Finite Group Databasebeta

← #28501536

Subgroups — #28501536

Order 15104.

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

normal subgroups only

This group has 336 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#21432084C211yesmaximal, characteristic
2#21423138C212yesmaximal
3#21422122C212yesmaximal
4#21422030C211yesmaximal, characteristic
5C59×#2028C211yesmaximal, characteristic
6#16837735—22no—
7#16837341C2212yes—
8#16837334—22no—
9#16837309—21no—
10#16837300—21no—
11#16837289—22no—
12#16837139—22no—
13#16837128C2212yes—
14C59×#544C2211yescharacteristic
15C59×#419—21no—
16C59×#391C2211yescharacteristic
17C59×#343C2211yescharacteristic
18C59×#339—21no—
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59D412yes—
23C2×Dic59⋊C4—22no—
24C2×C4×Dic59—21no—
25#14273660—42no—
26#14273651—42no—
27C22⋊(C59⋊C8)—22no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C59⋊#95—22no—
31C59⋊#95D412yes—
32C22×Q8×C59D411yescharacteristic
33C2×M16×C59—21no—
34C4⋊Q8×C59—41no—
35#125×C59—41no—
36C22⋊Q8×C59—22no—
37#117×C59—21no—
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59D411yescharacteristic
40#103×C59—41no—
41#103×C59—41no—
42#95×C59—22no—
43C2×Q8×C59—21no—
44C2×Q8×C59—22no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59Q1612yes—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D412yes—
52C22×Dic59—22no—
53C22×Dic59—22no—
54M16×C59—41no—
55M16×C59—41no—
56C2×C8×C59—41no—
57C4⋊C4×C59—21no—
58C4⋊C4×C59—21no—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—42no—
62C22⋊C4×C59—41no—
63C42×C59—41no—
64C42×C59—41no—
65C4⋊Dic59—42no—
66C4⋊Dic59—42no—
67C4⋊Dic59—81no—
68C4⋊Dic59—81no—
69Dic59⋊C4—44no—
70C4×Dic59—81no—
71C2×C59⋊C8—42no—
72C23×C59#12011yescharacteristic
73Q8×C59—42no—
74Q8×C59—81no—
75Q8×C59—82no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59—41no—
84C2×C4×C59—41no—
85C2×C4×C59—42no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59C2×Q1611yescharacteristic
89C2×C4×C59#13711yescharacteristic
90C2×Dic59—42no—
91C2×Dic59—42no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C8×C59—81no—
97C59⋊C8—82no—
98#12225—591nomaximal, Sylow
99C22×C59—21no—
100C22×C59—21no—
101C22×C59#45111yescharacteristic
102C22×C59#45511yescharacteristic
103C22×C59B(4,𝔽2)11yescharacteristic
104C4×C59—21no—
105C4×C59—21no—
106C4×C59—42no—
107C4×C59—42no—
108C4×C59—42no—
109C4×C59—81no—
110C4×C59—81no—
111Dic59—82no—
112Dic59—82no—
113#2809—591no—
114#2028D5911yescharacteristic
115#1617—592no—
116#1353—592no—
117#1319—591no—
118C2×C59—21no—
119C2×C59—21no—
120C2×C59#160211yescharacteristic
121C2×C59#220111yescharacteristic
122C2×C59#220411yescharacteristic
123#544C2×D5911yescharacteristic
124#484—1182no—
125#419—21no—
126#400—592no—
127#398—1182no—
128#391—1181no—
129#391C2×D5911yescharacteristic
130#389—1181no—
131#387—1182no—
132#343C2×D5911yescharacteristic
133#339—21no—
134#328—1182no—
135#324—592no—
136C59#1222511yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D5911yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—592no—
144C2×C4⋊C4—1181no—
145C2×C4⋊C4—1181no—
146C2×C4⋊C4—1182no—
147C2×C4⋊C4—1181no—
148C2×C4⋊C4C22×D5911yescharacteristic
149C2×C42—1181no—
150C2×C42C22⋊D5911yescharacteristic
151#107—2362no—
152#103—41no—
153#103—41no—
154#103—2362no—
155C22⋊C8—1182no—
156#95—22no—
157#95—592no—
158#95—1182no—
159#95—1182no—
160#95—1182no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1174567212yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1182no—
169C22×C4—1182no—
170C22×C4D4×D5912yes—
171C22×C4C2×C22⋊D5911yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—2362no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—4721no—
182C4⋊C4—4721no—
183C4⋊C4—2362no—
184C4⋊C4—2364no—
185C4⋊C4—2362no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—4721no—
190C23#1427369511yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—2362no—
207C2×C4—2362no—
208C2×C4—2362no—
209C2×C4—2362no—
210C2×C4—2362no—
211C2×C4—2362no—
212C2×C4C2×#1174567211yescharacteristic
213C2×C4#1427375711yescharacteristic
214C8—81no—
215C8—4722no—
216C22—21no—
217C22—21no—
218C22#1683755711yescharacteristic
219C22#1683757911yescharacteristic
220C22#1683758811yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—4722no—
229C4—4722no—
230C2—21no—
231C2—21no—
232C2#2142305211yescharacteristic
233C2#2142610811yescharacteristic
234C2#2142612111yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2028C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2028D59
hypocenterC59#12225
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
136. C59#12225semidirect

Maximal subgroups

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

subgroupindex
#214320842
#214231382
#214221222
#214220302
C59×#20282
#1222559