Finite Group Databasebeta

← #27276858

Subgroups — #27276858

Order 13568.

456 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 117 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#20277270C211yesmaximal, characteristic
2#20274569C212yesmaximal
3C2×#16810364C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#1295C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7C2×#13133146—22no—
8C2×#13133128C2212yes—
9#16810519C412yes—
10#16810505—22no—
11C4⋊C4×Dic53C412yes—
12#16810474—22no—
13C42×Dic53C2211yescharacteristic
14#16810377—22no—
15#16810364—24no—
16C2×#97×C53C2212yes—
17C53×#382C2211yescharacteristic
18C22×#88322—22no—
19#13133209—22no—
20#13133209D412yes—
21#13133209Q812yes—
22C2×C4⋊Dic53C2×C412yes—
23C2×#11719919—22no—
24C2×#11719919—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53D412yes—
30#13133146—28no—
31#13133128—22no—
32#13133128—22no—
33#13133124—22no—
34#13133124—22no—
35C22×C8×C53—22no—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C42×C53C2311yescharacteristic
38#97×C53—22no—
39#97×C53—22no—
40#95×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53C22×C411yescharacteristic
43C22×C4×C53C2×D412yes—
44C22×Dic53—22no—
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C22⋊C412yes—
47C22×Dic53C22⋊C411yescharacteristic
48C2×C8×C53—24no—
49C2×C8×C53—42no—
50C2×C8×C53—42no—
51C4⋊C4×C53—42no—
52C42×C53C2×D411yescharacteristic
53C42×C53C2×Q811yescharacteristic
54C42×C53#4112yes—
55C4⋊Dic53—22no—
56C4⋊Dic53C4⋊C414yes—
57#11719919—24no—
58#11719919—24no—
59#11719919—44no—
60C4×Dic53—22no—
61C4×Dic53—24no—
62C4×Dic53—41no—
63C4×Dic53—42no—
64C4×Dic53—44no—
65C2×#88322—24no—
66C2×#88322—42no—
67C2×#88322—42no—
68C23×C53C2×C22⋊C411yescharacteristic
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53C2×C4⋊C411yescharacteristic
77C2×C4×C53#11711yescharacteristic
78C2×C4×C53#12112yes—
79C2×C4×C53C22⋊Q812yes—
80C2×Dic53—24no—
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53#10112yes—
89C2×Dic53#10414yes—
90C8×C53—44no—
91#88322—44no—
92#8580—531nomaximal, Sylow
93C22×C53#38011yescharacteristic
94C22×C53C2×#10111yescharacteristic
95C22×C53#41311yescharacteristic
96C22×C53C2×#10412yes—
97C22×C53#42112yes—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—82no—
103Dic53—41no—
104Dic53—44no—
105Dic53—42no—
106Dic53#33014yescomplemented
107C4×#125—531no—
108#1767—532no—
109C2×#330—532no—
110#1295—531no—
111#1295D5311yescharacteristic
112C2×C53C2×#33011yescharacteristic
113C2×C53#152912yes—
114C2×C53#153711yescharacteristic
115C2×C53#180111yescharacteristic
116C2×C53#180612yes—
117C2×#125—531no—
118C2×C4⋊C8—1062no—
119C2×#97—532no—
120C2×#97C2×D5312yes—
121#389—532no—
122#387—1062no—
123#382C2×D5311yescharacteristic
124C4×C4⋊C4—532no—
125C4×C4⋊C4—1062no—
126C43—531no—
127#336—1062no—
128#330—1064no—
129C53#858011yescharacteristic, complemented, Sylow
130C22×C8—22no—
131C22×C8—1062no—
132#125—532no—
133#125—532no—
134#125—1062no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4—1062no—
138C2×C4⋊C4C4×D5312yes—
139C2×C42—531no—
140C2×C42—532no—
141C2×C42—531no—
142C2×C42—532no—
143C2×C42—1062no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—1068no—
146#97—22no—
147#97—22no—
148#97—1062no—
149#97—1062no—
150#95—22no—
151#95—1062no—
152#95—1062no—
153C22×C4—22no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—532no—
157C22×C4—1062no—
158C22×C4C2×C4×D5311yescharacteristic
159C22×C4D4×D5312yes—
160C2×C8—24no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—1064no—
164C2×C8—2122no—
165C2×C8—2122no—
166C4⋊C4—42no—
167C4⋊C4—534no—
168C4⋊C4—1062no—
169C4⋊C4—1064no—
170C4⋊C4—1064no—
171C4⋊C4—2124no—
172C42—1064no—
173C42—1062no—
174C42—2121no—
175C42—2122no—
176C42—2124no—
177C42#1171994512yes—
178C42D4×D5311yescharacteristic
179C42Q8×D5311yescharacteristic
180C23C22⋊C4×D5311yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—532no—
189C2×C4—534no—
190C2×C4—1061no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1064no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4C4⋊C4×D5311yescharacteristic
199C2×C4#1313317411yescharacteristic
200C2×C4#1313319012yes—
201C2×C4#1313319512yes—
202C8—44no—
203C8—2124no—
204C22#1681048211yescharacteristic
205C22#101×D5311yescharacteristic
206C22#1681060411yescharacteristic
207C22#104×D5312yes—
208C22#1681063812yes—
209C4—21no—
210C4—21no—
211C4—22no—
212C4—22no—
213C4—82no—
214C4—534no—
215C4—2124no—
216C4—2122no—
217C4—2121no—
218C2#330×D5311yescharacteristic
219C2#2027331912yes—
220C2#2027335111yescharacteristic
221C2#2027471011yescharacteristic
222C2#2027473412yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1295D53
hypocenterC53#8580
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
106. Dic53#330semidirect
129. C53#8580semidirect

Maximal subgroups

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

subgroupindex
#202772702
#202745692
C2×#168103642
#202725962
C53×#12952
#858053