Finite Group Databasebeta

← #28509041

Subgroups — #28509041

Order 15104.

326 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 59 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#21434495C211yesmaximal, characteristic
2#21425758C212yesmaximal
3#21422267C211yesmaximal, characteristic, complemented
4#21422266C212yesmaximal, complemented
5C59×#1412C211yesmaximal, characteristic, complemented
6#16837950C2211yescharacteristic
7#125×D59C411yescharacteristic, complemented
8#16837772C411yescharacteristic, complemented
9#16837771C411yescharacteristic, complemented
10#16837766C2211yescharacteristic
11#16837755—22no—
12#16837400—22no—
13#16837374—22no—
14#16837354C411yescharacteristic, complemented
15#16837196—22no—
16#16837195C2212yes—
17C59×#490C2211yescharacteristic
18C59×#353C2212yes—
19C2×#11745687D412yes—
20C2×#11745659—21no—
21#14273727C2×C411yescharacteristic
22#14273719C2×C411yescharacteristic
23#14273718—21no—
24#14273717—41no—
25#14273714D412yes—
26#14273698—41no—
27D4×Dic59—41no—
28#14273683—42no—
29C4⋊C4×D59—21no—
30C4⋊C4×D59—41no—
31C42×D59C2×C411yescharacteristic
32#14273652—22no—
33#14273650—22no—
34#14273648—22no—
35#14273642—22no—
36#14273641—42no—
37#14273638C2×C411yescharacteristic
38C8×Dic59—21no—
39#127×C59C2311yescharacteristic
40#125×C59C2×C411yescharacteristic
41#102×C59—22no—
42#99×C59—22no—
43#97×C59C2×C411yescharacteristic
44C2×D4×C59—41no—
45C2×D4×C59C2×D412yes—
46C2×C22⋊D59—22no—
47C22×Dic59—22no—
48#11745687—42no—
49#11745687—42no—
50C2×C4×D59—21no—
51C2×C4×D59—41no—
52C2×C4×D59C22⋊C411yescharacteristic
53C2×Dic118C22⋊C412yes—
54C2×C8×C59—21no—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—41no—
57C22⋊C4×C59—42no—
58C42×C59C22×C411yescharacteristic
59C22⋊Dic59—42no—
60C22⋊Dic59—42no—
61C4⋊Dic59—21no—
62C4⋊Dic59—41no—
63C4⋊Dic59—41no—
64Dic59⋊C4—41no—
65Dic59⋊C4—41no—
66Dic59⋊C4—41no—
67Dic59⋊C4—82no—
68C4×Dic59—21no—
69C4×Dic59—82no—
70C4×Dic59C22⋊C411yescharacteristic
71C2×C59⋊C8—21no—
72#11745659—42no—
73C23×C59—22no—
74C22×D59#9911yescharacteristic
75D4×C59—42no—
76D4×C59—82no—
77C2×C4×C59—21no—
78C2×C4×C59—41no—
79C2×C4×C59—82no—
80C2×C4×C59C2×C22⋊C411yescharacteristic
81C22⋊D59—82no—
82C2×Dic59—21no—
83C2×Dic59—22no—
84C2×Dic59—42no—
85C2×Dic59—42no—
86C2×Dic59—41no—
87C2×Dic59—82no—
88C2×Dic59#9911yescharacteristic
89C4×D59—22no—
90C4×D59—81no—
91C4×D59—82no—
92Dic118—42no—
93C8×C59—41no—
94C59⋊C8—41no—
95#13596—591nomaximal, Sylow
96C22×C59—42no—
97C22×C59—82no—
98C22×C59C2×#9911yescharacteristic
99C2×D59—21no—
100C2×D59—41no—
101C4×C59—21no—
102C4×C59—41no—
103C4×C59—81no—
104C4×C59—162no—
105Dic59—41no—
106Dic59—42no—
107Dic59—81no—
108Dic59—162no—
109Dic59#35412yes—
110#3142—591no—
111#2126—592no—
112#1412—591no—
113#1412—592no—
114#1412D5911yescharacteristic, complemented
115C2×C59—21no—
116C2×C59—82no—
117C2×C59C2×#35411yescharacteristic
118D59—41no—
119#532—591no—
120C2×#125—591no—
121#491—591no—
122#491—591no—
123#490—591no—
124#490C2×D5911yescharacteristic
125#488—1182no—
126#417—1182no—
127#410—1182no—
128C4×M16—591no—
129#354—1182no—
130#353—592no—
131#353C2×D5912yes—
132C59#1359611yescharacteristic, complemented, Sylow
133C2×#41—592no—
134C2×M16—1181no—
135C4⋊Q8—591no—
136#127C22×D5911yescharacteristic
137#125—591no—
138#125—1181no—
139#125—2361no—
140#125C4×D5911yescharacteristic
141#124—592no—
142#121—2361no—
143C4×D4—2361no—
144#117—2362no—
145C2×C4⋊C4—1181no—
146C2×C4⋊C4—2361no—
147C2×C42—591no—
148#103—1182no—
149#103—1182no—
150#102—22no—
151#102—1182no—
152#99—22no—
153#99—1182no—
154#99—2362no—
155#97—591no—
156#97C4×D5911yescharacteristic
157C4×C8—1181no—
158#41—2362no—
159#41—2362no—
160C2×Q8—592no—
161C2×D4—41no—
162C2×D4—1182no—
163C2×D4D4×D5912yes—
164C22×C4—591no—
165C22×C4—1181no—
166C22×C4—1182no—
167C22×C4—2361no—
168M16—2362no—
169C2×C8—21no—
170C2×C8—1181no—
171C4⋊C4—21no—
172C4⋊C4—41no—
173C4⋊C4—1181no—
174C4⋊C4—4722no—
175C4⋊C4—2361no—
176C4⋊C4—2361no—
177C4⋊C4—2361no—
178C4⋊C4—2361no—
179C4⋊C4—2361no—
180C22⋊C4—42no—
181C22⋊C4—2362no—
182C22⋊C4—2362no—
183C42—591no—
184C42—1181no—
185C42—4722no—
186C42C2×C4×D5911yescharacteristic
187C23—22no—
188C23—591no—
189Q8—2362no—
190D4—42no—
191D4—82no—
192D4—4722no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—82no—
196C2×C4—591no—
197C2×C4—1182no—
198C2×C4—1181no—
199C2×C4—1182no—
200C2×C4—4721no—
201C2×C4—4722no—
202C2×C4—4722no—
203C2×C4—2362no—
204C2×C4—2362no—
205C2×C4—2361no—
206C2×C4C22⋊C4×D5911yescharacteristic
207C8—41no—
208C8—2361no—
209C22—42no—
210C22—82no—
211C22—1181no—
212C22—2361no—
213C22#99×D5911yescharacteristic
214C4—21no—
215C4—41no—
216C4—81no—
217C4—162no—
218C4—592no—
219C4—9442no—
220C4—4721no—
221C4—2361no—
222C4—2362no—
223C2—21no—
224C2—82no—
225C2—2361no—
226C2#354×D5911yescharacteristic
subgroupG/H
centerC2#354×D59
commutatorC42×C59C22×C4
Frattini#127C22×D59
FittingC59×#1412C2
socleC2×C59C2×#354
prime radical#16837950C22
cosocle#127×C59C23
hypercenter#1412D59
hypocenterC59#13596
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
3. #21422267C2semidirect
4. #21422266C2semidirect
5. C59×#1412C2semidirect
7. #125×D59C4semidirect
8. #16837772C4semidirect
9. #16837771C4semidirect
14. #16837354C4semidirect
114. #1412D59semidirect
132. C59#13596semidirect

Maximal subgroups

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

subgroupindex
#214344952
#214257582
#214222672
#214222662
C59×#14122
#1359659