Finite Group Databasebeta

← C59×#11771 #28423880

Subgroups — C59×#11771 #28423880

Order 15104.

292 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3177C211yesmaximal, characteristic
2C59×#2166C211yesmaximal, characteristic
3C59×#1747C211yesmaximal, characteristic
4C59×#1342C212yesmaximal
5C59×#1340C212yesmaximal
6C59×#541—21no—
7C59×#499C411yescharacteristic, complemented
8C59×#494C411yescharacteristic, complemented
9C59×#491—21no—
10C59×#470C2212yes—
11C59×#467—21no—
12C59×#441C412yescomplemented
13C4×Q16×C59—21no—
14C59×#405—21no—
15C59×#405C2211yescharacteristic
16C59×#364—21no—
17C59×#349C2212yes—
18C59×#347—21no—
19C59×#338C2211yescharacteristic
20C59×H(3,ℤ/4)—21no—
21C59×H(3,ℤ/4)C2211yescharacteristic
22C2×Q16×C59—21no—
23C2×Q16×C59C2×C412yes—
24C2×M16×C59—21no—
25C2×M16×C59C2311yescharacteristic
26C22×C8×C59D411yescharacteristic
27C4⋊Q8×C59—21no—
28#126×C59—42no—
29#125×C59—21no—
30#125×C59—42no—
31C22⋊Q8×C59—22no—
32C4×Q8×C59—22no—
33#117×C59—21no—
34#117×C59D411yescharacteristic
35M32×C59—22no—
36#107×C59—21no—
37#106×C59C2×C412yes—
38#103×C59—22no—
39#103×C59—22no—
40#97×C59—22no—
41#97×C59C2×C411yescharacteristic
42#97×C59D411yescharacteristic
43C4×C8×C59—22no—
44C4×C8×C59C2×C411yescharacteristic
45C4×C8×C59D411yescharacteristic
46C2×Q8×C59—22no—
47C22×C4×C59C2×D411yescharacteristic
48Q16×C59—42no—
49Q16×C59—41no—
50Q16×C59—41no—
51M16×C59—41no—
52M16×C59—41no—
53M16×C59—41no—
54C2×C8×C59—21no—
55C2×C8×C59—21no—
56C2×C8×C59—22no—
57C2×C8×C59C22⋊C411yescharacteristic
58C2×C8×C59C22⋊C411yescharacteristic
59C2×C8×C59C22×C411yescharacteristic
60C2×C8×C59C2×D411yescharacteristic
61C4⋊C4×C59—21no—
62C4⋊C4×C59—22no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—42no—
66C4⋊C4×C59—42no—
67C4⋊C4×C59—42no—
68C22⋊C4×C59—21no—
69C22⋊C4×C59—41no—
70C22⋊C4×C59—42no—
71C42×C59—22no—
72C42×C59—42no—
73C42×C59C22⋊C411yescharacteristic
74C42×C59C22⋊C411yescharacteristic
75C16×C59—42no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77Q8×C59—42no—
78Q8×C59—42no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59C2×C22⋊C411yescharacteristic
87C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
88C8×C59—21no—
89C8×C59—21no—
90C8×C59—21no—
91C8×C59—42no—
92C8×C59#10411yescharacteristic
93C8×C59#10411yescharacteristic
94#11771C5911yesmaximal, characteristic, direct product factor, Sylow
95C22×C59—21no—
96C22×C59—41no—
97C22×C59#41811yescharacteristic
98C4×C59—21no—
99C4×C59—41no—
100C4×C59—41no—
101C4×C59—82no—
102C4×C59—82no—
103C4×C59—82no—
104C4×C59#41011yescharacteristic
105C4×C59C2×#10411yescharacteristic
106#3177C2×C5911yescharacteristic
107#2166C2×C5911yescharacteristic
108#1747C2×C5911yescharacteristic
109#1342C2×C5912yes—
110#1340C2×C5912yes—
111C2×C59—21no—
112C2×C59—41no—
113C2×C59#150511yescharacteristic
114#541—21no—
115#499C4×C5911yescharacteristic, complemented
116#494C4×C5911yescharacteristic, complemented
117#491—21no—
118#470C22×C5912yes—
119#467—21no—
120#441C4×C5912yescomplemented
121C4×Q16—21no—
122#405—21no—
123#405C22×C5911yescharacteristic
124#364—21no—
125#349C22×C5912yes—
126#347—21no—
127#338C22×C5911yescharacteristic
128H(3,ℤ/4)—21no—
129H(3,ℤ/4)C22×C5911yescharacteristic
130C59#1177111yescharacteristic, direct product factor, Sylow
131C2×Q16—21no—
132C2×Q16C2×C4×C5912yes—
133C2×M16—21no—
134C2×M16C23×C5911yescharacteristic
135C22×C8D4×C5911yescharacteristic
136C4⋊Q8—21no—
137#126—42no—
138#125—21no—
139#125—42no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142#117—21no—
143#117D4×C5911yescharacteristic
144M32—22no—
145#107—21no—
146#106C2×C4×C5912yes—
147#103—22no—
148#103—22no—
149#97—22no—
150#97C2×C4×C5911yescharacteristic
151#97D4×C5911yescharacteristic
152C4×C8—22no—
153C4×C8C2×C4×C5911yescharacteristic
154C4×C8D4×C5911yescharacteristic
155C2×Q8—22no—
156C22×C4C2×D4×C5911yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8C22⋊C4×C5911yescharacteristic
167C2×C8C22⋊C4×C5911yescharacteristic
168C2×C8C22×C4×C5911yescharacteristic
169C2×C8C2×D4×C5911yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—42no—
180C42—22no—
181C42—42no—
182C42C22⋊C4×C5911yescharacteristic
183C42C22⋊C4×C5911yescharacteristic
184C16—42no—
185C23AGL(1,ℤ/8)×C5911yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4C2×C22⋊C4×C5911yescharacteristic
196C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8#104×C5911yescharacteristic
202C8#104×C5911yescharacteristic
203C22—21no—
204C22—41no—
205C22C59×#41811yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—82no—
212C4C59×#41011yescharacteristic
213C4C2×#104×C5911yescharacteristic
214C2—21no—
215C2—41no—
216C2C59×#150511yescharacteristic
subgroupG/H
center, socleC2×C59#1505
commutatorC2×C8C22×C4×C59
Frattini, cosocleC2×M16C23×C59
prime radicalC23×C59AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C2×C4×C59 ◁ C59×#405 ◁ G

Complements

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

NH ≅ G/Ntype
7. C59×#499C4semidirect
8. C59×#494C4semidirect
12. C59×#441C4semidirect
94. #11771C59direct (N × H)
115. #499C4×C59semidirect
116. #494C4×C59semidirect
120. #441C4×C59semidirect
130. C59#11771direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31772
C59×#21662
C59×#17472
C59×#13422
C59×#13402
#1177159