Finite Group Databasebeta

← #28498122

Subgroups — #28498122

Order 15104.

284 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 69 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#21432731C211yesmaximal, characteristic
2#21425905C212yesmaximal, complemented
3#21422046C212yesmaximal, complemented
4#21422032C211yesmaximal, characteristic
5C59×#1320C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic
7#16837460—22no—
8#16837460—22no—
9#16837425C411yescharacteristic
10#16837425C2211yescharacteristic
11#16837366C411yescharacteristic
12M16×Dic59C411yescharacteristic
13M16×Dic59C2211yescharacteristic
14#16837222—22no—
15#16837184C2212yes—
16#16837179—22no—
17C59×#407C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745664C2×C411yescharacteristic
20C2×#11745664C2×C411yescharacteristic
21#14273686—21no—
22#14273686—21no—
23#14273686—22no—
24C2×C4⋊Dic59—21no—
25C2×C22⋊Dic59—21no—
26C2×C4×Dic59C2×C411yescharacteristic
27C2×C4×Dic59C2×C411yescharacteristic
28#14273667—22no—
29Dic59⋊C8—22no—
30Dic59⋊C8—22no—
31C22⋊(C59⋊C8)—22no—
32#14273637—21no—
33#14273637D412yes—
34C8×Dic59—21no—
35C8×Dic59D412yes—
36C23×C4×C59C2×C411yescharacteristic
37C2×M16×C59C2×C411yescharacteristic
38C2×M16×C59C2311yescharacteristic
39M32×C59—22no—
40C22⋊C8×C59—22no—
41C24×C59M1611yescharacteristic
42C22×C4×C59—21no—
43C22×C4×C59—21no—
44C22×C4×C59—41no—
45C22×C4×C59—41no—
46C22×C4×C59M1611yescharacteristic
47C22×C4×C59C22×C411yescharacteristic
48C22×Dic59—21no—
49C22×Dic59—21no—
50M16×C59—41no—
51M16×C59—41no—
52C2×C8×C59—21no—
53C2×C8×C59C2×D412yes—
54C22⋊Dic59—22no—
55C22⋊Dic59—22no—
56C4⋊Dic59—22no—
57C4⋊Dic59—22no—
58C4×Dic59—21no—
59C4×Dic59—21no—
60C4×Dic59C22⋊C411yescharacteristic
61C4×Dic59C22⋊C411yescharacteristic
62C4×Dic59M1612yes—
63#11745664—41no—
64#11745664—41no—
65C2×C59⋊C8—21no—
66C2×C59⋊C8C22⋊C412yes—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59—21no—
70C23×C59—41no—
71C23×C59—81no—
72C23×C59C2×M1611yescharacteristic
73C2×C4×C59—21no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—42no—
77C2×C4×C59—41no—
78C2×C4×C59—41no—
79C2×C4×C59—41no—
80C2×C4×C59C2×C22⋊C411yescharacteristic
81C2×C4×C59C2×M1611yescharacteristic
82C2×Dic59—21no—
83C2×Dic59—21no—
84C2×Dic59—22no—
85C2×Dic59—81no—
86C2×Dic59—81no—
87C8×C59—22no—
88C8×C59—41no—
89C59⋊C8—22no—
90C59⋊C8—41no—
91#11522—591nomaximal, Sylow
92C22×C59—21no—
93C22×C59—41no—
94C22×C59—42no—
95C22×C59—41no—
96C22×C59—82no—
97C22×C59#40711yescharacteristic
98C4×C59—21no—
99C4×C59—41no—
100C4×C59#41011yescharacteristic
101C4×C59#41311yescharacteristic
102Dic59—41no—
103Dic59—41no—
104Dic59—42no—
105#2921—591no—
106#2165—592no—
107#1327—592no—
108#1320—591no—
109#1320D5911yescharacteristic
110C2×C59—21no—
111C2×C59—41no—
112C2×C59—81no—
113C2×C59#146411yescharacteristic
114C2×#117—591no—
115#433—1182no—
116#433—1182no—
117#423—591no—
118#423—591no—
119#407—591no—
120#407C2×D5911yescharacteristic
121C4×M16—591no—
122C4×M16—591no—
123#364—1182no—
124#349—592no—
125#349C2×D5912yes—
126#347—1182no—
127C59#1152211yescharacteristic, complemented, Sylow
128C23×C4C4×D5911yescharacteristic
129C2×M16—591no—
130C2×M16—591no—
131C2×M16C4×D5911yescharacteristic
132C2×M16C22×D5911yescharacteristic
133#117—1182no—
134#117—1181no—
135#117—1181no—
136C2×C4⋊C4—1181no—
137C2×C22⋊C4—1181no—
138C2×C42—591no—
139C2×C42—591no—
140M32—22no—
141M32—1182no—
142C4⋊C8—1182no—
143C4⋊C8—1182no—
144C22⋊C8—22no—
145C22⋊C8—1182no—
146#97—592no—
147#97—1181no—
148C4×C8—592no—
149C4×C8—1181no—
150C24#1174565911yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—1181no—
156C22×C4—1181no—
157C22×C4#1174565911yescharacteristic
158C22×C4C2×C4×D5911yescharacteristic
159M16—41no—
160M16—41no—
161M16—2361no—
162M16—2361no—
163C2×C8—21no—
164C2×C8—592no—
165C2×C8—1181no—
166C2×C8D4×D5912yes—
167C4⋊C4—1182no—
168C4⋊C4—1182no—
169C22⋊C4—1182no—
170C22⋊C4—1182no—
171C42—591no—
172C42—591no—
173C42—592no—
174C42—1181no—
175C42—1181no—
176C16—42no—
177C16—2362no—
178C23—21no—
179C23—41no—
180C23—81no—
181C23C2×#1174565911yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1181no—
190C2×C4—1181no—
191C2×C4—1182no—
192C2×C4—4721no—
193C2×C4—4721no—
194C2×C4C22⋊C4×D5911yescharacteristic
195C2×C4M16×D5911yescharacteristic
196C8—22no—
197C8—41no—
198C8—1182no—
199C8—2361no—
200C22—21no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—82no—
205C22#1683736411yescharacteristic
206C4—21no—
207C4—41no—
208C4—2361no—
209C4—2361no—
210C4—2362no—
211C4#1683737411yescharacteristic
212C4#1683738311yescharacteristic
213C2—21no—
214C2—41no—
215C2—81no—
216C2#2142242711yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1320C2
socleC2×C59#1464
prime radicalC24×C59M16
cosocleC2×M16×C59C23
hypercenter#1320D59
hypocenterC59#11522
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #21425905C2semidirect
3. #21422046C2semidirect
127. C59#11522semidirect

Maximal subgroups

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

subgroupindex
#214327312
#214259052
#214220462
#214220322
C59×#13202
#1152259