Finite Group Databasebeta

← C101×#13351 #38795502

Subgroups — C101×#13351 #38795502

Order 25856.

710 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C101×#3200C211yesmaximal, characteristic, complemented
2C101×#3028C211yesmaximal, characteristic, complemented
3C101×#3027C211yesmaximal, characteristic
4C101×#2129C211yesmaximal, characteristic, complemented
5C101×#2128C211yesmaximal, characteristic, complemented
6C101×#1794C211yesmaximal, characteristic, complemented
7C101×#1391C211yesmaximal, characteristic, complemented
8#585×C101C2211yescharacteristic
9#536×C101—21no—
10#536×C101C2211yescharacteristic
11#535×C101—21no—
12#535×C101C2211yescharacteristic
13#485×C101—21no—
14#482×C101—21no—
15#468×C101—21no—
16#465×C101—21no—
17#458×C101—22no—
18#458×C101—22no—
19B(4,𝔽2)×C101—22no—
20B(4,𝔽2)×C101—22no—
21#450×C101—21no—
22#448×C101—21no—
23#421×C101—21no—
24#416×C101—21no—
25#410×C101—21no—
26#410×C101—21no—
27#410×C101C2211yescharacteristic
28#410×C101C2211yescharacteristic
29#410×C101C2211yescharacteristic
30#408×C101C2211yescharacteristic
31#354×C101—22no—
32#353×C101—22no—
332−1+4×C101—21no—
342−1+4×C101—41no—
35H(5,𝔽2)×C101—21no—
36H(5,𝔽2)×C101—21no—
37H(5,𝔽2)×C101—41no—
38H(5,𝔽2)×C101D412yes—
39C2×#41×C101—21no—
40C2×#41×C101—21no—
41C2×#41×C101—21no—
42C2×#41×C101—42no—
43C2×#41×C101C2311yescharacteristic
44C22×Q8×C101D411yescharacteristic, complemented
45C22×D4×C101D411yescharacteristic
46C2×SD16×C101—21no—
47C2×SD16×C101—21no—
48C2×M16×C101D411yescharacteristic, complemented
49C22×C8×C101D411yescharacteristic, complemented
50C4⋊Q8×C101—21no—
51C4⋊Q8×C101—41no—
52#127×C101—21no—
53#127×C101—41no—
54#124×C101—21no—
55#124×C101—21no—
56#124×C101—41no—
57#124×C101—41no—
58#123×C101—24no—
59#123×C101—21no—
60#123×C101—21no—
61C22⋊Q8×C101—21no—
62C22⋊Q8×C101—21no—
63C22⋊Q8×C101—42no—
64#121×C101—21no—
65#121×C101—21no—
66#121×C101—42no—
67#120×C101—24no—
68#120×C101—21no—
69#120×C101—21no—
70#117×C101—21no—
71#117×C101—21no—
72#117×C101—21no—
73#117×C101D411yescharacteristic
74#117×C101D411yescharacteristic, complemented
75#107×C101—21no—
76#106×C101—21no—
77#104×C101—42no—
78#103×C101—21no—
79#103×C101—21no—
80#103×C101—41no—
81#102×C101—21no—
82#102×C101—21no—
83#102×C101—41no—
84#99×C101—24no—
85#99×C101—24no—
86#99×C101—21no—
87#99×C101—21no—
88#99×C101—21no—
89#99×C101—21no—
90#99×C101D412yes—
91#99×C101D412yescomplemented
92C22⋊C8×C101—21no—
93C22⋊C8×C101—21no—
94C24×C101—21no—
95#41×C101—22no—
96#41×C101—22no—
97#41×C101—41no—
98#41×C101—41no—
99#41×C101—42no—
100#41×C101—41no—
101#41×C101—41no—
102#41×C101—41no—
103#41×C101—41no—
104#41×C101—41no—
105#41×C101—42no—
106#41×C101—42no—
107#41×C101—42no—
108#41×C101—41no—
109#41×C101—41no—
110#41×C101—41no—
111C2×Q8×C101—21no—
112C2×Q8×C101—21no—
113C2×Q8×C101—21no—
114C2×Q8×C101—21no—
115C2×Q8×C101—21no—
116C2×Q8×C101—41no—
117C2×Q8×C101—41no—
118C2×Q8×C101—42no—
119C2×Q8×C101#4111yescharacteristic
120C2×D4×C101—21no—
121C2×D4×C101—21no—
122C2×D4×C101—21no—
123C2×D4×C101—22no—
124C2×D4×C101—22no—
125C2×D4×C101—21no—
126C2×D4×C101—21no—
127C2×D4×C101—41no—
128C2×D4×C101—41no—
129C2×D4×C101—42no—
130C2×D4×C101—42no—
131C2×D4×C101—42no—
132C2×D4×C101—41no—
133C2×D4×C101—44no—
134C2×D4×C101—41no—
135C2×D4×C101—41no—
136C2×D4×C101C2×D411yescharacteristic
137C2×D4×C101C2×D411yescharacteristic
138C2×D4×C101C2×D411yescharacteristic
139C22×C4×C101—22no—
140C22×C4×C101—22no—
141C22×C4×C101—21no—
142C22×C4×C101—41no—
143C22×C4×C101C2×D411yescharacteristic
144C22×C4×C101C2×D411yescharacteristic
145C22×C4×C101C2×D411yescharacteristic
146SD16×C101—81no—
147SD16×C101—81no—
148M16×C101—41no—
149C2×C8×C101—21no—
150C2×C8×C101—21no—
151C2×C8×C101—41no—
152C4⋊C4×C101—21no—
153C4⋊C4×C101—21no—
154C4⋊C4×C101—41no—
155C4⋊C4×C101—41no—
156C4⋊C4×C101—44no—
157C4⋊C4×C101—41no—
158C4⋊C4×C101—41no—
159C4⋊C4×C101—41no—
160C22⋊C4×C101—21no—
161C22⋊C4×C101—21no—
162C22⋊C4×C101—22no—
163C22⋊C4×C101—22no—
164C22⋊C4×C101—44no—
165C22⋊C4×C101—44no—
166C22⋊C4×C101—41no—
167C22⋊C4×C101—41no—
168C22⋊C4×C101—41no—
169C22⋊C4×C101—41no—
170C22⋊C4×C101—41no—
171C42×C101—21no—
172C42×C101—21no—
173C42×C101—41no—
174C42×C101—41no—
175C42×C101—41no—
176C23×C101—22no—
177C23×C101—22no—
178C23×C101—41no—
179C23×C101—42no—
180C23×C101—41no—
181C23×C101—161no—
182C23×C101#12011yescharacteristic
183C23×C101#12011yescharacteristic
184C23×C101#12711yescharacteristic
185Q8×C101—22no—
186Q8×C101—41no—
187Q8×C101—41no—
188Q8×C101—41no—
189Q8×C101—41no—
190Q8×C101—41no—
191Q8×C101—41no—
192Q8×C101—81no—
193D4×C101—22no—
194D4×C101—41no—
195D4×C101—41no—
196D4×C101—41no—
197D4×C101—41no—
198D4×C101—41no—
199D4×C101—42no—
200D4×C101—42no—
201D4×C101—42no—
202D4×C101—42no—
203D4×C101—41no—
204D4×C101—41no—
205D4×C101—41no—
206D4×C101—81no—
207D4×C101—161no—
208D4×C101—161no—
209D4×C101—164no—
210C2×C4×C101—21no—
211C2×C4×C101—21no—
212C2×C4×C101—21no—
213C2×C4×C101—21no—
214C2×C4×C101—21no—
215C2×C4×C101—21no—
216C2×C4×C101—22no—
217C2×C4×C101—22no—
218C2×C4×C101—42no—
219C2×C4×C101—42no—
220C2×C4×C101—42no—
221C2×C4×C101—41no—
222C2×C4×C101—42no—
223C2×C4×C101—42no—
224C2×C4×C101—42no—
225C2×C4×C101—42no—
226C2×C4×C101—41no—
227C2×C4×C101—41no—
228C2×C4×C101—41no—
229C2×C4×C101—81no—
230C2×C4×C101#12011yescharacteristic
231C2×C4×C101#12111yescharacteristic
232C2×C4×C101#12111yescharacteristic
233C2×C4×C101#12111yescharacteristic
234C8×C101—41no—
235C8×C101—41no—
236C22×C101—21no—
237C22×C101—21no—
238C22×C101—21no—
239C22×C101—41no—
240C22×C101—41no—
241C22×C101—41no—
242C22×C101—42no—
243C22×C101—81no—
244C22×C101—81no—
245C22×C101—82no—
246C22×C101—82no—
247C22×C101—84no—
248C22×C101—81no—
249C22×C101—161no—
250C22×C101#39211yescharacteristic
251C4×C101—21no—
252C4×C101—21no—
253C4×C101—21no—
254C4×C101—42no—
255C4×C101—81no—
256C4×C101—81no—
257C4×C101—82no—
258C4×C101—82no—
259C4×C101—81no—
260C4×C101—161no—
261C4×C101B(4,𝔽2)11yescharacteristic
262C4×C101B(4,𝔽2)11yescharacteristic
263#13351C10111yesmaximal, characteristic, direct product factor, Sylow
264C2×C101—21no—
265C2×C101—41no—
266C2×C101—41no—
267C2×C101—41no—
268C2×C101—82no—
269C2×C101—161no—
270C2×C101#202111yescharacteristic
271#3200C2×C10111yescharacteristic, complemented
272#3028C2×C10111yescharacteristic, complemented
273#3027C2×C10111yescharacteristic
274#2129C2×C10111yescharacteristic, complemented
275#2128C2×C10111yescharacteristic, complemented
276#1794C2×C10111yescharacteristic, complemented
277#1391C2×C10111yescharacteristic, complemented
278C101#1335111yescharacteristic, direct product factor, Sylow
279#585C22×C10111yescharacteristic
280#536—21no—
281#536C22×C10111yescharacteristic
282#535—21no—
283#535C22×C10111yescharacteristic
284#485—21no—
285#482—21no—
286#468—21no—
287#465—21no—
288#458—22no—
289#458—22no—
290B(4,𝔽2)—22no—
291B(4,𝔽2)—22no—
292#450—21no—
293#448—21no—
294#421—21no—
295#416—21no—
296#410—21no—
297#410—21no—
298#410C22×C10111yescharacteristic
299#410C22×C10111yescharacteristic
300#410C22×C10111yescharacteristic
301#408C22×C10111yescharacteristic
302#354—22no—
303#353—22no—
3042−1+4—21no—
3052−1+4—41no—
306H(5,𝔽2)—21no—
307H(5,𝔽2)—21no—
308H(5,𝔽2)—41no—
309H(5,𝔽2)D4×C10112yes—
310C2×#41—21no—
311C2×#41—21no—
312C2×#41—21no—
313C2×#41—42no—
314C2×#41C23×C10111yescharacteristic
315C22×Q8D4×C10111yescharacteristic, complemented
316C22×D4D4×C10111yescharacteristic
317C2×SD16—21no—
318C2×SD16—21no—
319C2×M16D4×C10111yescharacteristic, complemented
320C22×C8D4×C10111yescharacteristic, complemented
321C4⋊Q8—21no—
322C4⋊Q8—41no—
323#127—21no—
324#127—41no—
325#124—21no—
326#124—21no—
327#124—41no—
328#124—41no—
329#123—24no—
330#123—21no—
331#123—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—42no—
335#121—21no—
336#121—21no—
337#121—42no—
338#120—24no—
339#120—21no—
340#120—21no—
341#117—21no—
342#117—21no—
343#117—21no—
344#117D4×C10111yescharacteristic
345#117D4×C10111yescharacteristic, complemented
346#107—21no—
347#106—21no—
348#104—42no—
349#103—21no—
350#103—21no—
351#103—41no—
352#102—21no—
353#102—21no—
354#102—41no—
355#99—24no—
356#99—24no—
357#99—21no—
358#99—21no—
359#99—21no—
360#99—21no—
361#99D4×C10112yes—
362#99D4×C10112yescomplemented
363C22⋊C8—21no—
364C22⋊C8—21no—
365C24—21no—
366#41—22no—
367#41—22no—
368#41—41no—
369#41—41no—
370#41—42no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—42no—
390C2×Q8#41×C10111yescharacteristic
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—44no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4C2×D4×C10111yescharacteristic
408C2×D4C2×D4×C10111yescharacteristic
409C2×D4C2×D4×C10111yescharacteristic
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—41no—
414C22×C4C2×D4×C10111yescharacteristic
415C22×C4C2×D4×C10111yescharacteristic
416C22×C4C2×D4×C10111yescharacteristic
417SD16—81no—
418SD16—81no—
419M16—41no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—41no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—161no—
453C23#120×C10111yescharacteristic
454C23#120×C10111yescharacteristic
455C23#127×C10111yescharacteristic
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—81no—
464D4—22no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—81no—
478D4—161no—
479D4—161no—
480D4—164no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—81no—
501C2×C4#120×C10111yescharacteristic
502C2×C4#121×C10111yescharacteristic
503C2×C4#121×C10111yescharacteristic
504C2×C4#121×C10111yescharacteristic
505C8—41no—
506C8—41no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—81no—
520C22—161no—
521C22#392×C10111yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—82no—
529C4—82no—
530C4—81no—
531C4—161no—
532C4B(4,𝔽2)×C10111yescharacteristic
533C4B(4,𝔽2)×C10111yescharacteristic
534C2—21no—
535C2—41no—
536C2—41no—
537C2—41no—
538C2—82no—
539C2—161no—
540C2C101×#202111yescharacteristic
subgroupG/H
center, socleC2×C101#2021
commutator, Frattini, cosocleC2×#41C23×C101
prime radicalC101×#3027C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C101 ◁ C2×C4×C101 ◁ C2×#41×C101 ◁ G

Complements

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

NH ≅ G/Ntype
1. C101×#3200C2semidirect
2. C101×#3028C2semidirect
4. C101×#2129C2semidirect
5. C101×#2128C2semidirect
6. C101×#1794C2semidirect
7. C101×#1391C2semidirect
44. C22×Q8×C101D4semidirect
48. C2×M16×C101D4semidirect
49. C22×C8×C101D4semidirect
74. #117×C101D4semidirect
91. #99×C101D4semidirect
263. #13351C101direct (N × H)
271. #3200C2×C101semidirect
272. #3028C2×C101semidirect
274. #2129C2×C101semidirect
275. #2128C2×C101semidirect
276. #1794C2×C101semidirect
277. #1391C2×C101semidirect
278. C101#13351direct (N × H)
315. C22×Q8D4×C101semidirect
319. C2×M16D4×C101semidirect
320. C22×C8D4×C101semidirect
345. #117D4×C101semidirect
362. #99D4×C101semidirect

Maximal subgroups

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

subgroupindex
C101×#32002
C101×#30282
C101×#30272
C101×#21292
C101×#21282
C101×#17942
C101×#13912
#13351101