Finite Group Databasebeta

← C59×#9357 #28421466

Subgroups — C59×#9357 #28421466

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3079C211yesmaximal, characteristic
2C59×#2990C211yesmaximal, characteristic
3C59×#1966C211yesmaximal, characteristic, complemented
4C59×#1549C211yesmaximal, characteristic, complemented
5C59×#1489C211yesmaximal, characteristic
6C59×#1309C211yesmaximal, characteristic, complemented
7C59×#1299C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C59C2211yescharacteristic
9C4×#41×C59C2211yescharacteristic
10C59×#477—21no—
11C59×#476—21no—
12C59×#475—21no—
13C59×#474—21no—
14C59×#446—21no—
15C59×#435—22no—
16C8×D4×C59—21no—
17C59×#432—21no—
18C59×#431C2211yescharacteristic
19C59×#428C412yes—
20C59×#423C2211yescharacteristic
21C59×#423C2211yescharacteristic
22C59×#423C2211yescharacteristic
23C2×C4⋊C8×C59—21no—
24C59×#417C412yes—
25C2×#101×C59—22no—
26C4×M16×C59C2211yescharacteristic
27C59×#332—21no—
28C59×#332—21no—
29C59×Q8⋊C8—21no—
30C59×D4⋊C8—21no—
31C2×#41×C59C2×C411yescharacteristic
32C22×Q8×C59—21no—
33C2×M16×C59—21no—
34C2×M16×C59—21no—
35C2×M16×C59C2×C412yes—
36C2×M16×C59D412yes—
37C22×C8×C59—22no—
38C4⋊Q8×C59—21no—
39C4⋊Q8×C59—21no—
40C4⋊Q8×C59—21no—
41C4⋊Q8×C59—21no—
42C4×Q8×C59D411yescharacteristic
43C4×Q8×C59D411yescharacteristic
44C4×D4×C59—21no—
45C4×D4×C59—21no—
46C4×D4×C59D411yescharacteristic
47C4×D4×C59D411yescharacteristic
48#117×C59—21no—
49#117×C59C2×C411yescharacteristic
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59C2×C412yes—
52C2×C42×C59—21no—
53C2×C42×C59C2311yescharacteristic
54#107×C59—22no—
55#106×C59—22no—
56C4⋊C8×C59—22no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D411yescharacteristic
64C4⋊C8×C59D411yescharacteristic
65#103×C59—22no—
66#102×C59—22no—
67#101×C59—44no—
68C22⋊C8×C59—22no—
69C22⋊C8×C59—21no—
70#97×C59—22no—
71#97×C59—21no—
72C4×C8×C59—21no—
73C4×C8×C59—21no—
74#41×C59—22no—
75#41×C59—41no—
76C2×Q8×C59—42no—
77C2×Q8×C59—42no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59C22⋊C411yescharacteristic
81C2×D4×C59—21no—
82C2×D4×C59C22⋊C411yescharacteristic
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—22no—
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59#4111yescharacteristic
90M16×C59—22no—
91M16×C59—41no—
92M16×C59—42no—
93M16×C59—42no—
94M16×C59—81no—
95C2×C8×C59—22no—
96C2×C8×C59—21no—
97C2×C8×C59—21no—
98C2×C8×C59—22no—
99C2×C8×C59—41no—
100C2×C8×C59—41no—
101C2×C8×C59—82no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59C22⋊C411yescharacteristic
109C4⋊C4×C59C22⋊C411yescharacteristic
110C22⋊C4×C59—21no—
111C22⋊C4×C59—21no—
112C22⋊C4×C59—21no—
113C42×C59—21no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59C2×D411yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120C23×C59—21no—
121C23×C59C4×D411yescharacteristic
122Q8×C59—21no—
123Q8×C59—21no—
124Q8×C59—82no—
125Q8×C59—81no—
126Q8×C59—81no—
127D4×C59—21no—
128D4×C59—42no—
129D4×C59#10112yescomplemented
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
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—42no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—44no—
145C2×C4×C59—42no—
146C2×C4×C59C2×C22⋊C411yescharacteristic
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59#12011yescharacteristic
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12311yescharacteristic
152C8×C59—42no—
153C8×C59—42no—
154C8×C59—41no—
155C8×C59—42no—
156C8×C59—41no—
157C8×C59—81no—
158#9357C5911yesmaximal, characteristic, direct product factor, Sylow
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—22no—
162C22×C59—21no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59#38611yescharacteristic
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—41no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—82no—
173C4×C59C2×#10111yescharacteristic
174C4×C59#41311yescharacteristic
175C4×C59#45311yescharacteristic
176C4×C59#45411yescharacteristic
177C4×C59#47111yescharacteristic
178C4×C59#47311yescharacteristic
179#3079C2×C5911yescharacteristic
180#2990C2×C5911yescharacteristic
181#1966C2×C5911yescharacteristic, complemented
182#1549C2×C5911yescharacteristic, complemented
183#1489C2×C5911yescharacteristic
184#1309C2×C5911yescharacteristic, complemented
185#1299C2×C5911yescharacteristic, complemented
186C2×C59—41no—
187C2×C59—42no—
188C2×C59#188311yescharacteristic
189C2×C59#190111yescharacteristic
190C2×C59#190811yescharacteristic
191C2×C4⋊Q8C22×C5911yescharacteristic
192C4×#41C22×C5911yescharacteristic
193#477—21no—
194#476—21no—
195#475—21no—
196#474—21no—
197#446—21no—
198#435—22no—
199C8×D4—21no—
200#432—21no—
201#431C22×C5911yescharacteristic
202#428C4×C5912yes—
203#423C22×C5911yescharacteristic
204#423C22×C5911yescharacteristic
205#423C22×C5911yescharacteristic
206C2×C4⋊C8—21no—
207#417C4×C5912yes—
208C2×#101—22no—
209C4×M16C22×C5911yescharacteristic
210#332—21no—
211#332—21no—
212Q8⋊C8—21no—
213D4⋊C8—21no—
214C59#935711yescharacteristic, direct product factor, Sylow
215C2×#41C2×C4×C5911yescharacteristic
216C22×Q8—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16C2×C4×C5912yes—
220C2×M16D4×C5912yes—
221C22×C8—22no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4×Q8D4×C5911yescharacteristic
227C4×Q8D4×C5911yescharacteristic
228C4×D4—21no—
229C4×D4—21no—
230C4×D4D4×C5911yescharacteristic
231C4×D4D4×C5911yescharacteristic
232#117—21no—
233#117C2×C4×C5911yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4C2×C4×C5912yes—
236C2×C42—21no—
237C2×C42C23×C5911yescharacteristic
238#107—22no—
239#106—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C5911yescharacteristic
248C4⋊C8D4×C5911yescharacteristic
249#103—22no—
250#102—22no—
251#101—44no—
252C22⋊C8—22no—
253C22⋊C8—21no—
254#97—22no—
255#97—21no—
256C4×C8—21no—
257C4×C8—21no—
258#41—22no—
259#41—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8C22⋊C4×C5911yescharacteristic
265C2×D4—21no—
266C2×D4C22⋊C4×C5911yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4C22×C4×C5911yescharacteristic
272C22×C4C2×D4×C5911yescharacteristic
273C22×C4#41×C5911yescharacteristic
274M16—22no—
275M16—41no—
276M16—42no—
277M16—42no—
278M16—81no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—82no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4C22⋊C4×C5911yescharacteristic
293C4⋊C4C22⋊C4×C5911yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42C2×D4×C5911yescharacteristic
301C42C2×D4×C5911yescharacteristic
302C42C2×D4×C5911yescharacteristic
303C42#41×C5911yescharacteristic
304C23—21no—
305C23C4×D4×C5911yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—82no—
309Q8—81no—
310Q8—81no—
311D4—21no—
312D4—42no—
313D4#101×C5912yescomplemented
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4C2×C22⋊C4×C5911yescharacteristic
331C2×C4C4×D4×C5911yescharacteristic
332C2×C4#120×C5911yescharacteristic
333C2×C4#121×C5911yescharacteristic
334C2×C4#121×C5911yescharacteristic
335C2×C4#123×C5911yescharacteristic
336C8—42no—
337C8—42no—
338C8—41no—
339C8—42no—
340C8—41no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22C59×#38611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—82no—
356C4C2×#101×C5911yescharacteristic
357C4C59×#41311yescharacteristic
358C4C59×#45311yescharacteristic
359C4C59×#45411yescharacteristic
360C4C59×#47111yescharacteristic
361C4C59×#47311yescharacteristic
362C2—41no—
363C2—42no—
364C2C59×#188311yescharacteristic
365C2C59×#190111yescharacteristic
366C2C59×#190811yescharacteristic
subgroupG/H
center, socleC22×C59#386
commutatorC22×C4C22×C4×C59
Frattini, cosocleC2×C42C23×C59
prime radicalC2×#41×C59C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
3. C59×#1966C2semidirect
4. C59×#1549C2semidirect
6. C59×#1309C2semidirect
7. C59×#1299C2semidirect
129. D4×C59#101semidirect
158. #9357C59direct (N × H)
181. #1966C2×C59semidirect
182. #1549C2×C59semidirect
184. #1309C2×C59semidirect
185. #1299C2×C59semidirect
214. C59#9357direct (N × H)
313. D4#101×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#30792
C59×#29902
C59×#19662
C59×#15492
C59×#14892
C59×#13092
C59×#12992
#935759