Finite Group Databasebeta

← #28486291

Subgroups — #28486291

Order 15104.

400 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 103 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#21434478C211yesmaximal, characteristic
2#21433198C211yesmaximal, characteristic
3#21422782C212yesmaximal
4#21421985C212yesmaximal
5C59×#1561C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837772C414yes—
8#16837754—22no—
9#16837533—22no—
10Q8×C59⋊C8—22no—
11#16837463—22no—
12#16837455C2212yes—
13#16837410—22no—
14C4×#11745664C2211yescharacteristic
15#16837147—22no—
16#16837147—22no—
17C59×#533C2211yescharacteristic
18C59×#423C2212yes—
19C59×#332—22no—
20C22×Dic118—21no—
21C2×#11745664—21no—
22#14273727—22no—
23#14273727—21no—
24#14273727C2×C412yes—
25C2×C4⋊Dic59—22no—
26C4⋊(C59⋊C8)—24no—
27C4⋊(C59⋊C8)D412yes—
28C4⋊(C59⋊C8)D412yes—
29#14273650—28no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638C2×C412yes—
33C4×C59⋊C8—22no—
34C4×C59⋊C8—22no—
35C4×C59⋊C8—21no—
36C2×M16×C59—22no—
37C4⋊Q8×C59D412yes—
38#125×C59C2×C412yes—
39C22⋊Q8×C59—22no—
40C4×Q8×C59—22no—
41#117×C59D412yes—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59—22no—
44C4⋊C8×C59—22no—
45C2×Q8×C59—22no—
46C22×C4×C59C2×D411yescharacteristic
47C22×C4×C59#4112yes—
48C22×Dic59—41no—
49C2×Dic118—22no—
50C2×Dic118—44no—
51C2×Dic118—41no—
52M16×C59—82no—
53C2×C8×C59—42no—
54C4⋊C4×C59—24no—
55C4⋊C4×C59—22no—
56C4⋊C4×C59C22⋊C414yes—
57C22⋊C4×C59—22no—
58C42×C59—22no—
59C42×C59C22×C411yescharacteristic
60C42×C59C2×D411yescharacteristic
61C42×C59C2×D412yes—
62C4⋊Dic59—24no—
63C4⋊Dic59—42no—
64#11745664—42no—
65C2×C59⋊C8—22no—
66C2×C59⋊C8—22no—
67C2×C59⋊C8—22no—
68C23×C59#12311yescharacteristic
69Q8×C59—42no—
70Q8×C59—42no—
71C2×C4×C59—22no—
72C2×C4×C59—22no—
73C2×C4×C59—22no—
74C2×C4×C59—24no—
75C2×C4×C59C2×C22⋊C411yescharacteristic
76C2×C4×C59C4×D412yes—
77C2×C4×C59#12011yescharacteristic
78C2×C4×C59#12112yes—
79C2×Dic59—42no—
80C2×Dic59—82no—
81Dic118—44no—
82Dic118—82no—
83C8×C59—82no—
84C59⋊C8—42no—
85C59⋊C8—42no—
86C59⋊C8—42no—
87#9436—591nomaximal, Sylow
88C22×C59—21no—
89C22×C59—22no—
90C22×C59#38611yescharacteristic
91C4×C59—21no—
92C4×C59—22no—
93C4×C59—44no—
94C4×C59#45412yes—
95C4×C59#47112yes—
96C4×C59#47312yes—
97Dic59—82no—
98#3139—591no—
99#2986—591no—
100#1561D5911yescharacteristic
101#1549—592no—
102#1299—592no—
103C2×C59—41no—
104C2×C59#188811yescharacteristic
105C2×C59#190812yes—
106#533C2×D5911yescharacteristic
107C2×C4⋊Q8—591no—
108#491—594no—
109#487—1182no—
110#446—1182no—
111C8×Q8—1182no—
112#433—1182no—
113#431—592no—
114#423C2×D5912yes—
115#419—1182no—
116C4×M16—591no—
117#332—22no—
118#332—1182no—
119#332—1182no—
120C59#943611yescharacteristic, complemented, Sylow
121C22×Q8—1181no—
122C2×M16—22no—
123C2×M16—1181no—
124C4⋊Q8—592no—
125C4⋊Q8—1182no—
126C4⋊Q8—1181no—
127C4⋊Q8C22⋊D5912yes—
128#125C4×D5912yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D23612yes—
132C2×C4⋊C4—1182no—
133C2×C42C22×D5911yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—592no—
137C4⋊C8—592no—
138C4⋊C8—1184no—
139#103—1188no—
140C22⋊C8—1182no—
141#97—592no—
142#97—1182no—
143C4×C8—1182no—
144C4×C8—1182no—
145C4×C8—1181no—
146C2×Q8—22no—
147C2×Q8—1182no—
148C2×Q8—2364no—
149C2×Q8—2361no—
150C22×C4—2361no—
151C22×C4C2×D23611yescharacteristic
152C22×C4#1174568712yes—
153M16—82no—
154M16—2362no—
155C2×C8—42no—
156C2×C8—1182no—
157C2×C8—1182no—
158C2×C8—1182no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—1184no—
162C4⋊C4—2362no—
163C4⋊C4(C2×D59)⋊C414yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D5911yescharacteristic
167C42D4×D5912yes—
168C42C2×C22⋊D5911yescharacteristic
169C23#1427371111yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—4722no—
173Q8—2364no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—4722no—
179C2×C4—2362no—
180C2×C4C2×(C2×D59)⋊C411yescharacteristic
181C2×C4(C22⋊D59)⋊C412yes—
182C2×C4#1427369711yescharacteristic
183C2×C4#1427369912yes—
184C8—82no—
185C8—2362no—
186C8—2362no—
187C8—2362no—
188C22—21no—
189C22—22no—
190C22#1683727911yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—4722no—
195C4#1683757712yes—
196C4#1683766312yes—
197C4#1683767512yes—
198C2—41no—
199C2#2142449711yescharacteristic
200C2#2142460912yes—
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1561C2
socleC22×C59#386
prime radicalC23×C59#123
cosocleC2×C42×C59C23
hypercenter#1561D59
hypocenterC59#9436
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
120. C59#9436semidirect

Maximal subgroups

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

subgroupindex
#214344782
#214331982
#214227822
#214219852
C59×#15612
#943659