Finite Group Databasebeta

← C59×#7800 #28419909

Subgroups — C59×#7800 #28419909

Order 15104.

516 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C59×#2682C211yesmaximal, characteristic
2C59×#2093C211yesmaximal, characteristic, complemented
3C59×#2084C211yesmaximal, characteristic, complemented
4C59×#2071C211yesmaximal, characteristic, complemented
5C59×#1271C211yesmaximal, characteristic, complemented
6C59×#1270C211yesmaximal, characteristic, complemented
7C59×#1270C211yesmaximal, characteristic, complemented
8C2×#123×C59—21no—
9C2×#121×C59—21no—
10C2×#107×C59—21no—
11C2×#106×C59—21no—
12C2×#102×C59—21no—
13C2×#102×C59—21no—
14C59×#396—21no—
15C59×#394—21no—
16C59×#392—21no—
17C59×#390C2211yescharacteristic
18C59×#390C2211yescharacteristic
19C59×#389C2211yescharacteristic
20C59×#385—21no—
21C59×#384C2211yescharacteristic
22C59×#340—21no—
23C59×#340—21no—
24C59×#336C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×D4×C59—21no—
28C22×D4×C59—21no—
29C22×D4×C59D411yescharacteristic
30C22×D4×C59D411yescharacteristic
31C23×C4×C59—21no—
32C22×C8×C59—21no—
33C22×C8×C59—21no—
34C22×C8×C59—21no—
35#123×C59—42no—
36#123×C59—42no—
37#121×C59—42no—
38#121×C59—42no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56#107×C59—42no—
57#106×C59—42no—
58#102×C59—42no—
59#102×C59—42no—
60#95×C59—21no—
61#95×C59—21no—
62C24×C59—21no—
63C24×C59—21no—
64C2×D4×C59—21no—
65C2×D4×C59—21no—
66C2×D4×C59—21no—
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—81no—
75C2×D4×C59—81no—
76C2×D4×C59—81no—
77C2×D4×C59SD1612yes—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—42no—
83C22×C4×C59—42no—
84C22×C4×C59—41no—
85C22×C4×C59—81no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C2×C8×C59—22no—
90C2×C8×C59—22no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—42no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59D812yes—
112C22⋊C4×C59—42no—
113C22⋊C4×C59—42no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C22⋊C4×C59—41no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—81no—
123C42×C59—41no—
124C23×C59—22no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59—41no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59#12011yescharacteristic
139D4×C59—42no—
140D4×C59—42no—
141D4×C59—81no—
142D4×C59—81no—
143D4×C59—84no—
144D4×C59—84no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—22no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—84no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59C2×D811yescharacteristic
168C2×C4×C59C2×SD1611yescharacteristic
169C2×C4×C59#13511yescharacteristic
170C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
173C8×C59—42no—
174C8×C59—42no—
175C8×C59—81no—
176#7800C5911yesmaximal, characteristic, direct product factor, Sylow
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59#44711yescharacteristic
192C22×C59#44911yescharacteristic
193C22×C59#45011yescharacteristic
194C22×C59#45311yescharacteristic
195C22×C59#45511yescharacteristic
196C22×C59#45611yescharacteristic
197C22×C59#45811yescharacteristic
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—42no—
205C4×C59—81no—
206C4×C59—81no—
207C4×C59—81no—
208C4×C59—161no—
209#2682C2×C5911yescharacteristic
210#2093C2×C5911yescharacteristic, complemented
211#2084C2×C5911yescharacteristic, complemented
212#2071C2×C5911yescharacteristic, complemented
213#1271C2×C5911yescharacteristic, complemented
214#1270C2×C5911yescharacteristic, complemented
215#1270C2×C5911yescharacteristic, complemented
216C2×C59—82no—
217C2×C59—161no—
218C2×C59#161011yescharacteristic
219C2×C59#161211yescharacteristic
220C2×C59#161311yescharacteristic
221C2×C59#162111yescharacteristic
222C2×C59#163911yescharacteristic
223C2×C59#165411yescharacteristic
224C2×C59#166111yescharacteristic
225C2×#123—21no—
226C2×#121—21no—
227C2×#107—21no—
228C2×#106—21no—
229C2×#102—21no—
230C2×#102—21no—
231#396—21no—
232#394—21no—
233#392—21no—
234#390C22×C5911yescharacteristic
235#390C22×C5911yescharacteristic
236#389C22×C5911yescharacteristic
237#385—21no—
238#384C22×C5911yescharacteristic
239#340—21no—
240#340—21no—
241#336C22×C5911yescharacteristic
242#336C22×C5911yescharacteristic
243#336C22×C5911yescharacteristic
244C59#780011yescharacteristic, direct product factor, Sylow
245C22×D4—21no—
246C22×D4—21no—
247C22×D4D4×C5911yescharacteristic
248C22×D4D4×C5911yescharacteristic
249C23×C4—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253#123—42no—
254#123—42no—
255#121—42no—
256#121—42no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4D4×C5911yescharacteristic
263C2×C4⋊C4D4×C5911yescharacteristic
264C2×C4⋊C4D4×C5911yescharacteristic
265C2×C4⋊C4D4×C5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42C23×C5911yescharacteristic
274#107—42no—
275#106—42no—
276#102—42no—
277#102—42no—
278#95—21no—
279#95—21no—
280C24—21no—
281C24—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4SD16×C5912yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—81no—
304C22×C4C2×D4×C5911yescharacteristic
305C22×C4C2×D4×C5911yescharacteristic
306C22×C4C2×D4×C5911yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4D8×C5912yes—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—81no—
341C42—41no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23#120×C5911yescharacteristic
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—84no—
362D4—84no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4C2×D8×C5911yescharacteristic
386C2×C4C2×SD16×C5911yescharacteristic
387C2×C4#135×C5911yescharacteristic
388C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
389C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
390C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
391C8—42no—
392C8—42no—
393C8—81no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C59×#44711yescharacteristic
409C22C59×#44911yescharacteristic
410C22C59×#45011yescharacteristic
411C22C59×#45311yescharacteristic
412C22C59×#45511yescharacteristic
413C22C59×#45611yescharacteristic
414C22C59×#45811yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—161no—
426C2—82no—
427C2—161no—
428C2C59×#161011yescharacteristic
429C2C59×#161211yescharacteristic
430C2C59×#161311yescharacteristic
431C2C59×#162111yescharacteristic
432C2C59×#163911yescharacteristic
433C2C59×#165411yescharacteristic
434C2C59×#166111yescharacteristic
subgroupG/H
center, socleC23×C59#120
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#2682C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2093C2semidirect
3. C59×#2084C2semidirect
4. C59×#2071C2semidirect
5. C59×#1271C2semidirect
6. C59×#1270C2semidirect
7. C59×#1270C2semidirect
176. #7800C59direct (N × H)
210. #2093C2×C59semidirect
211. #2084C2×C59semidirect
212. #2071C2×C59semidirect
213. #1271C2×C59semidirect
214. #1270C2×C59semidirect
215. #1270C2×C59semidirect
244. C59#7800direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#26822
C59×#20932
C59×#20842
C59×#20712
C59×#12712
C59×#12702
#780059