Finite Group Databasebeta

← C59×#13604 #28425713

Subgroups — C59×#13604 #28425713

Order 15104.

322 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3142C211yesmaximal, characteristic
2C59×#2132C211yesmaximal, characteristic
3C59×#2126C211yesmaximal, characteristic
4C59×#1414C214yesmaximal, complemented
5C59×#532C2211yescharacteristic
6C2×#125×C59C411yescharacteristic, complemented
7C59×#491—21no—
8C59×#490—21no—
9C59×#488C412yescomplemented
10C59×#488C2212yes—
11C59×#417—21no—
12C59×#417—21no—
13C59×#413—21no—
14C59×#410—21no—
15C4×M16×C59C411yescharacteristic, complemented
16C59×#356—21no—
17C59×#355C2212yes—
18C59×#354C2212yes—
19C59×#353—21no—
20C2×#41×C59D411yescharacteristic
21C2×#41×C59D411yescharacteristic
22C2×M16×C59—21no—
23C2×M16×C59—41no—
24C4⋊Q8×C59D411yescharacteristic
25#127×C59D411yescharacteristic
26#125×C59—21no—
27#125×C59—41no—
28#125×C59C2×C412yes—
29#124×C59C2×C411yescharacteristic
30#124×C59C2311yescharacteristic
31#121×C59—41no—
32C4×D4×C59—41no—
33#117×C59—41no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—41no—
36C2×C42×C59C2×C411yescharacteristic
37#103×C59—22no—
38#103×C59—22no—
39#102×C59—22no—
40#102×C59—22no—
41#101×C59—22no—
42#100×C59—41no—
43#99×C59—22no—
44#99×C59—41no—
45#97×C59—21no—
46C4×C8×C59C2×C412yes—
47#41×C59—41no—
48#41×C59—41no—
49#41×C59—41no—
50#41×C59—41no—
51C2×Q8×C59C22⋊C411yescharacteristic
52C2×Q8×C59C2×D411yescharacteristic
53C2×D4×C59—21no—
54C2×D4×C59—21no—
55C2×D4×C59—41no—
56C2×D4×C59C22⋊C411yescharacteristic
57C2×D4×C59C2×D411yescharacteristic
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63M16×C59—41no—
64M16×C59—41no—
65M16×C59—42no—
66M16×C59—81no—
67C2×C8×C59—22no—
68C2×C8×C59—81no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—81no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—42no—
78C42×C59—21no—
79C42×C59—81no—
80C42×C59C22⋊C411yescharacteristic
81C42×C59C22×C411yescharacteristic
82C23×C59—21no—
83C23×C59—21no—
84C23×C59#9911yescharacteristic
85Q8×C59—41no—
86Q8×C59—41no—
87D4×C59—41no—
88D4×C59—41no—
89D4×C59—81no—
90D4×C59—81no—
91D4×C59—82no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—42no—
103C2×C4×C59—81no—
104C2×C4×C59—82no—
105C2×C4×C59—82no—
106C2×C4×C59#9911yescharacteristic
107C2×C4×C59C2×C22⋊C411yescharacteristic
108C8×C59—42no—
109C8×C59—82no—
110#13604C5911yesmaximal, characteristic, direct product factor, Sylow
111C22×C59—21no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—41no—
115C22×C59—81no—
116C22×C59—81no—
117C22×C59C2×#9911yescharacteristic
118C4×C59—21no—
119C4×C59—41no—
120C4×C59—41no—
121C4×C59—41no—
122C4×C59—41no—
123C4×C59—82no—
124C4×C59—162no—
125C4×C59#35411yescharacteristic
126C4×C59#35411yescharacteristic
127#3142C2×C5911yescharacteristic
128#2132C2×C5911yescharacteristic
129#2126C2×C5911yescharacteristic
130#1414C2×C5914yescomplemented
131C2×C59—21no—
132C2×C59—41no—
133C2×C59—81no—
134C2×C59—81no—
135C2×C59C2×#35411yescharacteristic
136#532C22×C5911yescharacteristic
137C2×#125C4×C5911yescharacteristic, complemented
138#491—21no—
139#490—21no—
140#488C4×C5912yescomplemented
141#488C22×C5912yes—
142#417—21no—
143#417—21no—
144#413—21no—
145#410—21no—
146C4×M16C4×C5911yescharacteristic, complemented
147#356—21no—
148#355C22×C5912yes—
149#354C22×C5912yes—
150#353—21no—
151C59#1360411yescharacteristic, direct product factor, Sylow
152C2×#41D4×C5911yescharacteristic
153C2×#41D4×C5911yescharacteristic
154C2×M16—21no—
155C2×M16—41no—
156C4⋊Q8D4×C5911yescharacteristic
157#127D4×C5911yescharacteristic
158#125—21no—
159#125—41no—
160#125C2×C4×C5912yes—
161#124C2×C4×C5911yescharacteristic
162#124C23×C5911yescharacteristic
163#121—41no—
164C4×D4—41no—
165#117—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C42C2×C4×C5911yescharacteristic
169#103—22no—
170#103—22no—
171#102—22no—
172#102—22no—
173#101—22no—
174#100—41no—
175#99—22no—
176#99—41no—
177#97—21no—
178C4×C8C2×C4×C5912yes—
179#41—41no—
180#41—41no—
181#41—41no—
182#41—41no—
183C2×Q8C22⋊C4×C5911yescharacteristic
184C2×Q8C2×D4×C5911yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4C22⋊C4×C5911yescharacteristic
189C2×D4C2×D4×C5911yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4C22⋊C4×C5911yescharacteristic
195M16—41no—
196M16—41no—
197M16—42no—
198M16—81no—
199C2×C8—22no—
200C2×C8—81no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C42—21no—
211C42—81no—
212C42C22⋊C4×C5911yescharacteristic
213C42C22×C4×C5911yescharacteristic
214C23—21no—
215C23—21no—
216C23#99×C5911yescharacteristic
217Q8—41no—
218Q8—41no—
219D4—41no—
220D4—41no—
221D4—81no—
222D4—81no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#99×C5911yescharacteristic
239C2×C4C2×C22⋊C4×C5911yescharacteristic
240C8—42no—
241C8—82no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22C2×#99×C5911yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—162no—
256C4C59×#35411yescharacteristic
257C4C59×#35411yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—81no—
262C2C2×C59×#35411yescharacteristic
subgroupG/H
center, socleC2×C59C2×#354
commutatorC42C22×C4×C59
Frattini, cosocle#124C23×C59
prime radicalC59×#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C22×C4×C59 ◁ C59×#532 ◁ G

Complements

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

NH ≅ G/Ntype
4. C59×#1414C2semidirect
6. C2×#125×C59C4semidirect
9. C59×#488C4semidirect
15. C4×M16×C59C4semidirect
110. #13604C59direct (N × H)
130. #1414C2×C59semidirect
137. C2×#125C4×C59semidirect
140. #488C4×C59semidirect
146. C4×M16C4×C59semidirect
151. C59#13604direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31422
C59×#21322
C59×#21262
C59×#14142
#1360459