Finite Group Databasebeta

← #27281696

Subgroups — #27281696

Order 13568.

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#20285097C211yesmaximal, characteristic
2#20273453C211yesmaximal, characteristic
3#20273399C212yesmaximal
4#20272606C212yesmaximal
5C53×#2986C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810993C414yes—
8#16810975—22no—
9#16810647C2212yes—
10#16810631—22no—
11C4×#11719917C2211yescharacteristic
12#16810370—22no—
13#16810370—22no—
14#16810369—22no—
15C53×#533C2211yescharacteristic
16C53×#446—22no—
17C8×Q8×C53—22no—
18C53×#433—22no—
19C53×#431C2212yes—
20C22×Dic106—21no—
21C2×#11719917—21no—
22C2×#11719917—22no—
23#13133217—21no—
24#13133217—22no—
25#13133217C2×C412yes—
26C2×C4⋊Dic53—22no—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29#13133140—28no—
30#13133128C2×C412yes—
31C4×#88322—21no—
32C4⋊Q8×C53D412yes—
33#125×C53C2×C412yes—
34C22⋊Q8×C53—22no—
35C4×Q8×C53—22no—
36#117×C53D412yes—
37C2×C42×C53C2311yescharacteristic
38C4⋊C8×C53—24no—
39C4⋊C8×C53D412yes—
40C4⋊C8×C53D412yes—
41C22⋊C8×C53—22no—
42#97×C53—22no—
43C4×C8×C53—22no—
44C4×C8×C53—22no—
45C2×Q8×C53—22no—
46C22×C4×C53C2×D411yescharacteristic
47C22×C4×C53#4112yes—
48C22×Dic53—41no—
49C2×Dic106—22no—
50C2×Dic106—44no—
51C2×Dic106—41no—
52C2×C8×C53—22no—
53C2×C8×C53—22no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53C22⋊C414yes—
57C22⋊C4×C53—22no—
58C42×C53—22no—
59C42×C53C22×C411yescharacteristic
60C42×C53C2×D411yescharacteristic
61C42×C53C2×D412yes—
62C4⋊Dic53—24no—
63C4⋊Dic53—42no—
64#11719917—42no—
65#11719917—82no—
66C2×#88322—22no—
67C2×#88322—42no—
68C23×C53#12311yescharacteristic
69Q8×C53—42no—
70Q8×C53—42no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53C2×C22⋊C411yescharacteristic
76C2×C4×C53C4×D412yes—
77C2×C4×C53#12011yescharacteristic
78C2×C4×C53#12112yes—
79C2×Dic53—42no—
80C2×Dic53—82no—
81Dic106—44no—
82Dic106—82no—
83C8×C53—42no—
84C8×C53—42no—
85#88322—42no—
86#88322—82no—
87#9436—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—22no—
90C22×C53#38611yescharacteristic
91C4×C53—21no—
92C4×C53—22no—
93C4×C53—44no—
94C4×C53#45412yes—
95C4×C53#47112yes—
96C4×C53#47312yes—
97Dic53—82no—
98#3139—531no—
99#2986D5311yescharacteristic
100#1561—531no—
101#1549—532no—
102#1299—532no—
103C2×C53—41no—
104C2×C53#188811yescharacteristic
105C2×C53#190812yes—
106#533C2×D5311yescharacteristic
107C2×C4⋊Q8—531no—
108#491—534no—
109#487—1062no—
110#446—22no—
111C8×Q8—22no—
112#433—22no—
113#431C2×D5312yes—
114#423—532no—
115#419—1062no—
116C4×M16—531no—
117#332—1062no—
118#332—1062no—
119#332—1062no—
120C53#943611yescharacteristic, complemented, Sylow
121C22×Q8—1061no—
122C2×M16—1061no—
123C2×M16—1062no—
124C4⋊Q8—532no—
125C4⋊Q8—1061no—
126C4⋊Q8—1062no—
127C4⋊Q8D21212yes—
128#125C4×D5312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D5312yes—
132C2×C4⋊C4—1062no—
133C2×C42C22×D5311yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—1062no—
136C4⋊C8—1062no—
137C4⋊C8C22⋊D5312yes—
138C4⋊C8C22⋊D5312yes—
139#103—1068no—
140C22⋊C8—22no—
141#97—22no—
142#97—532no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—1061no—
146C2×Q8—22no—
147C2×Q8—1062no—
148C2×Q8—2124no—
149C2×Q8—2121no—
150C22×C4—2121no—
151C22×C4#1171994512yes—
152C22×C4C2×C22⋊D5311yescharacteristic
153M16—4242no—
154M16—2122no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—1062no—
158C2×C8—2122no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—1064no—
162C4⋊C4—2122no—
163C4⋊C4#1171992114yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D5311yescharacteristic
167C42C2×D21211yescharacteristic
168C42C2×C22⋊D5312yes—
169C23#1313320011yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—4242no—
173Q8—2124no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—4242no—
179C2×C4—2122no—
180C2×C4C2×#1171992111yescharacteristic
181C2×C4C4×C22⋊D5312yes—
182C2×C4#1313318611yescharacteristic
183C2×C4#1313319112yes—
184C8—42no—
185C8—42no—
186C8—4242no—
187C8—2122no—
188C22—22no—
189C22—21no—
190C22#1681050211yescharacteristic
191C4—22no—
192C4—21no—
193C4—44no—
194C4—4242no—
195C4#1681079612yes—
196C4#1681088612yes—
197C4#1681089812yes—
198C2—41no—
199C2#2027511811yescharacteristic
200C2#2027523012yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2986C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#2986D53
hypocenterC53#9436
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
120. C53#9436semidirect

Maximal subgroups

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

subgroupindex
#202850972
#202734532
#202733992
#202726062
C53×#29862
#943653