Finite Group Databasebeta

← C59×#12220 #28424329

Subgroups — C59×#12220 #28424329

Order 15104.

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

normal subgroups only

This group has 460 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×#2683C211yesmaximal, characteristic
2C59×#2027C211yesmaximal, characteristic
3C59×#1605C212yesmaximal, complemented
4C59×#1354C212yesmaximal, complemented
5C59×#1318C211yesmaximal, characteristic
6C59×#544C2211yescharacteristic
7C2×C22⋊Q8×C59—22no—
8C59×#451—22no—
9C59×#418—21no—
10C59×#396—22no—
11C59×#395C2212yes—
12C59×#390C2211yescharacteristic
13C59×#389—21no—
14C59×#386—21no—
15C59×#343C2211yescharacteristic
16C59×#339—21no—
17C59×#328—22no—
18C59×#324C2212yes—
19C22×Q8×C59D412yes—
20C22×D4×C59D411yescharacteristic
21C23×C4×C59—21no—
22C2×Q16×C59—42no—
23C2×M16×C59—21no—
24C4⋊Q8×C59—41no—
25#125×C59—41no—
26C22⋊Q8×C59—22no—
27C22⋊Q8×C59—48no—
28#117×C59—21no—
29C2×C4⋊C4×C59—21no—
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59—22no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59C2311yescharacteristic
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—21no—
37C2×C42×C59D411yescharacteristic
38#103×C59—42no—
39#102×C59—41no—
40#102×C59—41no—
41C22⋊C8×C59—22no—
42#95×C59—22no—
43#95×C59D412yes—
44C24×C59—21no—
45C2×Q8×C59—22no—
46C2×Q8×C59—44no—
47C2×Q8×C59—42no—
48C2×D4×C59—21no—
49C2×D4×C59—41no—
50C2×D4×C59—41no—
51C2×D4×C59Q1612yescomplemented
52C22×C4×C59—22no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—44no—
56C22×C4×C59—81no—
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D412yes—
59Q16×C59—84no—
60M16×C59—41no—
61M16×C59—41no—
62C2×C8×C59—42no—
63C2×C8×C59—41no—
64C4⋊C4×C59—24no—
65C4⋊C4×C59—21no—
66C4⋊C4×C59—21no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—44no—
70C4⋊C4×C59—44no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—81no—
73C22⋊C4×C59—44no—
74C22⋊C4×C59—44no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—81no—
77C42×C59—41no—
78C42×C59—41no—
79C23×C59—22no—
80C23×C59—42no—
81C23×C59—42no—
82C23×C59—41no—
83C23×C59—41no—
84C23×C59#12011yescharacteristic
85Q8×C59—44no—
86Q8×C59—82no—
87Q8×C59—82no—
88D4×C59—42no—
89D4×C59—81no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—24no—
94C2×C4×C59—22no—
95C2×C4×C59—42no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—84no—
106C2×C4×C59C2×Q1611yescharacteristic
107C2×C4×C59#13711yescharacteristic
108C8×C59—81no—
109C8×C59—82no—
110#12220C5911yesmaximal, characteristic, direct product factor, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59—44no—
114C22×C59—42no—
115C22×C59—82no—
116C22×C59—81no—
117C22×C59—81no—
118C22×C59#45111yescharacteristic
119C22×C59#45511yescharacteristic
120C22×C59B(4,𝔽2)11yescharacteristic
121C4×C59—21no—
122C4×C59—21no—
123C4×C59—42no—
124C4×C59—42no—
125C4×C59—44no—
126C4×C59—82no—
127C4×C59—81no—
128C4×C59—81no—
129#2683C2×C5911yescharacteristic
130#2027C2×C5911yescharacteristic
131#1605C2×C5912yescomplemented
132#1354C2×C5912yescomplemented
133#1318C2×C5911yescharacteristic
134C2×C59—21no—
135C2×C59—21no—
136C2×C59—82no—
137C2×C59#160211yescharacteristic
138C2×C59#219611yescharacteristic
139C2×C59#220311yescharacteristic
140#544C22×C5911yescharacteristic
141C2×C22⋊Q8—22no—
142#451—22no—
143#418—21no—
144#396—22no—
145#395C22×C5912yes—
146#390C22×C5911yescharacteristic
147#389—21no—
148#386—21no—
149#343C22×C5911yescharacteristic
150#339—21no—
151#328—22no—
152#324C22×C5912yes—
153C59#1222011yescharacteristic, direct product factor, Sylow
154C22×Q8D4×C5912yes—
155C22×D4D4×C5911yescharacteristic
156C23×C4—21no—
157C2×Q16—42no—
158C2×M16—21no—
159C4⋊Q8—41no—
160#125—41no—
161C22⋊Q8—22no—
162C22⋊Q8—48no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C23×C5911yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42D4×C5911yescharacteristic
173#103—42no—
174#102—41no—
175#102—41no—
176C22⋊C8—22no—
177#95—22no—
178#95D4×C5912yes—
179C24—21no—
180C2×Q8—22no—
181C2×Q8—44no—
182C2×Q8—42no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4Q16×C5912yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×D4×C5911yescharacteristic
193C22×C4C2×D4×C5912yes—
194Q16—84no—
195M16—41no—
196M16—41no—
197C2×C8—42no—
198C2×C8—41no—
199C4⋊C4—24no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—81no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—41no—
211C22⋊C4—81no—
212C42—41no—
213C42—41no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#120×C5911yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—82no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—84no—
241C2×C4C2×Q16×C5911yescharacteristic
242C2×C4#137×C5911yescharacteristic
243C8—81no—
244C8—82no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22C59×#45111yescharacteristic
253C22C59×#45511yescharacteristic
254C22C59×B(4,𝔽2)11yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2C59×#160211yescharacteristic
267C2C59×#219611yescharacteristic
268C2C59×#220311yescharacteristic
subgroupG/H
center, socleC22×C59B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C59
prime radicalC22×D4×C59D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C59 ◁ C2×C4⋊C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
3. C59×#1605C2semidirect
4. C59×#1354C2semidirect
51. C2×D4×C59Q16semidirect
110. #12220C59direct (N × H)
131. #1605C2×C59semidirect
132. #1354C2×C59semidirect
153. C59#12220direct (N × H)
186. C2×D4Q16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#26832
C59×#20272
C59×#16052
C59×#13542
C59×#13182
#1222059