Finite Group Databasebeta

← #28502070

Subgroups — #28502070

Order 15104.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#21426313C212yesmaximal
2#21425671C211yesmaximal, characteristic
3#21422084C212yesmaximal
4#21422047C211yesmaximal, characteristic
5C59×#3255C211yesmaximal, characteristic
6#16837451C2211yescharacteristic
7#16837224—22no—
8#16837218—22no—
9#16837184C2212yes—
10#16837174—21no—
11#16837162—21no—
12#16837159C2211yescharacteristic
13C59×#533—21no—
14C59×#475—22no—
15C59×#470—21no—
16C59×#470C2212yes—
17C59×#462—22no—
18C59×#423C2211yescharacteristic
19C2×#11745664—21no—
20#14273686—21no—
21#14273667—22no—
22#14273663—22no—
23C2×Q16×C59—21no—
24C2×Q16×C59D412yes—
25C2×M16×C59D411yescharacteristic
26C2×M16×C59C2311yescharacteristic
27C4⋊Q8×C59—21no—
28C4⋊Q8×C59—21no—
29#125×C59—41no—
30C22⋊Q8×C59—22no—
31C22⋊Q8×C59—41no—
32C4×Q8×C59—22no—
33C4×Q8×C59—41no—
34#117×C59—41no—
35C2×C42×C59D411yescharacteristic
36#106×C59—21no—
37#106×C59D412yes—
38C4⋊C8×C59—22no—
39#103×C59—22no—
40#103×C59—22no—
41C2×Q8×C59—22no—
42C2×Q8×C59—41no—
43C22×C4×C59—21no—
44C22×C4×C59C2×D411yescharacteristic
45Q16×C59—42no—
46Q16×C59—81no—
47M16×C59—41no—
48M16×C59—41no—
49C2×C8×C59—21no—
50C2×C8×C59C2×D412yes—
51C4⋊C4×C59—22no—
52C4⋊C4×C59—41no—
53C4⋊C4×C59—41no—
54C4⋊C4×C59—42no—
55C4⋊C4×C59—41no—
56C4⋊C4×C59—42no—
57C22⋊C4×C59—42no—
58C42×C59—21no—
59C42×C59—42no—
60C42×C59SD1611yescharacteristic
61C42×C59SD1611yescharacteristic
62C22⋊Dic59—41no—
63C4⋊Dic59—41no—
64C4×Dic59—41no—
65#11745664—41no—
66#11745664—41no—
67C2×C59⋊C8—41no—
68C59⋊C16—42no—
69C23×C59AGL(1,ℤ/8)11yescharacteristic
70Q8×C59—42no—
71Q8×C59—42no—
72Q8×C59—82no—
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—42no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59#12011yescharacteristic
81C2×C4×C59C2×SD1611yescharacteristic
82C2×Dic59—81no—
83C8×C59—22no—
84C8×C59—41no—
85C59⋊C8—81no—
86#12308—591nomaximal, Sylow
87C22×C59—21no—
88C22×C59—41no—
89C22×C59#45011yescharacteristic
90C4×C59—22no—
91C4×C59—21no—
92C4×C59—41no—
93C4×C59—82no—
94C4×C59—82no—
95C4×C59#45411yescharacteristic
96C4×C59#45811yescharacteristic
97Dic59—161no—
98#3255D5911yescharacteristic
99#2239—592no—
100#2102—591no—
101#1340—592no—
102#1327—591no—
103C2×C59—21no—
104C2×C59—41no—
105C2×C59#161611yescharacteristic
106#533—21no—
107#475—22no—
108#470—21no—
109#470C2×D5912yes—
110#462—22no—
111#430—591no—
112#423C2×D5911yescharacteristic
113#365—1182no—
114#362—1182no—
115#349—592no—
116#344—1181no—
117#339—1181no—
118H(3,ℤ/4)—591no—
119C59#1230811yescharacteristic, complemented, Sylow
120C2×Q16—21no—
121C2×Q16C22⋊D5912yes—
122C2×M16—1181no—
123C2×M16C22⋊D5911yescharacteristic
124C2×M16C22×D5911yescharacteristic
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—41no—
132#117—41no—
133#117—1181no—
134C2×C42C22⋊D5911yescharacteristic
135M32—1182no—
136#108—1182no—
137#106—21no—
138#106C22⋊D5912yes—
139C4⋊C8—22no—
140#103—22no—
141#103—22no—
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×C22⋊D5911yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—2361no—
151M16—2361no—
152C2×C8—21no—
153C2×C8—2361no—
154C2×C8C2×C22⋊D5912yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—2361no—
162C22⋊C4—42no—
163C22⋊C4—2361no—
164C42—21no—
165C42—42no—
166C42—2361no—
167C42#1174567011yescharacteristic
168C42#1174567011yescharacteristic
169C16—2362no—
170C23#1427375311yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—4721no—
182C2×C4#1427369611yescharacteristic
183C2×C4C2×#1174567011yescharacteristic
184C8—22no—
185C8—41no—
186C8—4721no—
187C22—21no—
188C22—41no—
189C22#1683755311yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—9441no—
196C4#1683757511yescharacteristic
197C4#1683759211yescharacteristic
198C2—21no—
199C2—41no—
200C2#2142313311yescharacteristic
subgroupG/H
centerC2#21423133
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#3255C2
socleC2×C59#1616
prime radicalC23×C59AGL(1,ℤ/8)
hypercenter#3255D59
hypocenterC59#12308
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3255

Complements

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

NH ≅ G/Ntype
119. C59#12308semidirect

Maximal subgroups

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

subgroupindex
#214263132
#214256712
#214220842
#214220472
C59×#32552
#1230859