265#107—1071no—
266C4⋊C8—1071no—
267#103—1071no—
268#103—1071no—
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102—1071no—
273C22⋊C8—1071no—
274C22⋊C8—1071no—
275#97—1071no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—4281no—
280#41—4281no—
281#41—2142no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8C22×D10711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4C22×D10711yescharacteristic
299C2×D4C22×D10711yescharacteristic
300C2×D4C22×D10711yescharacteristic
301C22×C4—21no—
302C22×C4—1071no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4—2141no—
306C22×C4C22×D10711yescharacteristic
307Q16—2142no—
308SD16—22no—
309SD16—4281no—
310SD16—2142no—
311SD16C22⋊D10714yescomplemented
312D8—22no—
313D8—22no—
314D8—4281no—
315D8—2142no—
316M16—22no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—2142no—
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—1071no—
325C4⋊C4—2141no—
326C4⋊C4—2141no—
327C22⋊C4—1071no—
328C22⋊C4—1071no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C42—1071no—
337C42—2141no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—1071no—
345C23—2141no—
346C23D4×D10711yescharacteristic
347Q8—21no—
348Q8—2141no—
349Q8—2141no—
350Q8C2×C22⋊D10712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—8561no—
359D4—8561no—
360D4—8561no—
361D4—8561no—
362D4—4282no—
363D4—4282no—
364D4—4281no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4C2×C22⋊D10712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—8561no—
374C2×C4—1071no—
375C2×C4—1071no—
376C2×C4—1071no—
377C2×C4—4282no—
378C2×C4—4282no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4D4×D10711yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—2141no—
388C8—2141no—
389C8C2×C22⋊D10712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—4281no—
404C22—4281no—
405C22—4281no—
406C22—2141no—
407C22—2141no—
408C22—2141no—
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—22no—
412C4—4281no—
413C4—4281no—
414C4—4281no—
415C4—2141no—
416C4—2141no—
417C4—2141no—
418C4C22×C22⋊D10711yescharacteristic
419C4#1681215411yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—8561no—
425C2—4281no—
426C2D4×C22⋊D10711yescharacteristic
427C2#2029342311yescharacteristic
428C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3268
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
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×#16812130C2semidirect
2. #20293296C2semidirect
3. #20293271C2semidirect
4. #20293015C2semidirect
5. #20292873C2semidirect
6. #20292860C2semidirect
7. #20292847C2semidirect
8. #20292820C2semidirect
9. #20292789C2semidirect
10. #20292771C2semidirect
11. #20292769C2semidirect
12. #20292761C2semidirect
13. #20292728C2semidirect
14. #20292696C2semidirect
15. C2×AGL(1,ℤ/8)×C107C2semidirect
16. C2×#13133819C22semidirect
19. C2×#13133808C22semidirect
20. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
24. C2×C8×D107C22semidirect
26. #16812105C22semidirect
35. #16812080C22semidirect
37. Q8×Dic107C22semidirect
42. #16812043C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812031C22semidirect
49. #16812030C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. #16812019C22semidirect
117. SD16×C107D4semidirect
215. C107#3268semidirect
216. C2×AGL(1,ℤ/8)D107semidirect
311. SD16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202932962
#202932712
#202930152
#202928732
#202928602
#202928472
#202928202
#202927892
#202927712
#202927692
#202927612
#202927282
#202926962
C2×AGL(1,ℤ/8)×C1072
#3268107