Finite Group Databasebeta

← #27281697

Subgroups — #27281697

Order 13568.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283817C211yesmaximal, characteristic
2#20273453C211yesmaximal, characteristic
3#20273400C212yesmaximal
4#20272605C212yesmaximal
5C53×#3139C211yesmaximal, characteristic
6#16810754—22no—
7Q8×#88322—22no—
8#16810684—22no—
9#16810676C2212yes—
10#16810647C2212yes—
11#16810369—22no—
12#16810369—22no—
13#16810369—22no—
14C53×#533C2211yescharacteristic
15C2×C4⋊Q8×C53C2211yescharacteristic
16C53×#491C414yes—
17C53×#487—22no—
18C53×#419—22no—
19C4×M16×C53C2211yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—24no—
22C4⋊#88322—22no—
23C4⋊#88322—22no—
24C4⋊#88322D412yes—
25C4⋊#88322D412yes—
26C22⋊#88322—22no—
27#13133128—22no—
28C4×#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53—21no—
31C2×M16×C53—21no—
32C4⋊Q8×C53—21no—
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53C2×C412yes—
35C4⋊Q8×C53D412yes—
36#125×C53C2×C412yes—
37C22⋊Q8×C53—22no—
38C4×Q8×C53—22no—
39#117×C53D412yes—
40C2×C4⋊C4×C53—22no—
41C2×C42×C53C2311yescharacteristic
42#103×C53—28no—
43#97×C53C2×C412yes—
44C4×C8×C53—21no—
45C2×Q8×C53—22no—
46C2×Q8×C53—22no—
47C2×Q8×C53—44no—
48C2×Q8×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53#4112yes—
52M16×C53—42no—
53C2×C8×C53—22no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53C22⋊C414yes—
59C22⋊C4×C53—22no—
60C42×C53—22no—
61C42×C53C22×C411yescharacteristic
62C42×C53C2×D411yescharacteristic
63C42×C53C2×D412yes—
64#11719917—82no—
65C2×#88322—22no—
66C2×#88322—22no—
67C2×#88322—42no—
68C23×C53#12311yescharacteristic
69Q8×C53—44no—
70Q8×C53—42no—
71Q8×C53—42no—
72Q8×C53—82no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—24no—
77C2×C4×C53—42no—
78C2×C4×C53—82no—
79C2×C4×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53C4×D412yes—
81C2×C4×C53#12011yescharacteristic
82C2×C4×C53#12112yes—
83C8×C53—42no—
84#88322—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—82no—
95C4×C53#45412yes—
96C4×C53#47112yes—
97C4×C53#47312yes—
98#3139D5311yescharacteristic
99#2986—531no—
100#1561—531no—
101#1549—532no—
102#1299—532no—
103C2×C53—41no—
104C2×C53#188811yescharacteristic
105C2×C53#190812yes—
106#533C2×D5311yescharacteristic
107C2×C4⋊Q8C2×D5311yescharacteristic
108#491Dic5314yes—
109#487—22no—
110#446—1062no—
111C8×Q8—1062no—
112#433—1062no—
113#431—532no—
114#423—532no—
115#419—22no—
116C4×M16C2×D5311yescharacteristic
117#332—1062no—
118#332—1062no—
119#332—1062no—
120C53#943611yescharacteristic, complemented, Sylow
121C22×Q8—21no—
122C2×M16—21no—
123C2×M16—1062no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8C2×Dic5312yes—
127C4⋊Q8C22⋊D5312yes—
128#125C2×Dic5312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D5312yes—
132C2×C4⋊C4—22no—
133C2×C42C22×D5311yescharacteristic
134C4⋊C8—532no—
135C4⋊C8—532no—
136C4⋊C8—1064no—
137C4⋊C8—1062no—
138C4⋊C8—1062no—
139#103—28no—
140C22⋊C8—1062no—
141#97—1062no—
142#97C2×Dic5312yes—
143C4×C8—21no—
144C4×C8—1062no—
145C4×C8—1062no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—44no—
149C2×Q8—41no—
150C22×C4—41no—
151C22×C4#1171994712yes—
152C22×C4C2×C22⋊D5311yescharacteristic
153M16—42no—
154M16—4242no—
155C2×C8—22no—
156C2×C8—1062no—
157C2×C8—1062no—
158C2×C8—2122no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4C22⋊Dic5314yes—
164C22⋊C4—22no—
165C42—22no—
166C42D4×D5312yes—
167C42C22×Dic5311yescharacteristic
168C42C2×C22⋊D5311yescharacteristic
169C23#1313319811yescharacteristic
170Q8—44no—
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4C2×C22⋊Dic5311yescharacteristic
181C2×C4D4×Dic5312yes—
182C2×C4#1313318511yescharacteristic
183C2×C4#1313318912yes—
184C8—42no—
185C8—4242no—
186C8—2122no—
187C8—2122no—
188C22—22no—
189C22—21no—
190C22#1681050311yescharacteristic
191C4—22no—
192C4—21no—
193C4—44no—
194C4—82no—
195C4#1681079512yes—
196C4#1681088812yes—
197C4#1681089612yes—
198C2—41no—
199C2#2027511911yescharacteristic
200C2#2027523212yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3139C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#3139D53
hypocenterC53#9436
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

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
#202838172
#202734532
#202734002
#202726052
C53×#31392
#943653