Finite Group Databasebeta

← C59×#10258 #28422367

Subgroups — C59×#10258 #28422367

Order 15104.

526 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3249C211yesmaximal, characteristic, complemented
2C59×#3077C212yesmaximal
3C59×#1954C211yesmaximal, characteristic
4C59×#1493C212yesmaximal, complemented
5C59×#1292C211yesmaximal, characteristic, complemented
6C59×#533C2212yes—
7C2×C4×D4×C59C2211yescharacteristic
8C59×#478—24no—
9C59×#477—22no—
10C59×#476—22no—
11C59×#474—22no—
12C59×#467—22no—
13C59×#462—22no—
14C59×#428—22no—
15C59×#423C2212yes—
16C2×C4⋊C8×C59C2211yescharacteristic
17C59×#418—22no—
18C2×C22⋊C8×C59—22no—
19C2×C4×C8×C59C2211yescharacteristic
20C59×Q8⋊C8—24no—
21C22×D4×C59D411yescharacteristic
22C23×C4×C59—21no—
23C2×Q16×C59—22no—
24C2×M16×C59—22no—
25C22×C8×C59—21no—
26C22×C8×C59D412yes—
27C4⋊Q8×C59D412yes—
28C4⋊Q8×C59D412yes—
29#125×C59—22no—
30C22⋊Q8×C59—22no—
31C22⋊Q8×C59—22no—
32C4×Q8×C59—22no—
33C4×Q8×C59—22no—
34C4×D4×C59—22no—
35C4×D4×C59—22no—
36#117×C59—22no—
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C22⋊C4×C59—21no—
39C2×C42×C59C2311yescharacteristic
40#107×C59—24no—
41#107×C59—22no—
42#106×C59—24no—
43C4⋊C8×C59—22no—
44C4⋊C8×C59—22no—
45C4⋊C8×C59D414yes—
46#103×C59—24no—
47#102×C59—24no—
48#102×C59—24no—
49C22⋊C8×C59—44no—
50C4×C8×C59—22no—
51C24×C59—21no—
52C2×Q8×C59—22no—
53C2×Q8×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—21no—
56C2×D4×C59—22no—
57C2×D4×C59Q1612yescomplemented
58C22×C4×C59—21no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59#4111yescharacteristic
66Q16×C59—82no—
67M16×C59—42no—
68M16×C59—42no—
69C2×C8×C59—22no—
70C2×C8×C59—24no—
71C2×C8×C59—22no—
72C2×C8×C59—41no—
73C2×C8×C59—41no—
74C2×C8×C59—42no—
75C4⋊C4×C59—22no—
76C4⋊C4×C59—22no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—42no—
82C22⋊C4×C59—42no—
83C22⋊C4×C59—42no—
84C42×C59—42no—
85C42×C59C2×D412yes—
86C42×C59C2×D412yes—
87C23×C59—22no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59#12111yescharacteristic
93Q8×C59—44no—
94Q8×C59—42no—
95Q8×C59—42no—
96D4×C59—24no—
97D4×C59—42no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—82no—
106C2×C4×C59—81no—
107C2×C4×C59—81no—
108C2×C4×C59#12012yes—
109C2×C4×C59#12011yescharacteristic
110C2×C4×C59#12112yes—
111C2×C4×C59#12711yescharacteristic
112C2×C4×C59C2×Q1611yescharacteristic
113C2×C4×C59#13511yescharacteristic
114C2×C4×C59#13712yes—
115C8×C59—42no—
116C8×C59—44no—
117C8×C59—42no—
118#10258C5911yesmaximal, characteristic, direct product factor, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—44no—
122C22×C59—42no—
123C22×C59—82no—
124C22×C59—81no—
125C22×C59—81no—
126C22×C59#39211yescharacteristic
127C22×C59#46711yescharacteristic
128C22×C59#47011yescharacteristic
129C4×C59—21no—
130C4×C59—22no—
131C4×C59—81no—
132C4×C59—82no—
133C4×C59—82no—
134C4×C59#45112yes—
135C4×C59#45212yes—
136C4×C59#45311yescharacteristic
137C4×C59#45411yescharacteristic
138#3249C2×C5911yescharacteristic, complemented
139#3077C2×C5912yes—
140#1954C2×C5911yescharacteristic
141#1493C2×C5912yescomplemented
142#1292C2×C5911yescharacteristic, complemented
143C2×C59—21no—
144C2×C59—21no—
145C2×C59—82no—
146C2×C59#200111yescharacteristic
147C2×C59#200611yescharacteristic
148C2×C59#201711yescharacteristic
149#533C22×C5912yes—
150C2×C4×D4C22×C5911yescharacteristic
151#478—24no—
152#477—22no—
153#476—22no—
154#474—22no—
155#467—22no—
156#462—22no—
157#428—22no—
158#423C22×C5912yes—
159C2×C4⋊C8C22×C5911yescharacteristic
160#418—22no—
161C2×C22⋊C8—22no—
162C2×C4×C8C22×C5911yescharacteristic
163Q8⋊C8—24no—
164C59#1025811yescharacteristic, direct product factor, Sylow
165C22×D4D4×C5911yescharacteristic
166C23×C4—21no—
167C2×Q16—22no—
168C2×M16—22no—
169C22×C8—21no—
170C22×C8D4×C5912yes—
171C4⋊Q8D4×C5912yes—
172C4⋊Q8D4×C5912yes—
173#125—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—22no—
181C2×C4⋊C4D4×C5911yescharacteristic
182C2×C22⋊C4—21no—
183C2×C42C23×C5911yescharacteristic
184#107—24no—
185#107—22no—
186#106—24no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8D4×C5914yes—
190#103—24no—
191#102—24no—
192#102—24no—
193C22⋊C8—44no—
194C4×C8—22no—
195C24—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4Q16×C5912yescomplemented
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C2×D4×C5911yescharacteristic
208C22×C4C2×D4×C5911yescharacteristic
209C22×C4#41×C5911yescharacteristic
210Q16—82no—
211M16—42no—
212M16—42no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—42no—
229C42C2×D4×C5912yes—
230C42C2×D4×C5912yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23#121×C5911yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—24no—
241D4—42no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4#120×C5912yes—
253C2×C4#120×C5911yescharacteristic
254C2×C4#121×C5912yes—
255C2×C4#127×C5911yescharacteristic
256C2×C4C2×Q16×C5911yescharacteristic
257C2×C4#135×C5911yescharacteristic
258C2×C4#137×C5912yes—
259C8—42no—
260C8—44no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—42no—
266C22—82no—
267C22—81no—
268C22—81no—
269C22C59×#39211yescharacteristic
270C22C59×#46711yescharacteristic
271C22C59×#47011yescharacteristic
272C4—21no—
273C4—22no—
274C4—81no—
275C4—82no—
276C4—82no—
277C4C59×#45112yes—
278C4C59×#45212yes—
279C4C59×#45311yescharacteristic
280C4C59×#45411yescharacteristic
281C2—21no—
282C2—21no—
283C2—82no—
284C2C59×#200111yescharacteristic
285C2C59×#200611yescharacteristic
286C2C59×#201711yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC22×D4×C59D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3249C2semidirect
4. C59×#1493C2semidirect
5. C59×#1292C2semidirect
57. C2×D4×C59Q16semidirect
118. #10258C59direct (N × H)
138. #3249C2×C59semidirect
141. #1493C2×C59semidirect
142. #1292C2×C59semidirect
164. C59#10258direct (N × H)
201. C2×D4Q16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32492
C59×#30772
C59×#19542
C59×#14932
C59×#12922
#1025859