Finite Group Databasebeta

← #28501537

Subgroups — #28501537

Order 15104.

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

normal subgroups only

This group has 336 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#21425257C211yesmaximal, characteristic
2#21423137C212yesmaximal
3#21422123C212yesmaximal
4#21422030C211yesmaximal, characteristic
5C59×#2809C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837737—22no—
8#16837411—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C59×#400C2212yes—
14C59×#398—22no—
15C59×#391—21no—
16C59×#391C2211yescharacteristic
17C59×#389—21no—
18C59×#387—22no—
19C2×#11745664—21no—
20#14273726—41no—
21#14273718—41no—
22#14273706—22no—
23#14273686—21no—
24#14273660—42no—
25#14273651—41no—
26#14273651—41no—
27#14273650—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59D411yescharacteristic
30C2×C4⋊C4×C59—22no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59D412yes—
35C2×C4⋊C4×C59C2311yescharacteristic
36C2×C42×C59—21no—
37C2×C42×C59D411yescharacteristic
38#95×C59—22no—
39#95×C59—22no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59D412yes—
43C2×Q8×C59—21no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×Q8×C59Q1612yes—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59C2×D411yescharacteristic
52C22×C4×C59C2×D412yes—
53C2×Dic118—22no—
54C4⋊C4×C59—41no—
55C4⋊C4×C59—44no—
56C4⋊C4×C59—42no—
57C4⋊C4×C59—42no—
58C4⋊C4×C59—81no—
59C4⋊C4×C59—81no—
60C42×C59—41no—
61C42×C59—81no—
62C22⋊Dic59—41no—
63C4⋊Dic59—21no—
64C4⋊Dic59—21no—
65C4⋊Dic59—22no—
66Dic59⋊C4—42no—
67C4×Dic59—41no—
68#11745664—41no—
69#11745664—41no—
70C2×C59⋊C8—42no—
71C2×C59⋊C8—41no—
72C23×C59#12011yescharacteristic
73Q8×C59—42no—
74Q8×C59—81no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59—41no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59C2×Q1611yescharacteristic
92C2×C4×C59#13711yescharacteristic
93C2×Dic59—41no—
94C2×Dic59—41no—
95Dic118—82no—
96C59⋊C8—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—82no—
110C4×C59—82no—
111Dic59—81no—
112Dic59—81no—
113#2809D5911yescharacteristic
114#2028—591no—
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#544—591no—
124#484—1182no—
125#419—1181no—
126#400C2×D5912yes—
127#398—22no—
128#391—21no—
129#391C2×D5911yescharacteristic
130#389—21no—
131#387—22no—
132#343—591no—
133#339—1181no—
134#328—1182no—
135#324—592no—
136C59#1222511yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D5911yescharacteristic
138C2×M16—1181no—
139C4⋊Q8—2361no—
140#125—2361no—
141C22⋊Q8—1182no—
142#117—1181no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D5912yes—
148C2×C4⋊C4C22×D5911yescharacteristic
149C2×C42—21no—
150C2×C42C22⋊D5911yescharacteristic
151#107—2362no—
152#103—2362no—
153#103—2361no—
154#103—2361no—
155C22⋊C8—1182no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95C22⋊D5912yes—
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—1182no—
165C2×Q8#1174567212yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C22⋊D5911yescharacteristic
171C22×C4C2×C22⋊D5912yes—
172M16—2361no—
173M16—2361no—
174C2×C8—2362no—
175C2×C8—2361no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—1181no—
183C4⋊C4—1181no—
184C4⋊C4—1182no—
185C4⋊C4—2362no—
186C22⋊C4—2361no—
187C42—41no—
188C42—81no—
189C42—2361no—
190C23#1427369611yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—4722no—
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—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—2361no—
211C2×C4—2361no—
212C2×C4C2×#1174567211yescharacteristic
213C2×C4#1427375811yescharacteristic
214C8—4721no—
215C8—4722no—
216C22—21no—
217C22—21no—
218C22#1683755811yescharacteristic
219C22#1683758011yescharacteristic
220C22#1683758911yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—4721no—
229C4—4721no—
230C2—21no—
231C2—21no—
232C2#2142305311yescharacteristic
233C2#2142610911yescharacteristic
234C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2809C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2809D59
hypocenterC59#12225
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

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
#214252572
#214231372
#214221232
#214220302
C59×#28092
#1222559