Finite Group Databasebeta

← C59×#13419 #28425528

Subgroups — C59×#13419 #28425528

Order 15104.

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

normal subgroups only

This group has 776 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×#2679C211yesmaximal, characteristic
2C2×C59×AΣL(1,𝔽16)C212yesmaximal, complemented
3C59×#1889C212yesmaximal, complemented
4C59×#1856C211yesmaximal, characteristic, complemented
5C59×#1393C211yesmaximal, characteristic, complemented
6C2×#120×C59—22no—
7C2×#120×C59C2212yes—
8C59×#518C2211yescharacteristic
9C59×#428—21no—
10C2×#100×C59C2211yescharacteristic
11C2×#99×C59C2212yes—
12C59×#394—21no—
13C59×#394—22no—
14C59×#390—21no—
15C59×#390C2211yescharacteristic
16C59×#379—22no—
17C59×AΣL(1,𝔽16)—24no—
18C59×#342—22no—
19C59×#339—21no—
20C25×C59D412yes—
21C22×D4×C59—21no—
22C22×D4×C59—21no—
23C22×D4×C59—22no—
24C22×D4×C59C2311yescharacteristic
25C2×M16×C59—21no—
26#120×C59—22no—
27#120×C59—22no—
28#120×C59—22no—
29#120×C59—22no—
30#120×C59—48no—
31C4×D4×C59—41no—
32C4×D4×C59—41no—
33#117×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59D411yescharacteristic
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—42no—
43C2×C22⋊C4×C59—42no—
44C2×C22⋊C4×C59D412yes—
45C2×C42×C59D411yescharacteristic
46#107×C59—41no—
47#106×C59—41no—
48#100×C59—21no—
49#100×C59D412yescomplemented
50#99×C59—22no—
51#99×C59—22no—
52#95×C59—21no—
53C24×C59—22no—
54C24×C59—22no—
55C24×C59—22no—
56C24×C59—42no—
57C24×C59—44no—
58C24×C59—42no—
59C24×C59—82no—
60C24×C59C2×D412yes—
61C2×D4×C59—24no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—44no—
65C2×D4×C59—44no—
66C2×D4×C59—42no—
67C2×D4×C59—42no—
68C2×D4×C59—81no—
69C2×D4×C59—81no—
70C2×D4×C59—82no—
71C2×D4×C59—81no—
72C2×D4×C59C2×D411yescharacteristic
73C2×D4×C59#4111yescharacteristic
74C2×D4×C59#4112yes—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—22no—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59—42no—
81C22×C4×C59C2×D411yescharacteristic
82M16×C59—22no—
83M16×C59—41no—
84C2×C8×C59—41no—
85C4⋊C4×C59—21no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—22no—
90C22⋊C4×C59—22no—
91C22⋊C4×C59—44no—
92C22⋊C4×C59—44no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—44no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—42no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—42no—
99C22⋊C4×C59—42no—
100C22⋊C4×C59—84no—
101C22⋊C4×C59—84no—
102C42×C59—41no—
103C42×C59—41no—
104C23×C59—22no—
105C23×C59—22no—
106C23×C59—24no—
107C23×C59—42no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—42no—
111C23×C59—42no—
112C23×C59—44no—
113C23×C59—44no—
114C23×C59—44no—
115C23×C59—42no—
116C23×C59—42no—
117C23×C59—42no—
118C23×C59—82no—
119C23×C59—84no—
120C23×C59—82no—
121C23×C59—84no—
122C23×C59—88no—
123C23×C59—82no—
124C23×C59#12011yescharacteristic
125C23×C59#12112yes—
126C23×C59#12312yes—
127D4×C59—41no—
128D4×C59—41no—
129D4×C59—42no—
130D4×C59—84no—
131D4×C59—84no—
132D4×C59—84no—
133D4×C59—84no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—22no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—82no—
147C2×C4×C59—82no—
148C2×C4×C59—82no—
149C2×C4×C59—82no—
150C2×C4×C59—82no—
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59#12411yescharacteristic
153C8×C59—42no—
154#13419C5911yesmaximal, characteristic, direct product factor, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—22no—
158C22×C59—22no—
159C22×C59—42no—
160C22×C59—42no—
161C22×C59—44no—
162C22×C59—44no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—48no—
166C22×C59—44no—
167C22×C59—82no—
168C22×C59—82no—
169C22×C59—82no—
170C22×C59—84no—
171C22×C59—84no—
172C22×C59—84no—
173C22×C59—82no—
174C22×C59—82no—
175C22×C59#39411yescharacteristic
176C22×C59B(4,𝔽2)11yescharacteristic
177C22×C59#45811yescharacteristic
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59—82no—
182C4×C59—81no—
183C4×C59—81no—
184C4×C59—81no—
185C4×C59—162no—
186#2679C2×C5911yescharacteristic
187C2×AΣL(1,𝔽16)C2×C5912yescomplemented
188#1889C2×C5912yescomplemented
189#1856C2×C5911yescharacteristic, complemented
190#1393C2×C5911yescharacteristic, complemented
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—42no—
194C2×C59—42no—
195C2×C59—84no—
196C2×C59#204411yescharacteristic
197C2×C59#219611yescharacteristic
198C2×C59#219911yescharacteristic
199C2×#120—22no—
200C2×#120C22×C5912yes—
201#518C22×C5911yescharacteristic
202#428—21no—
203C2×#100C22×C5911yescharacteristic
204C2×#99C22×C5912yes—
205#394—21no—
206#394—22no—
207#390—21no—
208#390C22×C5911yescharacteristic
209#379—22no—
210AΣL(1,𝔽16)—24no—
211#342—22no—
212#339—21no—
213C59#1341911yescharacteristic, direct product factor, Sylow
214C25D4×C5912yes—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4C23×C5911yescharacteristic
219C2×M16—21no—
220#120—22no—
221#120—22no—
222#120—22no—
223#120—22no—
224#120—48no—
225C4×D4—41no—
226C4×D4—41no—
227#117—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C5911yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4D4×C5912yes—
239C2×C42D4×C5911yescharacteristic
240#107—41no—
241#106—41no—
242#100—21no—
243#100D4×C5912yescomplemented
244#99—22no—
245#99—22no—
246#95—21no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—42no—
251C24—44no—
252C24—42no—
253C24—82no—
254C24C2×D4×C5912yes—
255C2×D4—24no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—82no—
265C2×D4—81no—
266C2×D4C2×D4×C5911yescharacteristic
267C2×D4#41×C5911yescharacteristic
268C2×D4#41×C5912yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4C2×D4×C5911yescharacteristic
276M16—22no—
277M16—41no—
278C2×C8—41no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—41no—
297C42—41no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—82no—
313C23—84no—
314C23—82no—
315C23—84no—
316C23—88no—
317C23—82no—
318C23#120×C5911yescharacteristic
319C23#121×C5912yes—
320C23#123×C5912yes—
321D4—41no—
322D4—41no—
323D4—42no—
324D4—84no—
325D4—84no—
326D4—84no—
327D4—84no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#121×C5911yescharacteristic
346C2×C4#124×C5911yescharacteristic
347C8—42no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—22no—
352C22—42no—
353C22—42no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—48no—
359C22—44no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22C59×#39411yescharacteristic
369C22C59×B(4,𝔽2)11yescharacteristic
370C22C59×#45811yescharacteristic
371C4—21no—
372C4—21no—
373C4—42no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—162no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2—84no—
384C2C59×#204411yescharacteristic
385C2C59×#219611yescharacteristic
386C2C59×#219911yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C59
prime radicalC59×#2679C2
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
2. C2×C59×AΣL(1,𝔽16)C2semidirect
3. C59×#1889C2semidirect
4. C59×#1856C2semidirect
5. C59×#1393C2semidirect
49. #100×C59D4semidirect
154. #13419C59direct (N × H)
187. C2×AΣL(1,𝔽16)C2×C59semidirect
188. #1889C2×C59semidirect
189. #1856C2×C59semidirect
190. #1393C2×C59semidirect
213. C59#13419direct (N × H)
243. #100D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#26792
C2×C59×AΣL(1,𝔽16)2
C59×#18892
C59×#18562
C59×#13932
#1341959