Finite Group Databasebeta

← #28348006

Subgroups — #28348006

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×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812131C211yesmaximal, characteristic, complemented
3#20293298C211yesmaximal, characteristic, complemented
4#20293273C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292873C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292850C211yesmaximal, characteristic, complemented
9#20292823C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292774C211yesmaximal, characteristic, complemented
12#20292774C211yesmaximal, characteristic, complemented
13#20292763C211yesmaximal, characteristic, complemented
14#20292729C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C2×#13133818C2211yescharacteristic, complemented
19C22×D428C2211yescharacteristic
20#16812137—24no—
21#16812131—24no—
22C2×Dic428C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133807C2211yescharacteristic, complemented
27C2×D856C2211yescharacteristic
28C2×D856C2211yescharacteristic
29C2×D856C2211yescharacteristic
30C2×D856C2211yescharacteristic
31#16812106C2211yescharacteristic, complemented
32#16812105—21no—
33#16812097C2211yescharacteristic
34#16812094—21no—
35#16812088C2211yescharacteristic
36#16812083C2211yescharacteristic
37#16812083C2211yescharacteristic
38#16812083C2211yescharacteristic, complemented
39#16812082—21no—
40#16812082—21no—
41#16812082—21no—
42#16812077—21no—
43C4×Dic214C2211yescharacteristic, complemented
44C4×D428C2211yescharacteristic
45#16812071—21no—
46C22⋊C4×D107—21no—
47#16812043C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52#16812031C2211yescharacteristic, complemented
53#16812031C2211yescharacteristic, complemented
54C2×M16×C107C2211yescharacteristic
55C22×C8×C107C2211yescharacteristic, complemented
56C4×D4×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C22⋊C8×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133818—22no—
63#13133818—22no—
64#13133818—22no—
65#13133818—41no—
66#13133818—41no—
67#13133818—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×D428—22no—
71C2×D428—21no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—41no—
75C2×D428C2311yescharacteristic
76C2×D428C2311yescharacteristic
77C2×D428C2311yescharacteristic
78C2×D428C2311yescharacteristic
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81C2×C22⋊D107—21no—
82C2×C22⋊D107—21no—
83C2×C22⋊D107—21no—
84C2×C22⋊D107—21no—
85C2×C4×D107—21no—
86C2×C4×D107—21no—
87C2×C4×D107—21no—
88C2×C4×D107—21no—
89Dic428—22no—
90#13133807—22no—
91#13133807—22no—
92#13133807—41no—
93#13133807D414yescomplemented
94D856—22no—
95D856—22no—
96D856—22no—
97D856—41no—
98C4⋊Dic107C2311yescharacteristic
99C4⋊Dic107C2311yescharacteristic
100Dic107⋊C4—21no—
101Dic107⋊C4—21no—
102C22⋊Dic107—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109C4×Dic107—21no—
110C2×D4×C107D411yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113M16×C107—22no—
114C2×C8×C107—22no—
115C2×C8×C107C2311yescharacteristic
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107C2311yescharacteristic
123C23×C107C2×D411yescharacteristic
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—81no—
132D4×C107—41no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—42no—
143C22⋊D107—42no—
144C22⋊D107—42no—
145C22⋊D107—81no—
146C22⋊D107—81no—
147C22⋊D107—81no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152D428—21no—
153D428—21no—
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—21no—
159D428—42no—
160D428—42no—
161D428—81no—
162D428C2×D412yes—
163C4×D107—42no—
164C4×D107—42no—
165C4×D107—42no—
166C4×D107—81no—
167Dic214—21no—
168Dic214—21no—
169Dic214—21no—
170Dic214C2×D412yes—
171C8×C107—21no—
172C8×C107—21no—
173C8×C107—21no—
174C8×C107C2×D412yes—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107C22×D411yescharacteristic
182C2×D107—22no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C4×C107—21no—
196C4×C107—21no—
197C4×C107—21no—
198C4×C107—21no—
199C4×C107C22×D411yescharacteristic
200C4×C107H(5,𝔽2)11yescharacteristic
201Dic107—22no—
202Dic107—41no—
203Dic107—41no—
204Dic107—41no—
205C2×C107—41no—
206C2×C107—41no—
207C2×C107D4211yescharacteristic
208C2×C107#57511yescharacteristic
209C2×C107#57611yescharacteristic
210D107—42no—
211D107—81no—
212D107—81no—
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#449—1071no—
228#447—1071no—
229#440—1071no—
230#435D10711yescharacteristic, complemented
231C2×#41—1071no—
232C2×#41—1071no—
233C2×#41—1071no—
234C22×D4—1071no—
235AGL(1,ℤ/8)—2144no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16—1071no—
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8—1071no—
246C2×M16C2×D10711yescharacteristic
247C22×C8C2×D10711yescharacteristic, complemented
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—2141no—
263C4×D4C2×D10711yescharacteristic
264C2×C22⋊C4—2141no—
265#107—1071no—
266C4⋊C8C2×D10711yescharacteristic, complemented
267#103—1071no—
268#103—1071no—
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102—1071no—
273C22⋊C8C2×D10711yescharacteristic
274C22⋊C8C2×D10711yescharacteristic, complemented
275#97C2×D10711yescharacteristic, complemented
276C24—2141no—
277#41—4281no—
278#41—4281no—
279#41—4281no—
280#41—2142no—
281#41—2142no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8—1071no—
285C2×D4—1071no—
286C2×D4—1071no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—4281no—
290C2×D4—2142no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4D42811yescharacteristic
301C22×C4—2141no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4C22×D10711yescharacteristic
306C22×C4C22×D10711yescharacteristic
307Q16—2142no—
308SD16—1074no—
309SD16—4281no—
310SD16—2142no—
311SD16—2142no—
312D8—4281no—
313D8—2142no—
314D8—2142no—
315D8—2142no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D10711yescharacteristic
319C2×C8C22×D10711yescharacteristic
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4—2141no—
326C4⋊C4D42811yescharacteristic
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4D42811yescharacteristic
335C22⋊C4D42811yescharacteristic
336C42—2141no—
337C42C22×D10711yescharacteristic
338C23—8561no—
339C23—4282no—
340C23—4282no—
341C23—2141no—
342C23—2141no—
343C23—2141no—
344C23—2141no—
345C23C2×D42811yescharacteristic
346C23C2×D42811yescharacteristic
347Q8—1072no—
348Q8—2141no—
349Q8—2141no—
350Q8—2141no—
351D4—41no—
352D4—8561no—
353D4—8561no—
354D4—8561no—
355D4—8561no—
356D4—1072no—
357D4—4282no—
358D4—4282no—
359D4—4282no—
360D4—4282no—
361D4—4282no—
362D4—2141no—
363D4—2141no—
364D4—2141no—
365D4—2141no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—8561no—
374C2×C4—4282no—
375C2×C4—4282no—
376C2×C4—4282no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4C2×D42811yescharacteristic
382C2×C4C2×D42811yescharacteristic
383C2×C4C2×D42811yescharacteristic
384C2×C4C2×D42811yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D10712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—8564no—
397C22—4281no—
398C22—4281no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—2142no—
409C22C22×D42811yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—4281no—
415C4—4281no—
416C4—4281no—
417C4—2142no—
418C4C2×D4×D10711yescharacteristic
419C4#1681215511yescharacteristic
420C2—41no—
421C2—41no—
422C2—8561no—
423C2—8561no—
424C2—8561no—
425C2—4282no—
426C2D4×D42811yescharacteristic
427C2#2029342711yescharacteristic
428C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
hypercenter#435D107
hypocenterC107#3268
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812131C2semidirect
3. #20293298C2semidirect
4. #20293273C2semidirect
5. #20293016C2semidirect
6. #20292873C2semidirect
7. #20292863C2semidirect
8. #20292850C2semidirect
9. #20292823C2semidirect
10. #20292791C2semidirect
11. #20292774C2semidirect
12. #20292774C2semidirect
13. #20292763C2semidirect
14. #20292729C2semidirect
15. #435×C107C2semidirect
18. C2×#13133818C22semidirect
22. C2×Dic428C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133807C22semidirect
31. #16812106C22semidirect
38. #16812083C22semidirect
43. C4×Dic214C22semidirect
47. #16812043C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
52. #16812031C22semidirect
53. #16812031C22semidirect
55. C22×C8×C107C22semidirect
57. C4⋊C8×C107C22semidirect
59. C22⋊C8×C107C22semidirect
60. #97×C107C22semidirect
93. #13133807D4semidirect
215. C107#3268semidirect
230. #435D107semidirect
247. C22×C8C2×D107semidirect
266. C4⋊C8C2×D107semidirect
274. C22⋊C8C2×D107semidirect
275. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121312
#202932982
#202932732
#202930162
#202928732
#202928632
#202928502
#202928232
#202927912
#202927742
#202927632
#202927292
#435×C1072
#3268107