Finite Group Databasebeta

← C59×#12692 #28424801

Subgroups — C59×#12692 #28424801

Order 15104.

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C59×#536C211yesmaximal, characteristic
2C2×C59×#352C214yesmaximal, complemented
3C59×#1785C211yesmaximal, characteristic
4C59×#1782C211yesmaximal, characteristic
5C22×#41×C59C2211yescharacteristic
6C23×Q8×C59C411yescharacteristic, complemented
7C59×#536—28no—
8C2×C4⋊Q8×C59—21no—
9C2×#124×C59—21no—
10C2×#123×C59C412yescomplemented
11C2×#123×C59C2212yes—
12C2×C22⋊Q8×C59—21no—
13C2×C22⋊Q8×C59—21no—
14C2×#117×C59C411yescharacteristic, complemented
15C2×#101×C59—21no—
16C2×#100×C59C2212yes—
17C2×#99×C59—21no—
18C2×#99×C59C2212yes—
19C59×#407—21no—
20C59×#381—21no—
21C59×#352—28no—
22C2×#41×C59—24no—
23C2×#41×C59—42no—
24C2×#41×C59—42no—
25C22×Q8×C59—21no—
26C22×Q8×C59—24no—
27C22×Q8×C59—21no—
28C22×Q8×C59—42no—
29C22×Q8×C59—42no—
30C22×Q8×C59D411yescharacteristic
31C22×D4×C59C2×C411yescharacteristic
32C22×D4×C59D411yescharacteristic
33C22×D4×C59C2311yescharacteristic
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59C2×C412yes—
36C23×C4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C2×M16×C59—22no—
39C4⋊Q8×C59—28no—
40#124×C59—28no—
41#123×C59—24no—
42#123×C59—24no—
43#123×C59—24no—
44#123×C59D418yes—
45C22⋊Q8×C59—28no—
46C22⋊Q8×C59—28no—
47#117×C59—24no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—22no—
51C2×C22⋊C4×C59—22no—
52C2×C22⋊C4×C59—22no—
53C2×C22⋊C4×C59—22no—
54C2×C22⋊C4×C59C2×C412yes—
55C2×C42×C59—21no—
56#101×C59—42no—
57#100×C59—24no—
58#99×C59—24no—
59#99×C59—42no—
60C22⋊C8×C59—42no—
61#95×C59—42no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22×C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65#41×C59—416no—
66C2×Q8×C59—24no—
67C2×Q8×C59—24no—
68C2×Q8×C59—22no—
69C2×Q8×C59—22no—
70C2×Q8×C59—41no—
71C2×Q8×C59—416no—
72C2×Q8×C59—42no—
73C2×Q8×C59—42no—
74C2×Q8×C59—42no—
75C2×Q8×C59—42no—
76C2×D4×C59—22no—
77C2×D4×C59—22no—
78C2×D4×C59—22no—
79C2×D4×C59—41no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59C2×D414yes—
85C22×C4×C59—22no—
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—22no—
94C22×C4×C59—22no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—42no—
98C22×C4×C59—42no—
99C22×C4×C59—42no—
100C22×C4×C59C22⋊C411yescharacteristic
101C22×C4×C59C22⋊C411yescharacteristic
102C22×C4×C59C22⋊C414yes—
103C22×C4×C59C22⋊C411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105M16×C59—42no—
106M16×C59—42no—
107C2×C8×C59—42no—
108C4⋊C4×C59—24no—
109C4⋊C4×C59—28no—
110C4⋊C4×C59—28no—
111C22⋊C4×C59—28no—
112C22⋊C4×C59—22no—
113C22⋊C4×C59—28no—
114C22⋊C4×C59—42no—
115C22⋊C4×C59—42no—
116C22⋊C4×C59C22⋊C414yes—
117C42×C59—24no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—42no—
128C23×C59C2×C22⋊C411yescharacteristic
129C23×C59C2×C22⋊C412yes—
130C23×C59#12012yes—
131Q8×C59—44no—
132Q8×C59—48no—
133Q8×C59—48no—
134D4×C59—44no—
135D4×C59—44no—
136D4×C59—44no—
137C2×C4×C59—28no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—24no—
145C2×C4×C59—24no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—44no—
150C2×C4×C59—44no—
151C2×C4×C59—44no—
152C2×C4×C59—44no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—44no—
156C2×C4×C59—82no—
157C2×C4×C59—84no—
158C2×C4×C59#9912yes—
159C2×C4×C59#9912yes—
160C2×C4×C59#12012yes—
161C8×C59—82no—
162#12692C5911yesmaximal, characteristic, direct product factor, Sylow
163C22×C59—22no—
164C22×C59—22no—
165C22×C59—22no—
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59—42no—
173C22×C59—41no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59#37911yescharacteristic
177C22×C59C2×#9911yescharacteristic
178C22×C59C2×#9911yescharacteristic
179C4×C59—22no—
180C4×C59—22no—
181C4×C59—22no—
182C4×C59—44no—
183C4×C59—41no—
184C4×C59—44no—
185C4×C59—162no—
186C2×#536C2×C5911yescharacteristic
187C2×#352C2×C5914yescomplemented
188#1785C2×C5911yescharacteristic
189#1782C2×C5911yescharacteristic
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—42no—
193C2×C59—42no—
194C2×C59—81no—
195C2×C59#178111yescharacteristic
196C2×C59#212312yes—
197C22×#41C22×C5911yescharacteristic
198C23×Q8C4×C5911yescharacteristic, complemented
199#536—28no—
200C2×C4⋊Q8—21no—
201C2×#124—21no—
202C2×#123C4×C5912yescomplemented
203C2×#123C22×C5912yes—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8—21no—
206C2×#117C4×C5911yescharacteristic, complemented
207C2×#101—21no—
208C2×#100C22×C5912yes—
209C2×#99—21no—
210C2×#99C22×C5912yes—
211#407—21no—
212#381—21no—
213#352—28no—
214C59#1269211yescharacteristic, direct product factor, Sylow
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8D4×C5911yescharacteristic
224C22×D4C2×C4×C5911yescharacteristic
225C22×D4D4×C5911yescharacteristic
226C22×D4C23×C5911yescharacteristic
227C23×C4C2×C4×C5911yescharacteristic
228C23×C4C2×C4×C5912yes—
229C23×C4D4×C5911yescharacteristic
230C23×C4D4×C5911yescharacteristic
231C2×M16—22no—
232C4⋊Q8—28no—
233#124—28no—
234#123—24no—
235#123—24no—
236#123—24no—
237#123D4×C5918yes—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240#117—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C2×C4×C5912yes—
248C2×C42—21no—
249#101—42no—
250#100—24no—
251#99—24no—
252#99—42no—
253C22⋊C8—42no—
254#95—42no—
255C24C22⋊C4×C5911yescharacteristic
256C24C22×C4×C5911yescharacteristic
257C24C2×D4×C5911yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C2×D4×C5914yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C22⋊C4×C5911yescharacteristic
294C22×C4C22⋊C4×C5911yescharacteristic
295C22×C4C22⋊C4×C5911yescharacteristic
296C22×C4C22⋊C4×C5914yes—
297C22×C4C2×D4×C5911yescharacteristic
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—28no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4C22⋊C4×C5914yes—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊C4×C5911yescharacteristic
322C23C2×C22⋊C4×C5912yes—
323C23#120×C5912yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4#99×C5912yes—
352C2×C4#99×C5912yes—
353C2×C4#120×C5912yes—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22C59×#37911yescharacteristic
369C22C2×#99×C5911yescharacteristic
370C22C2×#99×C5911yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—162no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2C59×#178111yescharacteristic
384C2C59×#212312yes—
subgroupG/H
center, socleC22×C59C2×#99
commutatorC24C22×C4×C59
Frattini, cosocleC22×D4C23×C59
prime radicalC22×#41×C59C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C22×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#352C2semidirect
6. C23×Q8×C59C4semidirect
10. C2×#123×C59C4semidirect
14. C2×#117×C59C4semidirect
162. #12692C59direct (N × H)
187. C2×#352C2×C59semidirect
198. C23×Q8C4×C59semidirect
202. C2×#123C4×C59semidirect
206. C2×#117C4×C59semidirect
214. C59#12692direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5362
C2×C59×#3522
C59×#17852
C59×#17822
#1269259