Finite Group Databasebeta

← C22×C59×#458 #28472502

Subgroups — C22×C59×#458 #28472502

Order 15104.

5294 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 1134 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
1C22×H(5,𝔽2)×C59C211yesmaximal, characteristic
2C22×#123×C59C213yesmaximal, complemented
3C2×C59×#458C2124yesmaximal, direct product factor
4C22×#99×C59C213yesmaximal, complemented
5C2×H(5,𝔽2)×C59—218no—
6C2×H(5,𝔽2)×C59C22112yes—
7C22×#41×C59—23no—
8C23×D4×C59—23no—
9C23×D4×C59C2213yes—
10C24×C4×C59C2211yescharacteristic, complemented
11C2×#123×C59—218no—
12C2×#123×C59C22136yescomplemented
13C22×C4⋊C4×C59—23no—
14C22×C22⋊C4×C59—23no—
15C22×C22⋊C4×C59C2213yescomplemented
16C59×#458C22164yesdirect product factor
17C2×#99×C59C22136yescomplemented
18C25×C59—21no—
19C25×C59D413yes—
20C25×C59C2311yescharacteristic
21H(5,𝔽2)×C59—248no—
22H(5,𝔽2)×C59—272no—
23H(5,𝔽2)×C59C23116yes—
24C2×#41×C59—236no—
25C2×#41×C59—236no—
26C22×Q8×C59—21no—
27C22×D4×C59—23no—
28C22×D4×C59—236no—
29C22×D4×C59—26no—
30C22×D4×C59—236no—
31C22×D4×C59—218no—
32C22×D4×C59D4118yes—
33C22×D4×C59C23118yes—
34C23×C4×C59—23no—
35C23×C4×C59—23no—
36C23×C4×C59—23no—
37C23×C4×C59—29no—
38C23×C4×C59D413yes—
39C23×C4×C59C2316yescomplemented
40#123×C59—272no—
41#123×C59C23148yescomplemented
42C2×C4⋊C4×C59—236no—
43C2×C22⋊C4×C59—236no—
44C2×C22⋊C4×C59—29no—
45C2×C22⋊C4×C59C23118yescomplemented
46#99×C59C23148yescomplemented
47C24×C59—212no—
48C24×C59—23no—
49C24×C59—29no—
50C24×C59—44no—
51C24×C59—412no—
52C24×C59—84no—
53C24×C59C2×D413yes—
54C24×C59C2×D4118yes—
55C24×C59C2×D419yes—
56C24×C59C2413yes—
57#41×C59—2144no—
58#41×C59—248no—
59C2×Q8×C59—212no—
60C2×D4×C59—236no—
61C2×D4×C59—2144no—
62C2×D4×C59—236no—
63C2×D4×C59—236no—
64C2×D4×C59—236no—
65C2×D4×C59—248no—
66C2×D4×C59—236no—
67C2×D4×C59C2×D4136yes—
68C2×D4×C59C24112yes—
69C22×C4×C59—236no—
70C22×C4×C59—29no—
71C22×C4×C59—23no—
72C22×C4×C59—218no—
73C22×C4×C59—218no—
74C22×C4×C59—218no—
75C22×C4×C59—412no—
76C22×C4×C59—49no—
77C22×C4×C59—46no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D4118yes—
80C22×C4×C59C2414yescomplemented
81C4⋊C4×C59—248no—
82C22⋊C4×C59—248no—
83C22⋊C4×C59—218no—
84C22⋊C4×C59C24112yescomplemented
85C23×C59—212no—
86C23×C59—216no—
87C23×C59—218no—
88C23×C59—218no—
89C23×C59—418no—
90C23×C59—46no—
91C23×C59—436no—
92C23×C59—412no—
93C23×C59—818no—
94C23×C59—824no—
95C23×C59#12019yes—
96C23×C59#12016yes—
97C23×C59#12011yescharacteristic
98C23×C59C22×D419yes—
99C23×C59C22×D4112yes—
100C23×C59C22×D419yes—
101C23×C59C2511yescharacteristic
102Q8×C59—216no—
103D4×C59—248no—
104D4×C59—272no—
105D4×C59—224no—
106C2×C4×C59—248no—
107C2×C4×C59—218no—
108C2×C4×C59—218no—
109C2×C4×C59—212no—
110C2×C4×C59—212no—
111C2×C4×C59—436no—
112C2×C4×C59—418no—
113C2×C4×C59—46no—
114C2×C4×C59C22×D4112yes—
115C22×#458C5911yesmaximal, characteristic, direct product factor, Sylow
116C22×C59—218no—
117C22×C59—218no—
118C22×C59—212no—
119C22×C59—46no—
120C22×C59—49no—
121C22×C59—41no—
122C22×C59—412no—
123C22×C59—436no—
124C22×C59—818no—
125C22×C59—816no—
126C22×C59#45814yesdirect product factor
127C22×C59C2×#12019yes—
128C22×C59C2×#12013yes—
129C22×C59C23×D413yes—
130C4×C59—212no—
131C4×C59—412no—
132C4×C59—44no—
133C22×H(5,𝔽2)C2×C5911yescharacteristic
134C22×#123C2×C5913yescomplemented
135C2×#458C2×C59124yesdirect product factor
136C22×#99C2×C5913yescomplemented
137C2×C59—23no—
138C2×C59—29no—
139C2×C59—412no—
140C2×C59C2×#45816yesdirect product factor
141C2×C59C22×#12011yescharacteristic
142C2×H(5,𝔽2)—218no—
143C2×H(5,𝔽2)C22×C59112yes—
144C22×#41—23no—
145C23×D4—23no—
146C23×D4C22×C5913yes—
147C24×C4C22×C5911yescharacteristic, complemented
148C2×#123—218no—
149C2×#123C22×C59136yescomplemented
150C22×C4⋊C4—23no—
151C22×C22⋊C4—23no—
152C22×C22⋊C4C22×C5913yescomplemented
153#458C22×C59164yesdirect product factor
154C2×#99C22×C59136yescomplemented
155C59C22×#45811yescharacteristic, direct product factor, Sylow
156C25—21no—
157C25D4×C5913yes—
158C25C23×C5911yescharacteristic
159H(5,𝔽2)—248no—
160H(5,𝔽2)—272no—
161H(5,𝔽2)C23×C59116yes—
162C2×#41—236no—
163C2×#41—236no—
164C22×Q8—21no—
165C22×D4—23no—
166C22×D4—236no—
167C22×D4—26no—
168C22×D4—236no—
169C22×D4—218no—
170C22×D4D4×C59118yes—
171C22×D4C23×C59118yes—
172C23×C4—23no—
173C23×C4—23no—
174C23×C4—23no—
175C23×C4—29no—
176C23×C4D4×C5913yes—
177C23×C4C23×C5916yescomplemented
178#123—272no—
179#123C23×C59148yescomplemented
180C2×C4⋊C4—236no—
181C2×C22⋊C4—236no—
182C2×C22⋊C4—29no—
183C2×C22⋊C4C23×C59118yescomplemented
184#99C23×C59148yescomplemented
185C24—212no—
186C24—23no—
187C24—29no—
188C24—44no—
189C24—412no—
190C24—84no—
191C24C2×D4×C5913yes—
192C24C2×D4×C59118yes—
193C24C2×D4×C5919yes—
194C24C24×C5913yes—
195#41—2144no—
196#41—248no—
197C2×Q8—212no—
198C2×D4—236no—
199C2×D4—2144no—
200C2×D4—236no—
201C2×D4—236no—
202C2×D4—236no—
203C2×D4—248no—
204C2×D4—236no—
205C2×D4C2×D4×C59136yes—
206C2×D4C24×C59112yes—
207C22×C4—236no—
208C22×C4—23no—
209C22×C4—29no—
210C22×C4—218no—
211C22×C4—218no—
212C22×C4—218no—
213C22×C4—412no—
214C22×C4—49no—
215C22×C4—46no—
216C22×C4—41no—
217C22×C4C2×D4×C59118yes—
218C22×C4C24×C5914yescomplemented
219C4⋊C4—248no—
220C22⋊C4—248no—
221C22⋊C4—218no—
222C22⋊C4C24×C59112yescomplemented
223C23—212no—
224C23—216no—
225C23—218no—
226C23—218no—
227C23—418no—
228C23—46no—
229C23—436no—
230C23—412no—
231C23—818no—
232C23—824no—
233C23#120×C5919yes—
234C23#120×C5916yes—
235C23#120×C5911yescharacteristic
236C23C22×D4×C5919yes—
237C23C22×D4×C59112yes—
238C23C22×D4×C5919yes—
239C23C25×C5911yescharacteristic
240Q8—216no—
241D4—248no—
242D4—272no—
243D4—224no—
244C2×C4—248no—
245C2×C4—218no—
246C2×C4—218no—
247C2×C4—212no—
248C2×C4—212no—
249C2×C4—436no—
250C2×C4—418no—
251C2×C4—46no—
252C2×C4C22×D4×C59112yes—
253C22—218no—
254C22—218no—
255C22—212no—
256C22—46no—
257C22—49no—
258C22—41no—
259C22—412no—
260C22—436no—
261C22—818no—
262C22—816no—
263C22C59×#45814yesdirect product factor
264C22C2×#120×C5919yes—
265C22C2×#120×C5913yes—
266C22C23×D4×C5913yes—
267C4—212no—
268C4—412no—
269C4—44no—
270C2—23no—
271C2—29no—
272C2—412no—
273C2C2×C59×#45816yesdirect product factor
274C2C22×#120×C5911yescharacteristic
subgroupG/H
center, socleC23×C59#120
commutator, Frattini, cosocleC23C25×C59
prime radicalC22×H(5,𝔽2)×C59C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C25×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C59C2semidirect
3. C2×C59×#458C2direct (N × H)
4. C22×#99×C59C2semidirect
10. C24×C4×C59C22semidirect
12. C2×#123×C59C22semidirect
15. C22×C22⋊C4×C59C22semidirect
16. C59×#458C22direct (N × H)
17. C2×#99×C59C22semidirect
39. C23×C4×C59C23semidirect
41. #123×C59C23semidirect
45. C2×C22⋊C4×C59C23semidirect
46. #99×C59C23semidirect
80. C22×C4×C59C24semidirect
84. C22⋊C4×C59C24semidirect
115. C22×#458C59direct (N × H)
126. C22×C59#458direct (N × H)
134. C22×#123C2×C59semidirect
135. C2×#458C2×C59direct (N × H)
136. C22×#99C2×C59semidirect
140. C2×C59C2×#458direct (N × H)
147. C24×C4C22×C59semidirect
149. C2×#123C22×C59semidirect
152. C22×C22⋊C4C22×C59semidirect
153. #458C22×C59direct (N × H)
154. C2×#99C22×C59semidirect
155. C59C22×#458direct (N × H)
177. C23×C4C23×C59semidirect
179. #123C23×C59semidirect
183. C2×C22⋊C4C23×C59semidirect
184. #99C23×C59semidirect
218. C22×C4C24×C59semidirect
222. C22⋊C4C24×C59semidirect
263. C22C59×#458direct (N × H)
273. C2C2×C59×#458direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C592
C22×#123×C592
C2×C59×#4582
C22×#99×C592
C22×#45859