Finite Group Databasebeta

← #28501767

Subgroups — #28501767

Order 15104.

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

normal subgroups only

This group has 342 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#21432368C211yesmaximal, characteristic
2#21423070C212yesmaximal
3#21422125C212yesmaximal
4#21422040C211yesmaximal, characteristic
5C59×#2061C211yesmaximal, characteristic
6#16837557—22no—
7#16837344—22no—
8#16837341—21no—
9#16837323—22no—
10#16837322C2212yes—
11#16837268—21no—
12#16837139—22no—
13#16837128C2212yes—
14C59×#544C2211yescharacteristic
15C59×#428—21no—
16C59×#382C2211yescharacteristic
17C59×#343C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118D412yes—
20C2×#11745672—42no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×Dic59⋊C4—22no—
25#14273650—42no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—22no—
28C59⋊#95—22no—
29C59⋊#95—21no—
30C59⋊#95—22no—
31C59⋊#95D412yes—
32C2×M16×C59—21no—
33C4⋊Q8×C59—41no—
34#125×C59—41no—
35C22⋊Q8×C59—22no—
36#117×C59—21no—
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C42×C59D411yescharacteristic
40#107×C59—41no—
41#106×C59—41no—
42#95×C59—22no—
43C2×Q8×C59—22no—
44C22×C4×C59—21no—
45C22×C4×C59—21no—
46C22×C4×C59C2×D411yescharacteristic
47C22×C4×C59C2×D412yes—
48C22×Dic59—22no—
49C22×Dic59—22no—
50C2×Dic118—42no—
51C2×Dic118—82no—
52M16×C59—22no—
53M16×C59—41no—
54C2×C8×C59—41no—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—21no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—41no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—41no—
61C22⋊C4×C59—41no—
62C42×C59—21no—
63C42×C59—41no—
64C42×C59Q1612yescomplemented
65#11745672—84no—
66C4⋊Dic59—81no—
67C4⋊Dic59—81no—
68C4⋊Dic59—81no—
69Dic59⋊C4—82no—
70C2×C59⋊C8—42no—
71C23×C59#12011yescharacteristic
72Q8×C59—82no—
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—41no—
78C2×C4×C59—41no—
79C2×C4×C59—42no—
80C2×C4×C59—42no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59—41no—
84C2×C4×C59C2×Q1611yescharacteristic
85C2×C4×C59#13711yescharacteristic
86C2×Dic59—42no—
87C2×Dic59—42no—
88C2×Dic59—42no—
89C2×Dic59—42no—
90C2×Dic59—42no—
91C2×Dic59—42no—
92Dic118—44no—
93Dic118—82no—
94C8×C59—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—81no—
107C4×C59—81no—
108C4×C59—81no—
109Dic59—82no—
110Dic59—82no—
111#2844—591no—
112#2061D5911yescharacteristic
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#544C2×D5911yescharacteristic
122#451—1182no—
123#428—21no—
124#400—1182no—
125#400—1181no—
126#395—592no—
127#395—1182no—
128#382—1181no—
129#382C2×D5911yescharacteristic
130#343C2×D5911yescharacteristic
131#339—21no—
132#328—1182no—
133#324—592no—
134C59#1226011yescharacteristic, complemented, Sylow
135C22×Q8—592no—
136C2×Q16—2362no—
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—1181no—
143C2×C4⋊C4—1181no—
144C2×C4⋊C4—1181no—
145C2×C4⋊C4—1182no—
146C2×C4⋊C4C22⋊D5911yescharacteristic
147C2×C4⋊C4C22×D5911yescharacteristic
148C2×C42C22⋊D5911yescharacteristic
149#107—41no—
150#106—41no—
151#103—2362no—
152C22⋊C8—1182no—
153#95—22no—
154#95—592no—
155#95—1182no—
156#95—1182no—
157#95—1181no—
158#95—1182no—
159C2×Q8—22no—
160C2×Q8—4722no—
161C2×Q8—2362no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1182no—
165C22×C4—1182no—
166C22×C4D4×D5912yes—
167C22×C4C2×C22⋊D5911yescharacteristic
168Q16—4724no—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—2362no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—4722no—
180C4⋊C4—4721no—
181C4⋊C4—4721no—
182C4⋊C4—4721no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42#1174567212yescomplemented
187C23#1427369511yescharacteristic
188Q8—82no—
189Q8—4722no—
190Q8—2364no—
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—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2362no—
203C2×C4—2362no—
204C2×C4—2362no—
205C2×C4—2362no—
206C2×C4—2362no—
207C2×C4—2362no—
208C2×C4C2×#1174567211yescharacteristic
209C2×C4#1427375711yescharacteristic
210C8—42no—
211C8—4722no—
212C22—21no—
213C22—21no—
214C22#1683755711yescharacteristic
215C22#1683758411yescharacteristic
216C22#1683758811yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—4722no—
225C4—4722no—
226C2—21no—
227C2—21no—
228C2#2142306911yescharacteristic
229C2#2142609611yescharacteristic
230C2#2142611611yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2061C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2061D59
hypocenterC59#12260
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
64. 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
#214323682
#214230702
#214221252
#214220402
C59×#20612
#1226059