Finite Group Databasebeta

← #28345733

Subgroups — #28345733

Order 13696.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812154C211yesmaximal, characteristic
2C2×#16812138C211yesmaximal, characteristic, complemented
3C2×SD16×D107C211yesmaximal, characteristic, complemented
4#20293200C211yesmaximal, characteristic, complemented
5#20292791C211yesmaximal, characteristic, complemented
6#20292790C211yesmaximal, characteristic, complemented
7#20292787C211yesmaximal, characteristic, complemented
8#20292786C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic, complemented
10#20292778C211yesmaximal, characteristic, complemented
11#20292776C211yesmaximal, characteristic, complemented
12#20292630C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14#20292592C211yesmaximal, characteristic, complemented
15#450×C107C211yesmaximal, characteristic, complemented
16#16812154—24no—
17#16812154—24no—
18C2×#13133818C2211yescharacteristic
19C2×#13133818C2211yescharacteristic
20C2×#13133817—21no—
21C2×#13133817C2211yescharacteristic
22C2×#13133817C2211yescharacteristic
23C2×Q8×D107C2211yescharacteristic, complemented
24C22×C22⋊D107—21no—
25C22×C22⋊D107—21no—
26C2×D4×D107—21no—
27C2×D4×D107C2211yescharacteristic
28C2×D4×D107C2211yescharacteristic
29#16812138—24no—
30C2×Dic428C2211yescharacteristic, complemented
31C2×#13133808C2211yescharacteristic, complemented
32C2×#13133807C2211yescharacteristic, complemented
33C2×#13133806C2211yescharacteristic, complemented
34C2×#13133806C2211yescharacteristic, complemented
35C2×#13133805C2211yescharacteristic, complemented
36SD16×D107—24no—
37C2×#13133801C2211yescharacteristic, complemented
38C2×C8×D107C2211yescharacteristic, complemented
39#16812108—21no—
40#16812095—21no—
41#16812094—21no—
42#16812091—21no—
43#16812089C2211yescharacteristic, complemented
44#16812088C2211yescharacteristic, complemented
45#16812085C2211yescharacteristic, complemented
46#16812067C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53C22⋊(C107⋊C8)C2211yescharacteristic, complemented
54(C2×D107)⋊C8C2211yescharacteristic, complemented
55(C2×D107)⋊C8C2211yescharacteristic, complemented
56C22×D4×C107C2211yescharacteristic
57C2×SD16×C107C2211yescharacteristic, complemented
58C2×SD16×C107C2211yescharacteristic, complemented
59C22⋊Q8×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62C22⋊C8×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64C23×D107—21no—
65#13133818—22no—
66#13133818—22no—
67#13133818—22no—
68#13133818—22no—
69#13133817—28no—
70#13133817—22no—
71#13133817—22no—
72#13133817—22no—
73#13133817D414yes—
74C2×Dic214—21no—
75C2×Dic214D411yescharacteristic
76C2×Dic214C2311yescharacteristic
77C2×Dic214C2311yescharacteristic
78Q8×D107—22no—
79Q8×D107—22no—
80C2×D428D411yescharacteristic
81C2×D428C2311yescharacteristic
82C2×C22⋊D107—28no—
83C2×C22⋊D107—28no—
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87C2×C22⋊D107—21no—
88C2×C22⋊D107—21no—
89C2×C22⋊D107D411yescharacteristic
90C2×C22⋊D107D411yescharacteristic
91D4×D107—28no—
92D4×D107—22no—
93D4×D107—22no—
94D4×D107—22no—
95D4×D107D414yes—
96C22×Dic107—21no—
97C22×Dic107—21no—
98C2×C4×D107—21no—
99C2×C4×D107—21no—
100C2×C4×D107C2311yescharacteristic
101C2×C4×D107C2311yescharacteristic
102Dic428—22no—
103#13133808—22no—
104#13133807—22no—
105#13133806—22no—
106#13133806—22no—
107#13133805—22no—
108#13133801—22no—
109C2×C107⋊C8C2311yescharacteristic
110C2×C107⋊C8C2311yescharacteristic
111C8×D107—22no—
112C4⋊Dic107—21no—
113C4⋊Dic107C2311yescharacteristic
114Dic107⋊C4—21no—
115Dic107⋊C4—21no—
116C22⋊Dic107—21no—
117(C2×D107)⋊C4—21no—
118(C2×D107)⋊C4—21no—
119(C2×D107)⋊C4—21no—
120C4×Dic107—21no—
121C24×C107—21no—
122C2×Q8×C107C2311yescharacteristic
123C2×D4×C107—22no—
124C2×D4×C107—22no—
125C2×D4×C107—21no—
126C2×D4×C107C2311yescharacteristic
127C2×D4×C107C2311yescharacteristic
128C22×C4×C107C2311yescharacteristic
129SD16×C107—22no—
130SD16×C107—22no—
131C2×C8×C107C2311yescharacteristic
132C2×C8×C107C2311yescharacteristic
133C4⋊C4×C107—21no—
134C4⋊C4×C107C2311yescharacteristic
135C22⋊C4×C107—21no—
136C23×C107—24no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107—81no—
140C23×C107—81no—
141C23×C107C2×D411yescharacteristic
142C22×D107—21no—
143C22×D107—24no—
144C22×D107—24no—
145C22×D107—81no—
146C22×D107—81no—
147C22×D107—81no—
148C22×D107—81no—
149C22×D107C2×D411yescharacteristic
150C22×D107C2×D411yescharacteristic
151Q8×C107—21no—
152Q8×C107—21no—
153D4×C107—24no—
154D4×C107—21no—
155D4×C107—21no—
156D4×C107C2×D412yes—
157D4×C107C2×D412yes—
158C2×C4×C107—21no—
159C2×C4×C107—21no—
160C2×C4×C107—21no—
161C2×C4×C107—21no—
162C2×C4×C107C2×D411yescharacteristic
163C2×C4×C107C2411yescharacteristic
164C22⋊D107—24no—
165C22⋊D107—24no—
166C22⋊D107—24no—
167C22⋊D107—24no—
168C22⋊D107—22no—
169C22⋊D107—24no—
170C22⋊D107—22no—
171C2×Dic107—24no—
172C2×Dic107—21no—
173C2×Dic107—24no—
174C2×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107C2×D411yescharacteristic
177C2×Dic107C2×D411yescharacteristic
178D428—22no—
179D428—21no—
180D428C2×D412yes—
181C4×D107—24no—
182C4×D107—21no—
183C4×D107—21no—
184C4×D107—42no—
185C4×D107C2×D412yes—
186C4×D107C2×D412yes—
187Dic214—21no—
188Dic214—22no—
189Dic214—21no—
190Dic214—21no—
191Dic214—42no—
192Dic214C2×D412yes—
193C8×C107—21no—
194C8×C107—21no—
195C107⋊C8—21no—
196C107⋊C8—21no—
197C22×C107—22no—
198C22×C107—22no—
199C22×C107—21no—
200C22×C107—21no—
201C22×C107—21no—
202C22×C107—41no—
203C22×C107—41no—
204C22×C107—41no—
205C22×C107—41no—
206C22×C107—84no—
207C22×C107C22×D411yescharacteristic
208C2×D107—21no—
209C2×D107—21no—
210C2×D107—21no—
211C2×D107—22no—
212C2×D107—21no—
213C2×D107—21no—
214C2×D107—41no—
215C2×D107—41no—
216C2×D107—84no—
217C2×D107—84no—
218C2×D107#12012yes—
219C4×C107—21no—
220C4×C107—41no—
221C4×C107—41no—
222C4×C107C22×D411yescharacteristic
223C4×C107C22×D411yescharacteristic
224Dic107—21no—
225Dic107—22no—
226Dic107—41no—
227Dic107—41no—
228Dic107#12012yes—
229C2×C107—42no—
230C2×C107—42no—
231C2×C107—41no—
232C2×C107C2×#12011yescharacteristic
233C2×C107#57711yescharacteristic
234C2×C107#57711yescharacteristic
235D107—22no—
236D107—41no—
237D107—42no—
238#3009—1071nomaximal, Sylow
239C107#300911yescharacteristic, complemented, Sylow
240C2×H(5,𝔽2)—1071no—
241C2×#137—1071no—
242C22×SD16—1071no—
243#536—1071no—
244#452—1071no—
245#452—1071no—
246#452—1071no—
247#452—1071no—
248#450—1071no—
249#450—1071no—
250#450—1071no—
251#450D10711yescharacteristic, complemented
252#418—1071no—
253#416—1071no—
254#408—1071no—
255H(5,𝔽2)—2144no—
256H(5,𝔽2)—2144no—
257C2×#41—1071no—
258C2×#41—1071no—
259C2×#41—1071no—
260C2×#41—1071no—
261C2×#41—2141no—
262C22×Q8—1071no—
263C22×D4—1071no—
264C22×D4—1071no—
265C22×D4—2141no—
266C22×D4—2141no—
267C22×D4—2141no—
268C22×D4C2×D10711yescharacteristic
269#137—2144no—
270C2×Q16—1071no—
271C2×Q16—1071no—
272C2×SD16—1071no—
273C2×SD16—1071no—
274C2×SD16—1071no—
275C2×SD16—1071no—
276C2×SD16—2144no—
277C2×SD16C2×D10711yescharacteristic, complemented
278C2×SD16C2×D10711yescharacteristic, complemented
279C2×M16—1071no—
280C22×C8—1071no—
281C4⋊Q8—2141no—
282#124—2141no—
283#123—2141no—
284#123—2141no—
285C22⋊Q8—1071no—
286C22⋊Q8—1071no—
287C22⋊Q8—1071no—
288C22⋊Q8C2×D10711yescharacteristic, complemented
289#117—1071no—
290#103—1071no—
291#103—1071no—
292#102—1071no—
293#102—1071no—
294#102—1071no—
295#102—1071no—
296#102C2×D10711yescharacteristic, complemented
297#102C2×D10711yescharacteristic, complemented
298C22⋊C8—1071no—
299C22⋊C8—1071no—
300C22⋊C8—1071no—
301C22⋊C8C2×D10711yescharacteristic, complemented
302C24—21no—
303C24—2141no—
304C24—2141no—
305#41—1074no—
306#41—2142no—
307#41—2142no—
308#41—2142no—
309#41—2142no—
310#41—2142no—
311#41—2142no—
312#41—2148no—
313#41—2142no—
314C2×Q8—1071no—
315C2×Q8—1071no—
316C2×Q8—1071no—
317C2×Q8—2141no—
318C2×Q8—2142no—
319C2×Q8—2142no—
320C2×Q8C22×D10711yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—1074no—
325C2×D4—1071no—
326C2×D4—1071no—
327C2×D4—1071no—
328C2×D4—1071no—
329C2×D4—2142no—
330C2×D4—2141no—
331C2×D4—2141no—
332C2×D4—2142no—
333C2×D4—2148no—
334C2×D4—2141no—
335C2×D4—2141no—
336C2×D4—2148no—
337C2×D4—2148no—
338C2×D4—2141no—
339C2×D4—2142no—
340C2×D4C22×D10711yescharacteristic
341C2×D4C22×D10711yescharacteristic
342C22×C4—1071no—
343C22×C4—1071no—
344C22×C4—2141no—
345C22×C4—2141no—
346C22×C4—2141no—
347C22×C4—2141no—
348C22×C4C22×D10711yescharacteristic
349Q16—2142no—
350Q16—2142no—
351SD16—22no—
352SD16—22no—
353SD16—2142no—
354SD16—2142no—
355SD16—2142no—
356SD16—2142no—
357M16—2142no—
358C2×C8—1071no—
359C2×C8—1071no—
360C2×C8—2142no—
361C2×C8C22×D10711yescharacteristic
362C2×C8C22×D10711yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—1071no—
365C4⋊C4—2141no—
366C4⋊C4—2141no—
367C4⋊C4—2141no—
368C4⋊C4C22×D10711yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—2141no—
371C22⋊C4—2141no—
372C22⋊C4—2141no—
373C22⋊C4—2141no—
374C42—2141no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—8561no—
381C23—8561no—
382C23—8561no—
383C23—8561no—
384C23—1071no—
385C23—1071no—
386C23—2144no—
387C23—2144no—
388C23—2141no—
389C23D4×D10711yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—1072no—
393Q8—4282no—
394Q8—2142no—
395Q8—2141no—
396Q8—2141no—
397Q8—2141no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—1072no—
402D4—2142no—
403D4—2142no—
404D4—2144no—
405D4—2142no—
406D4—2144no—
407D4—2144no—
408D4—2144no—
409D4—2144no—
410D4—2141no—
411D4D4×D10712yes—
412D4D4×D10712yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—1071no—
418C2×C4—1072no—
419C2×C4—1072no—
420C2×C4—1071no—
421C2×C4—4282no—
422C2×C4—2144no—
423C2×C4—2144no—
424C2×C4—2141no—
425C2×C4—2141no—
426C2×C4—2144no—
427C2×C4—2141no—
428C2×C4—2141no—
429C2×C4—2141no—
430C2×C4D4×D10711yescharacteristic
431C2×C4C23×D10711yescharacteristic
432C8—21no—
433C8—21no—
434C8—2141no—
435C8—2141no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—8564no—
447C22—8564no—
448C22—1072no—
449C22—4281no—
450C22—4281no—
451C22—2141no—
452C22—2141no—
453C22—2142no—
454C22—2141no—
455C22—2141no—
456C22—2141no—
457C22C2×D4×D10711yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—1072no—
462C4—4281no—
463C4—4281no—
464C4—2141no—
465C4—2142no—
466C4C2×D4×D10711yescharacteristic
467C4C2×D4×D10711yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—4282no—
472C2—4281no—
473C2—2142no—
474C2#120×D10711yescharacteristic
475C2#2029343311yescharacteristic
476C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812154C2
hypercenter#450D107
hypocenterC107#3009
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#16812138C2semidirect
3. C2×SD16×D107C2semidirect
4. #20293200C2semidirect
5. #20292791C2semidirect
6. #20292790C2semidirect
7. #20292787C2semidirect
8. #20292786C2semidirect
9. #20292779C2semidirect
10. #20292778C2semidirect
11. #20292776C2semidirect
12. #20292630C2semidirect
13. #20292620C2semidirect
14. #20292592C2semidirect
15. #450×C107C2semidirect
23. C2×Q8×D107C22semidirect
30. C2×Dic428C22semidirect
31. C2×#13133808C22semidirect
32. C2×#13133807C22semidirect
33. C2×#13133806C22semidirect
34. C2×#13133806C22semidirect
35. C2×#13133805C22semidirect
37. C2×#13133801C22semidirect
38. C2×C8×D107C22semidirect
43. #16812089C22semidirect
44. #16812088C22semidirect
45. #16812085C22semidirect
46. #16812067C22semidirect
47. #16812034C22semidirect
48. #16812032C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. C22⋊(C107⋊C8)C22semidirect
54. (C2×D107)⋊C8C22semidirect
55. (C2×D107)⋊C8C22semidirect
57. C2×SD16×C107C22semidirect
58. C2×SD16×C107C22semidirect
59. C22⋊Q8×C107C22semidirect
60. #102×C107C22semidirect
61. #102×C107C22semidirect
62. C22⋊C8×C107C22semidirect
239. C107#3009semidirect
251. #450D107semidirect
277. C2×SD16C2×D107semidirect
278. C2×SD16C2×D107semidirect
288. C22⋊Q8C2×D107semidirect
296. #102C2×D107semidirect
297. #102C2×D107semidirect
301. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
C2×#168121382
C2×SD16×D1072
#202932002
#202927912
#202927902
#202927872
#202927862
#202927792
#202927782
#202927762
#202926302
#202926202
#202925922
#450×C1072
#3009107