Finite Group Databasebeta

← #28345962

Subgroups — #28345962

Order 13696.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293195C211yesmaximal, characteristic, complemented
4#20293099C211yesmaximal, characteristic, complemented
5#20292845C211yesmaximal, characteristic, complemented
6#20292844C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292823C211yesmaximal, characteristic, complemented
9#20292822C211yesmaximal, characteristic, complemented
10#20292821C211yesmaximal, characteristic, complemented
11#20292820C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic, complemented
14#20292622C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812136—24no—
20C2×#13133805C2211yescharacteristic, complemented
21C2×D856C2211yescharacteristic
22C2×D856C2211yescharacteristic, complemented
23C2×#13133803C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic, complemented
26D8×D107—24no—
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812106C2211yescharacteristic, complemented
30#16812105C2211yescharacteristic, complemented
31#16812098C2211yescharacteristic, complemented
32#16812082—21no—
33#16812082—21no—
34#16812079—21no—
35#16812079—21no—
36C4×D428C2211yescharacteristic, complemented
37D4×Dic107C2211yescharacteristic, complemented
38#16812071C2211yescharacteristic, complemented
39C4×C22⋊D107—21no—
40(C22⋊D107)⋊C4—21no—
41#16812067C2211yescharacteristic
42#16812066C2211yescharacteristic
43C2×(C2×D107)⋊C4—21no—
44C22⋊C4×D107—21no—
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812033C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133819—22no—
64#13133819—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75D4×D107—22no—
76D4×D107—22no—
77D4×D107—22no—
78D4×D107D414yescomplemented
79C22×Dic107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107C2311yescharacteristic
84#13133805—22no—
85D856—22no—
86D856—22no—
87#13133803—22no—
88#13133803—22no—
89#13133803—22no—
90#13133801—22no—
91C2×C107⋊C8C2311yescharacteristic
92C2×C107⋊C8C2311yescharacteristic
93C8×D107—22no—
94C4⋊Dic107C2311yescharacteristic
95Dic107⋊C4D411yescharacteristic
96Dic107⋊C4D411yescharacteristic
97C22⋊Dic107—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4D411yescharacteristic
104(C2×D107)⋊C4D411yescharacteristic
105C4×Dic107—21no—
106C4×Dic107C2311yescharacteristic
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110C22×C4×C107—21no—
111SD16×C107—22no—
112D8×C107—22no—
113C2×C8×C107C2311yescharacteristic
114C2×C8×C107C2311yescharacteristic
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C23×C107—21no—
122C22×D107—21no—
123C22×D107—24no—
124C22×D107—21no—
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—81no—
128C22×D107—81no—
129C22×D107—81no—
130C22×D107—81no—
131C22×D107C2×D411yescharacteristic
132Q8×C107—21no—
133Q8×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107C2×D412yes—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—81no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2411yescharacteristic
144C22⋊D107—24no—
145C22⋊D107—42no—
146C22⋊D107—81no—
147C22⋊D107—81no—
148C2×Dic107—21no—
149C2×Dic107—81no—
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153D428—21no—
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—42no—
159D428—81no—
160D428—81no—
161D428C2×D412yes—
162C4×D107—42no—
163C4×D107—81no—
164C4×D107—81no—
165C4×D107C2×D412yes—
166C4×D107#4112yes—
167C8×C107—21no—
168C8×C107—21no—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—22no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—84no—
190C2×D107—84no—
191C2×D107#12112yes—
192C4×C107—21no—
193C4×C107—21no—
194C4×C107—41no—
195C4×C107—41no—
196C4×C107C22×D411yescharacteristic
197C4×C107C2×#4111yescharacteristic
198Dic107—21no—
199Dic107—21no—
200Dic107—41no—
201Dic107#12112yes—
202C2×C107—42no—
203C2×C107—81no—
204C2×C107C2×#12111yescharacteristic
205C2×C107#57611yescharacteristic
206C2×C107#57611yescharacteristic
207D107—22no—
208D107—42no—
209D107—81no—
210D107—81no—
211#3041—1071nomaximal, Sylow
212C107#304111yescharacteristic, complemented, Sylow
213C2×AGL(1,ℤ/8)—1071no—
214C22×D8—1071no—
215#535—1071no—
216#518—1071no—
217#463—1071no—
218#463—1071no—
219#463—1071no—
220#463D10711yescharacteristic, complemented
221#459—1071no—
222#459—1071no—
223#459—1071no—
224#459—1071no—
225#424—1071no—
226#418—1071no—
227#416—1071no—
228C2×#41—1071no—
229C22×D4—1071no—
230C22×D4—1071no—
231AGL(1,ℤ/8)—2144no—
232C2×SD16—1071no—
233C2×SD16C2×D10711yescharacteristic, complemented
234C2×D8—1071no—
235C2×D8—1071no—
236C2×D8—1071no—
237C2×D8—1071no—
238C2×D8—1071no—
239C2×D8—2144no—
240C2×D8C2×D10711yescharacteristic
241C2×M16—1071no—
242C22×C8—1071no—
243#127—1071no—
244#127—1071no—
245#124—1071no—
246#124C2×D10711yescharacteristic, complemented
247#121—2141no—
248#121—2141no—
249#120—2141no—
250#120—2141no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—1071no—
254C4×D4—2141no—
255C4×D4—2141no—
256C4×D4C2×D10711yescharacteristic, complemented
257#117—1071no—
258#117—1071no—
259C2×C22⋊C4—2141no—
260C2×C22⋊C4—2141no—
261C4⋊C8—1071no—
262C4⋊C8—1071no—
263C4⋊C8—1071no—
264C4⋊C8C2×D10711yescharacteristic, complemented
265#103—1071no—
266#103C2×D10711yescharacteristic, complemented
267#102—1071no—
268#102—1071no—
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102C2×D10711yescharacteristic, complemented
273C24—2141no—
274C24—2141no—
275#41—2142no—
276#41—2142no—
277C2×Q8C22×D10711yescharacteristic
278C2×D4—1074no—
279C2×D4—1071no—
280C2×D4—1071no—
281C2×D4—1071no—
282C2×D4—2141no—
283C2×D4—2142no—
284C2×D4—2142no—
285C2×D4—2142no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4—2141no—
291C2×D4—2141no—
292C2×D4C22×D10711yescharacteristic
293C2×D4C22×D10711yescharacteristic
294C22×C4—21no—
295C22×C4—1071no—
296C22×C4—2141no—
297C22×C4—2141no—
298C22×C4—2141no—
299C22×C4—2141no—
300SD16—22no—
301SD16—2142no—
302D8—22no—
303D8—2142no—
304D8—2142no—
305D8—2142no—
306D8—2142no—
307D8—2142no—
308M16—2142no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8—2142no—
312C2×C8C22×D10711yescharacteristic
313C2×C8C22×D10711yescharacteristic
314C4⋊C4—1071no—
315C4⋊C4—1071no—
316C4⋊C4—1071no—
317C4⋊C4C22×D10711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—1071no—
322C22⋊C4—1071no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C42—1071no—
330C42—2141no—
331C42C22×D10711yescharacteristic
332C23—21no—
333C23—21no—
334C23—8561no—
335C23—8561no—
336C23—8561no—
337C23—8561no—
338C23—1071no—
339C23—4282no—
340C23—2144no—
341C23—2141no—
342C23—2141no—
343C23—2141no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—8561no—
350D4—8561no—
351D4—8561no—
352D4—8561no—
353D4—1072no—
354D4—4282no—
355D4—4282no—
356D4—2144no—
357D4—2141no—
358D4—2141no—
359D4—2141no—
360D4—2141no—
361D4—2141no—
362D4D4×D10712yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—8561no—
367C2×C4—8561no—
368C2×C4—8561no—
369C2×C4—1071no—
370C2×C4—1072no—
371C2×C4—1072no—
372C2×C4—1071no—
373C2×C4—1071no—
374C2×C4—4282no—
375C2×C4—2141no—
376C2×C4D4×D10711yescharacteristic
377C2×C4D4×D10711yescharacteristic
378C2×C4C23×D10711yescharacteristic
379C8—21no—
380C8—21no—
381C8—2141no—
382C8—2141no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—8564no—
390C22—8564no—
391C22—1072no—
392C22—4281no—
393C22—4281no—
394C22—4281no—
395C22—4281no—
396C22—4281no—
397C22—4281no—
398C22—4281no—
399C22—4281no—
400C22—2142no—
401C22—2141no—
402C22—2141no—
403C22C2×D4×D10711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—1072no—
409C4—4281no—
410C4—2141no—
411C4—2141no—
412C4C2×D4×D10711yescharacteristic
413C4#41×D10711yescharacteristic
414C2—42no—
415C2—81no—
416C2—8561no—
417C2—8561no—
418C2—4282no—
419C2—2142no—
420C2#121×D10711yescharacteristic
421C2#2029342911yescharacteristic
422C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
hypercenter#463D107
hypocenterC107#3041
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×D8×D107C2semidirect
3. #20293195C2semidirect
4. #20293099C2semidirect
5. #20292845C2semidirect
6. #20292844C2semidirect
7. #20292839C2semidirect
8. #20292823C2semidirect
9. #20292822C2semidirect
10. #20292821C2semidirect
11. #20292820C2semidirect
12. #20292652C2semidirect
13. #20292630C2semidirect
14. #20292622C2semidirect
15. #463×C107C2semidirect
20. C2×#13133805C22semidirect
22. C2×D856C22semidirect
25. C2×#13133803C22semidirect
29. #16812106C22semidirect
30. #16812105C22semidirect
31. #16812098C22semidirect
36. C4×D428C22semidirect
37. D4×Dic107C22semidirect
38. #16812071C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812033C22semidirect
49. #16812031C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. #16812029C22semidirect
54. C2×SD16×C107C22semidirect
56. #124×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
78. D4×D107D4semidirect
212. C107#3041semidirect
220. #463D107semidirect
233. C2×SD16C2×D107semidirect
246. #124C2×D107semidirect
256. C4×D4C2×D107semidirect
264. C4⋊C8C2×D107semidirect
266. #103C2×D107semidirect
272. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×D8×D1072
#202931952
#202930992
#202928452
#202928442
#202928392
#202928232
#202928222
#202928212
#202928202
#202926522
#202926302
#202926222
#463×C1072
#3041107