Finite Group Databasebeta

← #28486293

Subgroups — #28486293

Order 15104.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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#21422834C211yesmaximal, characteristic
3#21422780C212yesmaximal
4#21421987C212yesmaximal
5C59×#2986C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837772C414yes—
8#16837754—22no—
9#16837426C2212yes—
10#16837410—22no—
11C4×#11745664C2211yescharacteristic
12#16837149—22no—
13#16837149—22no—
14#16837148—22no—
15C59×#533C2211yescharacteristic
16C59×#446—22no—
17C8×Q8×C59—22no—
18C59×#433—22no—
19C59×#431C2212yes—
20C22×Dic118—21no—
21C2×#11745664—21no—
22C2×#11745664—22no—
23#14273727—22no—
24#14273727—21no—
25#14273727C2×C412yes—
26C2×C4⋊Dic59—22no—
27C4⋊(C59⋊C8)—22no—
28C4⋊(C59⋊C8)—22no—
29#14273650—28no—
30#14273638C2×C412yes—
31C4×C59⋊C8—21no—
32C4⋊Q8×C59D412yes—
33#125×C59C2×C412yes—
34C22⋊Q8×C59—22no—
35C4×Q8×C59—22no—
36#117×C59D412yes—
37C2×C42×C59C2311yescharacteristic
38C4⋊C8×C59—24no—
39C4⋊C8×C59D412yes—
40C4⋊C8×C59D412yes—
41C22⋊C8×C59—22no—
42#97×C59—22no—
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—
52C2×C8×C59—22no—
53C2×C8×C59—22no—
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—
65#11745664—82no—
66C2×C59⋊C8—22no—
67C2×C59⋊C8—42no—
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—42no—
84C8×C59—42no—
85C59⋊C8—42no—
86C59⋊C8—82no—
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#2986D5911yescharacteristic
100#1561—591no—
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—22no—
111C8×Q8—22no—
112#433—22no—
113#431C2×D5912yes—
114#423—592no—
115#419—1182no—
116C4×M16—591no—
117#332—1182no—
118#332—1182no—
119#332—1182no—
120C59#943611yescharacteristic, complemented, Sylow
121C22×Q8—1181no—
122C2×M16—1181no—
123C2×M16—1182no—
124C4⋊Q8—592no—
125C4⋊Q8—1181no—
126C4⋊Q8—1182no—
127C4⋊Q8D23612yes—
128#125C4×D5912yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D5912yes—
132C2×C4⋊C4—1182no—
133C2×C42C22×D5911yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—1182no—
136C4⋊C8—1182no—
137C4⋊C8C22⋊D5912yes—
138C4⋊C8C22⋊D5912yes—
139#103—1188no—
140C22⋊C8—22no—
141#97—22no—
142#97—592no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—1181no—
146C2×Q8—22no—
147C2×Q8—1182no—
148C2×Q8—2364no—
149C2×Q8—2361no—
150C22×C4—2361no—
151C22×C4#1174568512yes—
152C22×C4C2×C22⋊D5911yescharacteristic
153M16—4722no—
154M16—2362no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—1182no—
158C2×C8—2362no—
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
167C42C2×D23611yescharacteristic
168C42C2×C22⋊D5912yes—
169C23#1427371011yescharacteristic
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×C4C4×C22⋊D5912yes—
182C2×C4#1427369611yescharacteristic
183C2×C4#1427370112yes—
184C8—42no—
185C8—42no—
186C8—4722no—
187C8—2362no—
188C22—21no—
189C22—22no—
190C22#1683728111yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—4722no—
195C4#1683757512yes—
196C4#1683766512yes—
197C4#1683767712yes—
198C2—41no—
199C2#2142449911yescharacteristic
200C2#2142461112yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2986C2
socleC22×C59#386
prime radicalC23×C59#123
cosocleC2×C42×C59C23
hypercenter#2986D59
hypocenterC59#9436
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

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
#214228342
#214227802
#214219872
C59×#29862
#943659