Finite Group Databasebeta

← #27281694

Subgroups — #27281694

Order 13568.

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#20285097C211yesmaximal, characteristic
2#20283817C211yesmaximal, characteristic
3#20273401C212yesmaximal
4#20272604C212yesmaximal
5C53×#1561C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810993C414yes—
8#16810975—22no—
9#16810754—22no—
10Q8×#88322—22no—
11#16810684—22no—
12#16810676C2212yes—
13#16810631—22no—
14C4×#11719917C2211yescharacteristic
15#16810368—22no—
16#16810368—22no—
17C53×#533C2211yescharacteristic
18C53×#423C2212yes—
19C53×#332—22no—
20C22×Dic106—21no—
21C2×#11719917—21no—
22#13133217—21no—
23#13133217—22no—
24#13133217C2×C412yes—
25C2×C4⋊Dic53—22no—
26C4⋊#88322—24no—
27C4⋊#88322D412yes—
28C4⋊#88322D412yes—
29#13133140—28no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128C2×C412yes—
33C4×#88322—22no—
34C4×#88322—22no—
35C4×#88322—21no—
36C2×M16×C53—22no—
37C4⋊Q8×C53D412yes—
38#125×C53C2×C412yes—
39C22⋊Q8×C53—22no—
40C4×Q8×C53—22no—
41#117×C53D412yes—
42C2×C42×C53C2311yescharacteristic
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—
52M16×C53—82no—
53C2×C8×C53—42no—
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—
65C2×#88322—22no—
66C2×#88322—22no—
67C2×#88322—22no—
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—82no—
84#88322—42no—
85#88322—42no—
86#88322—42no—
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#2986—531no—
100#1561D5311yescharacteristic
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—1062no—
111C8×Q8—1062no—
112#433—1062no—
113#431—532no—
114#423C2×D5312yes—
115#419—1062no—
116C4×M16—531no—
117#332—22no—
118#332—1062no—
119#332—1062no—
120C53#943611yescharacteristic, complemented, Sylow
121C22×Q8—1061no—
122C2×M16—22no—
123C2×M16—1061no—
124C4⋊Q8—532no—
125C4⋊Q8—1062no—
126C4⋊Q8—1061no—
127C4⋊Q8C22⋊D5312yes—
128#125C4×D5312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D21212yes—
132C2×C4⋊C4—1062no—
133C2×C42C22×D5311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—532no—
137C4⋊C8—532no—
138C4⋊C8—1064no—
139#103—1068no—
140C22⋊C8—1062no—
141#97—532no—
142#97—1062no—
143C4×C8—1062no—
144C4×C8—1062no—
145C4×C8—1061no—
146C2×Q8—22no—
147C2×Q8—1062no—
148C2×Q8—2124no—
149C2×Q8—2121no—
150C22×C4—2121no—
151C22×C4C2×D21211yescharacteristic
152C22×C4#1171994712yes—
153M16—82no—
154M16—2122no—
155C2×C8—42no—
156C2×C8—1062no—
157C2×C8—1062no—
158C2×C8—1062no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—1064no—
162C4⋊C4—2122no—
163C4⋊C4#1171992114yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D5311yescharacteristic
167C42D4×D5312yes—
168C42C2×C22⋊D5311yescharacteristic
169C23#1313320111yescharacteristic
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×C4#1313317712yes—
182C2×C4#1313318711yescharacteristic
183C2×C4#1313318912yes—
184C8—82no—
185C8—2122no—
186C8—2122no—
187C8—2122no—
188C22—22no—
189C22—21no—
190C22#1681050011yescharacteristic
191C4—22no—
192C4—21no—
193C4—44no—
194C4—4242no—
195C4#1681079812yes—
196C4#1681088412yes—
197C4#1681089612yes—
198C2—41no—
199C2#2027511611yescharacteristic
200C2#2027522812yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1561C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#1561D53
hypocenterC53#9436
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

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
#202838172
#202734012
#202726042
C53×#15612
#943653