Finite Group Databasebeta

← C2×C103×#2199 #39938822

Subgroups — C2×C103×#2199 #39938822

Order 26368.

1392 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1392 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×H(3,𝔽4)×C103C211yesmaximal, characteristic, complemented
2C2×B(4,𝔽2)×C103C212yesmaximal, complemented
3C2×#455×C103C211yesmaximal, characteristic, complemented
4C103×#2199C218yesmaximal, direct product factor
5C2×#354×C103C211yesmaximal, characteristic, complemented
6C2×AΣL(1,𝔽16)×C103C212yesmaximal, complemented
7C2×H(5,𝔽2)×C103C2211yescharacteristic
8C2×#137×C103—21no—
9H(3,𝔽4)×C103—22no—
10H(3,𝔽4)×C103—24no—
11H(3,𝔽4)×C103C2214yescomplemented
12C2×#124×C103—21no—
13C2×#124×C103—22no—
14C2×#124×C103—21no—
15C2×#124×C103C2211yescharacteristic
16C2×#120×C103—22no—
17C2×#120×C103—22no—
18C2×#120×C103C2212yes—
19B(4,𝔽2)×C103—24no—
20B(4,𝔽2)×C103C2218yescomplemented
21#455×C103—22no—
22#455×C103C2214yescomplemented
23C2×#104×C103—21no—
24C2×#100×C103C2211yescharacteristic
25C2×#99×C103—22no—
26C2×#99×C103C2212yes—
27#354×C103C2214yescomplemented
28AΣL(1,𝔽16)×C103C2218yescomplemented
29C25×C103D412yescomplemented
30H(5,𝔽2)×C103—22no—
31H(5,𝔽2)×C103—21no—
32H(5,𝔽2)×C103—42no—
33H(5,𝔽2)×C103C2312yes—
34C2×#41×C103—21no—
35C2×#41×C103—41no—
36C22×Q8×C103—21no—
37C22×Q8×C103D411yescharacteristic, complemented
38C22×D4×C103—22no—
39C22×D4×C103—21no—
40C22×D4×C103—21no—
41C22×D4×C103—22no—
42C22×D4×C103—41no—
43C22×D4×C103—42no—
44C22×D4×C103C2311yescharacteristic
45#137×C103—24no—
46#137×C103—42no—
47C2×Q16×C103—41no—
48C2×SD16×C103—41no—
49C2×M16×C103—21no—
50#124×C103—28no—
51#124×C103—24no—
52#124×C103—24no—
53#124×C103—24no—
54#124×C103—22no—
55#124×C103—28no—
56#124×C103—24no—
57#124×C103D412yescomplemented
58#124×C103C2312yes—
59#120×C103—28no—
60#120×C103—28no—
61#120×C103—22no—
62#120×C103—22no—
63#120×C103—28no—
64#120×C103—44no—
65#120×C103D414yescomplemented
66#120×C103C2314yes—
67C2×C22⋊C4×C103—22no—
68C2×C22⋊C4×C103—22no—
69C2×C22⋊C4×C103—22no—
70C2×C22⋊C4×C103—22no—
71C2×C22⋊C4×C103—22no—
72C2×C22⋊C4×C103—42no—
73C2×C22⋊C4×C103D412yescomplemented
74C2×C42×C103—21no—
75C2×C42×C103D411yescharacteristic, complemented
76#104×C103—24no—
77#100×C103—21no—
78#100×C103C2312yes—
79#99×C103—22no—
80#99×C103—28no—
81#99×C103C2314yes—
82C24×C103—22no—
83C24×C103—22no—
84C24×C103—22no—
85C24×C103—42no—
86C24×C103—42no—
87C24×C103—84no—
88C24×C103C2×D412yes—
89C24×C103C2×D414yescomplemented
90#41×C103—22no—
91#41×C103—22no—
92#41×C103—42no—
93#41×C103—41no—
94#41×C103—44no—
95#41×C103—42no—
96#41×C103—41no—
97C2×Q8×C103—24no—
98C2×Q8×C103—21no—
99C2×Q8×C103—21no—
100C2×Q8×C103—41no—
101C2×Q8×C103—41no—
102C2×Q8×C103—82no—
103C2×Q8×C103C2×D412yescomplemented
104C2×D4×C103—21no—
105C2×D4×C103—24no—
106C2×D4×C103—24no—
107C2×D4×C103—24no—
108C2×D4×C103—24no—
109C2×D4×C103—24no—
110C2×D4×C103—24no—
111C2×D4×C103—41no—
112C2×D4×C103—41no—
113C2×D4×C103—42no—
114C2×D4×C103—42no—
115C2×D4×C103—42no—
116C2×D4×C103—42no—
117C2×D4×C103—44no—
118C2×D4×C103—42no—
119C2×D4×C103—41no—
120C2×D4×C103—48no—
121C2×D4×C103—84no—
122C2×D4×C103—82no—
123C2×D4×C103—81no—
124C2×D4×C103—81no—
125C2×D4×C103—88no—
126C2×D4×C103C2×D412yes—
127C2×D4×C103C2×D411yescharacteristic
128C2×D4×C103C2411yescharacteristic
129C22×C4×C103—22no—
130C22×C4×C103—21no—
131C22×C4×C103—22no—
132C22×C4×C103—21no—
133C22×C4×C103—41no—
134C22×C4×C103—42no—
135C22×C4×C103C2×D411yescharacteristic
136Q16×C103—44no—
137SD16×C103—44no—
138M16×C103—22no—
139M16×C103—41no—
140C2×C8×C103—41no—
141C22⋊C4×C103—28no—
142C22⋊C4×C103—28no—
143C22⋊C4×C103—24no—
144C22⋊C4×C103—22no—
145C22⋊C4×C103—24no—
146C22⋊C4×C103—24no—
147C22⋊C4×C103—24no—
148C22⋊C4×C103—24no—
149C22⋊C4×C103—48no—
150C22⋊C4×C103—42no—
151C22⋊C4×C103C2×D414yescomplemented
152C42×C103—24no—
153C42×C103—21no—
154C42×C103C2×D412yescomplemented
155C23×C103—22no—
156C23×C103—22no—
157C23×C103—24no—
158C23×C103—24no—
159C23×C103—44no—
160C23×C103—44no—
161C23×C103—44no—
162C23×C103—44no—
163C23×C103—42no—
164C23×C103—48no—
165C23×C103—42no—
166C23×C103—41no—
167C23×C103—42no—
168C23×C103—82no—
169C23×C103—84no—
170C23×C103—84no—
171C23×C103—84no—
172C23×C103—88no—
173C23×C103—84no—
174C23×C103#12012yes—
175C23×C103#12011yescharacteristic
176C23×C103C22×D412yes—
177Q8×C103—22no—
178Q8×C103—42no—
179Q8×C103—41no—
180Q8×C103—81no—
181Q8×C103—84no—
182D4×C103—22no—
183D4×C103—41no—
184D4×C103—42no—
185D4×C103—41no—
186D4×C103—41no—
187D4×C103—44no—
188D4×C103—44no—
189D4×C103—42no—
190D4×C103—42no—
191D4×C103—84no—
192D4×C103—84no—
193D4×C103—84no—
194D4×C103—84no—
195D4×C103—816no—
196C2×C4×C103—21no—
197C2×C4×C103—21no—
198C2×C4×C103—24no—
199C2×C4×C103—22no—
200C2×C4×C103—24no—
201C2×C4×C103—22no—
202C2×C4×C103—42no—
203C2×C4×C103—42no—
204C2×C4×C103—42no—
205C2×C4×C103—44no—
206C2×C4×C103—42no—
207C2×C4×C103—41no—
208C2×C4×C103—41no—
209C2×C4×C103—41no—
210C2×C4×C103—44no—
211C2×C4×C103—84no—
212C2×C4×C103#12011yescharacteristic
213C2×C4×C103C22×D411yescharacteristic
214C8×C103—42no—
215C22×C103—22no—
216C22×C103—22no—
217C22×C103—22no—
218C22×C103—42no—
219C22×C103—42no—
220C22×C103—44no—
221C22×C103—44no—
222C22×C103—44no—
223C22×C103—48no—
224C22×C103—42no—
225C22×C103—44no—
226C22×C103—82no—
227C22×C103—84no—
228C22×C103—84no—
229C22×C103—84no—
230C22×C103—88no—
231C22×C103—84no—
232C22×C103—82no—
233C22×C103—162no—
234C22×C103B(4,𝔽2)11yescharacteristic
235C22×C103B(4,𝔽2)11yescharacteristic
236C22×C103C2×#12011yescharacteristic
237C4×C103—21no—
238C4×C103—21no—
239C4×C103—44no—
240C4×C103—42no—
241C4×C103—42no—
242C4×C103—42no—
243C4×C103—84no—
244C2×#2199C10311yesmaximal, characteristic, direct product factor, Sylow
245C2×C103—21no—
246C2×C103—21no—
247C2×C103—42no—
248C2×C103—42no—
249C2×C103—84no—
250C2×C103—82no—
251C2×C103#219912yesdirect product factor
252C2×C103C2×B(4,𝔽2)11yescharacteristic
253C2×H(3,𝔽4)C2×C10311yescharacteristic, complemented
254C2×B(4,𝔽2)C2×C10312yescomplemented
255C2×#455C2×C10311yescharacteristic, complemented
256#2199C2×C10318yesdirect product factor
257C2×#354C2×C10311yescharacteristic, complemented
258C2×AΣL(1,𝔽16)C2×C10312yescomplemented
259C103C2×#219911yescharacteristic, direct product factor, Sylow
260C2×H(5,𝔽2)C22×C10311yescharacteristic
261C2×#137—21no—
262H(3,𝔽4)—22no—
263H(3,𝔽4)—24no—
264H(3,𝔽4)C22×C10314yescomplemented
265C2×#124—21no—
266C2×#124—22no—
267C2×#124—21no—
268C2×#124C22×C10311yescharacteristic
269C2×#120—22no—
270C2×#120—22no—
271C2×#120C22×C10312yes—
272B(4,𝔽2)—24no—
273B(4,𝔽2)C22×C10318yescomplemented
274#455—22no—
275#455C22×C10314yescomplemented
276C2×#104—21no—
277C2×#100C22×C10311yescharacteristic
278C2×#99—22no—
279C2×#99C22×C10312yes—
280#354C22×C10314yescomplemented
281AΣL(1,𝔽16)C22×C10318yescomplemented
282C25D4×C10312yescomplemented
283H(5,𝔽2)—22no—
284H(5,𝔽2)—21no—
285H(5,𝔽2)—42no—
286H(5,𝔽2)C23×C10312yes—
287C2×#41—21no—
288C2×#41—41no—
289C22×Q8—21no—
290C22×Q8D4×C10311yescharacteristic, complemented
291C22×D4—22no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4—22no—
295C22×D4—41no—
296C22×D4—42no—
297C22×D4C23×C10311yescharacteristic
298#137—24no—
299#137—42no—
300C2×Q16—41no—
301C2×SD16—41no—
302C2×M16—21no—
303#124—28no—
304#124—24no—
305#124—24no—
306#124—24no—
307#124—22no—
308#124—28no—
309#124—24no—
310#124D4×C10312yescomplemented
311#124C23×C10312yes—
312#120—28no—
313#120—28no—
314#120—22no—
315#120—22no—
316#120—28no—
317#120—44no—
318#120D4×C10314yescomplemented
319#120C23×C10314yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4D4×C10312yescomplemented
327C2×C42—21no—
328C2×C42D4×C10311yescharacteristic, complemented
329#104—24no—
330#100—21no—
331#100C23×C10312yes—
332#99—22no—
333#99—28no—
334#99C23×C10314yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—84no—
341C24C2×D4×C10312yes—
342C24C2×D4×C10314yescomplemented
343#41—22no—
344#41—22no—
345#41—42no—
346#41—41no—
347#41—44no—
348#41—42no—
349#41—41no—
350C2×Q8—24no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—82no—
356C2×Q8C2×D4×C10312yescomplemented
357C2×D4—21no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—48no—
374C2×D4—84no—
375C2×D4—82no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—88no—
379C2×D4C2×D4×C10312yes—
380C2×D4C2×D4×C10311yescharacteristic
381C2×D4C24×C10311yescharacteristic
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4C2×D4×C10311yescharacteristic
389Q16—44no—
390SD16—44no—
391M16—22no—
392M16—41no—
393C2×C8—41no—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—48no—
403C22⋊C4—42no—
404C22⋊C4C2×D4×C10314yescomplemented
405C42—24no—
406C42—21no—
407C42C2×D4×C10312yescomplemented
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—48no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—82no—
422C23—84no—
423C23—84no—
424C23—84no—
425C23—88no—
426C23—84no—
427C23#120×C10311yescharacteristic
428C23#120×C10312yes—
429C23C22×D4×C10312yes—
430Q8—22no—
431Q8—42no—
432Q8—41no—
433Q8—81no—
434Q8—84no—
435D4—22no—
436D4—41no—
437D4—42no—
438D4—41no—
439D4—41no—
440D4—44no—
441D4—44no—
442D4—42no—
443D4—42no—
444D4—84no—
445D4—84no—
446D4—84no—
447D4—84no—
448D4—816no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—84no—
465C2×C4#120×C10311yescharacteristic
466C2×C4C22×D4×C10311yescharacteristic
467C8—42no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—42no—
478C22—44no—
479C22—82no—
480C22—84no—
481C22—84no—
482C22—84no—
483C22—88no—
484C22—84no—
485C22—82no—
486C22—162no—
487C22B(4,𝔽2)×C10311yescharacteristic
488C22B(4,𝔽2)×C10311yescharacteristic
489C22C2×#120×C10311yescharacteristic
490C4—21no—
491C4—21no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—84no—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—42no—
501C2—84no—
502C2—82no—
503C2C103×#219912yesdirect product factor
504C2C2×B(4,𝔽2)×C10311yescharacteristic
subgroupG/H
center, socleC22×C103B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C103
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C103 ◁ C23×C103 ◁ C22×D4×C103 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(3,𝔽4)×C103C2semidirect
2. C2×B(4,𝔽2)×C103C2semidirect
3. C2×#455×C103C2semidirect
4. C103×#2199C2direct (N × H)
5. C2×#354×C103C2semidirect
6. C2×AΣL(1,𝔽16)×C103C2semidirect
11. H(3,𝔽4)×C103C22semidirect
20. B(4,𝔽2)×C103C22semidirect
22. #455×C103C22semidirect
27. #354×C103C22semidirect
28. AΣL(1,𝔽16)×C103C22semidirect
29. C25×C103D4semidirect
37. C22×Q8×C103D4semidirect
57. #124×C103D4semidirect
65. #120×C103D4semidirect
73. C2×C22⋊C4×C103D4semidirect
75. C2×C42×C103D4semidirect
89. C24×C103C2×D4semidirect
103. C2×Q8×C103C2×D4semidirect
151. C22⋊C4×C103C2×D4semidirect
154. C42×C103C2×D4semidirect
244. C2×#2199C103direct (N × H)
251. C2×C103#2199direct (N × H)
253. C2×H(3,𝔽4)C2×C103semidirect
254. C2×B(4,𝔽2)C2×C103semidirect
255. C2×#455C2×C103semidirect
256. #2199C2×C103direct (N × H)
257. C2×#354C2×C103semidirect
258. C2×AΣL(1,𝔽16)C2×C103semidirect
259. C103C2×#2199direct (N × H)
264. H(3,𝔽4)C22×C103semidirect
273. B(4,𝔽2)C22×C103semidirect
275. #455C22×C103semidirect
280. #354C22×C103semidirect
281. AΣL(1,𝔽16)C22×C103semidirect
282. C25D4×C103semidirect
290. C22×Q8D4×C103semidirect
310. #124D4×C103semidirect
318. #120D4×C103semidirect
326. C2×C22⋊C4D4×C103semidirect
328. C2×C42D4×C103semidirect
342. C24C2×D4×C103semidirect
356. C2×Q8C2×D4×C103semidirect
404. C22⋊C4C2×D4×C103semidirect
407. C42C2×D4×C103semidirect
503. C2C103×#2199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(3,𝔽4)×C1032
C2×B(4,𝔽2)×C1032
C2×#455×C1032
C103×#21992
C2×#354×C1032
C2×AΣL(1,𝔽16)×C1032
C2×#2199103