Finite Group Databasebeta

← #28347994

Subgroups — #28347994

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×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293296C211yesmaximal, characteristic, complemented
4#20293274C211yesmaximal, characteristic, complemented
5#20293015C211yesmaximal, characteristic, complemented
6#20292872C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292847C211yesmaximal, characteristic, complemented
9#20292821C211yesmaximal, characteristic, complemented
10#20292792C211yesmaximal, characteristic, complemented
11#20292774C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292730C211yesmaximal, characteristic, complemented
14#20292695C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812136—24no—
20#16812129—24no—
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic
25C2×D856C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29C22×C107⋊C8C2211yescharacteristic, complemented
30#16812105—21no—
31#16812105C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic
33#16812091—21no—
34#16812089C2211yescharacteristic
35#16812083—21no—
36#16812083C2211yescharacteristic, complemented
37#16812080—21no—
38#16812080—21no—
39#16812080C2211yescharacteristic
40#16812077—21no—
41Q8×Dic107C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic
43C4×C22⋊D107—21no—
44C4×C22⋊D107C2211yescharacteristic
45C22⋊C4×D107—21no—
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic
50#16812030C2211yescharacteristic
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic, complemented
54C2×#41×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic
56C2×D8×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133818—22no—
65#13133818—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107D411yescharacteristic
77D4×D107—22no—
78D4×D107—41no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107C2311yescharacteristic
83#13133808—22no—
84#13133807—41no—
85#13133806—22no—
86#13133805D414yescomplemented
87D856—22no—
88#13133803—22no—
89#13133803—22no—
90#13133801—22no—
91C2×C107⋊C8—22no—
92C2×C107⋊C8C2311yescharacteristic
93C2×C107⋊C8C2311yescharacteristic
94C4⋊Dic107—21no—
95C4⋊Dic107—21no—
96C4⋊Dic107C2311yescharacteristic
97Dic107⋊C4D411yescharacteristic
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4D411yescharacteristic
106C4×Dic107—21no—
107C4×Dic107C2311yescharacteristic
108#41×C107—22no—
109#41×C107—41no—
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116C22×C4×C107C2311yescharacteristic
117SD16×C107—22no—
118D8×C107—41no—
119C2×C8×C107C2311yescharacteristic
120C2×C8×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107—21no—
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—81no—
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107C2×D412yes—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—21no—
138D4×C107—42no—
139D4×C107—81no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—42no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—41no—
148C22⋊D107—42no—
149C22⋊D107—42no—
150C22⋊D107—42no—
151C22⋊D107—81no—
152C22⋊D107—81no—
153C22⋊D107—81no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107C2×D411yescharacteristic
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159D428—21no—
160D428—21no—
161D428—21no—
162D428—42no—
163D428C2×D412yes—
164C4×D107—21no—
165C4×D107—21no—
166C4×D107—42no—
167C4×D107—42no—
168C4×D107—81no—
169Dic214—21no—
170Dic214—21no—
171C8×C107—21no—
172C8×C107—21no—
173C107⋊C8—21no—
174C107⋊C8C2×D412yes—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107C22×D411yescharacteristic
185C2×D107—21no—
186C2×D107—22no—
187C2×D107—21no—
188C2×D107—21no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C4×C107—21no—
196C4×C107—22no—
197C4×C107—41no—
198C4×C107C22×D411yescharacteristic
199C4×C107H(5,𝔽2)11yescharacteristic
200Dic107—21no—
201Dic107—21no—
202Dic107—21no—
203Dic107—41no—
204Dic107—41no—
205C2×C107—41no—
206C2×C107—81no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—42no—
213D107—81no—
214#3268—1071nomaximal, Sylow
215C107#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1071no—
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#449D10711yescharacteristic, complemented
228#447—1071no—
229#440—1071no—
230#435—1071no—
231C2×#41—1071no—
232C2×#41—1071no—
233C2×#41C2×D10711yescharacteristic, complemented
234C22×D4—1071no—
235AGL(1,ℤ/8)—2144no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16C2×D10711yescharacteristic
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8C2×D10711yescharacteristic
246C2×M16—1071no—
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—2141no—
256#121—2141no—
257#121—2141no—
258#121C2×D10711yescharacteristic
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#103C2×D10711yescharacteristic, complemented
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102C2×D10711yescharacteristic, complemented
273C22⋊C8—1071no—
274C22⋊C8C2×D10711yescharacteristic, complemented
275#97—1071no—
276C24—2141no—
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—21no—
286C2×D4—21no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—1071no—
290C2×D4—4281no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2141no—
295C2×D4—2142no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
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—1074no—
310SD16—4281no—
311SD16—2142no—
312D8—41no—
313D8—2142no—
314D8—2142no—
315D8—2142no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—2142no—
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4—2141no—
326C4⋊C4C22×D10711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—1071no—
329C22⋊C4—1071no—
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—8561no—
341C23—1071no—
342C23—4282no—
343C23—4282no—
344C23—2141no—
345C23—2141no—
346C23D4×D10711yescharacteristic
347Q8—21no—
348Q8—2141no—
349Q8—2141no—
350Q8D4×D10712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—8561no—
358D4—8561no—
359D4—8561no—
360D4—1072no—
361D4—4282no—
362D4—4282no—
363D4—4281no—
364D4—4282no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—8561no—
375C2×C4—1071no—
376C2×C4—1071no—
377C2×C4—1071no—
378C2×C4—4282no—
379C2×C4—4282no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4D4×D10711yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—21no—
388C8—1072no—
389C8—2141no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—8564no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—2141no—
406C22—2141no—
407C22—2142no—
408C22—2141no—
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—4281no—
414C4—4281no—
415C4—2141no—
416C4—2141no—
417C4—2141no—
418C4C2×D4×D10711yescharacteristic
419C4#1681215411yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—8561no—
424C2—4281no—
425C2—4282no—
426C2#2029325911yescharacteristic
427C2#2029342411yescharacteristic
428C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3268
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812129C2semidirect
3. #20293296C2semidirect
4. #20293274C2semidirect
5. #20293015C2semidirect
6. #20292872C2semidirect
7. #20292861C2semidirect
8. #20292847C2semidirect
9. #20292821C2semidirect
10. #20292792C2semidirect
11. #20292774C2semidirect
12. #20292759C2semidirect
13. #20292730C2semidirect
14. #20292695C2semidirect
15. #449×C107C2semidirect
21. C2×#13133808C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133806C22semidirect
29. C22×C107⋊C8C22semidirect
31. #16812105C22semidirect
36. #16812083C22semidirect
41. Q8×Dic107C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
51. #16812029C22semidirect
53. #16812019C22semidirect
54. C2×#41×C107C22semidirect
58. #103×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
86. #13133805D4semidirect
215. C107#3268semidirect
227. #449D107semidirect
233. C2×#41C2×D107semidirect
268. #103C2×D107semidirect
272. #102C2×D107semidirect
274. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121292
#202932962
#202932742
#202930152
#202928722
#202928612
#202928472
#202928212
#202927922
#202927742
#202927592
#202927302
#202926952
#449×C1072
#3268107