Finite Group Databasebeta

← C59×#10177 #28422286

Subgroups — C59×#10177 #28422286

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3205C211yesmaximal, characteristic
2C59×#3037C211yesmaximal, characteristic, complemented
3C2×C59×#460C211yesmaximal, characteristic, complemented
4C59×#1545C211yesmaximal, characteristic, complemented
5C59×#1480C211yesmaximal, characteristic, complemented
6C59×#1461C211yesmaximal, characteristic, complemented
7C59×#1301C211yesmaximal, characteristic, complemented
8C2×#137×C59—21no—
9C22×SD16×C59—21no—
10C59×#532C2211yescharacteristic
11C2×#127×C59C2211yescharacteristic
12C2×C4×Q8×C59C2211yescharacteristic
13C59×#493—21no—
14C59×#486—21no—
15C59×#464—22no—
16C59×#462—21no—
17C59×#460—22no—
18C59×#460—21no—
19C59×#460—22no—
20C59×#459—22no—
21C59×#447—22no—
22C59×#431C2211yescharacteristic
23C59×#431C2211yescharacteristic
24C59×#423C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C59×#417—21no—
27C2×#102×C59—21no—
28C59×#331—22no—
29C59×D4⋊C8—22no—
30C59×#324—22no—
31C2×#41×C59—21no—
32C22×Q8×C59D411yescharacteristic
33C22×D4×C59—21no—
34C22×D4×C59—21no—
35C22×D4×C59—21no—
36#137×C59—42no—
37#137×C59—42no—
38C2×Q16×C59—22no—
39C2×SD16×C59—24no—
40C2×SD16×C59—22no—
41C2×SD16×C59—42no—
42C2×SD16×C59—42no—
43C2×D8×C59—24no—
44C2×M16×C59—21no—
45C22×C8×C59—21no—
46C4⋊Q8×C59D411yescharacteristic
47#127×C59—21no—
48#127×C59—21no—
49#127×C59—21no—
50#127×C59D412yes—
51#127×C59D411yescharacteristic
52#124×C59—21no—
53#121×C59—21no—
54#121×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C4×Q8×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59—21no—
69C4⋊C8×C59D412yes—
70C4⋊C8×C59D412yes—
71#103×C59—22no—
72#102×C59—24no—
73#102×C59—22no—
74#102×C59—24no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59D412yes—
77C4×C8×C59—21no—
78C4×C8×C59—21no—
79C24×C59—41no—
80#41×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83#41×C59—41no—
84C2×Q8×C59—21no—
85C2×Q8×C59—21no—
86C2×Q8×C59—21no—
87C2×Q8×C59—21no—
88C2×Q8×C59—21no—
89C2×Q8×C59—21no—
90C2×Q8×C59D812yescomplemented
91C2×D4×C59—21no—
92C2×D4×C59—22no—
93C2×D4×C59—22no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—22no—
97C2×D4×C59—41no—
98C2×D4×C59—41no—
99C2×D4×C59—41no—
100C2×D4×C59—41no—
101C2×D4×C59—41no—
102C2×D4×C59—41no—
103C2×D4×C59—41no—
104C2×D4×C59—41no—
105C2×D4×C59—82no—
106C2×D4×C59—82no—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—41no—
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×D411yescharacteristic
112C22×C4×C59#4111yescharacteristic
113Q16×C59—44no—
114SD16×C59—48no—
115SD16×C59—44no—
116D8×C59—84no—
117M16×C59—41no—
118M16×C59—41no—
119C2×C8×C59—22no—
120C2×C8×C59—22no—
121C2×C8×C59—22no—
122C2×C8×C59—41no—
123C2×C8×C59—41no—
124C2×C8×C59—41no—
125C4⋊C4×C59—21no—
126C4⋊C4×C59—21no—
127C4⋊C4×C59—21no—
128C4⋊C4×C59—21no—
129C4⋊C4×C59—41no—
130C4⋊C4×C59—41no—
131C22⋊C4×C59—41no—
132C22⋊C4×C59—41no—
133C42×C59—41no—
134C42×C59—41no—
135C42×C59C2×D411yescharacteristic
136C42×C59C2×D411yescharacteristic
137C42×C59C2×D411yescharacteristic
138C42×C59C2×D411yescharacteristic
139C23×C59—41no—
140C23×C59—42no—
141C23×C59—82no—
142C23×C59—82no—
143C23×C59—81no—
144C23×C59—81no—
145C23×C59#12111yescharacteristic
146Q8×C59—22no—
147Q8×C59—22no—
148Q8×C59—41no—
149Q8×C59—41no—
150Q8×C59—41no—
151Q8×C59—41no—
152D4×C59—42no—
153D4×C59—44no—
154D4×C59—42no—
155D4×C59—41no—
156D4×C59—42no—
157D4×C59—41no—
158D4×C59—42no—
159D4×C59—82no—
160D4×C59—82no—
161D4×C59—81no—
162D4×C59—81no—
163D4×C59—81no—
164D4×C59—81no—
165D4×C59—82no—
166C2×C4×C59—22no—
167C2×C4×C59—22no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—81no—
174C2×C4×C59—81no—
175C2×C4×C59—81no—
176C2×C4×C59#12011yescharacteristic
177C2×C4×C59#12011yescharacteristic
178C2×C4×C59#12011yescharacteristic
179C2×C4×C59#12111yescharacteristic
180C2×C4×C59#12111yescharacteristic
181C2×C4×C59#12711yescharacteristic
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59#13511yescharacteristic
184C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
186C8×C59—42no—
187C8×C59—42no—
188C8×C59—42no—
189C8×C59—81no—
190#10177C5911yesmaximal, characteristic, direct product factor, Sylow
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—81no—
194C22×C59—82no—
195C22×C59—82no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—82no—
199C22×C59—162no—
200C22×C59—162no—
201C22×C59#39211yescharacteristic
202C22×C59#46611yescharacteristic
203C22×C59#46911yescharacteristic
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—42no—
208C4×C59—81no—
209C4×C59—81no—
210C4×C59#44711yescharacteristic
211C4×C59#44711yescharacteristic
212C4×C59#44911yescharacteristic
213C4×C59#44911yescharacteristic
214C4×C59#47112yes—
215#3205C2×C5911yescharacteristic
216#3037C2×C5911yescharacteristic, complemented
217C2×#460C2×C5911yescharacteristic, complemented
218#1545C2×C5911yescharacteristic, complemented
219#1480C2×C5911yescharacteristic, complemented
220#1461C2×C5911yescharacteristic, complemented
221#1301C2×C5911yescharacteristic, complemented
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—161no—
225C2×C59—162no—
226C2×C59#199911yescharacteristic
227C2×C59#200811yescharacteristic
228C2×C59#201911yescharacteristic
229C2×#137—21no—
230C22×SD16—21no—
231#532C22×C5911yescharacteristic
232C2×#127C22×C5911yescharacteristic
233C2×C4×Q8C22×C5911yescharacteristic
234#493—21no—
235#486—21no—
236#464—22no—
237#462—21no—
238#460—22no—
239#460—21no—
240#460—22no—
241#459—22no—
242#447—22no—
243#431C22×C5911yescharacteristic
244#431C22×C5911yescharacteristic
245#423C22×C5911yescharacteristic
246C2×C4⋊C8C22×C5911yescharacteristic
247#417—21no—
248C2×#102—21no—
249#331—22no—
250D4⋊C8—22no—
251#324—22no—
252C59#1017711yescharacteristic, direct product factor, Sylow
253C2×#41—21no—
254C22×Q8D4×C5911yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258#137—42no—
259#137—42no—
260C2×Q16—22no—
261C2×SD16—24no—
262C2×SD16—22no—
263C2×SD16—42no—
264C2×SD16—42no—
265C2×D8—24no—
266C2×M16—21no—
267C22×C8—21no—
268C4⋊Q8D4×C5911yescharacteristic
269#127—21no—
270#127—21no—
271#127—21no—
272#127D4×C5912yes—
273#127D4×C5911yescharacteristic
274#124—21no—
275#121—21no—
276#121—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×D4—21no—
282C4×D4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4D4×C5911yescharacteristic
285C2×C42—21no—
286C2×C42C23×C5911yescharacteristic
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8D4×C5912yes—
292C4⋊C8D4×C5912yes—
293#103—22no—
294#102—24no—
295#102—22no—
296#102—24no—
297C22⋊C8—21no—
298C22⋊C8D4×C5912yes—
299C4×C8—21no—
300C4×C8—21no—
301C24—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8D8×C5912yescomplemented
313C2×D4—21no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—82no—
328C2×D4—82no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4C2×D4×C5911yescharacteristic
333C22×C4C2×D4×C5911yescharacteristic
334C22×C4#41×C5911yescharacteristic
335Q16—44no—
336SD16—48no—
337SD16—44no—
338D8—84no—
339M16—41no—
340M16—41no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—41no—
356C42—41no—
357C42C2×D4×C5911yescharacteristic
358C42C2×D4×C5911yescharacteristic
359C42C2×D4×C5911yescharacteristic
360C42C2×D4×C5911yescharacteristic
361C23—41no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—81no—
366C23—81no—
367C23#121×C5911yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373Q8—41no—
374D4—42no—
375D4—44no—
376D4—42no—
377D4—41no—
378D4—42no—
379D4—41no—
380D4—42no—
381D4—82no—
382D4—82no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—82no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#120×C5911yescharacteristic
399C2×C4#120×C5911yescharacteristic
400C2×C4#120×C5911yescharacteristic
401C2×C4#121×C5911yescharacteristic
402C2×C4#121×C5911yescharacteristic
403C2×C4#127×C5911yescharacteristic
404C2×C4C2×D8×C5911yescharacteristic
405C2×C4#135×C5911yescharacteristic
406C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
407C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
408C8—42no—
409C8—42no—
410C8—42no—
411C8—81no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22—82no—
420C22—162no—
421C22—162no—
422C22C59×#39211yescharacteristic
423C22C59×#46611yescharacteristic
424C22C59×#46911yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—81no—
430C4—81no—
431C4C59×#44711yescharacteristic
432C4C59×#44711yescharacteristic
433C4C59×#44911yescharacteristic
434C4C59×#44911yescharacteristic
435C4C59×#47112yes—
436C2—21no—
437C2—21no—
438C2—162no—
439C2—161no—
440C2C59×#199911yescharacteristic
441C2C59×#200811yescharacteristic
442C2C59×#201911yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3037C2semidirect
3. C2×C59×#460C2semidirect
4. C59×#1545C2semidirect
5. C59×#1480C2semidirect
6. C59×#1461C2semidirect
7. C59×#1301C2semidirect
90. C2×Q8×C59D8semidirect
190. #10177C59direct (N × H)
216. #3037C2×C59semidirect
217. C2×#460C2×C59semidirect
218. #1545C2×C59semidirect
219. #1480C2×C59semidirect
220. #1461C2×C59semidirect
221. #1301C2×C59semidirect
252. C59#10177direct (N × H)
312. C2×Q8D8×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32052
C59×#30372
C2×C59×#4602
C59×#15452
C59×#14802
C59×#14612
C59×#13012
#1017759