Finite Group Databasebeta

← C59×#12206 #28424315

Subgroups — C59×#12206 #28424315

Order 15104.

500 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2815C211yesmaximal, characteristic
2C59×#2030C211yesmaximal, characteristic, complemented
3C59×#1596C212yesmaximal, complemented
4C59×#1343C212yesmaximal, complemented
5C59×#1317C211yesmaximal, characteristic, complemented
6C59×#543C2211yescharacteristic
7C2×#121×C59—21no—
8C59×#450—22no—
9C59×#397—22no—
10C59×#392—22no—
11C59×#392—21no—
12C59×#392C2212yes—
13C59×#381—21no—
14C59×#381C2211yescharacteristic
15C59×#342—21no—
16C59×#342C2211yescharacteristic
17C59×#328—22no—
18C59×#323C2212yes—
19C22×D4×C59—21no—
20C22×D4×C59—22no—
21C22×D4×C59—21no—
22C22×D4×C59D412yes—
23C23×C4×C59D411yescharacteristic
24C2×SD16×C59—42no—
25C22⋊Q8×C59—22no—
26C22⋊Q8×C59—42no—
27#121×C59—42no—
28#121×C59—42no—
29C2×C4⋊C4×C59—21no—
30C2×C22⋊C4×C59—22no—
31C2×C22⋊C4×C59—21no—
32C2×C22⋊C4×C59—21no—
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59D411yescharacteristic
36C2×C22⋊C4×C59C2311yescharacteristic
37#102×C59—42no—
38C22⋊C8×C59—22no—
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—42no—
42#95×C59D412yes—
43C24×C59—22no—
44C24×C59C2×D411yescharacteristic
45C2×Q8×C59—22no—
46C2×D4×C59—42no—
47C2×D4×C59—42no—
48C2×D4×C59—42no—
49C2×D4×C59—42no—
50C2×D4×C59—82no—
51C2×D4×C59—81no—
52C2×D4×C59—81no—
53C2×D4×C59—82no—
54C2×D4×C59—82no—
55C2×D4×C59—82no—
56C2×D4×C59—81no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59SD1612yescomplemented
66C22×C4×C59C2×D412yes—
67SD16×C59—84no—
68C2×C8×C59—42no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—42no—
72C22⋊C4×C59—41no—
73C22⋊C4×C59—44no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—82no—
80C22⋊C4×C59—82no—
81C23×C59—21no—
82C23×C59—21no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59#12011yescharacteristic
92C23×C59C2×SD1611yescharacteristic
93C23×C59AGL(1,ℤ/8)11yescharacteristic
94Q8×C59—82no—
95D4×C59—44no—
96D4×C59—82no—
97D4×C59—84no—
98D4×C59—84no—
99D4×C59—84no—
100D4×C59—84no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—81no—
114C2×C4×C59—82no—
115C2×C4×C59—81no—
116C2×C4×C59—82no—
117C8×C59—82no—
118#12206C5911yesmaximal, characteristic, direct product factor, Sylow
119C22×C59—22no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—42no—
125C22×C59—81no—
126C22×C59—82no—
127C22×C59—82no—
128C22×C59—82no—
129C22×C59—82no—
130C22×C59—82no—
131C22×C59—82no—
132C22×C59—82no—
133C22×C59#45011yescharacteristic
134C22×C59#45511yescharacteristic
135C22×C59B(4,𝔽2)11yescharacteristic
136C4×C59—42no—
137C4×C59—42no—
138C4×C59—82no—
139C4×C59—81no—
140C4×C59—82no—
141C4×C59—161no—
142#2815C2×C5911yescharacteristic
143#2030C2×C5911yescharacteristic, complemented
144#1596C2×C5912yescomplemented
145#1343C2×C5912yescomplemented
146#1317C2×C5911yescharacteristic, complemented
147C2×C59—21no—
148C2×C59—21no—
149C2×C59—41no—
150C2×C59—41no—
151C2×C59—162no—
152C2×C59#160111yescharacteristic
153C2×C59#220012yes—
154#543C22×C5911yescharacteristic
155C2×#121—21no—
156#450—22no—
157#397—22no—
158#392—22no—
159#392—21no—
160#392C22×C5912yes—
161#381—21no—
162#381C22×C5911yescharacteristic
163#342—21no—
164#342C22×C5911yescharacteristic
165#328—22no—
166#323C22×C5912yes—
167C59#1220611yescharacteristic, direct product factor, Sylow
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4D4×C5912yes—
172C23×C4D4×C5911yescharacteristic
173C2×SD16—42no—
174C22⋊Q8—22no—
175C22⋊Q8—42no—
176#121—42no—
177#121—42no—
178C2×C4⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4D4×C5911yescharacteristic
185C2×C22⋊C4C23×C5911yescharacteristic
186#102—42no—
187C22⋊C8—22no—
188#95—21no—
189#95—22no—
190#95—42no—
191#95D4×C5912yes—
192C24—22no—
193C24C2×D4×C5911yescharacteristic
194C2×Q8—22no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4SD16×C5912yescomplemented
215C22×C4C2×D4×C5912yes—
216SD16—84no—
217C2×C8—42no—
218C4⋊C4—22no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23#120×C5911yescharacteristic
241C23C2×SD16×C5911yescharacteristic
242C23AGL(1,ℤ/8)×C5911yescharacteristic
243Q8—82no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—84no—
249D4—84no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C8—82no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22C59×#45011yescharacteristic
282C22C59×#45511yescharacteristic
283C22C59×B(4,𝔽2)11yescharacteristic
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—82no—
289C4—161no—
290C2—21no—
291C2—21no—
292C2—41no—
293C2—41no—
294C2—162no—
295C2C59×#160111yescharacteristic
296C2C59×#220012yes—
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C59
prime radicalC59×#2815C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C22⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2030C2semidirect
3. C59×#1596C2semidirect
4. C59×#1343C2semidirect
5. C59×#1317C2semidirect
65. C22×C4×C59SD16semidirect
118. #12206C59direct (N × H)
143. #2030C2×C59semidirect
144. #1596C2×C59semidirect
145. #1343C2×C59semidirect
146. #1317C2×C59semidirect
167. C59#12206direct (N × H)
214. C22×C4SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#28152
C59×#20302
C59×#15962
C59×#13432
C59×#13172
#1220659