Finite Group Databasebeta

← #28346075

Subgroups — #28346075

Order 13696.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×M16×D107C211yesmaximal, characteristic, complemented
3#20293140C211yesmaximal, characteristic, complemented
4#121×D107C211yesmaximal, characteristic, complemented
5#20292880C211yesmaximal, characteristic, complemented
6#20292878C211yesmaximal, characteristic, complemented
7#20292875C211yesmaximal, characteristic, complemented
8#20292864C211yesmaximal, characteristic, complemented
9#20292862C211yesmaximal, characteristic, complemented
10#20292859C211yesmaximal, characteristic, complemented
11#20292853C211yesmaximal, characteristic, complemented
12#20292668C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic, complemented
14#20292627C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812136—24no—
21C2×#13133805C2211yescharacteristic, complemented
22C2×D856C2211yescharacteristic, complemented
23C2×#13133803C2211yescharacteristic, complemented
24C2×#13133802C2211yescharacteristic, complemented
25C2×#13133801C2211yescharacteristic
26C2×#13133801C2211yescharacteristic
27M16×D107—24no—
28C2×C8×D107C2211yescharacteristic, complemented
29C2×C8×D107C2211yescharacteristic, complemented
30#16812094—21no—
31#16812085C2211yescharacteristic, complemented
32#16812084C2211yescharacteristic
33#16812084C2211yescharacteristic, complemented
34#16812083C2211yescharacteristic
35#16812083C2211yescharacteristic, complemented
36#16812082—21no—
37#16812081—21no—
38#16812080C2211yescharacteristic
39#16812079—21no—
40#16812071—21no—
41(C22⋊D107)⋊C4—21no—
42#16812067C2211yescharacteristic
43C4⋊C4×D107C2211yescharacteristic
44C22⋊C4×D107—21no—
45#16812042C2211yescharacteristic, complemented
46#16812041C2211yescharacteristic, complemented
47#16812040C2211yescharacteristic, complemented
48#16812033C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107D411yescharacteristic
63#13133819—22no—
64#13133819—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74D4×D107—22no—
75D4×D107—22no—
76D4×D107—44no—
77C22×Dic107D411yescharacteristic
78C2×C4×D107—22no—
79C2×C4×D107—22no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107D411yescharacteristic
83C2×C4×D107D411yescharacteristic
84C2×C4×D107C2311yescharacteristic
85C2×C4×D107C2311yescharacteristic
86#13133805—22no—
87D856—22no—
88#13133803—22no—
89#13133802—22no—
90#13133801—22no—
91#13133801D414yescomplemented
92C2×C107⋊C8C2311yescharacteristic
93C2×C107⋊C8C2311yescharacteristic
94C8×D107—22no—
95C8×D107—22no—
96C4⋊Dic107C2311yescharacteristic
97C4⋊Dic107C2311yescharacteristic
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100C22⋊Dic107—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107C4×Dic107—21no—
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107—21no—
110C2×D4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112SD16×C107—22no—
113M16×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C4⋊C4×C107—21no—
117C4⋊C4×C107C2311yescharacteristic
118C4⋊C4×C107C2311yescharacteristic
119C22⋊C4×C107—21no—
120C22⋊C4×C107—21no—
121C23×C107—21no—
122C23×C107C2×D411yescharacteristic
123C22×D107—22no—
124C22×D107—22no—
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—41no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107C2×D411yescharacteristic
132C22×D107C2×D411yescharacteristic
133Q8×C107—21no—
134Q8×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—42no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2411yescharacteristic
145C22⋊D107—42no—
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C2×Dic107—22no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—42no—
159D428—42no—
160D428—42no—
161C4×D107—22no—
162C4×D107—21no—
163C4×D107—21no—
164C4×D107—22no—
165C4×D107—42no—
166C4×D107—42no—
167C4×D107C2×D412yes—
168C4×D107#4112yes—
169C8×C107—21no—
170C8×C107C2×D412yescomplemented
171C107⋊C8—21no—
172C107⋊C8C2×D412yescomplemented
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107C22×D411yescharacteristic
180C2×D107—21no—
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—44no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—84no—
192C2×D107#12112yes—
193C2×D107AGL(1,ℤ/8)12yescomplemented
194C4×C107—21no—
195C4×C107—41no—
196C4×C107—41no—
197C4×C107—41no—
198C4×C107C22×D411yescharacteristic
199C4×C107C2×#4111yescharacteristic
200Dic107—21no—
201Dic107—41no—
202Dic107—41no—
203Dic107#12112yes—
204C2×C107—41no—
205C2×C107—81no—
206C2×C107C2×#12111yescharacteristic
207C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C107#57511yescharacteristic
209D107—22no—
210D107—22no—
211D107—81no—
212D107—81no—
213#3055—1071nomaximal, Sylow
214C107#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—1071no—
216C22×M16—1071no—
217#525—1071no—
218C2×#121—1071no—
219#469—1071no—
220#469—1071no—
221#468—1071no—
222#468D10711yescharacteristic, complemented
223#466—1071no—
224#466—1071no—
225#465—1071no—
226#465—1071no—
227#428—1071no—
228#418—1071no—
229#417—1071no—
230C2×#41—1071no—
231C22×D4—1071no—
232C22×D4—2141no—
233C23×C4—1071no—
234AGL(1,ℤ/8)—2144no—
235C2×SD16—1071no—
236C2×SD16C2×D10711yescharacteristic, complemented
237C2×D8—1071no—
238C2×D8—1071no—
239C2×M16—1071no—
240C2×M16—1071no—
241C2×M16—1071no—
242C2×M16—2144no—
243C2×M16C2×D10711yescharacteristic, complemented
244C22×C8—1071no—
245C22×C8—1071no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8C2×D10711yescharacteristic, complemented
249#121—1071no—
250#121—1071no—
251#121—1071no—
252#121—1071no—
253#121—1071no—
254#121—2141no—
255#121—2141no—
256#121C2×D10711yescharacteristic
257#120—2141no—
258C4×D4—2141no—
259C4×D4—2141no—
260#117—1071no—
261C2×C4⋊C4—1071no—
262C2×C22⋊C4—2141no—
263#107—1071no—
264#107C2×D10711yescharacteristic, complemented
265#106—1071no—
266#106—1071no—
267#103—1071no—
268#103C2×D10711yescharacteristic, complemented
269#102—1071no—
270#102—1071no—
271#102—1071no—
272#102—1071no—
273#102—1071no—
274#102C2×D10711yescharacteristic, complemented
275C24—1071no—
276C24—2141no—
277#41—2142no—
278#41—2142no—
279C2×Q8C22×D10711yescharacteristic
280C2×D4—21no—
281C2×D4—1071no—
282C2×D4—1071no—
283C2×D4—4284no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2142no—
288C2×D4—2142no—
289C2×D4—2141no—
290C2×D4—2141no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4C22×D10711yescharacteristic
294C22×C4—1071no—
295C22×C4—1071no—
296C22×C4—1071no—
297C22×C4—1071no—
298C22×C4—1071no—
299C22×C4—2141no—
300C22×C4—2142no—
301C22×C4—2142no—
302C22×C4—2141no—
303C22×C4C22×D10711yescharacteristic
304SD16—22no—
305SD16—2142no—
306D8—2142no—
307D8—2142no—
308M16—22no—
309M16—1074no—
310M16—2142no—
311M16—2142no—
312C2×C8—1071no—
313C2×C8—1071no—
314C2×C8—2142no—
315C2×C8—2142no—
316C2×C8C22×D10711yescharacteristic
317C2×C8C22×D10711yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—1071no—
320C4⋊C4—1071no—
321C4⋊C4—2141no—
322C4⋊C4—2141no—
323C4⋊C4C22×D10711yescharacteristic
324C4⋊C4C22×D10711yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C42—2141no—
335C23—21no—
336C23—1071no—
337C23—1071no—
338C23—4281no—
339C23—4282no—
340C23—4282no—
341C23—4282no—
342C23—2142no—
343C23—2142no—
344C23—2141no—
345C23—2141no—
346C23D4×D10711yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—4282no—
353D4—4282no—
354D4—4282no—
355D4—4282no—
356D4—4282no—
357D4—4282no—
358D4—4282no—
359D4—2141no—
360D4—2141no—
361D4—2141no—
362D4—2141no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—1071no—
369C2×C4—1072no—
370C2×C4—1072no—
371C2×C4—1071no—
372C2×C4—4282no—
373C2×C4—4282no—
374C2×C4—2142no—
375C2×C4—2142no—
376C2×C4—2142no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4D4×D10711yescharacteristic
382C2×C4C23×D10711yescharacteristic
383C8—21no—
384C8—1072no—
385C8—2141no—
386C8D4×D10712yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—8564no—
394C22—1072no—
395C22—1072no—
396C22—4284no—
397C22—4281no—
398C22—4281no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—2141no—
404C22—2141no—
405C22—2141no—
406C22—2141no—
407C22C2×D4×D10711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—1072no—
413C4—4281no—
414C4—4281no—
415C4—2141no—
416C4C2×D4×D10711yescharacteristic
417C4#41×D10711yescharacteristic
418C2—41no—
419C2—81no—
420C2—8561no—
421C2—8561no—
422C2—2142no—
423C2—2142no—
424C2#121×D10711yescharacteristic
425C2AGL(1,ℤ/8)×D10711yescharacteristic
426C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3055
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×M16×D107C2semidirect
3. #20293140C2semidirect
4. #121×D107C2semidirect
5. #20292880C2semidirect
6. #20292878C2semidirect
7. #20292875C2semidirect
8. #20292864C2semidirect
9. #20292862C2semidirect
10. #20292859C2semidirect
11. #20292853C2semidirect
12. #20292668C2semidirect
13. #20292630C2semidirect
14. #20292627C2semidirect
15. #468×C107C2semidirect
21. C2×#13133805C22semidirect
22. C2×D856C22semidirect
23. C2×#13133803C22semidirect
24. C2×#13133802C22semidirect
28. C2×C8×D107C22semidirect
29. C2×C8×D107C22semidirect
31. #16812085C22semidirect
33. #16812084C22semidirect
35. #16812083C22semidirect
45. #16812042C22semidirect
46. #16812041C22semidirect
47. #16812040C22semidirect
48. #16812033C22semidirect
49. #16812031C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. #16812029C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×M16×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
91. #13133801D4semidirect
170. C8×C107C2×D4semidirect
172. C107⋊C8C2×D4semidirect
193. C2×D107AGL(1,ℤ/8)semidirect
214. C107#3055semidirect
222. #468D107semidirect
236. C2×SD16C2×D107semidirect
243. C2×M16C2×D107semidirect
248. C22⋊Q8C2×D107semidirect
264. #107C2×D107semidirect
268. #103C2×D107semidirect
274. #102C2×D107semidirect
386. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×M16×D1072
#202931402
#121×D1072
#202928802
#202928782
#202928752
#202928642
#202928622
#202928592
#202928532
#202926682
#202926302
#202926272
#468×C1072
#3055107