Finite Group Databasebeta

← #28503086

Subgroups — #28503086

Order 15104.

172 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 55 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#21426372C211yesmaximal, characteristic, complemented
2#21426273C211yesmaximal, characteristic
3#21426269C211yesmaximal, characteristic
4#21425932C211yesmaximal, characteristic
5#21422171C211yesmaximal, characteristic, complemented
6#21422158C211yesmaximal, characteristic, complemented
7C59×#1366C211yesmaximal, characteristic, complemented
8#16837820—21no—
9#16837804C2211yescharacteristic
10#16837520C2211yescharacteristic
11#16837229—21no—
12#16837227—21no—
13#16837225—21no—
14#16837224—21no—
15#16837223C2211yescharacteristic
16#16837213—21no—
17#16837211—21no—
18#16837187—21no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#499C2211yescharacteristic
21C59×#364C2211yescharacteristic
22C59×#364C2211yescharacteristic
23#14273749D411yescharacteristic
24#14273733—21no—
25#14273719—21no—
26#14273667—21no—
27#14273666—41no—
28C2×C59⋊C16D411yescharacteristic
29C2×C59⋊C16D411yescharacteristic
30C16×D59—41no—
31#14273663D411yescharacteristic
32#14273661—21no—
33#14273659—21no—
34#14273655—21no—
35#125×C59—21no—
36M32×C59—21no—
37M32×C59—21no—
38#107×C59—21no—
39#106×C59—21no—
40C4×C8×C59C2311yescharacteristic
41#11745685—21no—
42C2×C8×C59C2×D411yescharacteristic
43C2×C8×C59C2×D411yescharacteristic
44C4⋊C4×C59—41no—
45C4⋊C4×C59—81no—
46C42×C59#4111yescharacteristic
47C4⋊Dic59—41no—
48Dic59⋊C4—81no—
49#11745664—21no—
50Dic236Q1611yescharacteristic
51D472Q1611yescharacteristic
52#11745660—21no—
53#11745659—41no—
54C8×D59—41no—
55C16×C59—41no—
56C16×C59—41no—
57C59⋊C16—21no—
58C59⋊C16—41no—
59C59⋊C16SD1611yescharacteristic
60C59⋊C16SD1611yescharacteristic
61C2×C4×C59—21no—
62C2×C4×C59—81no—
63C2×C4×C59#12111yescharacteristic
64C22⋊D59—41no—
65C2×Dic59—81no—
66D236—21no—
67C4×D59—41no—
68Dic118—21no—
69C8×C59C2×SD1611yescharacteristic
70C8×C59C2×Q1611yescharacteristic
71C8×C59#13511yescharacteristic
72C8×C59#13711yescharacteristic
73C59⋊C8—41no—
74#12464—591nomaximal, Sylow
75C22×C59#47211yescharacteristic
76C2×D59—41no—
77C4×C59—41no—
78C4×C59—161no—
79C4×C59#46111yescharacteristic
80C4×C59#46711yescharacteristic
81Dic59—41no—
82Dic59—161no—
83#2253—591no—
84#2231—591no—
85#2231—591no—
86#2170—591no—
87#1373—591no—
88#1366—591no—
89#1366D5911yescharacteristic, complemented
90C2×C59—21no—
91C2×C59#167511yescharacteristic
92D59—81no—
93#504—1181no—
94#499—591no—
95#499C2×D5911yescharacteristic
96#443—591no—
97#367—1181no—
98#366—1181no—
99#365—1181no—
100#365—1181no—
101#364—591no—
102#364C2×D5911yescharacteristic
103#364C2×D5911yescharacteristic
104#360—1181no—
105#360—1181no—
106#350—1181no—
107C4×C16—591no—
108C59#1246411yescharacteristic, complemented, Sylow
109#135—591no—
110#131—1181no—
111#125—21no—
112#125—1181no—
113M32—21no—
114M32—21no—
115M32—1181no—
116M32—2361no—
117C2×C16—591no—
118C2×C16—591no—
119C2×C16—2361no—
120#108—591no—
121#107—21no—
122#107—1181no—
123#106—21no—
124#106—1181no—
125#104—1181no—
126C4×C8C22×D5911yescharacteristic
127#41—1181no—
128Q16—591no—
129SD16—1181no—
130D8—591no—
131M16—1181no—
132M16—2361no—
133C2×C8—2361no—
134C2×C8D4×D5911yescharacteristic
135C2×C8D4×D5911yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—81no—
138C4⋊C4—4721no—
139C4⋊C4—2361no—
140C42#1174568511yescharacteristic
141C16—41no—
142C16—41no—
143C16—591no—
144C16—591no—
145C16—1181no—
146C16—2361no—
147Q8—1181no—
148D4—1181no—
149D4—2361no—
150C2×C4—21no—
151C2×C4—81no—
152C2×C4—4721no—
153C2×C4—2361no—
154C2×C4#1427370011yescharacteristic
155C8—2361no—
156C8SD16×D5911yescharacteristic
157C8Q16×D5911yescharacteristic
158C8#1427374811yescharacteristic
159C8#1427375911yescharacteristic
160C22—2361no—
161C22#1683767111yescharacteristic
162C4—41no—
163C4—161no—
164C4—9441no—
165C4—2361no—
166C4#1683760611yescharacteristic
167C4#1683764211yescharacteristic
168C2—21no—
169C2—4721no—
170C2#2142349411yescharacteristic
subgroupG/H
centerC2#21423494
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1366C2
socleC2×C59#1675
prime radical#14273749D4
hypercenter#1366D59
hypocenterC59#12464
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
1. #21426372C2semidirect
5. #21422171C2semidirect
6. #21422158C2semidirect
7. C59×#1366C2semidirect
89. #1366D59semidirect
108. C59#12464semidirect

Maximal subgroups

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

subgroupindex
#214263722
#214262732
#214262692
#214259322
#214221712
#214221582
C59×#13662
#1246459