Finite Group Databasebeta

← C59×#13375 #28425484

Subgroups — C59×#13375 #28425484

Order 15104.

704 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2683C211yesmaximal, characteristic, complemented
2C59×#1792C211yesmaximal, characteristic
3C59×#1790C212yesmaximal
4C59×#1393C211yesmaximal, characteristic, complemented
5C59×#1390C212yesmaximal, complemented
6C2×H(5,𝔽2)×C59C2211yescharacteristic
7C2×C22⋊Q8×C59—22no—
8C2×#104×C59—21no—
9C59×#417—21no—
10C2×#100×C59C2211yescharacteristic
11C2×#99×C59—22no—
12C2×#99×C59C2212yes—
13C59×#396—22no—
14C59×#395—21no—
15C59×#390C2211yescharacteristic
16C59×#389—21no—
17C59×#386—22no—
18C59×#386C2212yes—
19H(5,𝔽2)×C59—22no—
20H(5,𝔽2)×C59—21no—
21H(5,𝔽2)×C59—42no—
22H(5,𝔽2)×C59Q812yescomplemented
23C2×#41×C59—21no—
24C2×#41×C59—41no—
25C22×Q8×C59—21no—
26C22×D4×C59—22no—
27C22×D4×C59—41no—
28C22×D4×C59C2311yescharacteristic
29C23×C4×C59D412yes—
30C2×M16×C59—21no—
31C22⋊Q8×C59—48no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59D411yescharacteristic
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—42no—
41C2×C22⋊C4×C59D412yes—
42C2×C42×C59D411yescharacteristic
43#104×C59—42no—
44#103×C59—41no—
45#102×C59—41no—
46#100×C59—22no—
47#99×C59—22no—
48#99×C59—22no—
49#99×C59—44no—
50#95×C59—21no—
51#95×C59—22no—
52#95×C59—42no—
53C24×C59—22no—
54C24×C59C2×D412yes—
55#41×C59—22no—
56#41×C59—22no—
57#41×C59—42no—
58#41×C59—41no—
59#41×C59—44no—
60#41×C59—42no—
61#41×C59—41no—
62C2×Q8×C59—21no—
63C2×Q8×C59—44no—
64C2×Q8×C59—81no—
65C2×D4×C59—21no—
66C2×D4×C59—24no—
67C2×D4×C59—24no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—42no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—44no—
75C2×D4×C59—41no—
76C2×D4×C59—42no—
77C2×D4×C59C2×Q811yescharacteristic
78C2×D4×C59#4111yescharacteristic
79C2×D4×C59#4112yes—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—21no—
83C22×C4×C59—22no—
84C22×C4×C59—21no—
85C22×C4×C59—41no—
86C22×C4×C59—44no—
87C22×C4×C59—42no—
88C22×C4×C59—42no—
89C22×C4×C59C2×D411yescharacteristic
90M16×C59—41no—
91M16×C59—41no—
92C2×C8×C59—41no—
93C4⋊C4×C59—24no—
94C4⋊C4×C59—44no—
95C4⋊C4×C59—44no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—81no—
98C22⋊C4×C59—44no—
99C22⋊C4×C59—44no—
100C22⋊C4×C59—42no—
101C22⋊C4×C59—42no—
102C22⋊C4×C59—42no—
103C22⋊C4×C59—84no—
104C42×C59—41no—
105C23×C59—22no—
106C23×C59—22no—
107C23×C59—24no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—41no—
112C23×C59—82no—
113C23×C59—82no—
114C23×C59#12011yescharacteristic
115C23×C59C22⋊Q812yes—
116C23×C59#12412yes—
117Q8×C59—22no—
118Q8×C59—41no—
119Q8×C59—84no—
120D4×C59—22no—
121D4×C59—41no—
122D4×C59—42no—
123D4×C59—41no—
124D4×C59—41no—
125D4×C59—44no—
126D4×C59—44no—
127D4×C59—42no—
128D4×C59—42no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—24no—
133C2×C4×C59—24no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—41no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—44no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—41no—
148C2×C4×C59—82no—
149C2×C4×C59—82no—
150C2×C4×C59—82no—
151C2×C4×C59—82no—
152C2×C4×C59C22⋊Q811yescharacteristic
153C2×C4×C59#12411yescharacteristic
154C8×C59—81no—
155#13375C5911yesmaximal, characteristic, direct product factor, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—22no—
159C22×C59—22no—
160C22×C59—42no—
161C22×C59—42no—
162C22×C59—42no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—82no—
166C22×C59—84no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—162no—
170C22×C59#39311yescharacteristic
171C22×C59B(4,𝔽2)11yescharacteristic
172C22×C59B(4,𝔽2)11yescharacteristic
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—42no—
176C4×C59—44no—
177C4×C59—42no—
178C4×C59—82no—
179C4×C59—81no—
180C4×C59—162no—
181#2683C2×C5911yescharacteristic, complemented
182#1792C2×C5911yescharacteristic
183#1790C2×C5912yes—
184#1393C2×C5911yescharacteristic, complemented
185#1390C2×C5912yescomplemented
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—42no—
189C2×C59—42no—
190C2×C59—82no—
191C2×C59#202911yescharacteristic
192C2×C59#219611yescharacteristic
193C2×C59#220111yescharacteristic
194C2×H(5,𝔽2)C22×C5911yescharacteristic
195C2×C22⋊Q8—22no—
196C2×#104—21no—
197#417—21no—
198C2×#100C22×C5911yescharacteristic
199C2×#99—22no—
200C2×#99C22×C5912yes—
201#396—22no—
202#395—21no—
203#390C22×C5911yescharacteristic
204#389—21no—
205#386—22no—
206#386C22×C5912yes—
207C59#1337511yescharacteristic, direct product factor, Sylow
208H(5,𝔽2)—22no—
209H(5,𝔽2)—21no—
210H(5,𝔽2)—42no—
211H(5,𝔽2)Q8×C5912yescomplemented
212C2×#41—21no—
213C2×#41—41no—
214C22×Q8—21no—
215C22×D4—22no—
216C22×D4—41no—
217C22×D4C23×C5911yescharacteristic
218C23×C4D4×C5912yes—
219C2×M16—21no—
220C22⋊Q8—48no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4D4×C5911yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4D4×C5912yes—
231C2×C42D4×C5911yescharacteristic
232#104—42no—
233#103—41no—
234#102—41no—
235#100—22no—
236#99—22no—
237#99—22no—
238#99—44no—
239#95—21no—
240#95—22no—
241#95—42no—
242C24—22no—
243C24C2×D4×C5912yes—
244#41—22no—
245#41—22no—
246#41—42no—
247#41—41no—
248#41—44no—
249#41—42no—
250#41—41no—
251C2×Q8—21no—
252C2×Q8—44no—
253C2×Q8—81no—
254C2×D4—21no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4C2×Q8×C5911yescharacteristic
267C2×D4#41×C5911yescharacteristic
268C2×D4#41×C5912yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×D4×C5911yescharacteristic
279M16—41no—
280M16—41no—
281C2×C8—41no—
282C4⋊C4—24no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—81no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—84no—
293C42—41no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—82no—
302C23—82no—
303C23#120×C5911yescharacteristic
304C23C22⋊Q8×C5912yes—
305C23#124×C5912yes—
306Q8—22no—
307Q8—41no—
308Q8—84no—
309D4—22no—
310D4—41no—
311D4—42no—
312D4—41no—
313D4—41no—
314D4—44no—
315D4—44no—
316D4—42no—
317D4—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4C22⋊Q8×C5911yescharacteristic
342C2×C4#124×C5911yescharacteristic
343C8—81no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—84no—
355C22—81no—
356C22—81no—
357C22—162no—
358C22C59×#39311yescharacteristic
359C22C59×B(4,𝔽2)11yescharacteristic
360C22C59×B(4,𝔽2)11yescharacteristic
361C4—21no—
362C4—21no—
363C4—42no—
364C4—42no—
365C4—44no—
366C4—82no—
367C4—81no—
368C4—162no—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—42no—
373C2—82no—
374C2C59×#202911yescharacteristic
375C2C59×#219611yescharacteristic
376C2C59×#220111yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C59
prime radicalC2×H(5,𝔽2)×C59C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C22×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#2683C2semidirect
4. C59×#1393C2semidirect
5. C59×#1390C2semidirect
22. H(5,𝔽2)×C59Q8semidirect
155. #13375C59direct (N × H)
181. #2683C2×C59semidirect
184. #1393C2×C59semidirect
185. #1390C2×C59semidirect
207. C59#13375direct (N × H)
211. H(5,𝔽2)Q8×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#26832
C59×#17922
C59×#17902
C59×#13932
C59×#13902
#1337559