Finite Group Databasebeta

← #28499543

Subgroups — #28499543

Order 15104.

280 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 71 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#21434841C211yesmaximal, characteristic
2#21424245C211yesmaximal, characteristic
3#21422091C212yesmaximal
4#21422085C212yesmaximal
5C59×#2165C211yesmaximal, characteristic
6#16837959—21no—
7#16837800C411yescharacteristic, complemented
8#16837782C411yescharacteristic, complemented
9#16837753—21no—
10#16837660—21no—
11#16837660C2212yes—
12#16837510—21no—
13#16837510C412yescomplemented
14#16837173—21no—
15#16837162—21no—
16C59⋊#338C2211yescharacteristic
17#16837159C2211yescharacteristic
18C4×M16×C59C2211yescharacteristic
19C59×#364—21no—
20C59×#349C2212yes—
21C59×#347—21no—
22C2×Dic236—21no—
23C2×Dic236C2×C412yes—
24C2×#11745664—21no—
25#14273727—21no—
26#14273727—21no—
27#14273719—41no—
28#14273706—22no—
29#14273706—41no—
30C4×Dic118—22no—
31C4×Dic118—41no—
32#14273686—21no—
33#14273686—41no—
34#14273659—21no—
35#14273659C2×C412yes—
36#14273652—22no—
37#14273652—22no—
38C2×M16×C59D411yescharacteristic
39C2×M16×C59C2311yescharacteristic
40C2×C42×C59D411yescharacteristic
41M32×C59—22no—
42#97×C59C2×C411yescharacteristic
43#97×C59D411yescharacteristic
44C4×C8×C59C2×C411yescharacteristic
45C4×C8×C59D411yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59C2×D411yescharacteristic
48C2×Dic118—22no—
49C2×Dic118—41no—
50M16×C59—41no—
51M16×C59—41no—
52C2×C8×C59C22⋊C411yescharacteristic
53C2×C8×C59C22⋊C411yescharacteristic
54C2×C8×C59C22×C411yescharacteristic
55C2×C8×C59C2×D411yescharacteristic
56C42×C59—21no—
57C42×C59C22⋊C411yescharacteristic
58C42×C59C22⋊C411yescharacteristic
59C22⋊Dic59—42no—
60C22⋊Dic59—41no—
61C4⋊Dic59—22no—
62C4⋊Dic59—41no—
63C4⋊Dic59—41no—
64C4⋊Dic59—41no—
65C4⋊Dic59—41no—
66Dic59⋊C4—42no—
67Dic59⋊C4—42no—
68C4×Dic59—42no—
69C4×Dic59—41no—
70#11745664—41no—
71#11745664—41no—
72C2×C59⋊C8—41no—
73Dic236—42no—
74Dic236—42no—
75C16×C59—42no—
76C23×C59AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—42no—
82C2×C4×C59C2×C22⋊C411yescharacteristic
83C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
84C2×Dic59—42no—
85C2×Dic59—42no—
86C2×Dic59—81no—
87Dic118—42no—
88Dic118—42no—
89Dic118—82no—
90C8×C59—21no—
91C8×C59—21no—
92C8×C59—21no—
93C8×C59—21no—
94C59⋊C8—81no—
95#11838—591nomaximal, Sylow
96C22×C59—21no—
97C22×C59—41no—
98C22×C59#41811yescharacteristic
99C4×C59—22no—
100C4×C59—21no—
101C4×C59—41no—
102C4×C59#41011yescharacteristic
103C4×C59#42111yescharacteristic
104Dic59—82no—
105Dic59—82no—
106Dic59—161no—
107#3185—591no—
108#2165D5911yescharacteristic
109#1841—591no—
110#1342—592no—
111#1340—592no—
112C2×C59—21no—
113C2×C59—41no—
114C2×C59#151411yescharacteristic
115#533—1181no—
116#498—591no—
117#494—591no—
118#487—1181no—
119#470—592no—
120#470—1181no—
121#441—592no—
122#441—1181no—
123C4×M16C2×D5911yescharacteristic
124#364—21no—
125#349C2×D5912yes—
126#347—21no—
127#344—1181no—
128#339—1181no—
129#338—591no—
130H(3,ℤ/4)—591no—
131C59#1183811yescharacteristic, complemented, Sylow
132C2×Q16—592no—
133C2×Q16—1181no—
134C2×M16—1181no—
135C2×M16D23611yescharacteristic
136C2×M16C22×D5911yescharacteristic
137C4⋊Q8—1181no—
138C4⋊Q8—1181no—
139#125—2361no—
140C22⋊Q8—1182no—
141C22⋊Q8—2361no—
142C4×Q8—1182no—
143C4×Q8—2361no—
144#117—1181no—
145#117—2361no—
146C2×C42D23611yescharacteristic
147M32—22no—
148#106—592no—
149#106—1181no—
150#103—1182no—
151#103—1182no—
152#97C4×D5911yescharacteristic
153#97C22⋊D5911yescharacteristic
154C4×C8C4×D5911yescharacteristic
155C4×C8C22⋊D5911yescharacteristic
156C2×Q8—1182no—
157C2×Q8—2361no—
158C22×C4—21no—
159C22×C4C2×D23611yescharacteristic
160Q16—2362no—
161Q16—2362no—
162M16—41no—
163M16—41no—
164M16—2361no—
165M16—2361no—
166C2×C8—2361no—
167C2×C8(C2×D59)⋊C411yescharacteristic
168C2×C8(C2×D59)⋊C411yescharacteristic
169C2×C8C2×C4×D5911yescharacteristic
170C2×C8C2×C22⋊D5911yescharacteristic
171C4⋊C4—1182no—
172C4⋊C4—2361no—
173C4⋊C4—2361no—
174C4⋊C4—2361no—
175C4⋊C4—2362no—
176C4⋊C4—2361no—
177C4⋊C4—2362no—
178C22⋊C4—2361no—
179C22⋊C4—2362no—
180C42—21no—
181C42—2361no—
182C42—2362no—
183C42(C2×D59)⋊C411yescharacteristic
184C42(C2×D59)⋊C411yescharacteristic
185C16—42no—
186C23#1427375511yescharacteristic
187Q8—4722no—
188Q8—2362no—
189Q8—2362no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—4721no—
196C2×C4—2362no—
197C2×C4—2362no—
198C2×C4C2×(C2×D59)⋊C411yescharacteristic
199C2×C4#1427375511yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—4721no—
205C22—21no—
206C22—41no—
207C22#1683740811yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—9441no—
212C4—4722no—
213C4—4722no—
214C4#1683737511yescharacteristic
215C4#1683741811yescharacteristic
216C2—21no—
217C2—41no—
218C2#2142264411yescharacteristic
subgroupG/H
centerC2#21422644
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2165C2
socleC2×C59#1514
prime radicalC23×C59AGL(1,ℤ/8)
cosocleC2×M16×C59C23
hypercenter#2165D59
hypocenterC59#11838
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
7. #16837800C4semidirect
8. #16837782C4semidirect
13. #16837510C4semidirect
131. C59#11838semidirect

Maximal subgroups

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

subgroupindex
#214348412
#214242452
#214220912
#214220852
C59×#21652
#1183859