Finite Group Databasebeta

← #28502013

Subgroups — #28502013

Order 15104.

376 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 55 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#21434896C211yesmaximal, characteristic
2#21426310C212yesmaximal, complemented
3#21422086C212yesmaximal, complemented
4#21422032C211yesmaximal, characteristic, complemented
5C59×#2091C211yesmaximal, characteristic, complemented
6C2×#14273701—21no—
7#16837721—22no—
8#16837655—21no—
9#16837655C2212yes—
10#16837538—22no—
11#16837226—22no—
12#16837214—22no—
13#16837184C2212yescomplemented
14C59×#430—21no—
15C59×#430C2211yescharacteristic, complemented
16C59×#407—21no—
17C59×#407C2211yescharacteristic
18C59×#338C2211yescharacteristic, complemented
19C22×D236—21no—
20C22×C22⋊D59—41no—
21C2×D472—21no—
22C2×D472D412yes—
23#14273701—22no—
24#14273701—22no—
25#14273701—41no—
26#14273701—41no—
27C2×C4⋊Dic59—21no—
28C2×(C2×D59)⋊C4—41no—
29#14273667—22no—
30#14273659—21no—
31#14273659D412yes—
32#14273649—22no—
33#14273649—22no—
34C23×C4×C59D411yescharacteristic
35C2×M16×C59—21no—
36C2×M16×C59—21no—
37C2×M16×C59D411yescharacteristic
38C2×M16×C59C2311yescharacteristic
39#108×C59—22no—
40#108×C59—42no—
41C22⋊C8×C59—22no—
42C22⋊C8×C59—42no—
43C24×C59SD1611yescharacteristic
44C23×D59—41no—
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59—41no—
48C22×C4×C59—41no—
49C22×C4×C59SD1611yescharacteristic
50C22×C4×C59C2×D411yescharacteristic
51C2×C22⋊D59—42no—
52C2×C22⋊D59—42no—
53C2×C22⋊D59—84no—
54C22×Dic59—41no—
55C2×D236—22no—
56C2×D236—41no—
57C2×D236—81no—
58C2×D236—81no—
59M16×C59—41no—
60M16×C59—41no—
61M16×C59—41no—
62M16×C59—41no—
63M16×C59—41no—
64M16×C59—41no—
65C2×C8×C59—22no—
66C2×C8×C59—21no—
67C2×C8×C59—41no—
68C2×C8×C59C2×D412yes—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71C4⋊Dic59—22no—
72C4⋊Dic59—41no—
73D472—42no—
74D472—81no—
75C59⋊C16—42no—
76C23×C59—21no—
77C23×C59—42no—
78C23×C59—41no—
79C23×C59C2×SD1611yescharacteristic
80C22×D59—42no—
81C22×D59—82no—
82C22×D59—82no—
83C22×D59—82no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—81no—
91C2×C4×C59#12011yescharacteristic
92C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
93C22⋊D59—84no—
94C22⋊D59—84no—
95C2×Dic59—42no—
96C2×Dic59—82no—
97D236—42no—
98D236—42no—
99D236—82no—
100C8×C59—22no—
101C8×C59—42no—
102C8×C59—41no—
103C8×C59—81no—
104#12298—591nomaximal, Sylow
105C22×C59—22no—
106C22×C59—21no—
107C22×C59—41no—
108C22×C59—41no—
109C22×C59—42no—
110C22×C59—41no—
111C22×C59—81no—
112C22×C59#45011yescharacteristic
113C2×D59—82no—
114C2×D59—82no—
115C2×D59—82no—
116C2×D59—164no—
117C4×C59—21no—
118C4×C59—41no—
119C4×C59#45311yescharacteristic
120C4×C59#45811yescharacteristic
121Dic59—82no—
122#3192—591no—
123#2238—592no—
124#2091D5911yescharacteristic, complemented
125#1341—592no—
126#1320—591no—
127C2×C59—21no—
128C2×C59—42no—
129C2×C59—41no—
130C2×C59#161311yescharacteristic
131D59—162no—
132C2×#121—1181no—
133#481—1182no—
134#469—592no—
135#469—1181no—
136#447—1182no—
137#430—21no—
138#430C2×D5911yescharacteristic, complemented
139#407—21no—
140#407C2×D5911yescharacteristic
141#365—1182no—
142#361—1182no—
143#349—592no—
144#338C2×D5911yescharacteristic, complemented
145C59#1229811yescharacteristic, complemented, Sylow
146C22×D4—1181no—
147C22×D4—2361no—
148C23×C4C22⋊D5911yescharacteristic
149C2×D8—592no—
150C2×D8—1181no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16C22⋊D5911yescharacteristic
154C2×M16C22×D5911yescharacteristic
155#121—1182no—
156#121—1182no—
157#121—2361no—
158#121—2361no—
159C2×C4⋊C4—1181no—
160C2×C22⋊C4—2361no—
161M32—1182no—
162#108—22no—
163#108—42no—
164#106—592no—
165#106—1181no—
166#102—1182no—
167#102—1182no—
168C22⋊C8—22no—
169C22⋊C8—42no—
170C24—2361no—
171C24#1174567111yescharacteristic
172C2×D4—1182no—
173C2×D4—4721no—
174C2×D4—4721no—
175C2×D4—4724no—
176C2×D4—2361no—
177C2×D4—2362no—
178C2×D4—2362no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—2361no—
184C22×C4#1174567111yescharacteristic
185C22×C4C2×C22⋊D5911yescharacteristic
186D8—4721no—
187D8—2362no—
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193M16—41no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8D4×D5912yes—
198C4⋊C4—1182no—
199C4⋊C4—2361no—
200C22⋊C4—2362no—
201C22⋊C4—2362no—
202C16—2362no—
203C23—21no—
204C23—41no—
205C23—42no—
206C23—4722no—
207C23—4722no—
208C23—4722no—
209C23—2362no—
210C23C2×#1174567111yescharacteristic
211D4—4722no—
212D4—4724no—
213D4—4724no—
214D4—2362no—
215D4—2362no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—4722no—
224C2×C4—2362no—
225C2×C4#1427369511yescharacteristic
226C2×C4#1427375211yescharacteristic
227C8—22no—
228C8—42no—
229C8—41no—
230C8—81no—
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—81no—
238C22—9444no—
239C22—4722no—
240C22—4722no—
241C22—4722no—
242C22#1683755211yescharacteristic
243C4—21no—
244C4—41no—
245C4—4722no—
246C4#1683756911yescharacteristic
247C4#1683759111yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—9442no—
252C2#2142311511yescharacteristic
subgroupG/H
centerC2#21423115
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#2091C2
socleC2×C59#1613
prime radical#21434896C2
hypercenter#2091D59
hypocenterC59#12298
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2091

Complements

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

NH ≅ G/Ntype
2. #21426310C2semidirect
3. #21422086C2semidirect
4. #21422032C2semidirect
5. C59×#2091C2semidirect
13. #16837184C22semidirect
15. C59×#430C22semidirect
18. C59×#338C22semidirect
124. #2091D59semidirect
138. #430C2×D59semidirect
144. #338C2×D59semidirect
145. C59#12298semidirect

Maximal subgroups

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

subgroupindex
#214348962
#214263102
#214220862
#214220322
C59×#20912
#1229859