Finite Group Databasebeta

← #28302873

Subgroups — #28302873

Order 13568.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290549C212yesmaximal
2#20289965C214yesmaximal
3C22×#13133215C211yesmaximal, characteristic
4C2×#13133278—24no—
5#16811593C2214yes—
6#16811469—28no—
7C23×Dic106C2211yescharacteristic
8C2×#13133215—24no—
9C2×#13133215—28no—
10C22×#11719919C412yes—
11C22×#11719919C2212yes—
12C22×C4×Dic53C411yescharacteristic
13C23×Q8×C53C411yescharacteristic
14C2×#11719940—24no—
15#13133278—48no—
16C22⋊#88324—28no—
17C22×Dic106—24no—
18C22×Dic106—44no—
19C22×Dic106D414yes—
20C23×Dic53C2×C412yes—
21C23×Dic53C2×C411yescharacteristic
22C23×Dic53C2311yescharacteristic
23#13133215—232no—
24C2×#11719919—24no—
25C2×#11719919—24no—
26C2×#11719919—24no—
27C2×#11719919—24no—
28C2×#11719919—24no—
29C2×#11719919D418yes—
30C2×C4×Dic53—22no—
31C2×C4×Dic53—24no—
32C22×Q8×C53—24no—
33C22×Q8×C53—22no—
34C22×Q8×C53—44no—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C412yes—
37C24×C53C22×C411yescharacteristic
38C2×Q8×C53—28no—
39C2×Q8×C53—416no—
40C2×Q8×C53—44no—
41C2×Q8×C53—44no—
42C22×C4×C53—22no—
43C22×C4×C53—24no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—44no—
47C22×C4×C53C22⋊C412yes—
48C22×C4×C53C22⋊C414yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—28no—
52C22×Dic53—22no—
53C22×Dic53—42no—
54C22×Dic53—42no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C22⋊C414yes—
57C22×Dic53C2×D412yes—
58C22×Dic53C2×D414yes—
59C2×Dic106—28no—
60C2×Dic106—28no—
61C2×Dic106—44no—
62C2×Dic106—416no—
63#11719940—48no—
64C2×#88324—44no—
65#11719919—216no—
66#11719919—216no—
67C4×Dic53—28no—
68C23×C53—22no—
69C23×C53—24no—
70C23×C53C2×C22⋊C411yescharacteristic
71C23×C53C2×C22⋊C412yes—
72Q8×C53—416no—
73C2×C4×C53—28no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—24no—
77C2×C4×C53—44no—
78C2×C4×C53—44no—
79C2×C4×C53—44no—
80C2×C4×C53#9914yes—
81C2×Dic53—28no—
82C2×Dic53—28no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—24no—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—44no—
89C2×Dic53—44no—
90C2×Dic53#10118yescomplemented
91C2×Dic53#12014yes—
92Dic106—416no—
93#88324—84no—
94#8551—531nomaximal, Sylow
95C22×C53—28no—
96C22×C53—22no—
97C22×C53—22no—
98C22×C53—22no—
99C22×C53#37911yescharacteristic
100C22×C53C2×#9912yes—
101C22×C53C2×#10114yes—
102C4×C53—24no—
103C4×C53—44no—
104Dic53—24no—
105Dic53—28no—
106Dic53—42no—
107C22×C4⋊Q8—531no—
108#1784—532no—
109#1463—534no—
110C2×C53—22no—
111C2×C53—24no—
112C2×C53#178111yescharacteristic
113C2×C53#178412yes—
114C23×Q8—531no—
115C23×Q8#615711yescharacteristic
116C2×C4⋊Q8—1068no—
117C2×C4⋊Q8—1064no—
118C22×C4⋊C4—532no—
119C22×C4⋊C4—532no—
120C22×C42—531no—
121C2×#101—1064no—
122#407—534no—
123#324—1068no—
124C53#855111yescharacteristic, complemented, Sylow
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—44no—
128C22×Q8—534no—
129C22×Q8—1064no—
130C22×Q8—2124no—
131C23×C4—531no—
132C23×C4—532no—
133C23×C4—531no—
134C23×C4C2×#615711yescharacteristic
135C23×C4C2×#615712yes—
136C2×M16—1064no—
137C4⋊Q8—10632no—
138C2×C4⋊C4—538no—
139C2×C4⋊C4—1064no—
140C2×C4⋊C4—1064no—
141C2×C4⋊C4—1064no—
142C2×C4⋊C4—1064no—
143C2×C4⋊C4—1064no—
144C2×C42—1064no—
145C2×C42—1062no—
146#101—2128no—
147C22⋊C8—1068no—
148C24C22×#615711yescharacteristic
149C2×Q8—28no—
150C2×Q8—416no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—1068no—
154C2×Q8—1068no—
155C2×Q8—2124no—
156C2×Q8—21216no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—534no—
163C22×C4—532no—
164C22×C4—534no—
165C22×C4—532no—
166C22×C4—1068no—
167C22×C4—1062no—
168C22×C4—1062no—
169C22×C4—1062no—
170C22×C4—2122no—
171C22×C4—2122no—
172C22×C4C22⋊#615712yes—
173C22×C4C22⋊#615714yes—
174M16—2128no—
175C2×C8—2124no—
176C4⋊C4—10616no—
177C4⋊C4—10616no—
178C42—1068no—
179C23—22no—
180C23—24no—
181C23C2×C22⋊#615711yescharacteristic
182C23C2×C22⋊#615712yes—
183Q8—416no—
184Q8—21216no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—534no—
193C2×C4—538no—
194C2×C4—1068no—
195C2×C4—1068no—
196C2×C4—1064no—
197C2×C4—1064no—
198C2×C4—1064no—
199C2×C4—1062no—
200C2×C4—1064no—
201C2×C4—2124no—
202C2×C4—2124no—
203C2×C4#1313327514yes—
204C8—4244no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22#1681156411yescharacteristic
210C22C2×#1313327512yes—
211C22D53⋊#10114yes—
212C4—24no—
213C4—44no—
214C4—1064no—
215C4—1068no—
216C4—2122no—
217C2—22no—
218C2—24no—
219C2#2029054111yescharacteristic
220C2#2029055012yes—
subgroupG/H
centerC22#16811564
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC23×Q8×C53C4
socleC22×C53#379
cosocleC23×Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#8551
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
90. C2×Dic53#101semidirect
124. C53#8551semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202899652
C22×#131332152
#855153