Finite Group Databasebeta

← C59×#13352 #28425461

Subgroups — C59×#13352 #28425461

Order 15104.

682 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3202C211yesmaximal, characteristic, complemented
2C59×#3028C212yesmaximal
3C59×#2129C212yesmaximal, complemented
4C59×#1794C211yesmaximal, characteristic
5C59×#1391C211yesmaximal, characteristic, complemented
6C59×#585C2211yescharacteristic
7C59×#536—22no—
8C59×#536C2212yes—
9C59×#485—22no—
10C59×#470—21no—
11C59×#467—21no—
12C59×#458—24no—
13C59×#458—24no—
14C59×#451—22no—
15C59×#421—21no—
16C59×#416—21no—
17C59×#410—22no—
18C59×#410C2211yescharacteristic
19C59×#410C2212yes—
20C59×#408C2211yescharacteristic
21C59×#354—24no—
222−1+4×C59—21no—
232−1+4×C59—41no—
24H(5,𝔽2)×C59—22no—
25H(5,𝔽2)×C59—41no—
26H(5,𝔽2)×C59D412yes—
27C2×#41×C59—22no—
28C2×#41×C59—21no—
29C2×#41×C59—42no—
30C2×#41×C59C2311yescharacteristic
31C22×Q8×C59D412yes—
32C2×Q16×C59—21no—
33C2×Q16×C59—21no—
34C2×M16×C59D411yescharacteristic
35C22×C8×C59D411yescharacteristic
36C4⋊Q8×C59—22no—
37C4⋊Q8×C59—42no—
38#124×C59—22no—
39#124×C59—42no—
40#123×C59—28no—
41#123×C59—22no—
42#123×C59—22no—
43C22⋊Q8×C59—22no—
44C22⋊Q8×C59—22no—
45C22⋊Q8×C59—44no—
46#117×C59—22no—
47#117×C59—21no—
48#117×C59D412yes—
49#107×C59—21no—
50#106×C59—21no—
51#104×C59—42no—
52#103×C59—22no—
53#103×C59—22no—
54#103×C59—41no—
55#102×C59—41no—
56#99×C59—28no—
57#99×C59—22no—
58#99×C59—22no—
59#99×C59D414yes—
60C22⋊C8×C59—22no—
61#41×C59—22no—
62#41×C59—22no—
63#41×C59—41no—
64#41×C59—41no—
65#41×C59—42no—
66#41×C59—42no—
67#41×C59—42no—
68#41×C59—41no—
69#41×C59—44no—
70#41×C59—42no—
71#41×C59—41no—
72#41×C59—42no—
73C2×Q8×C59—22no—
74C2×Q8×C59—21no—
75C2×Q8×C59—22no—
76C2×Q8×C59—22no—
77C2×Q8×C59—42no—
78C2×Q8×C59—42no—
79C2×Q8×C59—42no—
80C2×Q8×C59#4111yescharacteristic
81C2×D4×C59—21no—
82C2×D4×C59—22no—
83C2×D4×C59—24no—
84C2×D4×C59—42no—
85C2×D4×C59—44no—
86C2×D4×C59—41no—
87C2×D4×C59C2×D411yescharacteristic
88C2×D4×C59C2×D412yes—
89C22×C4×C59—24no—
90C22×C4×C59—22no—
91C22×C4×C59—41no—
92C22×C4×C59C2×D412yes—
93C22×C4×C59C2×D411yescharacteristic
94Q16×C59—81no—
95Q16×C59—81no—
96M16×C59—41no—
97C2×C8×C59—21no—
98C2×C8×C59—21no—
99C2×C8×C59—41no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—42no—
102C4⋊C4×C59—48no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—42no—
105C4⋊C4×C59—42no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—24no—
108C22⋊C4×C59—48no—
109C22⋊C4×C59—42no—
110C22⋊C4×C59—42no—
111C22⋊C4×C59—41no—
112C42×C59—22no—
113C42×C59—42no—
114C42×C59—41no—
115C23×C59—24no—
116C23×C59—41no—
117C23×C59#12012yes—
118C23×C59#12711yescharacteristic
119Q8×C59—22no—
120Q8×C59—41no—
121Q8×C59—42no—
122Q8×C59—41no—
123Q8×C59—42no—
124Q8×C59—42no—
125Q8×C59—82no—
126D4×C59—22no—
127D4×C59—41no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—44no—
131D4×C59—42no—
132D4×C59—42no—
133D4×C59—41no—
134C2×C4×C59—21no—
135C2×C4×C59—22no—
136C2×C4×C59—21no—
137C2×C4×C59—22no—
138C2×C4×C59—24no—
139C2×C4×C59—44no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—42no—
143C2×C4×C59—44no—
144C2×C4×C59—44no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—81no—
148C2×C4×C59#12011yescharacteristic
149C2×C4×C59#12112yes—
150C2×C4×C59#12111yescharacteristic
151C8×C59—41no—
152C8×C59—41no—
153#13352C5911yesmaximal, characteristic, direct product factor, Sylow
154C22×C59—22no—
155C22×C59—21no—
156C22×C59—42no—
157C22×C59—41no—
158C22×C59—42no—
159C22×C59—84no—
160C22×C59—161no—
161C22×C59#39211yescharacteristic
162C4×C59—21no—
163C4×C59—22no—
164C4×C59—42no—
165C4×C59—82no—
166C4×C59—84no—
167C4×C59—82no—
168C4×C59—161no—
169C4×C59B(4,𝔽2)11yescharacteristic
170C4×C59B(4,𝔽2)11yescharacteristic
171#3202C2×C5911yescharacteristic, complemented
172#3028C2×C5912yes—
173#2129C2×C5912yescomplemented
174#1794C2×C5911yescharacteristic
175#1391C2×C5911yescharacteristic, complemented
176C2×C59—21no—
177C2×C59—42no—
178C2×C59—41no—
179C2×C59—82no—
180C2×C59#202111yescharacteristic
181#585C22×C5911yescharacteristic
182#536—22no—
183#536C22×C5912yes—
184#485—22no—
185#470—21no—
186#467—21no—
187#458—24no—
188#458—24no—
189#451—22no—
190#421—21no—
191#416—21no—
192#410—22no—
193#410C22×C5911yescharacteristic
194#410C22×C5912yes—
195#408C22×C5911yescharacteristic
196#354—24no—
197C59#1335211yescharacteristic, direct product factor, Sylow
1982−1+4—21no—
1992−1+4—41no—
200H(5,𝔽2)—22no—
201H(5,𝔽2)—41no—
202H(5,𝔽2)D4×C5912yes—
203C2×#41—22no—
204C2×#41—21no—
205C2×#41—42no—
206C2×#41C23×C5911yescharacteristic
207C22×Q8D4×C5912yes—
208C2×Q16—21no—
209C2×Q16—21no—
210C2×M16D4×C5911yescharacteristic
211C22×C8D4×C5911yescharacteristic
212C4⋊Q8—22no—
213C4⋊Q8—42no—
214#124—22no—
215#124—42no—
216#123—28no—
217#123—22no—
218#123—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—44no—
222#117—22no—
223#117—21no—
224#117D4×C5912yes—
225#107—21no—
226#106—21no—
227#104—42no—
228#103—22no—
229#103—22no—
230#103—41no—
231#102—41no—
232#99—28no—
233#99—22no—
234#99—22no—
235#99D4×C5914yes—
236C22⋊C8—22no—
237#41—22no—
238#41—22no—
239#41—41no—
240#41—41no—
241#41—42no—
242#41—42no—
243#41—42no—
244#41—41no—
245#41—44no—
246#41—42no—
247#41—41no—
248#41—42no—
249C2×Q8—22no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8#41×C5911yescharacteristic
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—41no—
263C2×D4C2×D4×C5911yescharacteristic
264C2×D4C2×D4×C5912yes—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4C2×D4×C5912yes—
269C22×C4C2×D4×C5911yescharacteristic
270Q16—81no—
271Q16—81no—
272M16—41no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—48no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C42—22no—
289C42—42no—
290C42—41no—
291C23—24no—
292C23—41no—
293C23#120×C5912yes—
294C23#127×C5911yescharacteristic
295Q8—22no—
296Q8—41no—
297Q8—42no—
298Q8—41no—
299Q8—42no—
300Q8—42no—
301Q8—82no—
302D4—22no—
303D4—41no—
304D4—42no—
305D4—42no—
306D4—44no—
307D4—42no—
308D4—42no—
309D4—41no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4#120×C5911yescharacteristic
325C2×C4#121×C5912yes—
326C2×C4#121×C5911yescharacteristic
327C8—41no—
328C8—41no—
329C22—22no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—42no—
334C22—84no—
335C22—161no—
336C22C59×#39211yescharacteristic
337C4—21no—
338C4—22no—
339C4—42no—
340C4—82no—
341C4—84no—
342C4—82no—
343C4—161no—
344C4C59×B(4,𝔽2)11yescharacteristic
345C4C59×B(4,𝔽2)11yescharacteristic
346C2—21no—
347C2—42no—
348C2—41no—
349C2—82no—
350C2C59×#202111yescharacteristic
subgroupG/H
center, socleC2×C59#2021
commutator, Frattini, cosocleC2×#41C23×C59
prime radicalC59×#585C22
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C2×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3202C2semidirect
3. C59×#2129C2semidirect
5. C59×#1391C2semidirect
153. #13352C59direct (N × H)
171. #3202C2×C59semidirect
173. #2129C2×C59semidirect
175. #1391C2×C59semidirect
197. C59#13352direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32022
C59×#30282
C59×#21292
C59×#17942
C59×#13912
#1335259