Finite Group Databasebeta

← #28501264

Subgroups — #28501264

Order 15104.

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

normal subgroups only

This group has 372 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#21429338C211yesmaximal, characteristic, complemented
2#21423133C212yesmaximal
3#21422122C212yesmaximal, complemented
4#21422042C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic
6#16837718—22no—
7#16837341—21no—
8#16837341C2212yes—
9#16837301—22no—
10#16837267—22no—
11C4×C4⋊Dic59—21no—
12#16837136—22no—
13#16837128C2212yes—
14C59×#537C2211yescharacteristic
15C2×#104×C59—21no—
16C59×#417—21no—
17C59×#384C2211yescharacteristic
18C59×#343C2211yescharacteristic
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59D412yes—
23C2×C4×Dic59—22no—
24#14273658—42no—
25#14273648—42no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—21no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C59⋊#95—22no—
31C59⋊#95D412yes—
32C2×#41×C59—21no—
33C2×M16×C59—21no—
34C22⋊Q8×C59—41no—
35#121×C59—22no—
36#121×C59—41no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C4⋊C4×C59C2311yescharacteristic
40C2×C42×C59D411yescharacteristic
41#104×C59—42no—
42#103×C59—41no—
43#102×C59—41no—
44#41×C59—41no—
45#41×C59—41no—
46#41×C59—41no—
47#41×C59—41no—
48C2×Q8×C59—21no—
49C2×D4×C59—21no—
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—41no—
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59C2×D412yes—
57C22×Dic59—22no—
58C22×Dic59—22no—
59M16×C59—41no—
60M16×C59—41no—
61C2×C8×C59—41no—
62C4⋊C4×C59—22no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—44no—
65C4⋊C4×C59—41no—
66C22⋊C4×C59—42no—
67C42×C59—21no—
68C42×C59SD1612yes—
69C4⋊Dic59—42no—
70C4⋊Dic59—42no—
71C4⋊Dic59—42no—
72C4⋊Dic59—81no—
73C4×Dic59—44no—
74C2×C59⋊C8—42no—
75C23×C59—41no—
76C23×C59#12011yescharacteristic
77Q8×C59—41no—
78Q8×C59—41no—
79D4×C59—41no—
80D4×C59—41no—
81D4×C59—82no—
82D4×C59—81no—
83D4×C59—81no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—81no—
95C2×C4×C59—81no—
96C2×C4×C59C2×SD1611yescharacteristic
97C2×C4×C59#13711yescharacteristic
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C8×C59—81no—
105C59⋊C8—82no—
106#12181—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59—81no—
110C22×C59—81no—
111C22×C59—81no—
112C22×C59#44811yescharacteristic
113C22×C59#45411yescharacteristic
114C22×C59B(4,𝔽2)11yescharacteristic
115C4×C59—24no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—42no—
119C4×C59—81no—
120C4×C59—81no—
121Dic59—82no—
122Dic59—82no—
123#2571—591no—
124#2039D5911yescharacteristic
125#1616—592no—
126#1353—592no—
127#1325—591no—
128C2×C59—21no—
129C2×C59—21no—
130C2×C59—161no—
131C2×C59#159911yescharacteristic
132C2×C59#219811yescharacteristic
133C2×C59#220411yescharacteristic
134#537C2×D5911yescharacteristic
135#481—1182no—
136C2×#104—21no—
137#417—21no—
138#400—592no—
139#400—1181no—
140#389—1182no—
141#384C2×D5911yescharacteristic
142#382—1182no—
143C4×C4⋊C4—1181no—
144#343C2×D5911yescharacteristic
145#327—1182no—
146#324—592no—
147C59#1218111yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—592no—
155C2×C4⋊C4—1181no—
156C2×C4⋊C4—1181no—
157C2×C4⋊C4—1181no—
158C2×C4⋊C4C22⋊D5911yescharacteristic
159C2×C4⋊C4C22×D5911yescharacteristic
160C2×C42—1182no—
161C2×C42C22⋊D5911yescharacteristic
162#106—2362no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2362no—
167C22⋊C8—1182no—
168#95—592no—
169#95—1181no—
170#95—1182no—
171#95—1182no—
172#95—1182no—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—1182no—
185C22×C4—1182no—
186C22×C4D4×D5912yes—
187C22×C4C2×C22⋊D5911yescharacteristic
188M16—41no—
189M16—41no—
190C2×C8—41no—
191C2×C8—2362no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—44no—
196C4⋊C4—4721no—
197C4⋊C4—2362no—
198C4⋊C4—2362no—
199C4⋊C4—2362no—
200C22⋊C4—42no—
201C42—21no—
202C42—2364no—
203C42#1174567012yes—
204C23—41no—
205C23#1427369511yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—2362no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4—2362no—
231C2×C4C2×#1174567011yescharacteristic
232C2×C4#1427375711yescharacteristic
233C8—81no—
234C8—4722no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#1683754011yescharacteristic
241C22#1683757411yescharacteristic
242C22#1683758811yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—4722no—
250C4—4722no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2#2142303511yescharacteristic
255C2#2142609111yescharacteristic
256C2#2142612111yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2039C2
socleC22×C59B(4,𝔽2)
prime radicalC59×#537C22
hypercenter#2039D59
hypocenterC59#12181
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #21429338C2semidirect
3. #21422122C2semidirect
4. #21422042C2semidirect
147. C59#12181semidirect

Maximal subgroups

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

subgroupindex
#214293382
#214231332
#214221222
#214220422
C59×#20392
#1218159