Finite Group Databasebeta

← C2×C13×#2196 #15719882

Subgroups — C2×C13×#2196 #15719882

Order 3328.

1956 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42×C13C211yesmaximal, characteristic, complemented
2C2×C13×B(4,𝔽2)C212yesmaximal, complemented
3C2×C13×#453C211yesmaximal, characteristic, complemented
4C13×#2196C218yesmaximal, direct product factor
5C2×C13×#353C211yesmaximal, characteristic, complemented
6C2×C13×AΣL(1,𝔽16)C212yesmaximal, complemented
7C2×C13×H(5,𝔽2)C2211yescharacteristic, complemented
8C23×D4×C13—22no—
9C2×C13×AGL(1,ℤ/8)—21no—
10D42×C13—22no—
11D42×C13—24no—
12D42×C13C2214yescomplemented
13C2×C13×#127C2211yescharacteristic
14C2×C13×#121—22no—
15C2×C13×#120—21no—
16C2×C13×#120—22no—
17C2×C13×#120C2212yes—
18C2×C4×D4×C13—21no—
19C13×B(4,𝔽2)—24no—
20C13×B(4,𝔽2)C2218yescomplemented
21C13×#453—22no—
22C13×#453C2214yescomplemented
23C2×C13×#104—21no—
24C2×C13×#100C2211yescharacteristic, complemented
25C2×C13×#99—22no—
26C2×C13×#99C2212yescomplemented
27C13×#353C2214yescomplemented
28C13×AΣL(1,𝔽16)C2218yescomplemented
29C25×C13—21no—
30C25×C13D412yescomplemented
31C13×H(5,𝔽2)—22no—
32C13×H(5,𝔽2)—21no—
33C13×H(5,𝔽2)—42no—
34C13×H(5,𝔽2)C2312yescomplemented
35C2×C13×#41—21no—
36C2×C13×#41—41no—
37C22×D4×C13—21no—
38C22×D4×C13—22no—
39C22×D4×C13—28no—
40C22×D4×C13—22no—
41C22×D4×C13—22no—
42C22×D4×C13—22no—
43C22×D4×C13—28no—
44C22×D4×C13—41no—
45C22×D4×C13—416no—
46C22×D4×C13—42no—
47C22×D4×C13D411yescharacteristic, complemented
48C22×D4×C13C2311yescharacteristic
49C23×C4×C13—22no—
50C13×AGL(1,ℤ/8)—24no—
51C13×AGL(1,ℤ/8)—42no—
52C2×C13×SD16—41no—
53C2×C13×D8—41no—
54C2×C13×M16—21no—
55C13×#127—22no—
56C13×#127D412yescomplemented
57C13×#127C2312yes—
58C13×#121—28no—
59C13×#121—28no—
60C13×#120—24no—
61C13×#120—22no—
62C13×#120—22no—
63C13×#120—24no—
64C13×#120—28no—
65C13×#120—44no—
66C13×#120D414yescomplemented
67C13×#120C2314yes—
68C4×D4×C13—24no—
69C4×D4×C13—24no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C22⋊C4—21no—
72C2×C13×C22⋊C4—22no—
73C2×C13×C22⋊C4—22no—
74C2×C13×C22⋊C4—42no—
75C2×C13×C22⋊C4D412yescomplemented
76C2×C42×C13D411yescharacteristic, complemented
77C13×#104—24no—
78C13×#100—21no—
79C13×#100C2312yescomplemented
80C13×#99—22no—
81C13×#99—28no—
82C13×#99C2314yescomplemented
83C2×#2196C1311yesmaximal, characteristic, direct product factor, Sylow
84C24×C13—22no—
85C24×C13—24no—
86C24×C13—22no—
87C24×C13—21no—
88C24×C13—22no—
89C24×C13—42no—
90C24×C13—48no—
91C24×C13—48no—
92C24×C13—42no—
93C24×C13—81no—
94C24×C13—81no—
95C24×C13C2×D412yes—
96C24×C13C2×D414yescomplemented
97C13×#41—22no—
98C13×#41—22no—
99C13×#41—42no—
100C13×#41—41no—
101C13×#41—44no—
102C13×#41—42no—
103C13×#41—41no—
104C2×Q8×C13—21no—
105C2×D4×C13—24no—
106C2×D4×C13—28no—
107C2×D4×C13—21no—
108C2×D4×C13—28no—
109C2×D4×C13—24no—
110C2×D4×C13—24no—
111C2×D4×C13—24no—
112C2×D4×C13—24no—
113C2×D4×C13—24no—
114C2×D4×C13—24no—
115C2×D4×C13—21no—
116C2×D4×C13—41no—
117C2×D4×C13—41no—
118C2×D4×C13—48no—
119C2×D4×C13—41no—
120C2×D4×C13—41no—
121C2×D4×C13—42no—
122C2×D4×C13—432no—
123C2×D4×C13—432no—
124C2×D4×C13—42no—
125C2×D4×C13—48no—
126C2×D4×C13—48no—
127C2×D4×C13—42no—
128C2×D4×C13—42no—
129C2×D4×C13—44no—
130C2×D4×C13—42no—
131C2×D4×C13—41no—
132C2×D4×C13—48no—
133C2×D4×C13—88no—
134C2×D4×C13C2×D412yes—
135C2×D4×C13C2×D411yescharacteristic
136C2×D4×C13C2×D412yescomplemented
137C2×D4×C13C2411yescharacteristic
138C22×C4×C13—21no—
139C22×C4×C13—22no—
140C22×C4×C13—28no—
141C22×C4×C13—22no—
142C22×C4×C13—48no—
143C22×C4×C13—41no—
144C22×C4×C13—42no—
145C22×C4×C13C2×D411yescharacteristic
146C13×SD16—44no—
147C13×D8—44no—
148C13×M16—22no—
149C13×M16—41no—
150C2×C8×C13—41no—
151C13×C4⋊C4—24no—
152C13×C22⋊C4—24no—
153C13×C22⋊C4—22no—
154C13×C22⋊C4—24no—
155C13×C22⋊C4—24no—
156C13×C22⋊C4—24no—
157C13×C22⋊C4—48no—
158C13×C22⋊C4—42no—
159C13×C22⋊C4C2×D414yescomplemented
160C42×C13—21no—
161C42×C13C2×D412yescomplemented
162C2×D42C2×C1311yescharacteristic, complemented
163C2×B(4,𝔽2)C2×C1312yescomplemented
164C2×#453C2×C1311yescharacteristic, complemented
165#2196C2×C1318yesdirect product factor
166C2×#353C2×C1311yescharacteristic, complemented
167C2×AΣL(1,𝔽16)C2×C1312yescomplemented
168C23×C13—28no—
169C23×C13—24no—
170C23×C13—22no—
171C23×C13—22no—
172C23×C13—24no—
173C23×C13—22no—
174C23×C13—44no—
175C23×C13—44no—
176C23×C13—44no—
177C23×C13—48no—
178C23×C13—44no—
179C23×C13—48no—
180C23×C13—416no—
181C23×C13—42no—
182C23×C13—42no—
183C23×C13—42no—
184C23×C13—48no—
185C23×C13—41no—
186C23×C13—41no—
187C23×C13—41no—
188C23×C13—82no—
189C23×C13—84no—
190C23×C13—88no—
191C23×C13—84no—
192C23×C13—88no—
193C23×C13—88no—
194C23×C13—84no—
195C23×C13#12012yes—
196C23×C13#12011yescharacteristic
197C23×C13C22×D412yes—
198Q8×C13—22no—
199Q8×C13—41no—
200D4×C13—216no—
201D4×C13—22no—
202D4×C13—41no—
203D4×C13—42no—
204D4×C13—41no—
205D4×C13—48no—
206D4×C13—48no—
207D4×C13—41no—
208D4×C13—42no—
209D4×C13—48no—
210D4×C13—48no—
211D4×C13—416no—
212D4×C13—44no—
213D4×C13—44no—
214D4×C13—42no—
215D4×C13—42no—
216D4×C13—81no—
217D4×C13—816no—
218C2×C4×C13—24no—
219C2×C4×C13—22no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—24no—
223C2×C4×C13—24no—
224C2×C4×C13—416no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—44no—
228C2×C4×C13—42no—
229C2×C4×C13—41no—
230C2×C4×C13—44no—
231C2×C4×C13—84no—
232C2×C4×C13#12011yescharacteristic
233C2×C4×C13C22×D411yescharacteristic
234C8×C13—42no—
235C2×H(5,𝔽2)C22×C1311yescharacteristic, complemented
236C23×D4—22no—
237C2×AGL(1,ℤ/8)—21no—
238D42—22no—
239D42—24no—
240D42C22×C1314yescomplemented
241C2×#127C22×C1311yescharacteristic
242C2×#121—22no—
243C2×#120—21no—
244C2×#120—22no—
245C2×#120C22×C1312yes—
246C2×C4×D4—21no—
247B(4,𝔽2)—24no—
248B(4,𝔽2)C22×C1318yescomplemented
249#453—22no—
250#453C22×C1314yescomplemented
251C2×#104—21no—
252C2×#100C22×C1311yescharacteristic, complemented
253C2×#99—22no—
254C2×#99C22×C1312yescomplemented
255#353C22×C1314yescomplemented
256AΣL(1,𝔽16)C22×C1318yescomplemented
257C22×C13—22no—
258C22×C13—28no—
259C22×C13—22no—
260C22×C13—22no—
261C22×C13—42no—
262C22×C13—42no—
263C22×C13—44no—
264C22×C13—48no—
265C22×C13—44no—
266C22×C13—48no—
267C22×C13—44no—
268C22×C13—48no—
269C22×C13—44no—
270C22×C13—42no—
271C22×C13—42no—
272C22×C13—82no—
273C22×C13—82no—
274C22×C13—84no—
275C22×C13—88no—
276C22×C13—816no—
277C22×C13—84no—
278C22×C13—82no—
279C22×C13—82no—
280C22×C13—162no—
281C22×C13B(4,𝔽2)11yescharacteristic
282C22×C13B(4,𝔽2)11yescharacteristic
283C22×C13C2×#12011yescharacteristic
284C4×C13—24no—
285C4×C13—21no—
286C4×C13—21no—
287C4×C13—44no—
288C4×C13—42no—
289C4×C13—84no—
290C25—21no—
291C25D4×C1312yescomplemented
292H(5,𝔽2)—22no—
293H(5,𝔽2)—21no—
294H(5,𝔽2)—42no—
295H(5,𝔽2)C23×C1312yescomplemented
296C2×#41—21no—
297C2×#41—41no—
298C22×D4—21no—
299C22×D4—22no—
300C22×D4—28no—
301C22×D4—22no—
302C22×D4—22no—
303C22×D4—22no—
304C22×D4—28no—
305C22×D4—41no—
306C22×D4—416no—
307C22×D4—42no—
308C22×D4D4×C1311yescharacteristic, complemented
309C22×D4C23×C1311yescharacteristic
310C23×C4—22no—
311AGL(1,ℤ/8)—24no—
312AGL(1,ℤ/8)—42no—
313C2×SD16—41no—
314C2×D8—41no—
315C2×M16—21no—
316#127—22no—
317#127D4×C1312yescomplemented
318#127C23×C1312yes—
319#121—28no—
320#121—28no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—24no—
325#120—28no—
326#120—44no—
327#120D4×C1314yescomplemented
328#120C23×C1314yes—
329C4×D4—24no—
330C4×D4—24no—
331C2×C4⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4D4×C1312yescomplemented
337C2×C42D4×C1311yescharacteristic, complemented
338#104—24no—
339#100—21no—
340#100C23×C1312yescomplemented
341#99—22no—
342#99—28no—
343#99C23×C1314yescomplemented
344C2×C13—21no—
345C2×C13—21no—
346C2×C13—42no—
347C2×C13—48no—
348C2×C13—42no—
349C2×C13—82no—
350C2×C13—82no—
351C2×C13#219612yesdirect product factor
352C2×C13C2×B(4,𝔽2)11yescharacteristic
353C24—22no—
354C24—24no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—42no—
359C24—48no—
360C24—48no—
361C24—42no—
362C24—81no—
363C24—81no—
364C24C2×D4×C1312yes—
365C24C2×D4×C1314yescomplemented
366#41—22no—
367#41—22no—
368#41—42no—
369#41—41no—
370#41—44no—
371#41—42no—
372#41—41no—
373C2×Q8—21no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—21no—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—48no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—42no—
391C2×D4—432no—
392C2×D4—432no—
393C2×D4—42no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—48no—
402C2×D4—88no—
403C2×D4C2×D4×C1312yes—
404C2×D4C2×D4×C1311yescharacteristic
405C2×D4C2×D4×C1312yescomplemented
406C2×D4C24×C1311yescharacteristic
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—48no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4C2×D4×C1311yescharacteristic
415SD16—44no—
416D8—44no—
417M16—22no—
418M16—41no—
419C2×C8—41no—
420C4⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—48no—
427C22⋊C4—42no—
428C22⋊C4C2×D4×C1314yescomplemented
429C42—21no—
430C42C2×D4×C1312yescomplemented
431C13C2×#219611yescharacteristic, direct product factor, Sylow
432C23—28no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—48no—
442C23—44no—
443C23—48no—
444C23—416no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—82no—
453C23—84no—
454C23—88no—
455C23—84no—
456C23—88no—
457C23—88no—
458C23—84no—
459C23C13×#12011yescharacteristic
460C23C13×#12012yes—
461C23C22×D4×C1312yes—
462Q8—22no—
463Q8—41no—
464D4—216no—
465D4—22no—
466D4—41no—
467D4—42no—
468D4—41no—
469D4—42no—
470D4—48no—
471D4—48no—
472D4—41no—
473D4—48no—
474D4—48no—
475D4—416no—
476D4—44no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4—81no—
481D4—816no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—416no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—84no—
496C2×C4C13×#12011yescharacteristic
497C2×C4C22×D4×C1311yescharacteristic
498C8—42no—
499C22—22no—
500C22—28no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—48no—
507C22—44no—
508C22—48no—
509C22—44no—
510C22—44no—
511C22—48no—
512C22—42no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22—88no—
518C22—816no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—162no—
523C22C13×B(4,𝔽2)11yescharacteristic
524C22C13×B(4,𝔽2)11yescharacteristic
525C22C2×C13×#12011yescharacteristic
526C4—24no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—84no—
532C2—21no—
533C2—21no—
534C2—42no—
535C2—48no—
536C2—42no—
537C2—82no—
538C2—82no—
539C2C13×#219612yesdirect product factor
540C2C2×C13×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C13B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C13
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C13 ◁ C23×C13 ◁ C22×D4×C13 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×D42×C13C2semidirect
2. C2×C13×B(4,𝔽2)C2semidirect
3. C2×C13×#453C2semidirect
4. C13×#2196C2direct (N × H)
5. C2×C13×#353C2semidirect
6. C2×C13×AΣL(1,𝔽16)C2semidirect
7. C2×C13×H(5,𝔽2)C22semidirect
12. D42×C13C22semidirect
20. C13×B(4,𝔽2)C22semidirect
22. C13×#453C22semidirect
24. C2×C13×#100C22semidirect
26. C2×C13×#99C22semidirect
27. C13×#353C22semidirect
28. C13×AΣL(1,𝔽16)C22semidirect
30. C25×C13D4semidirect
34. C13×H(5,𝔽2)C23semidirect
47. C22×D4×C13D4semidirect
56. C13×#127D4semidirect
66. C13×#120D4semidirect
75. C2×C13×C22⋊C4D4semidirect
76. C2×C42×C13D4semidirect
79. C13×#100C23semidirect
82. C13×#99C23semidirect
83. C2×#2196C13direct (N × H)
96. C24×C13C2×D4semidirect
136. C2×D4×C13C2×D4semidirect
159. C13×C22⋊C4C2×D4semidirect
161. C42×C13C2×D4semidirect
162. C2×D42C2×C13semidirect
163. C2×B(4,𝔽2)C2×C13semidirect
164. C2×#453C2×C13semidirect
165. #2196C2×C13direct (N × H)
166. C2×#353C2×C13semidirect
167. C2×AΣL(1,𝔽16)C2×C13semidirect
235. C2×H(5,𝔽2)C22×C13semidirect
240. D42C22×C13semidirect
248. B(4,𝔽2)C22×C13semidirect
250. #453C22×C13semidirect
252. C2×#100C22×C13semidirect
254. C2×#99C22×C13semidirect
255. #353C22×C13semidirect
256. AΣL(1,𝔽16)C22×C13semidirect
291. C25D4×C13semidirect
295. H(5,𝔽2)C23×C13semidirect
308. C22×D4D4×C13semidirect
317. #127D4×C13semidirect
327. #120D4×C13semidirect
336. C2×C22⋊C4D4×C13semidirect
337. C2×C42D4×C13semidirect
340. #100C23×C13semidirect
343. #99C23×C13semidirect
351. C2×C13#2196direct (N × H)
365. C24C2×D4×C13semidirect
405. C2×D4C2×D4×C13semidirect
428. C22⋊C4C2×D4×C13semidirect
430. C42C2×D4×C13semidirect
431. C13C2×#2196direct (N × H)
539. C2C13×#2196direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D42×C132
C2×C13×B(4,𝔽2)2
C2×C13×#4532
C13×#21962
C2×C13×#3532
C2×C13×AΣL(1,𝔽16)2
C2×#219613