Finite Group Databasebeta

← #28345744

Subgroups — #28345744

Order 13696.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812156C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3C2×SD16×D107C211yesmaximal, characteristic, complemented
4#20293193C211yesmaximal, characteristic, complemented
5#20292773C211yesmaximal, characteristic, complemented
6#20292772C211yesmaximal, characteristic, complemented
7#20292769C211yesmaximal, characteristic, complemented
8#20292768C211yesmaximal, characteristic, complemented
9#20292767C211yesmaximal, characteristic, complemented
10#20292765C211yesmaximal, characteristic, complemented
11#20292764C211yesmaximal, characteristic, complemented
12#20292634C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14#20292592C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16#16812156—24no—
17#16812156—24no—
18C2×#13133818—21no—
19C2×#13133818C2211yescharacteristic
20C2×#13133817—21no—
21C2×#13133817—21no—
22C2×#13133817C2211yescharacteristic
23C2×#13133817C2211yescharacteristic, complemented
24C22×Dic214—21no—
25C22×Dic214C2211yescharacteristic, complemented
26C2×Q8×D107C2211yescharacteristic
27C2×Q8×D107C2211yescharacteristic, complemented
28C2×D4×D107C2211yescharacteristic
29#16812133—24no—
30C2×#13133807C2211yescharacteristic
31C2×#13133807C2211yescharacteristic
32C2×#13133806C2211yescharacteristic
33C2×#13133806C2211yescharacteristic
34C2×#13133805C2211yescharacteristic
35SD16×D107—24no—
36C2×#13133803C2211yescharacteristic
37C2×#13133801C2211yescharacteristic
38C2×C8×D107C2211yescharacteristic
39#16812105—21no—
40#16812095—21no—
41#16812084C2211yescharacteristic
42#16812083C2211yescharacteristic
43#16812080C2211yescharacteristic
44#16812079—21no—
45#16812077—21no—
46#16812067C2211yescharacteristic
47#16812034C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812033C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812032C2211yescharacteristic, complemented
52#16812030C2211yescharacteristic, complemented
53C22⋊(C107⋊C8)C2211yescharacteristic, complemented
54(C2×D107)⋊C8C2211yescharacteristic, complemented
55(C2×D107)⋊C8C2211yescharacteristic, complemented
56C2×#41×C107C2211yescharacteristic, complemented
57C2×SD16×C107C2211yescharacteristic
58C2×D8×C107C2211yescharacteristic
59#121×C107C2211yescharacteristic
60#103×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62C22⋊C8×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64#13133818—28no—
65#13133818—22no—
66#13133818—22no—
67#13133817—28no—
68#13133817—28no—
69#13133817—22no—
70#13133817—22no—
71#13133817—22no—
72#13133817D414yescomplemented
73C2×Dic214—28no—
74C2×Dic214—21no—
75C2×Dic214—21no—
76C2×Dic214—21no—
77C2×Dic214—22no—
78C2×Dic214—22no—
79C2×Dic214D411yescharacteristic
80C2×Dic214C2311yescharacteristic
81C2×Dic214C2311yescharacteristic
82Q8×D107—22no—
83Q8×D107—22no—
84Q8×D107—22no—
85Q8×D107D414yescomplemented
86C2×D428—21no—
87C2×D428D411yescharacteristic
88C2×D428C2311yescharacteristic
89C2×C22⋊D107—21no—
90C2×C22⋊D107—21no—
91C2×C22⋊D107—21no—
92C2×C22⋊D107—21no—
93C2×C22⋊D107—21no—
94C2×C22⋊D107D411yescharacteristic
95C2×C22⋊D107D411yescharacteristic
96D4×D107—22no—
97D4×D107—22no—
98C22×Dic107—21no—
99C22×Dic107—21no—
100C22×Dic107—21no—
101C2×C4×D107—21no—
102C2×C4×D107—21no—
103C2×C4×D107C2311yescharacteristic
104C2×C4×D107C2311yescharacteristic
105#13133807—22no—
106#13133807—22no—
107#13133806—22no—
108#13133806—22no—
109#13133805—22no—
110#13133803—22no—
111#13133801—22no—
112C2×C107⋊C8C2311yescharacteristic
113C2×C107⋊C8C2311yescharacteristic
114C8×D107—22no—
115C4⋊Dic107C2311yescharacteristic
116C22⋊Dic107—21no—
117(C2×D107)⋊C4—21no—
118(C2×D107)⋊C4—21no—
119(C2×D107)⋊C4—21no—
120C4×Dic107—21no—
121#41×C107—22no—
122#41×C107—22no—
123C2×Q8×C107C2311yescharacteristic
124C2×D4×C107—21no—
125C2×D4×C107—21no—
126C2×D4×C107C2311yescharacteristic
127C2×D4×C107C2311yescharacteristic
128C22×C4×C107—21no—
129C22×C4×C107C2311yescharacteristic
130SD16×C107—22no—
131D8×C107—22no—
132C2×C8×C107C2311yescharacteristic
133C2×C8×C107C2311yescharacteristic
134C4⋊C4×C107C2311yescharacteristic
135C22⋊C4×C107—21no—
136C23×C107—21no—
137C23×C107—21no—
138C23×C107C2×D411yescharacteristic
139C22×D107—21no—
140C22×D107—42no—
141C22×D107—81no—
142C22×D107—81no—
143C22×D107C2×D411yescharacteristic
144C22×D107C2×D411yescharacteristic
145Q8×C107—21no—
146Q8×C107C2×D412yes—
147D4×C107—24no—
148D4×C107—21no—
149D4×C107—21no—
150D4×C107—21no—
151D4×C107—81no—
152D4×C107C2×D412yes—
153C2×C4×C107—24no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107C2×D411yescharacteristic
159C2×C4×C107C2411yescharacteristic
160C22⋊D107—24no—
161C22⋊D107—24no—
162C22⋊D107—22no—
163C22⋊D107—22no—
164C22⋊D107—42no—
165C22⋊D107—81no—
166C22⋊D107—81no—
167C2×Dic107—24no—
168C2×Dic107—24no—
169C2×Dic107—21no—
170C2×Dic107—21no—
171C2×Dic107—24no—
172C2×Dic107—21no—
173C2×Dic107C2×D411yescharacteristic
174C2×Dic107C2×D411yescharacteristic
175D428—22no—
176D428—21no—
177D428—21no—
178D428—81no—
179C4×D107—24no—
180C4×D107—24no—
181C4×D107—21no—
182C4×D107—21no—
183C4×D107C2×D412yes—
184C4×D107C2×D412yes—
185Dic214—24no—
186Dic214—24no—
187Dic214—21no—
188Dic214—21no—
189Dic214—24no—
190Dic214—22no—
191Dic214C2×D412yes—
192Dic214C2×D412yes—
193C8×C107—21no—
194C8×C107—21no—
195C107⋊C8—21no—
196C107⋊C8—21no—
197C22×C107—21no—
198C22×C107—22no—
199C22×C107—21no—
200C22×C107—21no—
201C22×C107—41no—
202C22×C107—41no—
203C22×C107—41no—
204C22×C107—41no—
205C22×C107—41no—
206C22×C107C22×D411yescharacteristic
207C2×D107—21no—
208C2×D107—21no—
209C2×D107—21no—
210C2×D107—21no—
211C2×D107—21no—
212C2×D107—41no—
213C2×D107—41no—
214C2×D107—41no—
215C2×D107—84no—
216C2×D107#12012yes—
217C4×C107—21no—
218C4×C107—22no—
219C4×C107—41no—
220C4×C107C22×D411yescharacteristic
221C4×C107C22×D411yescharacteristic
222Dic107—21no—
223Dic107—22no—
224Dic107—22no—
225Dic107—41no—
226Dic107#12012yes—
227C2×C107—41no—
228C2×C107—42no—
229C2×C107—81no—
230C2×C107C2×#12011yescharacteristic
231C2×C107#57711yescharacteristic
232C2×C107#57711yescharacteristic
233D107—22no—
234D107—41no—
235D107—81no—
236#3010—1071nomaximal, Sylow
237C107#301011yescharacteristic, complemented, Sylow
238C2×2−1+4—1071no—
239C2×AGL(1,ℤ/8)—1071no—
240C22×SD16—1071no—
241#535—1071no—
242#449—1071no—
243#449—1071no—
244#449—1071no—
245#449D10711yescharacteristic, complemented
246#448—1071no—
247#448—1071no—
248#448—1071no—
249#448—1071no—
250#419—1071no—
251#416—1071no—
252#408—1071no—
2532−1+4—2144no—
2542−1+4—2144no—
255C2×#41—1071no—
256C2×#41—1071no—
257C2×#41—1071no—
258C2×#41—2141no—
259C2×#41—2141no—
260C2×#41—2141no—
261C2×#41C2×D10711yescharacteristic, complemented
262C22×Q8—1071no—
263C22×Q8—1071no—
264C22×Q8—1071no—
265C22×Q8—2141no—
266C22×D4—1071no—
267AGL(1,ℤ/8)—2144no—
268C2×SD16—1071no—
269C2×SD16—1071no—
270C2×SD16—1071no—
271C2×SD16—1071no—
272C2×SD16—1071no—
273C2×SD16—2144no—
274C2×SD16C2×D10711yescharacteristic
275C2×D8—1071no—
276C2×D8C2×D10711yescharacteristic
277C2×M16—1071no—
278C22×C8—1071no—
279#127—2141no—
280#124—2141no—
281#121—1071no—
282#121—1071no—
283#121—1071no—
284#121C2×D10711yescharacteristic
285#120—2141no—
286#120—2141no—
287#117—1071no—
288#103—1071no—
289#103—1071no—
290#103—1071no—
291#103—1071no—
292#103—1071no—
293#103C2×D10711yescharacteristic, complemented
294#102—1071no—
295#102C2×D10711yescharacteristic, complemented
296C22⋊C8—1071no—
297C22⋊C8—1071no—
298C22⋊C8—1071no—
299C22⋊C8C2×D10711yescharacteristic, complemented
300C24—2141no—
301#41—22no—
302#41—22no—
303#41—1074no—
304#41—2142no—
305#41—2142no—
306#41—2148no—
307#41—2142no—
308#41—2142no—
309#41—2148no—
310#41—2148no—
311#41—2142no—
312C2×Q8—1071no—
313C2×Q8—1074no—
314C2×Q8—1071no—
315C2×Q8—1071no—
316C2×Q8—2141no—
317C2×Q8—2141no—
318C2×Q8—2142no—
319C2×Q8—2142no—
320C2×Q8—2148no—
321C2×Q8—2142no—
322C2×Q8—2142no—
323C2×Q8—2142no—
324C2×Q8—2141no—
325C2×Q8C22×D10711yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—1071no—
329C2×D4—1071no—
330C2×D4—1071no—
331C2×D4—1071no—
332C2×D4—2141no—
333C2×D4—2142no—
334C2×D4—2142no—
335C2×D4—2141no—
336C2×D4—2141no—
337C2×D4—2141no—
338C2×D4—2141no—
339C2×D4—2141no—
340C2×D4C22×D10711yescharacteristic
341C2×D4C22×D10711yescharacteristic
342C22×C4—21no—
343C22×C4—1071no—
344C22×C4—1071no—
345C22×C4—2141no—
346C22×C4—2141no—
347C22×C4—2141no—
348C22×C4—2141no—
349C22×C4—2141no—
350C22×C4C22×D10711yescharacteristic
351SD16—22no—
352SD16—2142no—
353SD16—2142no—
354SD16—2142no—
355SD16—2142no—
356SD16—2142no—
357D8—22no—
358D8—2142no—
359M16—2142no—
360C2×C8—1071no—
361C2×C8—1071no—
362C2×C8—2142no—
363C2×C8C22×D10711yescharacteristic
364C2×C8C22×D10711yescharacteristic
365C4⋊C4—1071no—
366C4⋊C4C22×D10711yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—2141no—
369C22⋊C4—2141no—
370C22⋊C4—2141no—
371C22⋊C4—2141no—
372C42—2141no—
373C23—21no—
374C23—21no—
375C23—8561no—
376C23—8561no—
377C23—1071no—
378C23—1071no—
379C23—4282no—
380C23—2141no—
381C23D4×D10711yescharacteristic
382Q8—21no—
383Q8—1072no—
384Q8—1072no—
385Q8—2142no—
386Q8—2144no—
387Q8—2144no—
388Q8—2144no—
389Q8—2141no—
390Q8—2141no—
391Q8D4×D10712yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—8561no—
398D4—8561no—
399D4—8561no—
400D4—4282no—
401D4—2142no—
402D4—2142no—
403D4—2142no—
404D4—2144no—
405D4—2144no—
406D4—2141no—
407D4—2141no—
408D4D4×D10712yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—1071no—
415C2×C4—1072no—
416C2×C4—1072no—
417C2×C4—1071no—
418C2×C4—2144no—
419C2×C4—2144no—
420C2×C4—2144no—
421C2×C4—2141no—
422C2×C4—2144no—
423C2×C4—2144no—
424C2×C4—2141no—
425C2×C4—2141no—
426C2×C4—2141no—
427C2×C4—2141no—
428C2×C4D4×D10711yescharacteristic
429C2×C4C23×D10711yescharacteristic
430C8—21no—
431C8—21no—
432C8—2141no—
433C8—2141no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—8564no—
444C22—1072no—
445C22—4281no—
446C22—4281no—
447C22—4281no—
448C22—2141no—
449C22—2141no—
450C22—2141no—
451C22—2141no—
452C22—2141no—
453C22C2×D4×D10711yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—1072no—
458C4—4281no—
459C4—2141no—
460C4—2142no—
461C4—2142no—
462C4C2×D4×D10711yescharacteristic
463C4C2×D4×D10711yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—8561no—
468C2—4281no—
469C2—2142no—
470C2#120×D10711yescharacteristic
471C2#2029343411yescharacteristic
472C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3010
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×#16812156C2semidirect
2. C2×#16812133C2semidirect
3. C2×SD16×D107C2semidirect
4. #20293193C2semidirect
5. #20292773C2semidirect
6. #20292772C2semidirect
7. #20292769C2semidirect
8. #20292768C2semidirect
9. #20292767C2semidirect
10. #20292765C2semidirect
11. #20292764C2semidirect
12. #20292634C2semidirect
13. #20292620C2semidirect
14. #20292592C2semidirect
15. #449×C107C2semidirect
23. C2×#13133817C22semidirect
25. C22×Dic214C22semidirect
27. C2×Q8×D107C22semidirect
47. #16812034C22semidirect
48. #16812034C22semidirect
49. #16812033C22semidirect
50. #16812032C22semidirect
51. #16812032C22semidirect
52. #16812030C22semidirect
53. C22⋊(C107⋊C8)C22semidirect
54. (C2×D107)⋊C8C22semidirect
55. (C2×D107)⋊C8C22semidirect
56. C2×#41×C107C22semidirect
60. #103×C107C22semidirect
61. #102×C107C22semidirect
62. C22⋊C8×C107C22semidirect
72. #13133817D4semidirect
85. Q8×D107D4semidirect
237. C107#3010semidirect
245. #449D107semidirect
261. C2×#41C2×D107semidirect
293. #103C2×D107semidirect
295. #102C2×D107semidirect
299. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168121332
C2×SD16×D1072
#202931932
#202927732
#202927722
#202927692
#202927682
#202927672
#202927652
#202927642
#202926342
#202926202
#202925922
#449×C1072
#3010107