Finite Group Databasebeta

← C59×#7678 #28419787

Subgroups — C59×#7678 #28419787

Order 15104.

630 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#2854C211yesmaximal, characteristic
2C59×#1728C211yesmaximal, characteristic, complemented
3C2×C59×#336C211yesmaximal, characteristic, complemented
4C59×#1271C212yesmaximal, complemented
5C59×#1270C212yesmaximal, complemented
6C22×M16×C59—21no—
7C23×C8×C59—21no—
8C2×#123×C59—21no—
9C2×#121×C59C411yescharacteristic
10C22×C42×C59C2211yescharacteristic
11C59×#390C411yescharacteristic
12C59×#390C2211yescharacteristic
13C59×#389C411yescharacteristic
14C59×#389C2211yescharacteristic
15C59×#388—21no—
16C59×#388—21no—
17C59×#386—21no—
18C59×#380C411yescharacteristic
19C59×#336—21no—
20C59×#336—21no—
21C59×#336C2212yes—
22C59×#336C2212yes—
23C22×D4×C59—21no—
24C22×D4×C59C2×C411yescharacteristic
25C23×C4×C59C2×C411yescharacteristic
26C23×C4×C59D411yescharacteristic
27C23×C4×C59D411yescharacteristic
28C2×M16×C59—24no—
29C2×M16×C59—42no—
30C2×M16×C59—42no—
31C22×C8×C59—24no—
32C22×C8×C59—22no—
33C22×C8×C59—22no—
34C22×C8×C59—42no—
35C22×C8×C59—42no—
36#123×C59—42no—
37#123×C59—42no—
38#121×C59—42no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C42×C59—42no—
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59C2311yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58C24×C59—21no—
59C24×C59C22⋊C411yescharacteristic
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C2×D4×C59—42no—
64C2×D4×C59—42no—
65C2×D4×C59—42no—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—22no—
73C22×C4×C59—22no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59D812yes—
82C22×C4×C59SD1612yes—
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86M16×C59—48no—
87C2×C8×C59—24no—
88C2×C8×C59—24no—
89C2×C8×C59—48no—
90C2×C8×C59—42no—
91C2×C8×C59—42no—
92C2×C8×C59—42no—
93C2×C8×C59—42no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—42no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—41no—
100C22⋊C4×C59—42no—
101C22⋊C4×C59—42no—
102C22⋊C4×C59—42no—
103C22⋊C4×C59—42no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C42×C59—41no—
109C42×C59—41no—
110C42×C59—41no—
111C42×C59—41no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59#9912yes—
124C23×C59#10212yes—
125C23×C59#10412yes—
126C23×C59C2×C22⋊C411yescharacteristic
127D4×C59—82no—
128D4×C59—84no—
129C2×C4×C59—24no—
130C2×C4×C59—24no—
131C2×C4×C59—24no—
132C2×C4×C59—24no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—44no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59#10212yes—
157C2×C4×C59#10412yes—
158C2×C4×C59#13112yes—
159C2×C4×C59C2×D811yescharacteristic
160C2×C4×C59C2×SD1611yescharacteristic
161C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C59#13711yescharacteristic
163C8×C59—44no—
164C8×C59—44no—
165#7678C5911yesmaximal, characteristic, direct product factor, Sylow
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—22no—
169C22×C59—22no—
170C22×C59—22no—
171C22×C59—22no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59#32714yes—
180C22×C59#40811yescharacteristic
181C22×C59C2×#9911yescharacteristic
182C22×C59#41311yescharacteristic
183C22×C59C2×#10211yescharacteristic
184C22×C59#41711yescharacteristic
185C22×C59C2×#10411yescharacteristic
186C22×C59C2×#10411yescharacteristic
187C4×C59—22no—
188C4×C59—21no—
189C4×C59—22no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—22no—
193C4×C59—21no—
194C4×C59—41no—
195C4×C59—81no—
196C4×C59—81no—
197C4×C59—81no—
198#2854C2×C5911yescharacteristic
199#1728C2×C5911yescharacteristic, complemented
200C2×#336C2×C5911yescharacteristic, complemented
201#1271C2×C5912yescomplemented
202#1270C2×C5912yescomplemented
203C2×C59—24no—
204C2×C59—161no—
205C2×C59#146811yescharacteristic
206C2×C59#148311yescharacteristic
207C2×C59#149211yescharacteristic
208C2×C59C2×#32711yescharacteristic
209C2×C59#150411yescharacteristic
210C2×C59#152411yescharacteristic
211C2×C59#154711yescharacteristic
212C22×M16—21no—
213C23×C8—21no—
214C2×#123—21no—
215C2×#121C4×C5911yescharacteristic
216C22×C42C22×C5911yescharacteristic
217#390C4×C5911yescharacteristic
218#390C22×C5911yescharacteristic
219#389C4×C5911yescharacteristic
220#389C22×C5911yescharacteristic
221#388—21no—
222#388—21no—
223#386—21no—
224#380C4×C5911yescharacteristic
225#336—21no—
226#336—21no—
227#336C22×C5912yes—
228#336C22×C5912yes—
229C59#767811yescharacteristic, direct product factor, Sylow
230C22×D4—21no—
231C22×D4C2×C4×C5911yescharacteristic
232C23×C4C2×C4×C5911yescharacteristic
233C23×C4D4×C5911yescharacteristic
234C23×C4D4×C5911yescharacteristic
235C2×M16—24no—
236C2×M16—42no—
237C2×M16—42no—
238C22×C8—24no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—42no—
242C22×C8—42no—
243#123—42no—
244#123—42no—
245#121—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C2×C4×C5911yescharacteristic
249C2×C4⋊C4C2×C4×C5911yescharacteristic
250C2×C4⋊C4C2×C4×C5911yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—42no—
258C2×C42C2×C4×C5911yescharacteristic
259C2×C42D4×C5911yescharacteristic
260C2×C42D4×C5911yescharacteristic
261C2×C42C23×C5911yescharacteristic
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C24C22⋊C4×C5911yescharacteristic
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C22⋊C4×C5911yescharacteristic
286C22×C4C22⋊C4×C5911yescharacteristic
287C22×C4C22⋊C4×C5911yescharacteristic
288C22×C4D8×C5912yes—
289C22×C4SD16×C5912yes—
290C22×C4C22×C4×C5911yescharacteristic
291C22×C4C2×D4×C5911yescharacteristic
292C22×C4C2×D4×C5911yescharacteristic
293M16—48no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—48no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#99×C5912yes—
331C23#102×C5912yes—
332C23#104×C5912yes—
333C23C2×C22⋊C4×C5911yescharacteristic
334D4—82no—
335D4—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4#102×C5912yes—
364C2×C4#104×C5912yes—
365C2×C4#131×C5912yes—
366C2×C4C2×D8×C5911yescharacteristic
367C2×C4C2×SD16×C5911yescharacteristic
368C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
369C2×C4#137×C5911yescharacteristic
370C8—44no—
371C8—44no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C59×#32714yes—
386C22C59×#40811yescharacteristic
387C22C2×#99×C5911yescharacteristic
388C22C59×#41311yescharacteristic
389C22C2×#102×C5911yescharacteristic
390C22C59×#41711yescharacteristic
391C22C2×#104×C5911yescharacteristic
392C22C2×#104×C5911yescharacteristic
393C4—22no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C2—24no—
405C2—161no—
406C2C59×#146811yescharacteristic
407C2C59×#148311yescharacteristic
408C2C59×#149211yescharacteristic
409C2C2×C59×#32711yescharacteristic
410C2C59×#150411yescharacteristic
411C2C59×#152411yescharacteristic
412C2C59×#154711yescharacteristic
subgroupG/H
center, socleC23×C59C2×C22⋊C4
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC2×#121×C59C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C59 ◁ C22×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1728C2semidirect
3. C2×C59×#336C2semidirect
4. C59×#1271C2semidirect
5. C59×#1270C2semidirect
165. #7678C59direct (N × H)
199. #1728C2×C59semidirect
200. C2×#336C2×C59semidirect
201. #1271C2×C59semidirect
202. #1270C2×C59semidirect
229. C59#7678direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#28542
C59×#17282
C2×C59×#3362
C59×#12712
C59×#12702
#767859