Finite Group Databasebeta

← C101×#10070 #38792221

Subgroups — C101×#10070 #38792221

Order 25856.

790 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C101C211yesmaximal, characteristic, complemented
2C2×#474×C101C211yesmaximal, characteristic, complemented
3C101×#2541C211yesmaximal, characteristic
4C101×#1981C211yesmaximal, characteristic, complemented
5C2×#331×C101C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C101C211yesmaximal, characteristic, complemented
7C101×#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C101C2211yescharacteristic
9C2×#127×C101—21no—
10C2×#127×C101C2211yescharacteristic
11C2×#121×C101—22no—
12C2×C4×D4×C101C2211yescharacteristic
13#486×C101—22no—
14#486×C101—22no—
15#474×C101—22no—
16#474×C101—22no—
17C2×#107×C101—21no—
18C2×C4⋊C8×C101C2211yescharacteristic
19C2×C4⋊C8×C101C2211yescharacteristic
20C2×#103×C101—22no—
21C2×#102×C101—21no—
22C2×#102×C101—22no—
23C2×C4×C8×C101C2211yescharacteristic
24#392×C101—22no—
25#390×C101—21no—
26C4×C4⋊C4×C101C2211yescharacteristic
27#331×C101—22no—
28D4⋊C8×C101—22no—
29C22×Q8×C101—21no—
30C22×D4×C101—24no—
31C22×D4×C101—21no—
32C22×D4×C101—21no—
33C22×D4×C101D411yescharacteristic
34C22×D4×C101D412yes—
35C23×C4×C101—21no—
36C22×C8×C101—21no—
37C22×C8×C101—21no—
38C22×C8×C101—21no—
39C4⋊Q8×C101—21no—
40C4⋊Q8×C101—21no—
41C4⋊Q8×C101—21no—
42C4⋊Q8×C101D412yescomplemented
43#127×C101—21no—
44#127×C101—21no—
45#127×C101—21no—
46#127×C101—21no—
47#127×C101—42no—
48#127×C101—42no—
49#121×C101—44no—
50#121×C101—44no—
51C4×D4×C101—21no—
52C4×D4×C101—21no—
53C4×D4×C101—21no—
54C4×D4×C101—21no—
55C2×C4⋊C4×C101—21no—
56C2×C4⋊C4×C101—21no—
57C2×C4⋊C4×C101D411yescharacteristic
58C2×C4⋊C4×C101D412yes—
59C2×C22⋊C4×C101—24no—
60C2×C22⋊C4×C101—21no—
61C2×C42×C101—21no—
62C2×C42×C101C2311yescharacteristic
63#107×C101—24no—
64C4⋊C8×C101—21no—
65C4⋊C8×C101—21no—
66C4⋊C8×C101—21no—
67C4⋊C8×C101D412yescomplemented
68#103×C101—44no—
69#102×C101—24no—
70#102×C101—28no—
71C4×C8×C101—21no—
72C4×C8×C101D412yescomplemented
73#95×C101—21no—
74C24×C101—21no—
75C24×C101—22no—
76C2×Q8×C101—22no—
77C2×Q8×C101—41no—
78C2×Q8×C101—42no—
79C2×Q8×C101—42no—
80C2×D4×C101—21no—
81C2×D4×C101—22no—
82C2×D4×C101—21no—
83C2×D4×C101—22no—
84C2×D4×C101—21no—
85C2×D4×C101—21no—
86C2×D4×C101—21no—
87C2×D4×C101—44no—
88C2×D4×C101—48no—
89C2×D4×C101—48no—
90C2×D4×C101—41no—
91C2×D4×C101—41no—
92C2×D4×C101—41no—
93C2×D4×C101—41no—
94C2×D4×C101—42no—
95C2×D4×C101—41no—
96C2×D4×C101—41no—
97C2×D4×C101—42no—
98C2×D4×C101—41no—
99C2×D4×C101—41no—
100C2×D4×C101—84no—
101C2×D4×C101SD1612yes—
102C22×C4×C101—22no—
103C22×C4×C101—21no—
104C22×C4×C101—41no—
105C22×C4×C101—41no—
106C22×C4×C101—41no—
107C22×C4×C101—42no—
108C22×C4×C101—42no—
109C22×C4×C101C2×D411yescharacteristic
110C22×C4×C101C2×D411yescharacteristic
111C22×C4×C101C2×D411yescharacteristic
112C2×C8×C101—22no—
113C2×C8×C101—22no—
114C2×C8×C101—42no—
115C2×C8×C101—41no—
116C2×C8×C101—41no—
117C2×C8×C101—41no—
118C2×C8×C101—42no—
119C4⋊C4×C101—22no—
120C4⋊C4×C101—22no—
121C4⋊C4×C101—22no—
122C4⋊C4×C101—21no—
123C4⋊C4×C101—21no—
124C4⋊C4×C101—41no—
125C4⋊C4×C101—41no—
126C4⋊C4×C101D814yes—
127C22⋊C4×C101—48no—
128C22⋊C4×C101—41no—
129C22⋊C4×C101—41no—
130C42×C101—41no—
131C42×C101—41no—
132C42×C101C2×D411yescharacteristic
133C42×C101C2×D411yescharacteristic
134C42×C101C2×D411yescharacteristic
135C42×C101#4111yescharacteristic
136C23×C101—22no—
137C23×C101—41no—
138C23×C101—42no—
139C23×C101—41no—
140C23×C101—42no—
141C23×C101—42no—
142C23×C101—44no—
143C23×C101—42no—
144C23×C101—42no—
145C23×C101—44no—
146C23×C101#12011yescharacteristic
147Q8×C101—44no—
148Q8×C101—81no—
149Q8×C101—81no—
150D4×C101—22no—
151D4×C101—22no—
152D4×C101—44no—
153D4×C101—41no—
154D4×C101—41no—
155D4×C101—82no—
156D4×C101—816no—
157D4×C101—81no—
158D4×C101—81no—
159D4×C101—82no—
160D4×C101—81no—
161D4×C101—81no—
162D4×C101—81no—
163D4×C101—81no—
164C2×C4×C101—22no—
165C2×C4×C101—24no—
166C2×C4×C101—42no—
167C2×C4×C101—42no—
168C2×C4×C101—41no—
169C2×C4×C101—42no—
170C2×C4×C101—42no—
171C2×C4×C101—82no—
172C2×C4×C101—82no—
173C2×C4×C101—81no—
174C2×C4×C101—81no—
175C2×C4×C101#12011yescharacteristic
176C2×C4×C101#12011yescharacteristic
177C2×C4×C101#12111yescharacteristic
178C2×C4×C101#12111yescharacteristic
179C2×C4×C101#12111yescharacteristic
180C2×C4×C101#12711yescharacteristic
181C2×C4×C101C2×D812yes—
182C2×C4×C101C2×SD1611yescharacteristic
183C2×C4×C101#13511yescharacteristic
184C2×C4×C101AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C101AGL(1,ℤ/8)12yes—
186C2×C4×C101#13711yescharacteristic
187C8×C101—42no—
188C8×C101—42no—
189C8×C101—81no—
190C22×C101—44no—
191C22×C101—42no—
192C22×C101—82no—
193C22×C101—81no—
194C22×C101—81no—
195C22×C101—84no—
196C22×C101—84no—
197C22×C101—82no—
198C22×C101—82no—
199C22×C101—82no—
200C22×C101#39211yescharacteristic
201C22×C101#44712yes—
202C22×C101#45011yescharacteristic
203C22×C101#45211yescharacteristic
204C22×C101#45512yes—
205C4×C101—21no—
206C4×C101—21no—
207C4×C101—44no—
208C4×C101—81no—
209C4×C101—82no—
210C4×C101#44811yescharacteristic
211C4×C101#44911yescharacteristic
212C4×C101#45912yes—
213C4×C101#46511yescharacteristic
214C4×C101#46811yescharacteristic
215C4×C101#49311yescharacteristic
216C4×C101#49611yescharacteristic
217#10070C10111yesmaximal, characteristic, direct product factor, Sylow
218C2×C101—82no—
219C2×C101—162no—
220C2×C101#163912yes—
221C2×C101#164112yes—
222C2×C101#200011yescharacteristic
223C2×C101#200411yescharacteristic
224C2×C101#201111yescharacteristic
225C2×#486C2×C10111yescharacteristic, complemented
226C2×#474C2×C10111yescharacteristic, complemented
227#2541C2×C10111yescharacteristic
228#1981C2×C10111yescharacteristic, complemented
229C2×#331C2×C10111yescharacteristic, complemented
230C2×D4⋊C8C2×C10111yescharacteristic, complemented
231#1279C2×C10111yescharacteristic, complemented
232C101#1007011yescharacteristic, direct product factor, Sylow
233C2×C4⋊Q8C22×C10111yescharacteristic
234C2×#127—21no—
235C2×#127C22×C10111yescharacteristic
236C2×#121—22no—
237C2×C4×D4C22×C10111yescharacteristic
238#486—22no—
239#486—22no—
240#474—22no—
241#474—22no—
242C2×#107—21no—
243C2×C4⋊C8C22×C10111yescharacteristic
244C2×C4⋊C8C22×C10111yescharacteristic
245C2×#103—22no—
246C2×#102—22no—
247C2×#102—21no—
248C2×C4×C8C22×C10111yescharacteristic
249#392—22no—
250#390—21no—
251C4×C4⋊C4C22×C10111yescharacteristic
252#331—22no—
253D4⋊C8—22no—
254C22×Q8—21no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4D4×C10111yescharacteristic
259C22×D4D4×C10112yes—
260C23×C4—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8D4×C10112yescomplemented
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272#127—42no—
273#127—42no—
274#121—44no—
275#121—44no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4D4×C10111yescharacteristic
283C2×C4⋊C4D4×C10112yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—21no—
286C2×C42—21no—
287C2×C42C23×C10111yescharacteristic
288#107—24no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8D4×C10112yescomplemented
293#103—44no—
294#102—24no—
295#102—28no—
296C4×C8—21no—
297C4×C8D4×C10112yescomplemented
298#95—21no—
299C24—21no—
300C24—22no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—84no—
326C2×D4SD16×C10112yes—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×D4×C10111yescharacteristic
335C22×C4C2×D4×C10111yescharacteristic
336C22×C4C2×D4×C10111yescharacteristic
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4D8×C10114yes—
352C22⋊C4—48no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—41no—
356C42—41no—
357C42C2×D4×C10111yescharacteristic
358C42C2×D4×C10111yescharacteristic
359C42C2×D4×C10111yescharacteristic
360C42#41×C10111yescharacteristic
361C23—22no—
362C23—42no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23#120×C10111yescharacteristic
372Q8—44no—
373Q8—81no—
374Q8—81no—
375D4—22no—
376D4—22no—
377D4—44no—
378D4—41no—
379D4—41no—
380D4—82no—
381D4—816no—
382D4—81no—
383D4—81no—
384D4—82no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4#120×C10111yescharacteristic
401C2×C4#120×C10111yescharacteristic
402C2×C4#121×C10111yescharacteristic
403C2×C4#121×C10111yescharacteristic
404C2×C4#121×C10111yescharacteristic
405C2×C4#127×C10111yescharacteristic
406C2×C4C2×D8×C10112yes—
407C2×C4C2×SD16×C10111yescharacteristic
408C2×C4#135×C10111yescharacteristic
409C2×C4AGL(1,ℤ/8)×C10111yescharacteristic
410C2×C4AGL(1,ℤ/8)×C10112yes—
411C2×C4#137×C10111yescharacteristic
412C8—42no—
413C8—42no—
414C8—81no—
415C22—44no—
416C22—42no—
417C22—82no—
418C22—81no—
419C22—81no—
420C22—84no—
421C22—84no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22#392×C10111yescharacteristic
426C22#447×C10112yes—
427C22#450×C10111yescharacteristic
428C22#452×C10111yescharacteristic
429C22#455×C10112yes—
430C4—21no—
431C4—21no—
432C4—44no—
433C4—81no—
434C4—82no—
435C4#448×C10111yescharacteristic
436C4#449×C10111yescharacteristic
437C4#459×C10112yes—
438C4#465×C10111yescharacteristic
439C4#468×C10111yescharacteristic
440C4#493×C10111yescharacteristic
441C4#496×C10111yescharacteristic
442C2—82no—
443C2—162no—
444C2C101×#163912yes—
445C2C101×#164112yes—
446C2C101×#200011yescharacteristic
447C2C101×#200411yescharacteristic
448C2C101×#201111yescharacteristic
subgroupG/H
center, socleC23×C101#120
commutator, Frattini, cosocleC2×C42C23×C101
prime radicalC101×#2541C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C101 ◁ C2×C42×C101 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#486×C101C2semidirect
2. C2×#474×C101C2semidirect
4. C101×#1981C2semidirect
5. C2×#331×C101C2semidirect
6. C2×D4⋊C8×C101C2semidirect
7. C101×#1279C2semidirect
42. C4⋊Q8×C101D4semidirect
67. C4⋊C8×C101D4semidirect
72. C4×C8×C101D4semidirect
217. #10070C101direct (N × H)
225. C2×#486C2×C101semidirect
226. C2×#474C2×C101semidirect
228. #1981C2×C101semidirect
229. C2×#331C2×C101semidirect
230. C2×D4⋊C8C2×C101semidirect
231. #1279C2×C101semidirect
232. C101#10070direct (N × H)
267. C4⋊Q8D4×C101semidirect
292. C4⋊C8D4×C101semidirect
297. C4×C8D4×C101semidirect

Maximal subgroups

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

subgroupindex
C2×#486×C1012
C2×#474×C1012
C101×#25412
C101×#19812
C2×#331×C1012
C2×D4⋊C8×C1012
C101×#12792
#10070101