Finite Group Databasebeta

← #28500225

Subgroups — #28500225

Order 15104.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#21425908C212yesmaximal, complemented
2#21422137C212yesmaximal, complemented
3#21422135C212yesmaximal, complemented
4C59×#3152C211yesmaximal, characteristic
5#16837223C2214yes—
6#16837184—22no—
7#16837180—22no—
8C2×#137×C59—22no—
9C59×#532C411yescharacteristic
10C2×C4⋊Q8×C59C411yescharacteristic
11C59×#497—22no—
12C59×#494C411yescharacteristic
13C59×#494C2211yescharacteristic
14C59×#492C411yescharacteristic
15C59×#492C2211yescharacteristic
16C4×M16×C59C2211yescharacteristic
17#14273667—24no—
18C2×#41×C59—21no—
19C22×Q8×C59—21no—
20#137×C59—48no—
21C2×Q16×C59—22no—
22C2×Q16×C59—22no—
23C2×SD16×C59—22no—
24C2×SD16×C59—22no—
25C2×M16×C59D412yes—
26C4⋊Q8×C59—21no—
27C4⋊Q8×C59—41no—
28C4⋊Q8×C59C2×C411yescharacteristic
29C4⋊Q8×C59C2×C411yescharacteristic
30C4⋊Q8×C59C2×C411yescharacteristic
31#127×C59C2×C411yescharacteristic
32#124×C59—21no—
33#121×C59—41no—
34C4×D4×C59—41no—
35C2×C4⋊C4×C59—41no—
36C2×C42×C59C2×C411yescharacteristic
37#97×C59D412yes—
38C4×C8×C59C2×C411yescharacteristic
39C4×C8×C59C2311yescharacteristic
40#41×C59—44no—
41C2×Q8×C59—21no—
42C2×Q8×C59—21no—
43C2×Q8×C59—21no—
44C2×Q8×C59—44no—
45C2×Q8×C59—41no—
46C2×D4×C59—21no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C22×C4×C59—21no—
50C22×C4×C59—41no—
51C22×C4×C59—41no—
52C22×C4×C59C22⋊C411yescharacteristic
53Q16×C59—44no—
54Q16×C59—44no—
55SD16×C59—44no—
56SD16×C59—44no—
57M16×C59—24no—
58C2×C8×C59C22⋊C412yes—
59C2×C8×C59C2×D412yes—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C22⋊C4×C59—42no—
65C42×C59—21no—
66C42×C59C22⋊C411yescharacteristic
67C42×C59C22×C411yescharacteristic
68C59⋊C16—44no—
69C23×C59—41no—
70C23×C59#10011yescharacteristic
71Q8×C59—42no—
72Q8×C59—42no—
73Q8×C59—42no—
74Q8×C59—82no—
75D4×C59—42no—
76D4×C59—82no—
77D4×C59—82no—
78C2×C4×C59—21no—
79C2×C4×C59—22no—
80C2×C4×C59—21no—
81C2×C4×C59—42no—
82C2×C4×C59—41no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—82no—
86C2×C4×C59—82no—
87C2×C4×C59#10011yescharacteristic
88C2×C4×C59C2×C22⋊C411yescharacteristic
89C8×C59—22no—
90C8×C59AGL(1,ℤ/8)14yes—
91#11993—591nomaximal, Sylow
92C22×C59—21no—
93C22×C59—41no—
94C22×C59—82no—
95C22×C59—81no—
96C22×C59C2×#10011yescharacteristic
97C4×C59—22no—
98C4×C59—21no—
99C4×C59—41no—
100C4×C59—81no—
101C4×C59—81no—
102C4×C59—81no—
103C4×C59#41812yes—
104#3152D5911yescharacteristic
105#2165—592no—
106#1359—592no—
107#1358—592no—
108C2×C59—21no—
109C2×C59—41no—
110C2×C59—161no—
111C2×C59#155511yescharacteristic
112C2×#137—22no—
113#532Dic5911yescharacteristic
114C2×C4⋊Q8Dic5911yescharacteristic
115#497—22no—
116#494Dic5911yescharacteristic
117#494C2×D5911yescharacteristic
118#492Dic5911yescharacteristic
119#492C2×D5911yescharacteristic
120C4×M16C2×D5911yescharacteristic
121#364—594no—
122#349—1182no—
123#347—1182no—
124C59#1199311yescharacteristic, complemented, Sylow
125C2×#41—21no—
126C22×Q8—21no—
127#137—48no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×SD16—22no—
131C2×SD16—22no—
132C2×M16C22⋊D5912yes—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×Dic5911yescharacteristic
136C4⋊Q8C2×Dic5911yescharacteristic
137C4⋊Q8C2×Dic5911yescharacteristic
138#127C2×Dic5911yescharacteristic
139#124—21no—
140#121—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C42C2×Dic5911yescharacteristic
144M32—1184no—
145#97C22⋊D5912yes—
146C4×C8C2×Dic5911yescharacteristic
147C4×C8C22×D5911yescharacteristic
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—44no—
153C2×Q8—41no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊Dic5911yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊Dic5912yes—
167C2×C8C2×C22⋊D5912yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊Dic5911yescharacteristic
175C42C22×Dic5911yescharacteristic
176C16—2364no—
177C23—41no—
178C23#1427364411yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#1427364411yescharacteristic
196C2×C4C2×C22⋊Dic5911yescharacteristic
197C8—22no—
198C8#1427375314yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#1427364411yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4#1683740712yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2#2142281011yescharacteristic
subgroupG/H
centerC2#21422810
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#3152C2
socleC2×C59#1555
prime radicalC59×#532C4
cosocleC4×C8×C59C23
hypercenter#3152D59
hypocenterC59#11993
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #21425908C2semidirect
2. #21422137C2semidirect
3. #21422135C2semidirect
124. C59#11993semidirect

Maximal subgroups

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

subgroupindex
#214259082
#214221372
#214221352
C59×#31522
#1199359