Finite Group Databasebeta

← C101×#9981 #38792132

Subgroups — C101×#9981 #38792132

Order 25856.

764 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3007C211yesmaximal, characteristic, complemented
2C101×#3006C211yesmaximal, characteristic
3C101×#2816C211yesmaximal, characteristic, complemented
4C101×#1782C211yesmaximal, characteristic, complemented
5C101×#1511C211yesmaximal, characteristic, complemented
6C101×#1510C211yesmaximal, characteristic, complemented
7C101×#1306C211yesmaximal, characteristic, complemented
8C22×#41×C101C2211yescharacteristic
9C2×#137×C101—21no—
10C2×AGL(1,ℤ/8)×C101—21no—
11C2×C22⋊Q8×C101C2211yescharacteristic
12C2×#121×C101C2211yescharacteristic
13#452×C101—22no—
14#451×C101—21no—
15#450×C101—21no—
16#449×C101—22no—
17#448×C101—21no—
18#447×C101—21no—
19#417×C101—21no—
20#417×C101—21no—
21C2×#99×C101—22no—
22#407×C101C2211yescharacteristic
23#407×C101C2211yescharacteristic
24#399×C101—21no—
25#398×C101—21no—
26#394×C101—21no—
27#394×C101—22no—
28#393×C101—21no—
29#381×C101C2211yescharacteristic
30#381×C101C2211yescharacteristic
31#328×C101—22no—
32#327×C101—22no—
33C2×#41×C101—21no—
34C2×#41×C101—21no—
35C2×#41×C101—21no—
36C2×#41×C101—21no—
37C2×#41×C101—44no—
38C22×Q8×C101D411yescharacteristic
39C22×Q8×C101D411yescharacteristic
40C22×D4×C101—21no—
41C22×D4×C101—21no—
42C22×D4×C101D411yescharacteristic
43C22×D4×C101D411yescharacteristic
44C23×C4×C101—21no—
45C23×C4×C101C2311yescharacteristic
46#137×C101—44no—
47AGL(1,ℤ/8)×C101—44no—
48C2×Q16×C101—22no—
49C2×SD16×C101—22no—
50C2×SD16×C101—22no—
51C2×D8×C101—22no—
52C2×M16×C101—21no—
53C2×M16×C101—21no—
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101—21no—
56C22⋊Q8×C101—21no—
57C22⋊Q8×C101—21no—
58#121×C101—21no—
59#121×C101—21no—
60#121×C101—21no—
61#121×C101—21no—
62C2×C4⋊C4×C101—21no—
63C2×C4⋊C4×C101—22no—
64C2×C4⋊C4×C101D411yescharacteristic
65C2×C4⋊C4×C101D411yescharacteristic
66C2×C22⋊C4×C101—21no—
67C2×C22⋊C4×C101—21no—
68C2×C22⋊C4×C101—21no—
69C2×C22⋊C4×C101—22no—
70C2×C22⋊C4×C101—22no—
71C2×C22⋊C4×C101D412yes—
72#103×C101—22no—
73#103×C101—22no—
74#102×C101—22no—
75#102×C101—22no—
76#99×C101—44no—
77C22⋊C8×C101—21no—
78C22⋊C8×C101—21no—
79C22⋊C8×C101D412yes—
80C22⋊C8×C101D412yes—
81#95×C101—22no—
82#95×C101—21no—
83#95×C101—21no—
84#95×C101—21no—
85#95×C101—21no—
86C24×C101—21no—
87C24×C101—21no—
88C24×C101C2×D411yescharacteristic
89#41×C101—24no—
90#41×C101—24no—
91#41×C101—42no—
92#41×C101—42no—
93#41×C101—42no—
94#41×C101—42no—
95#41×C101—42no—
96#41×C101—42no—
97C2×Q8×C101—21no—
98C2×Q8×C101—21no—
99C2×Q8×C101—21no—
100C2×Q8×C101—21no—
101C2×Q8×C101—42no—
102C2×Q8×C101—42no—
103C2×D4×C101—24no—
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—21no—
107C2×D4×C101—21no—
108C2×D4×C101—42no—
109C2×D4×C101—42no—
110C2×D4×C101—42no—
111C2×D4×C101—41no—
112C2×D4×C101—41no—
113C2×D4×C101—41no—
114C2×D4×C101—41no—
115C2×D4×C101—82no—
116C22×C4×C101—21no—
117C22×C4×C101—22no—
118C22×C4×C101—21no—
119C22×C4×C101—21no—
120C22×C4×C101—21no—
121C22×C4×C101—42no—
122C22×C4×C101—42no—
123C22×C4×C101—42no—
124C22×C4×C101—41no—
125C22×C4×C101—41no—
126C22×C4×C101—41no—
127C22×C4×C101C2×D411yescharacteristic
128C22×C4×C101C2×D411yescharacteristic
129C22×C4×C101C2×D411yescharacteristic
130C22×C4×C101C2×D411yescharacteristic
131C22×C4×C101C2×D411yescharacteristic
132C22×C4×C101#4111yescharacteristic
133Q16×C101—44no—
134SD16×C101—44no—
135SD16×C101—44no—
136D8×C101—44no—
137M16×C101—42no—
138M16×C101—42no—
139C2×C8×C101—22no—
140C2×C8×C101—22no—
141C4⋊C4×C101—21no—
142C4⋊C4×C101—21no—
143C4⋊C4×C101—21no—
144C4⋊C4×C101—21no—
145C4⋊C4×C101—41no—
146C4⋊C4×C101—41no—
147C4⋊C4×C101—82no—
148C22⋊C4×C101—41no—
149C22⋊C4×C101—41no—
150C22⋊C4×C101—41no—
151C22⋊C4×C101—41no—
152C22⋊C4×C101—42no—
153C22⋊C4×C101—42no—
154C22⋊C4×C101—82no—
155C22⋊C4×C101—82no—
156C23×C101—22no—
157C23×C101—22no—
158C23×C101—41no—
159C23×C101—41no—
160C23×C101—41no—
161C23×C101—42no—
162C23×C101—42no—
163C23×C101—41no—
164C23×C101—41no—
165C23×C101—42no—
166C23×C101—81no—
167C23×C101—81no—
168C23×C101#12011yescharacteristic
169C23×C101#12111yescharacteristic
170C23×C101#12711yescharacteristic
171Q8×C101—22no—
172Q8×C101—22no—
173Q8×C101—41no—
174Q8×C101—41no—
175Q8×C101—42no—
176Q8×C101—81no—
177D4×C101—22no—
178D4×C101—22no—
179D4×C101—41no—
180D4×C101—41no—
181D4×C101—42no—
182D4×C101—44no—
183D4×C101—44no—
184D4×C101—81no—
185D4×C101—82no—
186D4×C101—82no—
187C2×C4×C101—22no—
188C2×C4×C101—22no—
189C2×C4×C101—22no—
190C2×C4×C101—41no—
191C2×C4×C101—41no—
192C2×C4×C101—41no—
193C2×C4×C101—41no—
194C2×C4×C101—44no—
195C2×C4×C101—44no—
196C2×C4×C101—41no—
197C2×C4×C101—41no—
198C2×C4×C101—41no—
199C2×C4×C101—42no—
200C2×C4×C101—41no—
201C2×C4×C101—42no—
202C2×C4×C101—42no—
203C2×C4×C101—41no—
204C2×C4×C101—41no—
205C2×C4×C101—41no—
206C2×C4×C101—42no—
207C2×C4×C101—42no—
208C2×C4×C101—81no—
209C2×C4×C101—81no—
210C2×C4×C101—81no—
211C2×C4×C101#12011yescharacteristic
212C2×C4×C101#12011yescharacteristic
213C2×C4×C101#12111yescharacteristic
214C2×C4×C101#12111yescharacteristic
215C8×C101—42no—
216C8×C101—42no—
217C22×C101—21no—
218C22×C101—21no—
219C22×C101—21no—
220C22×C101—21no—
221C22×C101—21no—
222C22×C101—21no—
223C22×C101—21no—
224C22×C101—21no—
225C22×C101—42no—
226C22×C101—82no—
227C22×C101—84no—
228C22×C101—82no—
229C22×C101—82no—
230C22×C101—81no—
231C22×C101—81no—
232C22×C101—81no—
233C22×C101—81no—
234C22×C101—81no—
235C22×C101#39211yescharacteristic
236C22×C101B(4,𝔽2)12yes—
237C4×C101—21no—
238C4×C101—21no—
239C4×C101—21no—
240C4×C101—21no—
241C4×C101—42no—
242C4×C101—82no—
243C4×C101—81no—
244C4×C101—81no—
245C4×C101—81no—
246C4×C101—161no—
247#9981C10111yesmaximal, characteristic, direct product factor, Sylow
248C2×C101—22no—
249C2×C101—41no—
250C2×C101—41no—
251C2×C101—82no—
252C2×C101—161no—
253C2×C101#200811yescharacteristic
254C2×C101#200911yescharacteristic
255C2×C101#202111yescharacteristic
256#3007C2×C10111yescharacteristic, complemented
257#3006C2×C10111yescharacteristic
258#2816C2×C10111yescharacteristic, complemented
259#1782C2×C10111yescharacteristic, complemented
260#1511C2×C10111yescharacteristic, complemented
261#1510C2×C10111yescharacteristic, complemented
262#1306C2×C10111yescharacteristic, complemented
263C101#998111yescharacteristic, direct product factor, Sylow
264C22×#41C22×C10111yescharacteristic
265C2×#137—21no—
266C2×AGL(1,ℤ/8)—21no—
267C2×C22⋊Q8C22×C10111yescharacteristic
268C2×#121C22×C10111yescharacteristic
269#452—22no—
270#451—21no—
271#450—21no—
272#449—22no—
273#448—21no—
274#447—21no—
275#417—21no—
276#417—21no—
277C2×#99—22no—
278#407C22×C10111yescharacteristic
279#407C22×C10111yescharacteristic
280#399—21no—
281#398—21no—
282#394—21no—
283#394—22no—
284#393—21no—
285#381C22×C10111yescharacteristic
286#381C22×C10111yescharacteristic
287#328—22no—
288#327—22no—
289C2×#41—21no—
290C2×#41—21no—
291C2×#41—21no—
292C2×#41—21no—
293C2×#41—44no—
294C22×Q8D4×C10111yescharacteristic
295C22×Q8D4×C10111yescharacteristic
296C22×D4—21no—
297C22×D4—21no—
298C22×D4D4×C10111yescharacteristic
299C22×D4D4×C10111yescharacteristic
300C23×C4—21no—
301C23×C4C23×C10111yescharacteristic
302#137—44no—
303AGL(1,ℤ/8)—44no—
304C2×Q16—22no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×D8—22no—
308C2×M16—21no—
309C2×M16—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314#121—21no—
315#121—21no—
316#121—21no—
317#121—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4D4×C10111yescharacteristic
321C2×C4⋊C4D4×C10111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4D4×C10112yes—
328#103—22no—
329#103—22no—
330#102—22no—
331#102—22no—
332#99—44no—
333C22⋊C8—21no—
334C22⋊C8—21no—
335C22⋊C8D4×C10112yes—
336C22⋊C8D4×C10112yes—
337#95—22no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342C24—21no—
343C24—21no—
344C24C2×D4×C10111yescharacteristic
345#41—24no—
346#41—24no—
347#41—42no—
348#41—42no—
349#41—42no—
350#41—42no—
351#41—42no—
352#41—42no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—82no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4C2×D4×C10111yescharacteristic
384C22×C4C2×D4×C10111yescharacteristic
385C22×C4C2×D4×C10111yescharacteristic
386C22×C4C2×D4×C10111yescharacteristic
387C22×C4C2×D4×C10111yescharacteristic
388C22×C4#41×C10111yescharacteristic
389Q16—44no—
390SD16—44no—
391SD16—44no—
392D8—44no—
393M16—42no—
394M16—42no—
395C2×C8—22no—
396C2×C8—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—82no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C23—22no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—81no—
423C23—81no—
424C23#120×C10111yescharacteristic
425C23#121×C10111yescharacteristic
426C23#127×C10111yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—42no—
432Q8—81no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437D4—42no—
438D4—44no—
439D4—44no—
440D4—81no—
441D4—82no—
442D4—82no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4#120×C10111yescharacteristic
468C2×C4#120×C10111yescharacteristic
469C2×C4#121×C10111yescharacteristic
470C2×C4#121×C10111yescharacteristic
471C8—42no—
472C8—42no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—42no—
482C22—82no—
483C22—84no—
484C22—82no—
485C22—82no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22#392×C10111yescharacteristic
492C22B(4,𝔽2)×C10112yes—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—42no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—161no—
503C2—22no—
504C2—41no—
505C2—41no—
506C2—82no—
507C2—161no—
508C2C101×#200811yescharacteristic
509C2C101×#200911yescharacteristic
510C2C101×#202111yescharacteristic
subgroupG/H
center, socleC22×C101#392
commutator, Frattini, cosocleC23×C4C23×C101
prime radicalC101×#3006C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C101 ◁ C23×C4×C101 ◁ G

Complements

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

NH ≅ G/Ntype
1. C101×#3007C2semidirect
3. C101×#2816C2semidirect
4. C101×#1782C2semidirect
5. C101×#1511C2semidirect
6. C101×#1510C2semidirect
7. C101×#1306C2semidirect
247. #9981C101direct (N × H)
256. #3007C2×C101semidirect
258. #2816C2×C101semidirect
259. #1782C2×C101semidirect
260. #1511C2×C101semidirect
261. #1510C2×C101semidirect
262. #1306C2×C101semidirect
263. C101#9981direct (N × H)

Maximal subgroups

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

subgroupindex
C101×#30072
C101×#30062
C101×#28162
C101×#17822
C101×#15112
C101×#15102
C101×#13062
#9981101