Finite Group Databasebeta

← #28501532

Subgroups — #28501532

Order 15104.

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

normal subgroups only

This group has 380 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#21425257C211yesmaximal, characteristic
2#21423069C212yesmaximal
3#21422126C212yesmaximal
4#21422030C211yesmaximal, characteristic
5C59×#2811C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837558—22no—
8#16837411—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C59×#395—22no—
14C59×#395C2212yes—
15C59×#391—21no—
16C59×#391—21no—
17C59×#391—22no—
18C59×#391C2211yescharacteristic
19C2×#11745672—42no—
20C2×#11745664—21no—
21#14273726—41no—
22#14273718—41no—
23#14273706—22no—
24#14273686—21no—
25#14273651—41no—
26#14273651—41no—
27#14273651—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59—21no—
30C22×Q8×C59D411yescharacteristic
31C22×Q8×C59D412yes—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59C2311yescharacteristic
36C2×C42×C59—21no—
37C2×C42×C59D411yescharacteristic
38#95×C59—22no—
39#95×C59—22no—
40#95×C59—22no—
41#95×C59—22no—
42#95×C59D412yes—
43C2×Q8×C59—24no—
44C2×Q8×C59—21no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59—42no—
48C2×Q8×C59—82no—
49C2×Q8×C59—81no—
50C2×Q8×C59—81no—
51C2×Q8×C59Q1612yescomplemented
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D412yes—
58C2×Dic118—22no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—82no—
61C4⋊C4×C59—81no—
62C4⋊C4×C59—81no—
63C42×C59—41no—
64C42×C59—81no—
65C22⋊Dic59—41no—
66#11745672—84no—
67C4⋊Dic59—21no—
68C4⋊Dic59—21no—
69C4⋊Dic59—22no—
70Dic59⋊C4—42no—
71C4×Dic59—41no—
72#11745664—41no—
73#11745664—41no—
74C2×C59⋊C8—42no—
75C2×C59⋊C8—41no—
76C23×C59#12011yescharacteristic
77Q8×C59—42no—
78Q8×C59—44no—
79Q8×C59—81no—
80Q8×C59—84no—
81Q8×C59—82no—
82C2×C4×C59—21no—
83C2×C4×C59—21no—
84C2×C4×C59—22no—
85C2×C4×C59—24no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—42no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59C2×Q1611yescharacteristic
99C2×C4×C59#13711yescharacteristic
100C2×Dic59—41no—
101C2×Dic59—41no—
102Dic118—82no—
103C59⋊C8—81no—
104C59⋊C8—82no—
105#12224—591nomaximal, Sylow
106C22×C59—21no—
107C22×C59—21no—
108C22×C59#45111yescharacteristic
109C22×C59#45511yescharacteristic
110C22×C59B(4,𝔽2)11yescharacteristic
111C4×C59—21no—
112C4×C59—21no—
113C4×C59—42no—
114C4×C59—44no—
115C4×C59—42no—
116C4×C59—42no—
117C4×C59—82no—
118Dic59—81no—
119Dic59—81no—
120#2811D5911yescharacteristic
121#2028—591no—
122#1605—592no—
123#1354—592no—
124#1319—591no—
125C2×C59—21no—
126C2×C59—21no—
127C2×C59#160211yescharacteristic
128C2×C59#219911yescharacteristic
129C2×C59#220511yescharacteristic
130#544—591no—
131#451—1182no—
132#419—1181no—
133#395—22no—
134#395C2×D5912yes—
135#391—21no—
136#391—21no—
137#391—22no—
138#391C2×D5911yescharacteristic
139#343—591no—
140#339—1181no—
141#328—1182no—
142#324—592no—
143C59#1222411yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C22×Q8C22⋊D5911yescharacteristic
146C22×Q8C22⋊D5912yes—
147C2×Q16—2362no—
148C2×M16—1181no—
149C4⋊Q8—2361no—
150#125—2361no—
151C22⋊Q8—1182no—
152#117—1181no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22×D5911yescharacteristic
157C2×C42—21no—
158C2×C42C22⋊D5911yescharacteristic
159#103—2361no—
160#103—2361no—
161#103—2362no—
162C22⋊C8—1182no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95C22⋊D5912yes—
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—1182no—
177C2×Q8#1174567212yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D5911yescharacteristic
183C22×C4C2×C22⋊D5912yes—
184Q16—4724no—
185M16—2361no—
186M16—2361no—
187C2×C8—2362no—
188C2×C8—2361no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—1181no—
194C4⋊C4—1181no—
195C4⋊C4—1182no—
196C4⋊C4—2362no—
197C22⋊C4—2361no—
198C42—41no—
199C42—81no—
200C42—2361no—
201C23#1427369611yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—4722no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—2361no—
225C2×C4—2361no—
226C2×C4C2×#1174567211yescharacteristic
227C2×C4#1427375811yescharacteristic
228C8—4721no—
229C8—4722no—
230C22—21no—
231C22—21no—
232C22#1683755811yescharacteristic
233C22#1683758011yescharacteristic
234C22#1683758911yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—4721no—
243C4—4721no—
244C2—21no—
245C2—21no—
246C2#2142305311yescharacteristic
247C2#2142609711yescharacteristic
248C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2811C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
hypercenter#2811D59
hypocenterC59#12224
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
51. C2×Q8×C59Q16semidirect
143. C59#12224semidirect
177. C2×Q8#11745672semidirect

Maximal subgroups

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

subgroupindex
#214252572
#214230692
#214221262
#214220302
C59×#28112
#1222459