Finite Group Databasebeta

← #28498118

Subgroups — #28498118

Order 15104.

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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#21432725C211yesmaximal, characteristic
2#21425905C212yesmaximal, complemented
3#21422046C212yesmaximal, complemented
4#21422032C211yesmaximal, characteristic
5C59×#1320C211yesmaximal, characteristic
6C22×C4×Dic59C411yescharacteristic
7#16837456—22no—
8#16837452—22no—
9#16837425C411yescharacteristic
10#16837425C2211yescharacteristic
11#16837366C411yescharacteristic
12M16×Dic59C411yescharacteristic
13M16×Dic59C2211yescharacteristic
14#16837222—22no—
15#16837184C2212yes—
16#16837179—22no—
17C59×#407C2211yescharacteristic
18C59×#349C2212yes—
19C23×Dic59—21no—
20C2×#11745664C2×C411yescharacteristic
21C2×#11745664C2×C411yescharacteristic
22C2×C4×Dic59—21no—
23C2×C4×Dic59—21no—
24C2×C4×Dic59—21no—
25C2×C4×Dic59—21no—
26C2×C4×Dic59—21no—
27C2×C4×Dic59C2×C411yescharacteristic
28C2×C4×Dic59C2×C411yescharacteristic
29#14273667—22no—
30Dic59⋊C8—22no—
31Dic59⋊C8—22no—
32C22⋊(C59⋊C8)—22no—
33#14273637—21no—
34#14273637D412yes—
35C8×Dic59—21no—
36C8×Dic59D412yes—
37C23×C4×C59C2×C411yescharacteristic
38C2×M16×C59C2×C411yescharacteristic
39C2×M16×C59C2311yescharacteristic
40M32×C59—22no—
41C22⋊C8×C59—22no—
42C24×C59M1611yescharacteristic
43C22×C4×C59—21no—
44C22×C4×C59—21no—
45C22×C4×C59—41no—
46C22×C4×C59—41no—
47C22×C4×C59M1611yescharacteristic
48C22×C4×C59C22×C411yescharacteristic
49C22×Dic59—21no—
50C22×Dic59—21no—
51C22×Dic59—21no—
52C22×Dic59—21no—
53C22×Dic59—21no—
54C22×Dic59—21no—
55C22×Dic59—44no—
56M16×C59—41no—
57M16×C59—41no—
58C2×C8×C59—21no—
59C2×C8×C59C2×D412yes—
60C4×Dic59—21no—
61C4×Dic59—21no—
62C4×Dic59—21no—
63C4×Dic59—21no—
64C4×Dic59—21no—
65C4×Dic59—21no—
66C4×Dic59C22⋊C411yescharacteristic
67C4×Dic59C22⋊C411yescharacteristic
68C4×Dic59M1611yescharacteristic
69C4×Dic59M1611yescharacteristic
70#11745664—41no—
71#11745664—41no—
72C2×C59⋊C8—21no—
73C2×C59⋊C8C22⋊C412yes—
74C16×C59—42no—
75C59⋊C16—42no—
76C23×C59—21no—
77C23×C59—42no—
78C23×C59—41no—
79C23×C59C2×M1611yescharacteristic
80C2×C4×C59—21no—
81C2×C4×C59—21no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59C2×C22⋊C411yescharacteristic
88C2×C4×C59C2×M1611yescharacteristic
89C2×Dic59—24no—
90C2×Dic59—21no—
91C2×Dic59—21no—
92C2×Dic59—21no—
93C2×Dic59—21no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—42no—
99C8×C59—22no—
100C8×C59—41no—
101C59⋊C8—22no—
102C59⋊C8—41no—
103#11521—591nomaximal, Sylow
104C22×C59—22no—
105C22×C59—21no—
106C22×C59—41no—
107C22×C59—41no—
108C22×C59—42no—
109C22×C59—42no—
110C22×C59—41no—
111C22×C59#40711yescharacteristic
112C4×C59—21no—
113C4×C59—41no—
114C4×C59#41011yescharacteristic
115C4×C59#41311yescharacteristic
116Dic59—22no—
117Dic59—22no—
118Dic59—41no—
119Dic59—41no—
120#2920—591no—
121#2165—592no—
122#1327—592no—
123#1320—591no—
124#1320D5911yescharacteristic
125C2×C59—21no—
126C2×C59—42no—
127C2×C59—41no—
128C2×C59#146411yescharacteristic
129C22×C42—591no—
130#432—1182no—
131#431—1182no—
132#423—591no—
133#423—591no—
134#407—591no—
135#407C2×D5911yescharacteristic
136C4×M16—591no—
137C4×M16—591no—
138#364—1182no—
139#349—592no—
140#349C2×D5912yes—
141#347—1182no—
142C59#1152111yescharacteristic, complemented, Sylow
143C23×C4—1181no—
144C23×C4C4×D5911yescharacteristic
145C2×M16—591no—
146C2×M16—591no—
147C2×M16C4×D5911yescharacteristic
148C2×M16C22×D5911yescharacteristic
149C2×C42—591no—
150C2×C42—591no—
151C2×C42—1181no—
152C2×C42—1181no—
153C2×C42—1181no—
154C2×C42—1181no—
155C2×C42—1181no—
156M32—22no—
157M32—1182no—
158C4⋊C8—1182no—
159C4⋊C8—1182no—
160C22⋊C8—22no—
161C22⋊C8—1182no—
162#97—592no—
163#97—1181no—
164C4×C8—592no—
165C4×C8—1181no—
166C24#1174565911yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—1181no—
172C22×C4—1181no—
173C22×C4—1181no—
174C22×C4—1181no—
175C22×C4—1181no—
176C22×C4—1181no—
177C22×C4—2364no—
178C22×C4#1174565911yescharacteristic
179C22×C4C2×C4×D5911yescharacteristic
180M16—41no—
181M16—41no—
182M16—2361no—
183M16—2361no—
184C2×C8—21no—
185C2×C8—592no—
186C2×C8—1181no—
187C2×C8D4×D5912yes—
188C42—591no—
189C42—591no—
190C42—591no—
191C42—591no—
192C42—1181no—
193C42—1181no—
194C42—1181no—
195C42—1181no—
196C42—1181no—
197C42—1181no—
198C16—42no—
199C16—2362no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×#1174565911yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1184no—
212C2×C4—1181no—
213C2×C4—1181no—
214C2×C4—1181no—
215C2×C4—1181no—
216C2×C4—2362no—
217C2×C4—2362no—
218C2×C4—2362no—
219C2×C4—2362no—
220C2×C4—2362no—
221C2×C4C22⋊C4×D5911yescharacteristic
222C2×C4M16×D5911yescharacteristic
223C8—22no—
224C8—41no—
225C8—1182no—
226C8—2361no—
227C22—22no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22#1683736411yescharacteristic
235C4—21no—
236C4—41no—
237C4—1182no—
238C4—1182no—
239C4—2361no—
240C4—2361no—
241C4#1683737411yescharacteristic
242C4#1683738311yescharacteristic
243C2—21no—
244C2—42no—
245C2—41no—
246C2#2142242711yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1320C2
socleC2×C59#1464
prime radicalC24×C59M16
cosocleC2×M16×C59C23
hypercenter#1320D59
hypocenterC59#11521
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #21425905C2semidirect
3. #21422046C2semidirect
142. C59#11521semidirect

Maximal subgroups

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

subgroupindex
#214327252
#214259052
#214220462
#214220322
C59×#13202
#1152159