Finite Group Databasebeta

← #28475907

Subgroups — #28475907

Order 15104.

324 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 75 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#21424394C211yesmaximal, characteristic
2#21422414C211yesmaximal, characteristic
3C59×#1460C211yesmaximal, characteristic
4C22×#11745664C411yescharacteristic
5C2×C22⋊(C59⋊C8)—21no—
6C2×C22⋊(C59⋊C8)—21no—
7C2×C22⋊(C59⋊C8)C411yescharacteristic
8#16837157—21no—
9#16837128C412yes—
10#16837126—21no—
11C2×C4×D4×C59C2211yescharacteristic
12C59×#431—21no—
13C59×#323—21no—
14C59×#323C412yes—
15C2×#11745664—42no—
16C2×#11745664—41no—
17C22×C59⋊C8—21no—
18C22×C59⋊C8—21no—
19C22×C59⋊C8—41no—
20C22⋊(C59⋊C8)—22no—
21C22⋊(C59⋊C8)—42no—
22C22⋊(C59⋊C8)—41no—
23C22⋊(C59⋊C8)—41no—
24C22×D4×C59Q811yescharacteristic
25C23×C4×C59C2×C411yescharacteristic
26C23×C4×C59D411yescharacteristic
27C4×D4×C59—41no—
28C4×D4×C59—41no—
29C2×C4⋊C4×C59C2×C411yescharacteristic
30C2×C22⋊C4×C59C2×C411yescharacteristic
31C2×C22⋊C4×C59D411yescharacteristic
32C2×C42×C59D411yescharacteristic
33C4⋊C8×C59—41no—
34C22⋊C8×C59—22no—
35C4×C8×C59—41no—
36C24×C59C4⋊C411yescharacteristic
37C24×C59C4⋊C411yescharacteristic
38C2×D4×C59—22no—
39C2×D4×C59—22no—
40C2×D4×C59—41no—
41C22×C4×C59—21no—
42C22×C4×C59—21no—
43C22×C4×C59—21no—
44C22×C4×C59—21no—
45C22×C4×C59—41no—
46C22×C4×C59—41no—
47C22×C4×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59C4211yescharacteristic
50C22×C4×C59C22⋊C411yescharacteristic
51C22×C4×C59C22⋊C411yescharacteristic
52C22×C4×C59C22⋊C411yescharacteristic
53C22×C4×C59C4⋊C411yescharacteristic
54C2×C8×C59—42no—
55C4⋊C4×C59—41no—
56C22⋊C4×C59—41no—
57C22⋊C4×C59—41no—
58C42×C59—21no—
59C42×C59—21no—
60#11745664—42no—
61#11745664—42no—
62C2×C59⋊C8—22no—
63C2×C59⋊C8—41no—
64C2×C59⋊C8—41no—
65C2×C59⋊C8—42no—
66C2×C59⋊C8—41no—
67C2×C59⋊C8—41no—
68C2×C59⋊C8—41no—
69C2×C59⋊C8—81no—
70C2×C59⋊C8—81no—
71C23×C59—21no—
72C23×C59—21no—
73C23×C59—21no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—41no—
77C23×C59—41no—
78C23×C59—41no—
79C23×C59—41no—
80C23×C59#9511yescharacteristic
81C23×C59#10812yes—
82D4×C59—44no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—42no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—81no—
99C2×C4×C59#9911yescharacteristic
100C2×C4×C59#9911yescharacteristic
101C2×C4×C59#10812yes—
102C8×C59—82no—
103C59⋊C8—42no—
104C59⋊C8—81no—
105C59⋊C8—82no—
106#7285—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59—22no—
110C22×C59—42no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—41no—
115C22×C59—41no—
116C22×C59—81no—
117C22×C59—81no—
118C22×C59—81no—
119C22×C59#34111yescharacteristic
120C22×C59#34111yescharacteristic
121C22×C59#34211yescharacteristic
122C4×C59—22no—
123C4×C59—21no—
124C4×C59—41no—
125C4×C59—41no—
126C4×C59—41no—
127C4×C59AΣL(1,𝔽16)11yescharacteristic
128C4×C59#35211yescharacteristic
129#1870—591no—
130#1460D5911yescharacteristic
131#1459—591no—
132C2×C59—21no—
133C2×C59—21no—
134C2×C59—42no—
135C2×C59—81no—
136C2×C59#129811yescharacteristic
137C2×C59#139011yescharacteristic
138C2×C59#139511yescharacteristic
139C22×M16—591no—
140C2×C4×D4C2×D5911yescharacteristic
141#431—21no—
142C2×C22⋊C8—591no—
143C2×C22⋊C8—1181no—
144C2×C22⋊C8—1181no—
145#336—1181no—
146#324—592no—
147#323—21no—
148#323—1181no—
149#323Dic5912yes—
150C59#728511yescharacteristic, complemented, Sylow
151C22×D4Dic11811yescharacteristic
152C23×C4C4×D5911yescharacteristic
153C23×C4D23611yescharacteristic
154C2×M16—2362no—
155C2×M16—2361no—
156C22×C8—1181no—
157C22×C8—1181no—
158C22×C8—2361no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C4⋊C4C4×D5911yescharacteristic
162C2×C22⋊C4C2×Dic5911yescharacteristic
163C2×C22⋊C4C22⋊D5911yescharacteristic
164C2×C42C22⋊D5911yescharacteristic
165C4⋊C8—41no—
166C22⋊C8—22no—
167C22⋊C8—1182no—
168C22⋊C8—2362no—
169C22⋊C8—2361no—
170C22⋊C8—2361no—
171C4×C8—41no—
172C24Dic59⋊C411yescharacteristic
173C24C4⋊Dic5911yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—41no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4C4×Dic5911yescharacteristic
186C22×C4Dic59⋊C411yescharacteristic
187C22×C4(C2×D59)⋊C411yescharacteristic
188C22×C4(C2×D59)⋊C411yescharacteristic
189C22×C4C22⋊Dic5911yescharacteristic
190M16—2362no—
191M16—2362no—
192C2×C8—42no—
193C2×C8—1182no—
194C2×C8—4721no—
195C2×C8—4721no—
196C2×C8—2362no—
197C2×C8—2361no—
198C2×C8—2361no—
199C2×C8—2361no—
200C2×C8—2361no—
201C2×C8—2361no—
202C4⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C42—21no—
206C42—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23C59⋊#9511yescharacteristic
217C23#1427366312yes—
218D4—44no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4#1427364111yescharacteristic
236C2×C4#1427364111yescharacteristic
237C2×C4#1427366212yes—
238C8—82no—
239C8—4721no—
240C8—4722no—
241C8—2362no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#1683716611yescharacteristic
255C22#1683716711yescharacteristic
256C22#1683716811yescharacteristic
257C4—21no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4#1683719011yescharacteristic
263C4#1683719311yescharacteristic
264C2—21no—
265C2—21no—
266C2—42no—
267C2—81no—
268C2#2142198411yescharacteristic
269C2#2142221511yescharacteristic
270C2#2142222411yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C42
FrattiniC2×C4×D4C2×D59
FittingC59×#1460C2
socleC22×C59#342
prime radicalC22×D4×C59Q8
cosocleC2×C4×D4×C59C22
hypercenter#1460D59
hypocenterC59#7285
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
150. C59#7285semidirect

Maximal subgroups

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

subgroupindex
#214243942
#214224142
C59×#14602
#728559