Finite Group Databasebeta

← #28501768

Subgroups — #28501768

Order 15104.

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

normal subgroups only

This group has 342 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#21425446C211yesmaximal, characteristic
2#21423069C212yesmaximal
3#21422126C212yesmaximal
4#21422040C211yesmaximal, characteristic
5C59×#2844C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837558—22no—
8#16837442—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C59×#400—21no—
14C59×#400—22no—
15C59×#395—22no—
16C59×#395C2212yes—
17C59×#382—21no—
18C59×#382C2211yescharacteristic
19C2×#11745672—42no—
20C2×#11745664—21no—
21#14273726—41no—
22#14273718—41no—
23#14273706—22no—
24#14273686—21no—
25#14273660—41no—
26#14273658—41no—
27#14273651—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59D412yes—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C4⋊C4×C59C2311yescharacteristic
36C2×C42×C59D411yescharacteristic
37#95×C59—22no—
38#95×C59—22no—
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59D412yes—
43C2×Q8×C59—42no—
44C2×Q8×C59—82no—
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59—22no—
48C22×C4×C59—22no—
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59C2×D412yes—
51C2×Dic118—22no—
52C4⋊C4×C59—41no—
53C4⋊C4×C59—41no—
54C4⋊C4×C59—81no—
55C4⋊C4×C59—81no—
56C4⋊C4×C59—82no—
57C4⋊C4×C59—81no—
58C42×C59—21no—
59C42×C59Q1612yescomplemented
60C22⋊Dic59—41no—
61#11745672—84no—
62C4⋊Dic59—22no—
63C4⋊Dic59—21no—
64C4⋊Dic59—21no—
65Dic59⋊C4—42no—
66C4×Dic59—41no—
67#11745664—22no—
68#11745664—41no—
69C2×C59⋊C8—41no—
70C2×C59⋊C8—42no—
71C23×C59#12011yescharacteristic
72Q8×C59—44no—
73Q8×C59—82no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—41no—
79C2×C4×C59—41no—
80C2×C4×C59—42no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59C2×Q1611yescharacteristic
90C2×C4×C59#13711yescharacteristic
91C2×Dic59—41no—
92C2×Dic59—41no—
93Dic118—82no—
94C59⋊C8—42no—
95C59⋊C8—82no—
96#12260—591nomaximal, Sylow
97C22×C59—21no—
98C22×C59—21no—
99C22×C59#45111yescharacteristic
100C22×C59#45611yescharacteristic
101C22×C59B(4,𝔽2)11yescharacteristic
102C4×C59—21no—
103C4×C59—21no—
104C4×C59—42no—
105C4×C59—42no—
106C4×C59—82no—
107C4×C59—82no—
108C4×C59—81no—
109Dic59—81no—
110Dic59—81no—
111#2844D5911yescharacteristic
112#2061—591no—
113#1605—592no—
114#1354—592no—
115#1324—591no—
116C2×C59—21no—
117C2×C59—21no—
118C2×C59#160511yescharacteristic
119C2×C59#219911yescharacteristic
120C2×C59#220311yescharacteristic
121#544—591no—
122#451—1182no—
123#428—1181no—
124#400—21no—
125#400—22no—
126#395—22no—
127#395C2×D5912yes—
128#382—21no—
129#382C2×D5911yescharacteristic
130#343—591no—
131#339—1181no—
132#328—1182no—
133#324—592no—
134C59#1226011yescharacteristic, complemented, Sylow
135C22×Q8C22⋊D5912yes—
136C2×Q16—2362no—
137C2×M16—1181no—
138C4⋊Q8—2361no—
139#125—2361no—
140C22⋊Q8—1182no—
141#117—1181no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊D5911yescharacteristic
147C2×C4⋊C4C22×D5911yescharacteristic
148C2×C42C22⋊D5911yescharacteristic
149#107—2361no—
150#106—2361no—
151#103—2362no—
152C22⋊C8—1182no—
153#95—22no—
154#95—22no—
155#95—21no—
156#95—22no—
157#95—22no—
158#95C22⋊D5912yes—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—1182no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D5911yescharacteristic
167C22×C4C2×C22⋊D5912yes—
168Q16—4724no—
169M16—1182no—
170M16—2361no—
171C2×C8—2362no—
172C2×C8—2361no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—1182no—
180C4⋊C4—1181no—
181C4⋊C4—1181no—
182C4⋊C4—2362no—
183C22⋊C4—2361no—
184C42—21no—
185C42—2361no—
186C42#1174567212yescomplemented
187C23#1427369611yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—4722no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—2361no—
207C2×C4—2361no—
208C2×C4C2×#1174567211yescharacteristic
209C2×C4#1427375811yescharacteristic
210C8—4722no—
211C8—2362no—
212C22—21no—
213C22—21no—
214C22#1683755811yescharacteristic
215C22#1683758511yescharacteristic
216C22#1683758911yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—4721no—
225C4—4721no—
226C2—21no—
227C2—21no—
228C2#2142307011yescharacteristic
229C2#2142609711yescharacteristic
230C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2844C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2844D59
hypocenterC59#12260
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
59. C42×C59Q16semidirect
134. C59#12260semidirect
186. C42#11745672semidirect

Maximal subgroups

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

subgroupindex
#214254462
#214230692
#214221262
#214220402
C59×#28442
#1226059