Finite Group Databasebeta

← #28499538

Subgroups — #28499538

Order 15104.

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

normal subgroups only

This group has 324 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#21434834C211yesmaximal, characteristic
2#21424245C211yesmaximal, characteristic, complemented
3#21422088C212yesmaximal, complemented
4#21422082C212yesmaximal, complemented
5C59×#2165C211yesmaximal, characteristic, complemented
6#16837952—21no—
7#16837800C411yescharacteristic, complemented
8#16837779C411yescharacteristic, complemented
9#16837749—21no—
10#16837655—21no—
11#16837655C2212yes—
12#16837515—21no—
13#16837515C412yescomplemented
14#16837173—21no—
15#16837162—21no—
16C59⋊#338C2211yescharacteristic
17#16837159C2211yescharacteristic
18C4×M16×C59C2211yescharacteristic
19C59×#364—21no—
20C59×#349C2212yes—
21C59×#347—21no—
22C2×#11745685—41no—
23C2×D472—21no—
24C2×D472C2×C412yes—
25C2×#11745664—21no—
26#14273727—21no—
27#14273724—21no—
28#14273715—41no—
29#14273701—22no—
30#14273701—41no—
31C4×D236—22no—
32C4×D236—41no—
33#14273686—21no—
34#14273659—21no—
35#14273659C2×C412yes—
36#14273649—22no—
37#14273649—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×C22⋊D59—42no—
49#11745685—84no—
50C2×D236—22no—
51C2×D236—41no—
52C2×D236—41no—
53C2×C4×D59—42no—
54C2×Dic118—41no—
55M16×C59—41no—
56M16×C59—41no—
57C2×C8×C59C22⋊C411yescharacteristic
58C2×C8×C59C22⋊C411yescharacteristic
59C2×C8×C59C22×C411yescharacteristic
60C2×C8×C59C2×D411yescharacteristic
61C42×C59—21no—
62C42×C59C22⋊C411yescharacteristic
63C42×C59C22⋊C411yescharacteristic
64C22⋊Dic59—41no—
65(C2×D59)⋊C4—42no—
66(C2×D59)⋊C4—42no—
67C4⋊Dic59—22no—
68C4⋊Dic59—41no—
69C4⋊Dic59—41no—
70C4×Dic59—41no—
71#11745664—41no—
72#11745664—41no—
73C2×C59⋊C8—41no—
74D472—42no—
75D472—42no—
76C16×C59—42no—
77C23×C59AGL(1,ℤ/8)11yescharacteristic
78C22×D59—42no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59C2×C22⋊C411yescharacteristic
85C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
86C22⋊D59—84no—
87C2×Dic59—42no—
88C2×Dic59—81no—
89D236—42no—
90D236—42no—
91D236—82no—
92D236—82no—
93C4×D59—84no—
94Dic118—82no—
95C8×C59—21no—
96C8×C59—21no—
97C8×C59—21no—
98C8×C59—21no—
99C59⋊C8—81no—
100#11837—591nomaximal, Sylow
101C22×C59—21no—
102C22×C59—41no—
103C22×C59#41811yescharacteristic
104C2×D59—82no—
105C2×D59—82no—
106C2×D59—82no—
107C4×C59—22no—
108C4×C59—21no—
109C4×C59—41no—
110C4×C59#41011yescharacteristic
111C4×C59#42111yescharacteristic
112Dic59—82no—
113Dic59—161no—
114#3184—591no—
115#2165D5911yescharacteristic, complemented
116#1841—591no—
117#1341—592no—
118#1339—592no—
119C2×C59—21no—
120C2×C59—41no—
121C2×C59#151411yescharacteristic
122D59—162no—
123#532—1181no—
124#498—591no—
125#493—591no—
126#486—1181no—
127#469—592no—
128#469—1181no—
129#442—592no—
130#442—1181no—
131C4×M16C2×D5911yescharacteristic
132#364—21no—
133#349C2×D5912yes—
134#347—21no—
135#344—1181no—
136#339—1181no—
137#338—591no—
138H(3,ℤ/4)—591no—
139C59#1183711yescharacteristic, complemented, Sylow
140C2×#41—2361no—
141C2×D8—592no—
142C2×D8—1181no—
143C2×M16—1181no—
144C2×M16D23611yescharacteristic
145C2×M16C22×D5911yescharacteristic
146C4⋊Q8—1181no—
147#127—1181no—
148#124—2361no—
149#121—1182no—
150#121—2361no—
151C4×D4—1182no—
152C4×D4—2361no—
153#117—1181no—
154C2×C42D23611yescharacteristic
155M32—22no—
156#106—592no—
157#106—1181no—
158#102—1182no—
159#102—1182no—
160#97C4×D5911yescharacteristic
161#97C22⋊D5911yescharacteristic
162C4×C8C4×D5911yescharacteristic
163C4×C8C22⋊D5911yescharacteristic
164#41—4724no—
165C2×Q8—2361no—
166C2×D4—1182no—
167C2×D4—2361no—
168C2×D4—2361no—
169C2×D4—2362no—
170C22×C4—21no—
171C22×C4—2362no—
172C22×C4C2×D23611yescharacteristic
173D8—2362no—
174D8—2362no—
175M16—41no—
176M16—41no—
177M16—2361no—
178M16—2361no—
179C2×C8—2361no—
180C2×C8(C2×D59)⋊C411yescharacteristic
181C2×C8(C2×D59)⋊C411yescharacteristic
182C2×C8C2×C4×D5911yescharacteristic
183C2×C8C2×C22⋊D5911yescharacteristic
184C4⋊C4—1182no—
185C4⋊C4—2361no—
186C4⋊C4—2361no—
187C22⋊C4—2362no—
188C22⋊C4—2361no—
189C22⋊C4—2362no—
190C42—21no—
191C42—2361no—
192C42(C2×D59)⋊C411yescharacteristic
193C42(C2×D59)⋊C411yescharacteristic
194C16—42no—
195C23—2362no—
196C23#1427375511yescharacteristic
197Q8—4722no—
198D4—4722no—
199D4—4722no—
200D4—4724no—
201D4—2362no—
202D4—2362no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—4724no—
209C2×C4—4721no—
210C2×C4—2362no—
211C2×C4C2×(C2×D59)⋊C411yescharacteristic
212C2×C4#1427375511yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—4721no—
218C22—21no—
219C22—41no—
220C22—4722no—
221C22—4722no—
222C22—4722no—
223C22#1683740811yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—9441no—
228C4—4722no—
229C4#1683737511yescharacteristic
230C4#1683741811yescharacteristic
231C2—21no—
232C2—41no—
233C2—9442no—
234C2#2142264411yescharacteristic
subgroupG/H
centerC2#21422644
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2165C2
socleC2×C59#1514
prime radical#21434834C2
cosocleC2×M16×C59C23
hypercenter#2165D59
hypocenterC59#11837
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
2. #21424245C2semidirect
3. #21422088C2semidirect
4. #21422082C2semidirect
5. C59×#2165C2semidirect
7. #16837800C4semidirect
8. #16837779C4semidirect
13. #16837515C4semidirect
115. #2165D59semidirect
139. C59#11837semidirect

Maximal subgroups

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

subgroupindex
#214348342
#214242452
#214220882
#214220822
C59×#21652
#1183759