Finite Group Databasebeta

← #28501152

Subgroups — #28501152

Order 15104.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426029C212yesmaximal
2#21425419C211yesmaximal, characteristic
3#21422108C212yesmaximal
4#21422036C211yesmaximal, characteristic
5C59×#3249C211yesmaximal, characteristic
6C22×Dic236C2211yescharacteristic
7C2×#14273673—22no—
8C2×#14273652—21no—
9#16837308—21no—
10#16837205—22no—
11C22⋊(C59⋊C16)C2212yes—
12#16837165C2211yescharacteristic
13C59×#533—21no—
14C59×#477—22no—
15C59×#467—21no—
16C59×#467C2212yes—
17C59×#462—22no—
18C2×C4⋊C8×C59C2211yescharacteristic
19C22×Dic118—21no—
20C2×Dic236—22no—
21C2×Dic236—42no—
22C2×C4⋊Dic59—21no—
23#14273673—44no—
24C2×C59⋊C16—22no—
25#14273652—42no—
26C59⋊#95—41no—
27C59⋊#95—41no—
28C2×Q16×C59—21no—
29C2×Q16×C59D412yes—
30C22×C8×C59D411yescharacteristic
31C22×C8×C59C2311yescharacteristic
32C4⋊Q8×C59—21no—
33C4⋊Q8×C59—21no—
34#125×C59—41no—
35C22⋊Q8×C59—22no—
36C22⋊Q8×C59—41no—
37C4×Q8×C59—22no—
38C4×Q8×C59—41no—
39#117×C59—41no—
40C2×C42×C59D411yescharacteristic
41#107×C59—21no—
42#107×C59D412yes—
43C4⋊C8×C59—22no—
44#103×C59—22no—
45#103×C59—22no—
46C2×Q8×C59—22no—
47C2×Q8×C59—41no—
48C22×C4×C59D811yescharacteristic
49C22×C4×C59D811yescharacteristic
50C22×C4×C59C2×D411yescharacteristic
51C22×Dic59—41no—
52C22×Dic59—41no—
53C2×Dic118—22no—
54C2×Dic118—41no—
55C2×Dic118—41no—
56C2×Dic118—41no—
57C2×Dic118—41no—
58C2×Dic118—41no—
59Q16×C59—42no—
60Q16×C59—81no—
61C2×C8×C59—21no—
62C2×C8×C59—41no—
63C2×C8×C59—41no—
64C2×C8×C59C2×D412yes—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C22⋊C4×C59—42no—
72C42×C59—21no—
73C42×C59—21no—
74C42×C59—42no—
75C4⋊Dic59—41no—
76C4⋊Dic59—41no—
77Dic236—24no—
78Dic236—42no—
79C59⋊C16—42no—
80C23×C59C2×D811yescharacteristic
81Q8×C59—42no—
82Q8×C59—42no—
83Q8×C59—82no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59Q3212yescomplemented
93C2×C4×C59#12011yescharacteristic
94C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
95C2×Dic59—42no—
96C2×Dic59—81no—
97C2×Dic59—81no—
98C2×Dic59—81no—
99C2×Dic59—81no—
100C2×Dic59—81no—
101Dic118—42no—
102Dic118—42no—
103Dic118—81no—
104Dic118—81no—
105C8×C59—22no—
106C8×C59—41no—
107#12162—591nomaximal, Sylow
108C22×C59—21no—
109C22×C59—21no—
110C22×C59#44711yescharacteristic
111C22×C59C2×Q3211yescharacteristic
112C22×C59#51011yescharacteristic
113C4×C59—21no—
114C4×C59—22no—
115C4×C59—41no—
116C4×C59—82no—
117C4×C59—82no—
118C4×C59#45411yescharacteristic
119C4×C59B(4,𝔽2)11yescharacteristic
120Dic59—82no—
121Dic59—161no—
122#3249D5911yescharacteristic
123#2188—592no—
124#2056—591no—
125#1348—592no—
126#1322—591no—
127C2×C59—21no—
128C2×C59—21no—
129C2×C59#159811yescharacteristic
130C2×C59#218811yescharacteristic
131C2×C59#219111yescharacteristic
132C22×Q16—591no—
133#533—21no—
134C2×Q32—1182no—
135#477—22no—
136#467—21no—
137#467C2×D5912yes—
138#462—22no—
139C2×C4⋊C8C2×D5911yescharacteristic
140C2×#103—1181no—
141#391—1181no—
142#358—1182no—
143C22⋊C16—592no—
144#340—591no—
145C59#1216211yescharacteristic, complemented, Sylow
146C22×Q8—1181no—
147C2×Q16—21no—
148C2×Q16—1182no—
149C2×Q16—2362no—
150C2×Q16C22⋊D5912yes—
151C22×C8C22⋊D5911yescharacteristic
152C22×C8C22×D5911yescharacteristic
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155#125—41no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—41no—
160#117—41no—
161C2×C4⋊C4—1181no—
162C2×C42C22⋊D5911yescharacteristic
163Q32—2364no—
164C2×C16—1182no—
165#107—21no—
166#107C22⋊D5912yes—
167C4⋊C8—22no—
168#103—22no—
169#103—22no—
170#103—2362no—
171#95—2361no—
172#95—2361no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—1182no—
176C2×Q8—2361no—
177C2×Q8—2361no—
178C2×Q8—2361no—
179C2×Q8—2361no—
180C2×Q8—2361no—
181C22×C4—2361no—
182C22×C4—2361no—
183C22×C4#1174566911yescharacteristic
184C22×C4#1174566911yescharacteristic
185C22×C4C2×C22⋊D5911yescharacteristic
186Q16—42no—
187Q16—81no—
188Q16—1184no—
189Q16—2362no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8C2×C22⋊D5912yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—2361no—
201C4⋊C4—2361no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—42no—
206C16—2362no—
207C23C2×#1174566911yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—82no—
211Q8—4721no—
212Q8—4721no—
213Q8—2362no—
214Q8—2362no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4721no—
224C2×C4—4721no—
225C2×C4—4721no—
226C2×C4—4721no—
227C2×C4—4721no—
228C2×C4—2362no—
229C2×C4#1427367312yescomplemented
230C2×C4#1427369611yescharacteristic
231C2×C4#1427375311yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#1683753611yescharacteristic
237C22C2×#1427367311yescharacteristic
238C22#1683784311yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4—9441no—
245C4—4722no—
246C4#1683757511yescharacteristic
247C4#1683758911yescharacteristic
248C2—21no—
249C2—21no—
250C2#2142303111yescharacteristic
251C2#2142603011yescharacteristic
252C2#2142604711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3249C2
socleC22×C59#447
prime radicalC23×C59C2×D8
hypercenter#3249D59
hypocenterC59#12162
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
92. C2×C4×C59Q32semidirect
145. C59#12162semidirect
229. C2×C4#14273673semidirect

Maximal subgroups

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

subgroupindex
#214260292
#214254192
#214221082
#214220362
C59×#32492
#1216259