Finite Group Databasebeta

← #28477959

Subgroups — #28477959

Order 15104.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#21431026C211yesmaximal, characteristic
2#21425660C212yesmaximal, complemented
3#21425629C211yesmaximal, characteristic
4#21421918C212yesmaximal, complemented
5C59×#1270C211yesmaximal, characteristic
6C2×#14273708—21no—
7C2×#14273658—21no—
8C2×#14273648—21no—
9#16837332—21no—
10#16837327—22no—
11#16837300C2212yes—
12#16837277—22no—
13#16837272C2211yescharacteristic
14#16837165—22no—
15#16837165—22no—
16#16837157C2211yescharacteristic
17C59×#390C2211yescharacteristic
18C59×#336C2212yes—
19C23×Dic59—21no—
20C22×C59⋊C8—21no—
21#14273708—42no—
22#14273708—42no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59D412yes—
27C2×C4⋊Dic59D412yes—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—21no—
31#14273658—42no—
32#14273648—42no—
33C59⋊#95—22no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C22×D4×C59D411yescharacteristic
37C22×C8×C59—22no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C42×C59C2311yescharacteristic
41C24×C59—21no—
42C2×D4×C59—21no—
43C2×D4×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59SD1612yes—
46C22×C4×C59—21no—
47C22×C4×C59C2×D411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—22no—
50C22×Dic59—22no—
51C22×Dic59—42no—
52C22×Dic59—41no—
53C22×Dic59—81no—
54C22×Dic59—81no—
55C2×C8×C59—42no—
56C2×C8×C59—42no—
57C2×C8×C59—42no—
58C4⋊C4×C59—41no—
59C22⋊C4×C59—42no—
60C22⋊C4×C59—42no—
61C42×C59—41no—
62C22⋊Dic59—42no—
63C22⋊Dic59—44no—
64C4⋊Dic59—22no—
65C4⋊Dic59—22no—
66C4⋊Dic59—42no—
67C4⋊Dic59—41no—
68C4⋊Dic59—41no—
69C4⋊Dic59—44no—
70C4⋊Dic59—81no—
71Dic59⋊C4—42no—
72Dic59⋊C4—42no—
73C2×C59⋊C8—22no—
74C2×C59⋊C8—41no—
75C2×C59⋊C8—41no—
76C23×C59—22no—
77C23×C59—42no—
78C23×C59—41no—
79C23×C59—42no—
80C23×C59—41no—
81C23×C59#12011yescharacteristic
82D4×C59—42no—
83D4×C59—81no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—42no—
89C2×C4×C59—41no—
90C2×C4×C59C2×SD1611yescharacteristic
91C2×C4×C59#13512yes—
92C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C59#13712yes—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—42no—
97C2×Dic59—42no—
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—84no—
101C2×Dic59—81no—
102C2×Dic59—82no—
103C8×C59—82no—
104C59⋊C8—42no—
105#7824—591nomaximal, Sylow
106C22×C59—42no—
107C22×C59—44no—
108C22×C59—82no—
109C22×C59—81no—
110C22×C59—81no—
111C22×C59#45011yescharacteristic
112C22×C59#45212yes—
113C22×C59#45311yescharacteristic
114C22×C59#45412yes—
115C22×C59#45811yescharacteristic
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—22no—
119C4×C59—22no—
120C4×C59—81no—
121Dic59—82no—
122Dic59—82no—
123Dic59—161no—
124#2724—591no—
125#2100—592no—
126#2094—591no—
127#1271—592no—
128#1270D5911yescharacteristic
129C2×C59—82no—
130C2×C59#161311yescharacteristic
131C2×C59#161512yes—
132C2×C59#163011yescharacteristic
133C2×C59#163512yes—
134C2×C59#165711yescharacteristic
135C2×#123—1181no—
136C2×#106—1181no—
137C2×#102—1181no—
138#398—1181no—
139#397—1182no—
140#390C2×D5911yescharacteristic
141#389—592no—
142#385—1182no—
143#384—591no—
144#340—1182no—
145#340—1182no—
146#336—591no—
147#336C2×D5912yes—
148C59#782411yescharacteristic, complemented, Sylow
149C22×D4D23611yescharacteristic
150C23×C4—1181no—
151C22×C8—22no—
152C22×C8—1181no—
153#123—2362no—
154#123—2362no—
155C2×C4⋊C4—592no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—1181no—
158C2×C4⋊C4—1182no—
159C2×C4⋊C4—1181no—
160C2×C4⋊C4—1182no—
161C2×C4⋊C4D23611yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—1182no—
164C2×C22⋊C4—1181no—
165C2×C42C22×D5911yescharacteristic
166#106—2362no—
167#102—2362no—
168#95—1182no—
169#95—1181no—
170#95—1182no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1174566012yes—
176C22×C4—21no—
177C22×C4—1182no—
178C22×C4—1182no—
179C22×C4—4721no—
180C22×C4—4721no—
181C22×C4—2361no—
182C22×C4—2362no—
183C22×C4C2×D23611yescharacteristic
184C22×C4D4×D5912yes—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—1182no—
189C2×C8—2361no—
190C2×C8—2361no—
191C4⋊C4—41no—
192C4⋊C4—1182no—
193C4⋊C4—1182no—
194C4⋊C4—4721no—
195C4⋊C4—2361no—
196C4⋊C4—2361no—
197C4⋊C4—2364no—
198C4⋊C4—2362no—
199C4⋊C4—2362no—
200C4⋊C4—2362no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—2362no—
204C22⋊C4—2364no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1427369711yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—4721no—
221C2×C4—4722no—
222C2×C4—4724no—
223C2×C4—2362no—
224C2×C4—2362no—
225C2×C4—2362no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4C2×#1174566011yescharacteristic
230C2×C4#1427374612yes—
231C2×C4#1427375511yescharacteristic
232C2×C4#1427375612yes—
233C8—82no—
234C8—2362no—
235C22—42no—
236C22—44no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22#1683755511yescharacteristic
241C22#1683756512yes—
242C22#1683757211yescharacteristic
243C22#1683757612yes—
244C22#1683759311yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—81no—
250C4—9441no—
251C4—4722no—
252C4—4722no—
253C2—82no—
254C2#2142311811yescharacteristic
255C2#2142312812yes—
256C2#2142321811yescharacteristic
257C2#2142324912yes—
258C2#2142339511yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#1270D59
hypocenterC59#7824
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21425660C2semidirect
4. #21421918C2semidirect
148. C59#7824semidirect

Maximal subgroups

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

subgroupindex
#214310262
#214256602
#214256292
#214219182
C59×#12702
#782459