Finite Group Databasebeta

← #27272393

Subgroups — #27272393

Order 13568.

490 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 123 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#20274783C212yesmaximal
2#20272538C214yesmaximal, complemented
3C53×#2296C211yesmaximal, characteristic
4C2×C22⋊#88322—22no—
5#16810378—22no—
6#16810378C2214yes—
7C2×#117×C53C411yescharacteristic
8C2×#117×C53C411yescharacteristic
9C22×C42×C53C2211yescharacteristic
10C53×#389C412yes—
11C53×#389C2212yes—
12C53×#385—22no—
13C4×C4⋊C4×C53—22no—
14C22×#88322—24no—
15C22⋊#88322—44no—
16C23×C4×C53C2×C411yescharacteristic
17C23×C4×C53D412yes—
18#117×C53—24no—
19#117×C53—42no—
20C2×C4⋊C4×C53—22no—
21C2×C4⋊C4×C53—22no—
22C2×C4⋊C4×C53C2×C412yes—
23C2×C4⋊C4×C53C2×C412yes—
24C2×C22⋊C4×C53—21no—
25C2×C22⋊C4×C53—21no—
26C2×C42×C53—21no—
27C2×C42×C53—21no—
28C2×C42×C53—42no—
29C2×C42×C53C2×C411yescharacteristic
30C2×C42×C53D412yes—
31C2×C42×C53C2311yescharacteristic
32#95×C53—22no—
33#95×C53—22no—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—24no—
39C22×C4×C53—22no—
40C22×C4×C53—24no—
41C22×C4×C53—22no—
42C22×C4×C53—42no—
43C22×C4×C53C22⋊C412yes—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22×C411yescharacteristic
46C22×C4×C53C2×D412yes—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—44no—
49C4⋊C4×C53—42no—
50C4⋊C4×C53—44no—
51C4⋊C4×C53M1614yes—
52C22⋊C4×C53—24no—
53C22⋊C4×C53—42no—
54C42×C53—24no—
55C42×C53—41no—
56C42×C53—42no—
57C42×C53—41no—
58C42×C53—42no—
59C2×#88322—44no—
60C2×#88322—44no—
61C2×#88322—44no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53#9912yes—
67C23×C53C2×C22⋊C411yescharacteristic
68C2×C4×C53—28no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—24no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—48no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53#10112yes—
86C2×C4×C53C2×M1612yes—
87C2×C4×C53#13514yes—
88#88322—84no—
89#7646—531nomaximal, Sylow
90C22×C53—22no—
91C22×C53—24no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—42no—
95C22×C53#40711yescharacteristic
96C22×C53C2×#9911yescharacteristic
97C22×C53C2×#10111yescharacteristic
98C22×C53#41612yes—
99C22×C53#42112yes—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—24no—
103C4×C53—44no—
104C4×C53—42no—
105C4×C53—82no—
106#2296D5311yescharacteristic
107#1817—532no—
108#1271—534no—
109C2×C53—42no—
110C2×C53#146311yescharacteristic
111C2×C53#148812yes—
112C2×C53#151612yes—
113C2×C53#153711yescharacteristic
114C2×C53#154411yescharacteristic
115C2×#117Dic5311yescharacteristic
116C2×#117Dic5311yescharacteristic
117C22×C42C2×D5311yescharacteristic
118C2×C22⋊C8—1062no—
119#389Dic5312yes—
120#389C2×D5312yes—
121#385—22no—
122C4×C4⋊C4—22no—
123#336—534no—
124#336—1062no—
125C53#764611yescharacteristic, complemented, Sylow
126C23×C4C2×Dic5311yescharacteristic
127C23×C4C22⋊D5312yes—
128C22×C8—1064no—
129#117—24no—
130#117—42no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×Dic5312yes—
134C2×C4⋊C4C2×Dic5312yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2×Dic5311yescharacteristic
141C2×C42C22⋊D5312yes—
142C2×C42C22×D5311yescharacteristic
143C22⋊C8—2124no—
144#95—22no—
145#95—22no—
146C24C22⋊Dic5311yescharacteristic
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4C22⋊Dic5312yes—
156C22×C4C22⋊Dic5311yescharacteristic
157C22×C4C22×Dic5311yescharacteristic
158C22×C4C2×C22⋊D5312yes—
159C2×C8—2124no—
160C2×C8—2124no—
161C2×C8—2124no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4#1171991714yes—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C42—24no—
170C42—41no—
171C42—42no—
172C42—41no—
173C42—42no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23#1313313212yes—
179C23C2×C22⋊Dic5311yescharacteristic
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4#1313313612yes—
198C2×C4C2×#1171991712yes—
199C2×C4#1313323714yes—
200C8—4244no—
201C22—24no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22#1681058711yescharacteristic
207C22C2×#1313313211yescharacteristic
208C22C2×#1313313611yescharacteristic
209C22#1681061912yes—
210C22#1681064012yes—
211C4—24no—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—44no—
216C4—82no—
217C2—42no—
218C2#2027304511yescharacteristic
219C2#2027314912yes—
220C2#2027327112yes—
221C2#2027335311yescharacteristic
222C2#2027338011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2296C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2296D53
hypocenterC53#7646
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #20272538C2semidirect
125. C53#7646semidirect

Maximal subgroups

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

subgroupindex
#202747832
#202725382
C53×#22962
#764653