Finite Group Databasebeta

← #28475363

Subgroups — #28475363

Order 15104.

338 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 49 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#21425094C211yesmaximal, characteristic
2#21421964C211yesmaximal, characteristic, complemented
3C59×#1314C211yesmaximal, characteristic, complemented
4C22×D472C411yescharacteristic, complemented
5C2×#14273701—21no—
6C2×#14273649—21no—
7#16837262—21no—
8#16837165C411yescharacteristic, complemented
9C2×C22⋊C8×C59C2211yescharacteristic
10C59×C22⋊C16—21no—
11C22×D236—21no—
12C22×C22⋊D59—41no—
13C2×D472—41no—
14C2×D472—42no—
15#14273701—42no—
16#14273701—42no—
17C2×C4⋊Dic59—21no—
18C2×C4⋊Dic59—41no—
19C2×C22⋊Dic59—41no—
20C2×(C2×D59)⋊C4—41no—
21#14273649—42no—
22C59⋊#95—22no—
23C23×C4×C59D411yescharacteristic
24C22×C8×C59C2×C411yescharacteristic
25C22×C8×C59D411yescharacteristic
26C2×C16×C59—41no—
27C22⋊C8×C59—41no—
28C24×C59D811yescharacteristic
29C23×D59—41no—
30C22×C4×C59—21no—
31C22×C4×C59—21no—
32C22×C4×C59—41no—
33C22×C4×C59—41no—
34C22×C4×C59C22⋊C411yescharacteristic
35C22×C4×C59SD1611yescharacteristic
36C2×C22⋊D59—44no—
37C2×C22⋊D59—82no—
38C2×C22⋊D59—81no—
39C2×C22⋊D59—81no—
40C22×Dic59—41no—
41C22×Dic59—42no—
42C2×D236—22no—
43C2×D236—41no—
44C2×D236—41no—
45C2×D236—41no—
46C2×D236—41no—
47C2×D236—41no—
48C2×C8×C59—21no—
49C2×C8×C59—21no—
50C2×C8×C59—41no—
51C2×C8×C59—41no—
52C22⋊Dic59—81no—
53C22⋊Dic59—81no—
54(C2×D59)⋊C4—44no—
55C4⋊Dic59—22no—
56C4⋊Dic59—41no—
57C4⋊Dic59—81no—
58C4⋊Dic59—81no—
59D472—42no—
60D472—42no—
61C16×C59—81no—
62C23×C59—21no—
63C23×C59—21no—
64C23×C59—41no—
65C23×C59—41no—
66C23×C59#10211yescharacteristic
67C23×C59D1611yescharacteristic
68C23×C59SD3211yescharacteristic
69C22×D59—42no—
70C22×D59—81no—
71C22×D59—81no—
72C22×D59—81no—
73C22×D59—81no—
74C22×D59—81no—
75C22×D59—81no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—81no—
83C2×C4×C59#9911yescharacteristic
84C2×C4×C59#10411yescharacteristic
85C22⋊D59—84no—
86C22⋊D59—84no—
87C2×Dic59—42no—
88C2×Dic59—81no—
89C2×Dic59—81no—
90C2×Dic59—82no—
91C2×Dic59—82no—
92C2×Dic59—82no—
93D236—42no—
94D236—42no—
95D236—81no—
96D236—81no—
97C8×C59—41no—
98C8×C59—41no—
99#7076—591nomaximal, Sylow
100C22×C59—21no—
101C22×C59—21no—
102C22×C59—21no—
103C22×C59—21no—
104C22×C59—41no—
105C22×C59—41no—
106C22×C59—41no—
107C22×C59—41no—
108C22×C59—81no—
109C22×C59#32711yescharacteristic
110C22×C59#35711yescharacteristic
111C22×C59#36011yescharacteristic
112C2×D59—82no—
113C2×D59—82no—
114C2×D59—82no—
115C2×D59—161no—
116C2×D59—161no—
117C2×D59—161no—
118C2×D59—161no—
119C4×C59—21no—
120C4×C59—41no—
121C4×C59AΣL(1,𝔽16)11yescharacteristic
122C4×C59#35311yescharacteristic
123Dic59—82no—
124Dic59—162no—
125#1999—591no—
126#1314D5911yescharacteristic, complemented
127#1289—591no—
128C2×C59—21no—
129C2×C59—21no—
130C2×C59—41no—
131C2×C59—41no—
132C2×C59#133911yescharacteristic
133C2×C59#134311yescharacteristic
134C2×C59#134711yescharacteristic
135D59—162no—
136C22×D8—591no—
137C2×#121—1181no—
138C2×#102—1181no—
139C2×C22⋊C8C2×D5911yescharacteristic
140#380—1181no—
141C22⋊C16—21no—
142#340—591no—
143C59#707611yescharacteristic, complemented, Sylow
144C22×D4—1181no—
145C22×D4—2361no—
146C23×C4D23611yescharacteristic
147C2×D8—2361no—
148C2×D8—2362no—
149C22×C8C4×D5911yescharacteristic
150C22×C8C22⋊D5911yescharacteristic
151#121—2362no—
152#121—2362no—
153C2×C4⋊C4—1181no—
154C2×C4⋊C4—2361no—
155C2×C22⋊C4—2361no—
156C2×C22⋊C4—2361no—
157C2×C16—41no—
158#102—2362no—
159C22⋊C8—41no—
160#95—1182no—
161C24—2361no—
162C24D47211yescharacteristic
163C2×D4—1182no—
164C2×D4—4722no—
165C2×D4—4721no—
166C2×D4—4721no—
167C2×D4—2361no—
168C2×D4—2361no—
169C2×D4—2361no—
170C2×D4—2364no—
171C2×D4—2361no—
172C2×D4—2361no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—2362no—
178C22×C4—2361no—
179C22×C4#1174566011yescharacteristic
180C22×C4(C2×D59)⋊C411yescharacteristic
181D8—2362no—
182D8—2362no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C4⋊C4—1182no—
188C4⋊C4—4721no—
189C4⋊C4—4721no—
190C4⋊C4—2361no—
191C22⋊C4—4721no—
192C22⋊C4—4721no—
193C22⋊C4—2364no—
194C16—81no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—4721no—
200C23—4721no—
201C23—4721no—
202C23—4721no—
203C23—4721no—
204C23—4721no—
205C23—2362no—
206C23#1427364911yescharacteristic
207C23D94411yescharacteristic
208C23#1427367211yescharacteristic
209D4—4721no—
210D4—4724no—
211D4—4724no—
212D4—4721no—
213D4—2362no—
214D4—2362no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—4722no—
223C2×C4—4722no—
224C2×C4—4722no—
225C2×C4—4721no—
226C2×C4—4721no—
227C2×C4—2362no—
228C2×C4#1427364111yescharacteristic
229C2×C4#1427365311yescharacteristic
230C8—41no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—9441no—
242C22—9441no—
243C22—9441no—
244C22—9441no—
245C22—4722no—
246C22—4722no—
247C22—4722no—
248C22#1683713711yescharacteristic
249C22#1683720411yescharacteristic
250C22#1683721311yescharacteristic
251C4—21no—
252C4—41no—
253C4—9442no—
254C4—4722no—
255C4#1683719011yescharacteristic
256C4#1683719411yescharacteristic
257C2—21no—
258C2—21no—
259C2—41no—
260C2—41no—
261C2—9442no—
262C2#2142208211yescharacteristic
263C2#2142209411yescharacteristic
264C2#2142210611yescharacteristic
subgroupG/H
centerC22#16837137
commutatorC22×C8×C59C2×C4
FrattiniC2×C22⋊C8C2×D59
FittingC59×#1314C2
socleC22×C59#327
prime radical#21425094C2
cosocleC2×C22⋊C8×C59C22
hypercenter#1314D59
hypocenterC59#7076
LCSG ▷ C22×C8×C59 ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21421964C2semidirect
3. C59×#1314C2semidirect
4. C22×D472C4semidirect
8. #16837165C4semidirect
126. #1314D59semidirect
143. C59#7076semidirect

Maximal subgroups

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

subgroupindex
#214250942
#214219642
C59×#13142
#707659