Finite Group Databasebeta

← #28509312

Subgroups — #28509312

Order 15104.

194 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 63 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#21426409C211yesmaximal, characteristic
2#21425983C211yesmaximal, characteristic, complemented
3#21422301C211yesmaximal, characteristic, complemented
4#21422286C211yesmaximal, characteristic, complemented
5#21422282C211yesmaximal, characteristic, complemented
6#21422232C211yesmaximal, characteristic, complemented
7C59×#2177C211yesmaximal, characteristic, complemented
8#16837834C2211yescharacteristic
9#16837520—21no—
10#16837240C412yescomplemented
11#16837237C412yescomplemented
12#16837236C2211yescharacteristic
13#16837236C2211yescharacteristic
14#16837232C2211yescharacteristic
15#16837210—21no—
16C2×D16×C59C2211yescharacteristic
17C4×D8×C59—21no—
18C59×#367C2211yescharacteristic
19C59×#357—21no—
20C59×#346C2211yescharacteristic
21#14273749—21no—
22#14273733—41no—
23Dic472C2×C411yescharacteristic, complemented
24#14273672—21no—
25D944C2×C411yescharacteristic, complemented
26#14273663—21no—
27#14273655—41no—
28C59⋊C32—21no—
29C59⋊C32C2×C412yescomplemented
30C2×D8×C59—21no—
31C4×D4×C59—41no—
32D16×C59—21no—
33D16×C59C2×C412yes—
34C2×C16×C59D411yescharacteristic
35C2×C16×C59C2311yescharacteristic
36#107×C59—21no—
37#102×C59—41no—
38C4×C8×C59D411yescharacteristic
39C2×D4×C59—41no—
40C22×C4×C59—81no—
41#11745685—41no—
42D8×C59—22no—
43D8×C59—41no—
44C2×C8×C59D811yescharacteristic
45C2×C8×C59C2×D411yescharacteristic
46C4⋊C4×C59—41no—
47C22⋊C4×C59—81no—
48C42×C59D811yescharacteristic
49#11745664—41no—
50Dic236—21no—
51D472—21no—
52#11745660—41no—
53#11745659—81no—
54C8×D59—81no—
55C16×C59—21no—
56C16×C59C22×C411yescharacteristic
57C16×C59#4111yescharacteristic
58C23×C59—81no—
59D4×C59—42no—
60D4×C59—81no—
61C2×C4×C59—21no—
62C2×C4×C59—81no—
63C2×C4×C59—162no—
64C2×C4×C59C2×D811yescharacteristic
65C22⋊D59—81no—
66D236—41no—
67C4×D59—81no—
68Dic118—41no—
69C8×C59—21no—
70C8×C59C4×D411yescharacteristic
71C8×C59#13511yescharacteristic
72C59⋊C8—81no—
73#13658—591nomaximal, Sylow
74C22×C59—82no—
75C22×C59—161no—
76C22×C59—161no—
77C22×C59#50911yescharacteristic
78C2×D59—81no—
79C4×C59—41no—
80C4×C59—161no—
81C4×C59C4×D811yescharacteristic
82C4×C59#50911yescharacteristic
83Dic59—81no—
84#2263—591no—
85#2179—591no—
86#2177D5911yescharacteristic, complemented
87#1426—591no—
88#1419—591no—
89#1418—591no—
90#1398—591no—
91C2×C59—21no—
92C2×C59—162no—
93C2×C59#217711yescharacteristic
94D59—161no—
95#508—591no—
96C2×D16C2×D5911yescharacteristic
97#443—1181no—
98C4×D8—21no—
99SD64—592no—
100D32—592no—
101M64—591no—
102M64—591no—
103#368—591no—
104#367C2×D5911yescharacteristic
105#359—1181no—
106#357—21no—
107#346C2×D5911yescharacteristic
108C59#1365811yescharacteristic, complemented, Sylow
109#135—1181no—
110C2×D8—21no—
111#131—2361no—
112C4×D4—41no—
113Q32—591no—
114SD32—1181no—
115D16—21no—
116D16—591no—
117D16C4×D5912yes—
118C2×C16C22⋊D5911yescharacteristic
119C2×C16C22×D5911yescharacteristic
120#108—1181no—
121#107—21no—
122#104—2361no—
123#102—41no—
124C4×C8C22⋊D5911yescharacteristic
125C32—592no—
126C32—1181no—
127#41—2361no—
128C2×D4—41no—
129C22×C4—81no—
130Q16—1181no—
131SD16—2361no—
132D8—22no—
133D8—41no—
134D8—1181no—
135M16—4721no—
136M16—2361no—
137C2×C8—4721no—
138C2×C8#1174566911yescharacteristic
139C2×C8C2×C22⋊D5911yescharacteristic
140C4⋊C4—41no—
141C22⋊C4—81no—
142C42#1174566911yescharacteristic
143C16—21no—
144C16C2×C4×D5911yescharacteristic
145C16#1174568511yescharacteristic
146C23—81no—
147Q8—2361no—
148D4—42no—
149D4—81no—
150D4—4721no—
151D4—2361no—
152C2×C4—21no—
153C2×C4—81no—
154C2×C4—162no—
155C2×C4—4721no—
156C2×C4C2×#1174566911yescharacteristic
157C8—21no—
158C8—4721no—
159C8C4×C22⋊D5911yescharacteristic
160C8#1427374711yescharacteristic
161C22—82no—
162C22—161no—
163C22—161no—
164C22—4721no—
165C22#1683783811yescharacteristic
166C4—41no—
167C4—161no—
168C4—4721no—
169C4C4×#1174566911yescharacteristic
170C4#1683783711yescharacteristic
171C2—21no—
172C2—162no—
173C2—9441no—
174C2#2142597011yescharacteristic
subgroupG/H
centerC2#21425970
commutatorC16×C59C22×C4
FrattiniC2×C16C22×D59
FittingC59×#2177C2
socleC2×C59#2177
prime radical#21426409C2
cosocleC2×C16×C59C23
hypercenter#2177D59
hypocenterC59#13658
LCSG ▷ C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #21425983C2semidirect
3. #21422301C2semidirect
4. #21422286C2semidirect
5. #21422282C2semidirect
6. #21422232C2semidirect
7. C59×#2177C2semidirect
10. #16837240C4semidirect
11. #16837237C4semidirect
23. Dic472C2×C4semidirect
25. D944C2×C4semidirect
29. C59⋊C32C2×C4semidirect
86. #2177D59semidirect
108. C59#13658semidirect

Maximal subgroups

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

subgroupindex
#214264092
#214259832
#214223012
#214222862
#214222822
#214222322
C59×#21772
#1365859