Finite Group Databasebeta

← #28737315

Subgroups — #28737315

Order 15104.

594 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 135 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#21425612C214yesmaximal, complemented
2#21425497C214yesmaximal, complemented
3#21425494C214yesmaximal, complemented
4#21424024C211yesmaximal, characteristic
5#21424024C211yesmaximal, characteristic
6C59×#1785C211yesmaximal, characteristic
7#16837451C2214yes—
8#16837450C2214yes—
9#16837450C2214yes—
10C2×#14273646C2212yes—
11C2×#14273646C2212yes—
12C2×#14273644C2212yes—
13C2×#14273644C2212yes—
14#16837366C2212yes—
15#16837366C2212yes—
16C59⋊#338C2214yes—
17C22×#41×C59C2211yescharacteristic
18C2×#101×C59C2212yes—
19C2×#100×C59C2212yes—
20C59×#407C2212yes—
21C2×#11745664C2314yes—
22C2×#11745664C2314yes—
23#14273663—44no—
24#14273662—44no—
25#14273662—44no—
26#14273646—42no—
27#14273646—42no—
28#14273644—42no—
29#14273644—42no—
30C22⋊(C59⋊C8)—42no—
31C22⋊(C59⋊C8)—42no—
32C2×#41×C59—42no—
33C2×#41×C59—42no—
34C2×#41×C59—42no—
35C22×Q8×C59C2311yescharacteristic
36C22×D4×C59Q812yes—
37C22×D4×C59C2311yescharacteristic
38C23×C4×C59Q812yes—
39C23×C4×C59C2311yescharacteristic
40C2×M16×C59C2314yes—
41#101×C59—42no—
42#100×C59—42no—
43C22⋊C8×C59—42no—
44C24×C59C2×Q812yes—
45C24×C59C2×Q811yescharacteristic
46#41×C59—416no—
47C2×Q8×C59—22no—
48C2×Q8×C59—22no—
49C2×Q8×C59—22no—
50C2×D4×C59—22no—
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—42no—
54C2×D4×C59—42no—
55C2×D4×C59—42no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C2×Q811yescharacteristic
63C22×C4×C59C2×Q812yes—
64C22×C4×C59C2411yescharacteristic
65M16×C59—24no—
66M16×C59—24no—
67C2×C8×C59—24no—
68#11745664—24no—
69#11745664—24no—
70#11745664—24no—
71#11745664—24no—
72C2×C59⋊C8—24no—
73C2×C59⋊C8—24no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59—21no—
77C23×C59—22no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—81no—
81C23×C59—82no—
82C23×C59C22×Q811yescharacteristic
83Q8×C59—44no—
84D4×C59—48no—
85D4×C59—44no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—48no—
93C2×C4×C59—44no—
94C2×C4×C59H(5,𝔽2)12yes—
95C2×C4×C59H(5,𝔽2)12yes—
96C2×C4×C59H(5,𝔽2)12yes—
97C8×C59—44no—
98C59⋊C8—44no—
99C59⋊C8—44no—
100#32877—591nomaximal, Sylow
101C22×C59—41no—
102C22×C59—42no—
103C22×C59—42no—
104C22×C59—42no—
105C22×C59—41no—
106C22×C59—42no—
107C22×C59—82no—
108C22×C59—82no—
109C22×C59—82no—
110C22×C59#54311yescharacteristic
111C22×C59#54311yescharacteristic
112C22×C59#54311yescharacteristic
113C4×C59—41no—
114C4×C59—42no—
115C4×C59#56014yes—
116#2091—594no—
117#2070—594no—
118#2069—594no—
119#1785—591no—
120#1785—591no—
121#1785D5911yescharacteristic
122C2×C59—21no—
123C2×C59—21no—
124C2×C59—21no—
125C2×C59—81no—
126C2×C59—82no—
127C2×C59#284211yescharacteristic
128C22×#41C2×D5911yescharacteristic
129#430—594no—
130#430—594no—
131#430—594no—
132C2×#101—592no—
133C2×#101—592no—
134C2×#101C2×D5912yes—
135C2×#100—592no—
136C2×#100—592no—
137C2×#100C2×D5912yes—
138#407—592no—
139#407—592no—
140#407C2×D5912yes—
141#338—594no—
142C59#3287711yescharacteristic, complemented, Sylow
143C2×#41—42no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8C22×D5911yescharacteristic
147C22×D4Dic11812yes—
148C22×D4C22×D5911yescharacteristic
149C23×C4Dic11812yes—
150C23×C4C22×D5911yescharacteristic
151C2×M16—594no—
152C2×M16—594no—
153C2×M16C22×D5914yes—
154#108—2364no—
155#108—2364no—
156#108—2364no—
157#101—42no—
158#101—2362no—
159#101—2362no—
160#100—42no—
161#100—2362no—
162#100—2362no—
163C22⋊C8—42no—
164C22⋊C8—2362no—
165C22⋊C8—2362no—
166C24C2×Dic11812yes—
167C24C2×Dic11811yescharacteristic
168#41—416no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×Dic11811yescharacteristic
185C22×C4C2×Dic11812yes—
186C22×C4C23×D5911yescharacteristic
187M16—24no—
188M16—24no—
189M16—1184no—
190M16—1184no—
191M16—1184no—
192M16—1184no—
193C2×C8—24no—
194C2×C8—1184no—
195C2×C8—1184no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—81no—
203C23—82no—
204C23C22×Dic11811yescharacteristic
205Q8—44no—
206D4—48no—
207D4—44no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—48no—
215C2×C4—44no—
216C2×C4#1427377212yes—
217C2×C4#1427377212yes—
218C2×C4#1427377212yes—
219C8—44no—
220C8—2364no—
221C8—2364no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—42no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22#1683802811yescharacteristic
232C22#1683802811yescharacteristic
233C22#1683802811yescharacteristic
234C4—41no—
235C4—42no—
236C4#1683814114yes—
237C2—21no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—82no—
242C2#2143236011yescharacteristic
subgroupG/H
centerC2#21432360
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1785C2
socleC2×C59#2842
prime radicalC22×#41×C59C22
hypercenter#1785D59
hypocenterC59#32877
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
1. #21425612C2semidirect
2. #21425497C2semidirect
3. #21425494C2semidirect
142. C59#32877semidirect

Maximal subgroups

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

subgroupindex
#214256122
#214254972
#214254942
#214240242
C59×#17852
#3287759