Finite Group Databasebeta

← #28302288

Subgroups — #28302288

Order 13568.

684 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290017C214yesmaximal, complemented
2#20289948C212yesmaximal
3C2×#16811234C211yesmaximal, characteristic
4#16811639C2214yes—
5Q8⋊#88324—28no—
6#16811469—22no—
7#16811467—22no—
8#16811234—28no—
9C2×#13133212—22no—
10C2×#13133209C411yescharacteristic
11C2×#13133207—22no—
12C2×#13133193C412yes—
13C2×Q8×Dic53C2212yes—
14C2×#13133174C2211yescharacteristic
15C2×C4⋊Q8×C53C411yescharacteristic
16C4⋊#88324—24no—
17C22⋊#88324—24no—
18C4×#88324—24no—
19C22×C4×D53C2×C411yescharacteristic
20#13133212—216no—
21#13133209—24no—
22#13133207—216no—
23#13133193—24no—
24#13133193—24no—
25Q8×Dic53—24no—
26Q8×Dic53D418yes—
27#13133174—24no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×#11719919D412yes—
33C2×#11719921—22no—
34C2×#11719921D412yes—
35C2×C4×Dic53—22no—
36C2×C4×Dic53C2311yescharacteristic
37C22×Q8×C53C2×C412yes—
38C4⋊Q8×C53—24no—
39C2×C4⋊C4×C53—22no—
40C2×C42×C53C2×C411yescharacteristic
41C23×D53C22⋊C411yescharacteristic
42C2×Q8×C53—22no—
43C2×Q8×C53—28no—
44C2×Q8×C53C22⋊C414yes—
45C22×C4×C53—22no—
46C22×C4×C53C22⋊C412yes—
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C2×D412yes—
51C2×C4×D53—22no—
52C2×C4×D53—24no—
53C2×#88324—44no—
54C4⋊C4×C53—28no—
55C42×C53—22no—
56#11719921—28no—
57#11719921—24no—
58C4⋊Dic53—22no—
59C4⋊Dic53—28no—
60C4⋊Dic53C22⋊C414yes—
61#11719919—28no—
62#11719919—28no—
63#11719919—24no—
64C4×Dic53—28no—
65C4×Dic53C2×D414yes—
66C23×C53C2×C22⋊C411yescharacteristic
67C22×D53—21no—
68C22×D53—22no—
69C22×D53—42no—
70Q8×C53—28no—
71Q8×C53—44no—
72C2×C4×C53—24no—
73C2×C4×C53—22no—
74C2×C4×C53—44no—
75C2×C4×C53—42no—
76C2×C4×C53C2×C22⋊C412yes—
77C2×C4×C53#13114yes—
78C2×Dic53—21no—
79C2×Dic53—24no—
80C2×Dic53—22no—
81C2×Dic53—42no—
82C2×Dic53—44no—
83C2×Dic53#12014yes—
84C4×D53—44no—
85#88324—84no—
86#8327—531nomaximal, Sylow
87C22×C53—22no—
88C22×C53#37911yescharacteristic
89C22×C53#40812yes—
90C2×D53—42no—
91C2×D53—41no—
92C2×D53—82no—
93C4×C53—42no—
94C4×C53—44no—
95C4×C53#40814yes—
96Dic53—24no—
97Dic53—44no—
98Dic53—41no—
99C2×#541—531no—
100#1479—534no—
101#1458—532no—
102C2×C53—22no—
103C2×C53#178011yescharacteristic
104C2×C53#179612yes—
105D53—81no—
106#541—1068no—
107C2×C4⋊Q8#615711yescharacteristic
108C2×#126—1062no—
109C2×#125—531no—
110C2×#125—1062no—
111C2×C22⋊Q8—532no—
112C2×C4×Q8—532no—
113C2×#117—531no—
114#431—534no—
115Q8⋊C8—1068no—
116#324—1062no—
117#323—1062no—
118C53#832711yescharacteristic, complemented, Sylow
119C22×Q8C2×#615712yes—
120C23×C4—531no—
121C4⋊Q8—24no—
122#126—10616no—
123#125—1064no—
124#125—10616no—
125C22⋊Q8—1064no—
126C22⋊Q8—1064no—
127C4×Q8—538no—
128C4×Q8—1064no—
129#117—1064no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1062no—
136C2×C22⋊C4—532no—
137C2×C22⋊C4—1062no—
138C2×C42—531no—
139C2×C42—1062no—
140C2×C42C2×#615711yescharacteristic
141C4⋊C8—1064no—
142C22⋊C8—1064no—
143C4×C8—1064no—
144C24—531no—
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊#615714yes—
148C22×C4—22no—
149C22×C4—532no—
150C22×C4—531no—
151C22×C4—1062no—
152C22×C4—1062no—
153C22×C4—1064no—
154C22×C4C22⋊#615712yes—
155C22×C4C22×#615711yescharacteristic
156C2×C8—2124no—
157C4⋊C4—28no—
158C4⋊C4—534no—
159C4⋊C4—1064no—
160C4⋊C4—1068no—
161C4⋊C4—1068no—
162C4⋊C4—1068no—
163C4⋊C4—1062no—
164C22⋊C4—1064no—
165C22⋊C4—1068no—
166C42—22no—
167C42—534no—
168C42—1068no—
169C23—1062no—
170C23—1061no—
171C23—2122no—
172C23C2×C22⋊#615711yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—534no—
180C2×C4—1064no—
181C2×C4—1061no—
182C2×C4—1062no—
183C2×C4—2122no—
184C2×C4—2124no—
185C2×C4—2124no—
186C2×C4C2×C22⋊#615712yes—
187C2×C4#1313330414yes—
188C8—4244no—
189C22—22no—
190C22—4242no—
191C22—2122no—
192C22—2121no—
193C22#1681156411yescharacteristic
194C22#1681159712yes—
195C4—42no—
196C4—44no—
197C4—1064no—
198C4—2124no—
199C4—2121no—
200C4#1681159614yes—
201C2—22no—
202C2—4241no—
203C2#2029053911yescharacteristic
204C2#2029056712yes—
subgroupG/H
centerC22#16811564
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53#379
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#8327
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20290017C2semidirect
118. C53#8327semidirect

Maximal subgroups

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

subgroupindex
#202900172
#202899482
C2×#168112342
#832753