Finite Group Databasebeta

← C59×#7824 #28419933

Subgroups — C59×#7824 #28419933

Order 15104.

418 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2724C211yesmaximal, characteristic
2C59×#2100C212yesmaximal, complemented
3C59×#2094C211yesmaximal, characteristic
4C59×#1271C212yesmaximal, complemented
5C59×#1270C211yesmaximal, characteristic
6C2×#123×C59—21no—
7C2×#106×C59—21no—
8C2×#102×C59—21no—
9C59×#398—21no—
10C59×#397—22no—
11C59×#390C2211yescharacteristic
12C59×#389C2212yes—
13C59×#385—22no—
14C59×#384C2211yescharacteristic
15C59×#340—22no—
16C59×#340—22no—
17C59×#336C2212yes—
18C59×#336C2211yescharacteristic
19C22×D4×C59D411yescharacteristic
20C23×C4×C59—21no—
21C22×C8×C59—21no—
22C22×C8×C59—22no—
23#123×C59—42no—
24#123×C59—42no—
25C2×C4⋊C4×C59—22no—
26C2×C4⋊C4×C59—22no—
27C2×C4⋊C4×C59—21no—
28C2×C4⋊C4×C59—21no—
29C2×C4⋊C4×C59D412yes—
30C2×C4⋊C4×C59D412yes—
31C2×C4⋊C4×C59D411yescharacteristic
32C2×C22⋊C4×C59—22no—
33C2×C22⋊C4×C59—21no—
34C2×C22⋊C4×C59—22no—
35C2×C42×C59C2311yescharacteristic
36#106×C59—42no—
37#102×C59—42no—
38#95×C59—22no—
39#95×C59—21no—
40#95×C59—22no—
41C24×C59—21no—
42C2×D4×C59—21no—
43C2×D4×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59SD1612yes—
46C22×C4×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59—22no—
49C22×C4×C59—42no—
50C22×C4×C59—41no—
51C22×C4×C59—81no—
52C22×C4×C59—81no—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D412yes—
55C2×C8×C59—22no—
56C2×C8×C59—41no—
57C2×C8×C59—41no—
58C2×C8×C59—42no—
59C2×C8×C59—42no—
60C2×C8×C59—42no—
61C4⋊C4×C59—22no—
62C4⋊C4×C59—22no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—44no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—81no—
71C22⋊C4×C59—42no—
72C22⋊C4×C59—44no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—41no—
76C23×C59—22no—
77C23×C59—42no—
78C23×C59—41no—
79C23×C59—42no—
80C23×C59—41no—
81C23×C59#12011yescharacteristic
82D4×C59—42no—
83D4×C59—81no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—84no—
97C2×C4×C59—81no—
98C2×C4×C59—82no—
99C2×C4×C59C2×SD1611yescharacteristic
100C2×C4×C59#13512yes—
101C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C59#13712yes—
103C8×C59—42no—
104C8×C59—82no—
105#7824C5911yesmaximal, characteristic, direct product factor, Sylow
106C22×C59—42no—
107C22×C59—44no—
108C22×C59—82no—
109C22×C59—81no—
110C22×C59—81no—
111C22×C59#45011yescharacteristic
112C22×C59#45212yes—
113C22×C59#45311yescharacteristic
114C22×C59#45412yes—
115C22×C59#45811yescharacteristic
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—22no—
119C4×C59—22no—
120C4×C59—82no—
121C4×C59—81no—
122C4×C59—82no—
123C4×C59—161no—
124#2724C2×C5911yescharacteristic
125#2100C2×C5912yescomplemented
126#2094C2×C5911yescharacteristic
127#1271C2×C5912yescomplemented
128#1270C2×C5911yescharacteristic
129C2×C59—82no—
130C2×C59#161311yescharacteristic
131C2×C59#161512yes—
132C2×C59#163011yescharacteristic
133C2×C59#163512yes—
134C2×C59#165711yescharacteristic
135C2×#123—21no—
136C2×#106—21no—
137C2×#102—21no—
138#398—21no—
139#397—22no—
140#390C22×C5911yescharacteristic
141#389C22×C5912yes—
142#385—22no—
143#384C22×C5911yescharacteristic
144#340—22no—
145#340—22no—
146#336C22×C5911yescharacteristic
147#336C22×C5912yes—
148C59#782411yescharacteristic, direct product factor, Sylow
149C22×D4D4×C5911yescharacteristic
150C23×C4—21no—
151C22×C8—21no—
152C22×C8—22no—
153#123—42no—
154#123—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4D4×C5912yes—
160C2×C4⋊C4D4×C5912yes—
161C2×C4⋊C4D4×C5911yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42C23×C5911yescharacteristic
166#106—42no—
167#102—42no—
168#95—22no—
169#95—21no—
170#95—22no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4SD16×C5912yes—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×D4×C5911yescharacteristic
184C22×C4C2×D4×C5912yes—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#120×C5911yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4C2×SD16×C5911yescharacteristic
230C2×C4#135×C5912yes—
231C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
232C2×C4#137×C5912yes—
233C8—42no—
234C8—82no—
235C22—42no—
236C22—44no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22C59×#45011yescharacteristic
241C22C59×#45212yes—
242C22C59×#45311yescharacteristic
243C22C59×#45412yes—
244C22C59×#45811yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—161no—
253C2—82no—
254C2C59×#161311yescharacteristic
255C2C59×#161512yes—
256C2C59×#163011yescharacteristic
257C2C59×#163512yes—
258C2C59×#165711yescharacteristic
subgroupG/H
center, socleC23×C59#120
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC22×D4×C59D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#2100C2semidirect
4. C59×#1271C2semidirect
105. #7824C59direct (N × H)
125. #2100C2×C59semidirect
127. #1271C2×C59semidirect
148. C59#7824direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#27242
C59×#21002
C59×#20942
C59×#12712
C59×#12702
#782459