Finite Group Databasebeta

← #28348000

Subgroups — #28348000

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×#16812133C211yesmaximal, characteristic, complemented
2C2×#16812130C211yesmaximal, characteristic, complemented
3#20293297C211yesmaximal, characteristic, complemented
4#20293270C211yesmaximal, characteristic, complemented
5#20292870C211yesmaximal, characteristic, complemented
6#20292860C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292819C211yesmaximal, characteristic, complemented
9#20292786C211yesmaximal, characteristic, complemented
10#20292771C211yesmaximal, characteristic, complemented
11#20292769C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292729C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#496×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×#13133817C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic
20#16812133—24no—
21#16812130—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133803C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic, complemented
29#16812105—21no—
30#16812091—21no—
31#16812085C2211yescharacteristic
32#16812082—21no—
33#16812081—21no—
34#16812081—21no—
35#16812080C2211yescharacteristic
36#16812080C2211yescharacteristic
37#16812080C2211yescharacteristic, complemented
38#16812079—21no—
39C4×Dic214C2211yescharacteristic, complemented
40C4×D428C2211yescharacteristic
41C4×D428C2211yescharacteristic
42D4×Dic107—21no—
43C22⋊C4×D107—21no—
44#16812043C2211yescharacteristic, complemented
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46#16812034C2211yescharacteristic, complemented
47#16812033C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812030C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic
57C2×D8×C107C2211yescharacteristic
58#127×C107C2211yescharacteristic, complemented
59#124×C107C2211yescharacteristic
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133817—22no—
65#13133817—22no—
66#13133817—41no—
67#13133817—41no—
68C2×Dic214C2311yescharacteristic
69C2×D428—21no—
70C2×D428D411yescharacteristic
71C2×D428C2311yescharacteristic
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78D4×D107—22no—
79D4×D107—41no—
80C22×Dic107—21no—
81C22×Dic107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85C2×C4×D107C2311yescharacteristic
86#13133808—22no—
87#13133807D414yescomplemented
88#13133806—22no—
89#13133803—22no—
90#13133803—22no—
91#13133801—22no—
92C2×C107⋊C8C2311yescharacteristic
93C2×C107⋊C8C2311yescharacteristic
94C8×D107—22no—
95C4⋊Dic107D411yescharacteristic
96C4⋊Dic107C2311yescharacteristic
97C4⋊Dic107C2311yescharacteristic
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4D411yescharacteristic
106(C2×D107)⋊C4D411yescharacteristic
107C4×Dic107—21no—
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107—21no—
110C2×D4×C107—21no—
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114SD16×C107—22no—
115SD16×C107—41no—
116D8×C107—22no—
117D8×C107—41no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C22⋊C4×C107—21no—
121C22⋊C4×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130C22×D107C2×D411yescharacteristic
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—42no—
141D4×C107—81no—
142C2×C4×C107—21no—
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2×D411yescharacteristic
145C2×C4×C107C2411yescharacteristic
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—81no—
150C22⋊D107—81no—
151C22⋊D107—81no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—42no—
156C2×Dic107—42no—
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159D428—21no—
160D428—41no—
161D428—42no—
162D428C2×D412yes—
163C4×D107—21no—
164C4×D107—21no—
165C4×D107—21no—
166C4×D107—21no—
167C4×D107—42no—
168C4×D107—81no—
169Dic214—21no—
170Dic214C2×D412yes—
171C8×C107—21no—
172C8×C107C2×D412yes—
173C107⋊C8—21no—
174C107⋊C8—21no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
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—21no—
187C2×D107—22no—
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—21no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C4×C107—21no—
196C4×C107—21no—
197C4×C107—41no—
198C4×C107C22×D411yescharacteristic
199C4×C107H(5,𝔽2)11yescharacteristic
200Dic107—22no—
201Dic107—21no—
202Dic107—21no—
203Dic107—41no—
204Dic107—41no—
205C2×C107—81no—
206C2×C107—81no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—41no—
213D107—42no—
214#3268—1071nomaximal, Sylow
215C107#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1071no—
217C2×#135—1071no—
218#550—1071no—
219#546—1071no—
220#496D10711yescharacteristic, complemented
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×#41—1071no—
234C22×D4—1071no—
235AGL(1,ℤ/8)—2144no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16C2×D10711yescharacteristic, complemented
241C2×SD16C2×D10711yescharacteristic, complemented
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8C2×D10711yescharacteristic
245C2×D8C2×D10711yescharacteristic
246C2×M16—1071no—
247C22×C8—1071no—
248#127—2141no—
249#127C2×D10711yescharacteristic, complemented
250#124C2×D10711yescharacteristic
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#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×Q8C22×D10711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—4281no—
290C2×D4—2141no—
291C2×D4—2141no—
292C2×D4—2142no—
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—1071no—
302C22×C4—1071no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4—2141no—
306C22×C4—2141no—
307Q16—2142no—
308SD16—22no—
309SD16—41no—
310SD16—1074no—
311SD16—2142no—
312D8—22no—
313D8—41no—
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—1071no—
325C4⋊C4—2141no—
326C4⋊C4—2141no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1071no—
330C22⋊C4—1071no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C42—2141no—
337C42C22×D10711yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—8561no—
342C23—1071no—
343C23—1071no—
344C23—4282no—
345C23—4282no—
346C23—2141no—
347Q8—21no—
348Q8—21no—
349Q8—1072no—
350Q8—2141no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—8561no—
360D4—8561no—
361D4—8561no—
362D4—1072no—
363D4—4281no—
364D4—4282no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—2141no—
369C2×C4—21no—
370C2×C4—8561no—
371C2×C4—1071no—
372C2×C4—1071no—
373C2×C4—4282no—
374C2×C4—4282no—
375C2×C4—4282no—
376C2×C4—2141no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4D4×D10711yescharacteristic
384C2×C4D4×D10711yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—2141no—
388C8—2141no—
389C8D4×D10712yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—8564no—
400C22—4281no—
401C22—4281no—
402C22—2141no—
403C22—2141no—
404C22—2141no—
405C22—2141no—
406C22—2142no—
407C22—2141no—
408C22—2141no—
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—4281no—
414C4—4281no—
415C4—2141no—
416C4—2141no—
417C4—2142no—
418C4C2×D4×D10711yescharacteristic
419C4#1681215411yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—4281no—
424C2—4282no—
425C2—4281no—
426C2#2029326211yescharacteristic
427C2#2029342311yescharacteristic
428C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3268
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×#16812130C2semidirect
3. #20293297C2semidirect
4. #20293270C2semidirect
5. #20292870C2semidirect
6. #20292860C2semidirect
7. #20292846C2semidirect
8. #20292819C2semidirect
9. #20292786C2semidirect
10. #20292771C2semidirect
11. #20292769C2semidirect
12. #20292759C2semidirect
13. #20292729C2semidirect
14. #20292700C2semidirect
15. #496×C107C2semidirect
18. C2×#13133817C22semidirect
22. C2×#13133808C22semidirect
28. C2×C8×D107C22semidirect
37. #16812080C22semidirect
39. C4×Dic214C22semidirect
44. #16812043C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. #16812034C22semidirect
47. #16812033C22semidirect
50. #16812030C22semidirect
51. #16812030C22semidirect
53. (C2×D107)⋊C8C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×SD16×C107C22semidirect
58. #127×C107C22semidirect
60. #97×C107C22semidirect
87. #13133807D4semidirect
215. C107#3268semidirect
220. #496D107semidirect
240. C2×SD16C2×D107semidirect
241. C2×SD16C2×D107semidirect
249. #127C2×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×#168121332
C2×#168121302
#202932972
#202932702
#202928702
#202928602
#202928462
#202928192
#202927862
#202927712
#202927692
#202927592
#202927292
#202927002
#496×C1072
#3268107