Finite Group Databasebeta

← #28501435

Subgroups — #28501435

Order 15104.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425263C211yesmaximal, characteristic
2#21423086C212yesmaximal
3#21422096C212yesmaximal
4#21422026C211yesmaximal, characteristic
5C59×#2817C211yesmaximal, characteristic
6#16838028C2211yescharacteristic
7#16837732—22no—
8#16837265—21no—
9#16837169C2211yescharacteristic
10#16837168—21no—
11#16837138—22no—
12#16837126C2212yes—
13C2×C22⋊Q8×C59—21no—
14C59×#400—22no—
15C59×#397—21no—
16C59×#397C2212yes—
17C59×#395—22no—
18C59×#381C2211yescharacteristic
19#14273706—22no—
20#14273704—42no—
21C2×C22⋊Dic59—21no—
22C2×C22⋊Dic59—21no—
23#14273658—42no—
24#14273650—42no—
25C22⋊(C59⋊C8)—22no—
26C59⋊#95—42no—
27C22×Q8×C59—21no—
28C23×C4×C59D411yescharacteristic
29C22⋊Q8×C59—42no—
30C22⋊Q8×C59—42no—
31C2×C4⋊C4×C59—22no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59D412yes—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59D411yescharacteristic
37C2×C22⋊C4×C59C2311yescharacteristic
38#95×C59—22no—
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59D412yes—
43C24×C59C2×D411yescharacteristic
44C2×Q8×C59—42no—
45C2×Q8×C59—81no—
46C2×Q8×C59—81no—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59SD1612yes—
55C22×C4×C59C2×D412yes—
56C22×Dic59—22no—
57C22×Dic59—41no—
58C2×Dic118—22no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—44no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—81no—
63C22⋊C4×C59—41no—
64C22⋊C4×C59—42no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—41no—
67C22⋊Dic59—41no—
68C22⋊Dic59—41no—
69C22⋊Dic59—41no—
70C22⋊Dic59—41no—
71C4⋊Dic59—22no—
72Dic59⋊C4—42no—
73C2×C59⋊C8—42no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—42no—
77C23×C59#12011yescharacteristic
78C23×C59C2×SD1611yescharacteristic
79C23×C59AGL(1,ℤ/8)11yescharacteristic
80Q8×C59—84no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—44no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—41no—
92C2×C4×C59—42no—
93C2×C4×C59—42no—
94C2×C4×C59—81no—
95C2×Dic59—42no—
96C2×Dic59—82no—
97C2×Dic59—81no—
98C2×Dic59—82no—
99Dic118—82no—
100C59⋊C8—82no—
101#12207—591nomaximal, Sylow
102C22×C59—22no—
103C22×C59—21no—
104C22×C59—21no—
105C22×C59—41no—
106C22×C59—41no—
107C22×C59—42no—
108C22×C59—81no—
109C22×C59#45011yescharacteristic
110C22×C59#45511yescharacteristic
111C22×C59B(4,𝔽2)11yescharacteristic
112C4×C59—42no—
113C4×C59—42no—
114C4×C59—82no—
115C4×C59—81no—
116C4×C59—82no—
117Dic59—82no—
118Dic59—161no—
119#2817D5911yescharacteristic
120#2030—591no—
121#1608—592no—
122#1344—592no—
123#1317—591no—
124C2×C59—21no—
125C2×C59—21no—
126C2×C59—41no—
127C2×C59—41no—
128C2×C59#160111yescharacteristic
129C2×C59#220112yes—
130#543—591no—
131C2×C22⋊Q8—21no—
132#483—1182no—
133#400—22no—
134#397—21no—
135#397C2×D5912yes—
136#395—22no—
137#381—1181no—
138#381C2×D5911yescharacteristic
139#342—591no—
140#342—1181no—
141#328—1182no—
142#323—592no—
143C59#1220711yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C23×C4C22⋊D5911yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—1182no—
149C22⋊Q8—2362no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊D5912yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—1181no—
156C2×C22⋊C4—1181no—
157C2×C22⋊C4C22⋊D5911yescharacteristic
158C2×C22⋊C4C22×D5911yescharacteristic
159#106—2362no—
160#103—2362no—
161C22⋊C8—1182no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—2362no—
167#95C22⋊D5912yes—
168C24C2×C22⋊D5911yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—1182no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—1182no—
181C22×C4—2361no—
182C22×C4#1174567012yes—
183C22×C4C2×C22⋊D5912yes—
184C2×C8—2362no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—1182no—
190C4⋊C4—2362no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—2361no—
196C22⋊C4—2361no—
197C22⋊C4—2361no—
198C22⋊C4—2361no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1427369611yescharacteristic
203C23C2×#1174567011yescharacteristic
204C23#1427375311yescharacteristic
205Q8—84no—
206Q8—4722no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—4721no—
222C2×C4—4722no—
223C2×C4—4722no—
224C2×C4—2362no—
225C8—4722no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1683755311yescharacteristic
234C22#1683758011yescharacteristic
235C22#1683758911yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—9441no—
242C4—4722no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2142304811yescharacteristic
248C2#2142610912yes—
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2817C2
socleC22×C59B(4,𝔽2)
prime radicalC24×C59C2×D4
hypercenter#2817D59
hypocenterC59#12207
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
143. C59#12207semidirect

Maximal subgroups

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

subgroupindex
#214252632
#214230862
#214220962
#214220262
C59×#28172
#1220759