Finite Group Databasebeta

← #28345682

Subgroups — #28345682

Order 13696.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic, complemented
2C2×#16812132C211yesmaximal, characteristic, complemented
3C2×#16812130C211yesmaximal, characteristic, complemented
4#20293140C211yesmaximal, characteristic, complemented
5#20292787C211yesmaximal, characteristic, complemented
6#20292781C211yesmaximal, characteristic, complemented
7#20292776C211yesmaximal, characteristic, complemented
8#20292774C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic, complemented
10#20292767C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292622C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14C22⋊C8×D107C211yesmaximal, characteristic, complemented
15#452×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133819C2211yescharacteristic, complemented
18C2×#13133818—21no—
19C2×#13133818C2211yescharacteristic
20C2×#13133817—21no—
21C2×#13133817C2211yescharacteristic
22#41×D107—24no—
23#41×D107—24no—
24C2×Q8×D107C2211yescharacteristic
25C2×D4×D107—21no—
26C2×D4×D107C2211yescharacteristic
27C22×C4×D107—21no—
28C22×C4×D107C2211yescharacteristic
29#16812132—24no—
30#16812130—24no—
31C2×#13133808C2211yescharacteristic, complemented
32C2×#13133807C2211yescharacteristic, complemented
33C2×#13133805C2211yescharacteristic, complemented
34C2×#13133805C2211yescharacteristic, complemented
35C2×D856C2211yescharacteristic, complemented
36C2×#13133803C2211yescharacteristic, complemented
37C2×C8×D107C2211yescharacteristic, complemented
38C2×C8×D107C2211yescharacteristic, complemented
39#16812094—21no—
40#16812085C2211yescharacteristic
41#16812084C2211yescharacteristic
42#16812083C2211yescharacteristic, complemented
43#16812079—21no—
44#16812071—21no—
45(C22⋊D107)⋊C4—21no—
46#16812067C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812033C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53C22⋊(C107⋊C8)C2211yescharacteristic, complemented
54(C2×D107)⋊C8C2211yescharacteristic
55(C2×D107)⋊C8C2211yescharacteristic
56C2×#41×C107C2211yescharacteristic
57C2×Q16×C107C2211yescharacteristic, complemented
58C2×SD16×C107C2211yescharacteristic, complemented
59C22⋊Q8×C107C2211yescharacteristic, complemented
60#103×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62C22⋊C8×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64C23×D107D411yescharacteristic
65#13133819—22no—
66#13133819—22no—
67#13133819—22no—
68#13133819D414yescomplemented
69#13133818—28no—
70#13133818—22no—
71#13133818—22no—
72#13133817—28no—
73#13133817—22no—
74#13133817D414yescomplemented
75C2×Dic214—21no—
76C2×Dic214C2311yescharacteristic
77Q8×D107—22no—
78Q8×D107—22no—
79C2×D428—21no—
80C2×D428—21no—
81C2×D428—21no—
82C2×D428C2311yescharacteristic
83C2×D428C2311yescharacteristic
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87C2×C22⋊D107—21no—
88D4×D107—28no—
89D4×D107—22no—
90D4×D107—22no—
91C22×Dic107—21no—
92C22×Dic107D411yescharacteristic
93C2×C4×D107—28no—
94C2×C4×D107—21no—
95C2×C4×D107—21no—
96C2×C4×D107—21no—
97C2×C4×D107—22no—
98C2×C4×D107—22no—
99C2×C4×D107—21no—
100C2×C4×D107D411yescharacteristic
101C2×C4×D107D411yescharacteristic
102C2×C4×D107C2311yescharacteristic
103C2×C4×D107C2311yescharacteristic
104#13133808—22no—
105#13133807—22no—
106#13133805—22no—
107#13133805—22no—
108D856—22no—
109#13133803—22no—
110C2×C107⋊C8C2311yescharacteristic
111C2×C107⋊C8C2311yescharacteristic
112C8×D107—22no—
113C8×D107—22no—
114C4⋊Dic107C2311yescharacteristic
115Dic107⋊C4—21no—
116(C2×D107)⋊C4—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×Q8×C107C2311yescharacteristic
125C2×D4×C107—21no—
126C2×D4×C107C2311yescharacteristic
127C22×C4×C107—21no—
128C22×C4×C107C2311yescharacteristic
129Q16×C107—22no—
130SD16×C107—22no—
131C2×C8×C107C2311yescharacteristic
132C2×C8×C107C2311yescharacteristic
133C4⋊C4×C107—21no—
134C4⋊C4×C107C2311yescharacteristic
135C22⋊C4×C107—21no—
136C23×C107—21no—
137C23×C107C2×D411yescharacteristic
138C22×D107—24no—
139C22×D107—21no—
140C22×D107—22no—
141C22×D107—22no—
142C22×D107—21no—
143C22×D107—41no—
144C22×D107—42no—
145C22×D107—42no—
146C22×D107C2×D411yescharacteristic
147C22×D107C2×D411yescharacteristic
148Q8×C107—21no—
149Q8×C107—21no—
150Q8×C107—21no—
151Q8×C107C2×D412yes—
152D4×C107—24no—
153D4×C107—21no—
154D4×C107C2×D412yes—
155C2×C4×C107—24no—
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—21no—
159C2×C4×C107—21no—
160C2×C4×C107—21no—
161C2×C4×C107C2×D411yescharacteristic
162C2×C4×C107C2411yescharacteristic
163C22⋊D107—24no—
164C22⋊D107—24no—
165C22⋊D107—24no—
166C22⋊D107—42no—
167C2×Dic107—21no—
168C2×Dic107—24no—
169C2×Dic107—22no—
170C2×Dic107—21no—
171C2×Dic107C2×D411yescharacteristic
172C2×Dic107C2×D411yescharacteristic
173D428—24no—
174D428—21no—
175D428—21no—
176D428—21no—
177D428—42no—
178D428—42no—
179D428C2×D412yes—
180C4×D107—24no—
181C4×D107—24no—
182C4×D107—22no—
183C4×D107—21no—
184C4×D107—21no—
185C4×D107—22no—
186C4×D107—24no—
187C4×D107—42no—
188C4×D107C2×D412yes—
189C4×D107C2×D412yes—
190Dic214—24no—
191Dic214—21no—
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×C107C22×D411yescharacteristic
204C2×D107—21no—
205C2×D107—21no—
206C2×D107—21no—
207C2×D107—21no—
208C2×D107—22no—
209C2×D107—44no—
210C2×D107—48no—
211C2×D107—41no—
212C2×D107—41no—
213C2×D107—41no—
214C2×D107—41no—
215C2×D107—41no—
216C2×D107#12012yes—
217C2×D107#13512yescomplemented
218C4×C107—22no—
219C4×C107—21no—
220C4×C107—41no—
221C4×C107—41no—
222C4×C107C22×D411yescharacteristic
223C4×C107C22×D411yescharacteristic
224Dic107—21no—
225Dic107—22no—
226Dic107—41no—
227Dic107#12012yes—
228C2×C107—41no—
229C2×C107—42no—
230C2×C107C2×#12011yescharacteristic
231C2×C107C2×#13511yescharacteristic
232C2×C107#57511yescharacteristic
233D107—22no—
234D107—22no—
235D107—42no—
236D107—81no—
237#3002—1071nomaximal, Sylow
238C107#300211yescharacteristic, complemented, Sylow
239C22×#41—1071no—
240C2×#135—1071no—
241C2×#135—1071no—
242#525—1071no—
243#452—1071no—
244#452D10711yescharacteristic, complemented
245#451—1071no—
246#450—1071no—
247#449—1071no—
248#449—1071no—
249#448—1071no—
250#447—1071no—
251#416—1071no—
252#416—1071no—
253C2×C22⋊C8—1071no—
254C2×#41—1071no—
255C2×#41—1071no—
256C2×#41—1071no—
257C2×#41—1071no—
258C2×#41—2141no—
259C2×#41—2141no—
260C2×#41—2144no—
261C2×#41—2144no—
262C2×#41C2×D10711yescharacteristic
263C22×Q8—1071no—
264C22×D4—1071no—
265C22×D4—2141no—
266C23×C4—1071no—
267C23×C4—2141no—
268#135—2144no—
269#135—2144no—
270C2×Q16—1071no—
271C2×Q16C2×D10711yescharacteristic, complemented
272C2×SD16—1071no—
273C2×SD16—1071no—
274C2×SD16—1071no—
275C2×SD16C2×D10711yescharacteristic, complemented
276C2×D8—1071no—
277C2×D8—1071no—
278C22×C8—1071no—
279C22×C8—1071no—
280#123—2141no—
281C22⋊Q8—1071no—
282C22⋊Q8C2×D10711yescharacteristic, complemented
283#121—1071no—
284#121—1071no—
285#120—2141no—
286C4×D4—2141no—
287C4×D4—2141no—
288#117—1071no—
289#103—1071no—
290#103—1071no—
291#103—1071no—
292#103C2×D10711yescharacteristic, complemented
293#102—1071no—
294#102—1071no—
295#102—1071no—
296#102C2×D10711yescharacteristic, complemented
297C22⋊C8—1071no—
298C22⋊C8—1071no—
299C22⋊C8—1071no—
300C22⋊C8C2×D10711yescharacteristic, complemented
301C24—1071no—
302C24—2141no—
303#41—22no—
304#41—22no—
305#41—1074no—
306#41—1074no—
307#41—2142no—
308#41—2142no—
309#41—2148no—
310#41—2142no—
311#41—2142no—
312#41—2142no—
313#41—2142no—
314#41—2148no—
315C2×Q8—1071no—
316C2×Q8—2141no—
317C2×Q8—2142no—
318C2×Q8—2142no—
319C2×Q8C22×D10711yescharacteristic
320C2×Q8C22×D10711yescharacteristic
321C2×D4—21no—
322C2×D4—1071no—
323C2×D4—1071no—
324C2×D4—2141no—
325C2×D4—2141no—
326C2×D4—2141no—
327C2×D4—2141no—
328C2×D4—2142no—
329C2×D4—2148no—
330C2×D4—2141no—
331C2×D4—2141no—
332C2×D4—2141no—
333C2×D4—2142no—
334C2×D4C22×D10711yescharacteristic
335C22×C4—21no—
336C22×C4—1071no—
337C22×C4—1071no—
338C22×C4—1071no—
339C22×C4—1071no—
340C22×C4—1071no—
341C22×C4—2141no—
342C22×C4—2142no—
343C22×C4—2142no—
344C22×C4—2141no—
345C22×C4—2141no—
346C22×C4—2148no—
347C22×C4—2141no—
348C22×C4—2141no—
349C22×C4C22×D10711yescharacteristic
350Q16—22no—
351Q16—2142no—
352SD16—22no—
353SD16—2142no—
354SD16—2142no—
355SD16—2142no—
356D8—2142no—
357D8—2142no—
358C2×C8—1071no—
359C2×C8—1071no—
360C2×C8—2142no—
361C2×C8—2142no—
362C2×C8C22×D10711yescharacteristic
363C2×C8C22×D10711yescharacteristic
364C4⋊C4—21no—
365C4⋊C4—1071no—
366C4⋊C4—2141no—
367C4⋊C4C22×D10711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—2141no—
370C22⋊C4—2141no—
371C22⋊C4—2141no—
372C22⋊C4—2141no—
373C42—2141no—
374C23—21no—
375C23—1071no—
376C23—1071no—
377C23—4281no—
378C23—4282no—
379C23—4282no—
380C23—2142no—
381C23—2142no—
382C23—2144no—
383C23—2141no—
384C23—2141no—
385C23D4×D10711yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—1072no—
390Q8—2144no—
391Q8—2141no—
392Q8D4×D10712yes—
393D4—24no—
394D4—21no—
395D4—1072no—
396D4—4282no—
397D4—4282no—
398D4—4282no—
399D4—2144no—
400D4—2144no—
401D4—2144no—
402D4—2144no—
403D4—2141no—
404D4—2141no—
405D4—2141no—
406D4D4×D10712yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—1071no—
414C2×C4—1072no—
415C2×C4—1072no—
416C2×C4—1071no—
417C2×C4—4282no—
418C2×C4—2142no—
419C2×C4—2144no—
420C2×C4—2142no—
421C2×C4—2144no—
422C2×C4—2144no—
423C2×C4—2142no—
424C2×C4—2144no—
425C2×C4—2141no—
426C2×C4—2141no—
427C2×C4—2141no—
428C2×C4—2141no—
429C2×C4D4×D10711yescharacteristic
430C2×C4C23×D10711yescharacteristic
431C8—21no—
432C8—21no—
433C8—2141no—
434C8—2141no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—1072no—
442C22—1072no—
443C22—4288no—
444C22—4284no—
445C22—4281no—
446C22—4281no—
447C22—4281no—
448C22—4281no—
449C22—4281no—
450C22—2142no—
451C22—2141no—
452C22—2141no—
453C22—2141no—
454C22—2141no—
455C22C2×D4×D10711yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—1072no—
461C4—4281no—
462C4—2141no—
463C4—2142no—
464C4C2×D4×D10711yescharacteristic
465C4C2×D4×D10711yescharacteristic
466C2—41no—
467C2—42no—
468C2—8561no—
469C2—4282no—
470C2—2142no—
471C2—2142no—
472C2#120×D10711yescharacteristic
473C2#135×D10711yescharacteristic
474C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#452×C107C2
socleC22×C107C22×D4
hypercenter#452D107
hypocenterC107#3002
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#41×D107C2semidirect
2. C2×#16812132C2semidirect
3. C2×#16812130C2semidirect
4. #20293140C2semidirect
5. #20292787C2semidirect
6. #20292781C2semidirect
7. #20292776C2semidirect
8. #20292774C2semidirect
9. #20292772C2semidirect
10. #20292767C2semidirect
11. #20292762C2semidirect
12. #20292622C2semidirect
13. #20292620C2semidirect
14. C22⋊C8×D107C2semidirect
15. #452×C107C2semidirect
17. C2×#13133819C22semidirect
31. C2×#13133808C22semidirect
32. C2×#13133807C22semidirect
33. C2×#13133805C22semidirect
34. C2×#13133805C22semidirect
35. C2×D856C22semidirect
36. C2×#13133803C22semidirect
37. C2×C8×D107C22semidirect
38. C2×C8×D107C22semidirect
42. #16812083C22semidirect
46. #16812067C22semidirect
47. #16812034C22semidirect
48. #16812033C22semidirect
49. #16812032C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. C22⋊(C107⋊C8)C22semidirect
57. C2×Q16×C107C22semidirect
58. C2×SD16×C107C22semidirect
59. C22⋊Q8×C107C22semidirect
60. #103×C107C22semidirect
61. #102×C107C22semidirect
62. C22⋊C8×C107C22semidirect
68. #13133819D4semidirect
74. #13133817D4semidirect
217. C2×D107#135semidirect
238. C107#3002semidirect
244. #452D107semidirect
271. C2×Q16C2×D107semidirect
275. C2×SD16C2×D107semidirect
282. C22⋊Q8C2×D107semidirect
292. #103C2×D107semidirect
296. #102C2×D107semidirect
300. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1072
C2×#168121322
C2×#168121302
#202931402
#202927872
#202927812
#202927762
#202927742
#202927722
#202927672
#202927622
#202926222
#202926202
C22⋊C8×D1072
#452×C1072
#3002107