Finite Group Databasebeta

← #28501289

Subgroups — #28501289

Order 15104.

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

normal subgroups only

This group has 360 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#21425325C211yesmaximal, characteristic
2#21423132C212yesmaximal
3#21422123C212yesmaximal, complemented
4#21422040C211yesmaximal, characteristic, complemented
5C59×#2844C211yesmaximal, characteristic, complemented
6#16837983C2211yescharacteristic
7#16837720—22no—
8C2×#14273655—21no—
9#16837401—21no—
10#16837171C2211yescharacteristic
11#16837135—22no—
12#16837128C2212yes—
13C59×#400—22no—
14C59×#400C2212yes—
15C59×#395—22no—
16C59×#395—21no—
17C59×#382—21no—
18C59×#382C2211yescharacteristic
19C2×#11745685—21no—
20C2×#11745664—21no—
21#14273707—41no—
22#14273702—41no—
23#14273701—22no—
24#14273658—42no—
25#14273655—42no—
26#14273650—41no—
27#14273647—41no—
28#14273647—42no—
29C22⋊(C59⋊C8)—22no—
30C22×Q8×C59—21no—
31C2×C4⋊C4×C59—21no—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59D411yescharacteristic
35C2×C4⋊C4×C59D412yes—
36C2×C4⋊C4×C59C2311yescharacteristic
37C2×C42×C59D411yescharacteristic
38#95×C59—22no—
39#95×C59—21no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59—22no—
43#95×C59D412yes—
44C2×Q8×C59—82no—
45C2×Q8×C59—81no—
46C22×C4×C59—21no—
47C22×C4×C59—22no—
48C22×C4×C59—22no—
49C22×C4×C59—21no—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D412yes—
52C2×C22⋊D59—41no—
53#11745685—41no—
54#11745685—41no—
55#11745685—41no—
56#11745685—41no—
57C2×D236—21no—
58C2×D236—22no—
59C2×C4×D59—41no—
60C2×Dic118—21no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—81no—
65C4⋊C4×C59—81no—
66C4⋊C4×C59—82no—
67C42×C59—21no—
68C42×C59SD1612yes—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—22no—
71#11745664—41no—
72#11745664—41no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8—41no—
75C23×C59#12011yescharacteristic
76C22×D59—41no—
77Q8×C59—84no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
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×C59—41no—
92C2×C4×C59—42no—
93C2×C4×C59C2×SD1611yescharacteristic
94C2×C4×C59#13711yescharacteristic
95C22⋊D59—81no—
96C22⋊D59—81no—
97C2×Dic59—41no—
98D236—41no—
99D236—41no—
100D236—82no—
101C4×D59—81no—
102C4×D59—81no—
103Dic118—41no—
104Dic118—41no—
105C59⋊C8—81no—
106C59⋊C8—82no—
107#12185—591nomaximal, Sylow
108C22×C59—21no—
109C22×C59—21no—
110C22×C59#44811yescharacteristic
111C22×C59#45411yescharacteristic
112C22×C59B(4,𝔽2)11yescharacteristic
113C2×D59—81no—
114C2×D59—81no—
115C2×D59—81no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—42no—
119C4×C59—42no—
120C4×C59—82no—
121C4×C59—81no—
122C4×C59—82no—
123Dic59—81no—
124#2844D5911yescharacteristic, complemented
125#2040—591no—
126#1616—592no—
127#1353—592no—
128#1324—591no—
129C2×C59—21no—
130C2×C59—21no—
131C2×C59#159911yescharacteristic
132C2×C59#220111yescharacteristic
133C2×C59#220211yescharacteristic
134D59—161no—
135#537—591no—
136#481—1182no—
137C2×#104—1181no—
138#417—1181no—
139#400—22no—
140#400C2×D5912yes—
141#395—22no—
142#395—21no—
143#382—21no—
144#382C2×D5911yescharacteristic
145#343—591no—
146#327—1182no—
147#324—592no—
148C59#1218511yescharacteristic, complemented, Sylow
149C2×#41—1181no—
150C22×Q8—21no—
151C2×M16—1181no—
152C22⋊Q8—2361no—
153#121—1182no—
154#121—2361no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊D5911yescharacteristic
159C2×C4⋊C4C22⋊D5912yes—
160C2×C4⋊C4C22×D5911yescharacteristic
161C2×C42C22⋊D5911yescharacteristic
162#106—2362no—
163#104—2362no—
164#103—2361no—
165#102—2361no—
166#102—2362no—
167C22⋊C8—1182no—
168#95—22no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C22⋊D5912yes—
174#41—2361no—
175#41—2361no—
176#41—2361no—
177#41—2361no—
178C2×Q8—82no—
179C2×Q8—81no—
180C2×Q8—1181no—
181C2×D4—1182no—
182C2×D4—1181no—
183C2×D4—2361no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—2361no—
189C22×C4C2×C22⋊D5911yescharacteristic
190C22×C4C2×C22⋊D5912yes—
191M16—2361no—
192M16—2361no—
193C2×C8—2362no—
194C2×C8—2361no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—1182no—
202C22⋊C4—2362no—
203C42—21no—
204C42#1174567112yes—
205C23—2361no—
206C23#1427369611yescharacteristic
207Q8—84no—
208Q8—2361no—
209Q8—2361no—
210D4—4722no—
211D4—4721no—
212D4—4721no—
213D4—2361no—
214D4—2361no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—4721no—
231C2×C4—4721no—
232C2×C4—2361no—
233C2×C4C2×#1174567111yescharacteristic
234C2×C4#1427375811yescharacteristic
235C8—4721no—
236C8—4722no—
237C22—21no—
238C22—21no—
239C22—4721no—
240C22—4721no—
241C22—4721no—
242C22#1683754111yescharacteristic
243C22#1683757511yescharacteristic
244C22#1683758911yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—4721no—
253C2—21no—
254C2—21no—
255C2—9441no—
256C2#2142303611yescharacteristic
257C2#2142610911yescharacteristic
258C2#2142611311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2844C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2844D59
hypocenterC59#12185
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
3. #21422123C2semidirect
4. #21422040C2semidirect
5. C59×#2844C2semidirect
124. #2844D59semidirect
148. C59#12185semidirect

Maximal subgroups

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

subgroupindex
#214253252
#214231322
#214221232
#214220402
C59×#28442
#1218559