Finite Group Databasebeta

← #27275017

Subgroups — #27275017

Order 13568.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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#20274463C212yesmaximal
3C2×#16810364C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#1295C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7C2×#13133128C2212yes—
8#16810519C412yes—
9#16810505—22no—
10C4⋊C4×Dic53C412yes—
11#16810474—22no—
12C42×Dic53C2211yescharacteristic
13#16810377—22no—
14#16810377—22no—
15#16810364C418yescomplemented
16C2×#97×C53C2212yes—
17C53×#382C2211yescharacteristic
18C22×#88322—22no—
19#13133209—21no—
20#13133209—21no—
21#13133209C2×C414yes—
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#13133128—22no—
31#13133128C2×C414yes—
32#13133124—22no—
33#13133124—22no—
34C22×C8×C53—22no—
35C2×C4⋊C4×C53C2×C412yes—
36C2×C42×C53C2311yescharacteristic
37#97×C53—22no—
38#97×C53C2×C414yes—
39#95×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C22×C411yescharacteristic
42C22×C4×C53C2×D412yes—
43C22×Dic53—22no—
44C22×Dic53C22⋊C411yescharacteristic
45C22×Dic53C22⋊C412yes—
46C22×Dic53C22⋊C411yescharacteristic
47C2×C8×C53—24no—
48C2×C8×C53—42no—
49C2×C8×C53—42no—
50C4⋊C4×C53—42no—
51C42×C53C22×C412yes—
52C42×C53#4111yescharacteristic
53C42×C53#4111yescharacteristic
54C4⋊Dic53—22no—
55C4⋊Dic53C4214yes—
56#11719919—24no—
57#11719919—24no—
58#11719919—44no—
59C4×Dic53—22no—
60C4×Dic53—41no—
61C4×Dic53—42no—
62C4×Dic53—42no—
63C4×Dic53—44no—
64C2×#88322—24no—
65C2×#88322—42no—
66C2×#88322—42no—
67C23×C53C2×C22⋊C411yescharacteristic
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53C2×C4211yescharacteristic
76C2×C4×C53#11711yescharacteristic
77C2×C4×C53C4×D412yes—
78C2×C4×C53C4×D412yes—
79C2×Dic53—22no—
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—
89C8×C53—44no—
90#88322—44no—
91#8225—531nomaximal, Sylow
92C22×C53C4×C22⋊C411yescharacteristic
93C22×C53C2×#10111yescharacteristic
94C22×C53#41311yescharacteristic
95C22×C53#42112yes—
96C22×C53#42112yes—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—82no—
102Dic53—22no—
103Dic53—41no—
104Dic53—44no—
105Dic53—42no—
106C4×#125—531no—
107#1745—532no—
108C2×#330—532no—
109#1295—531no—
110#1295D5311yescharacteristic
111C2×C53#153411yescharacteristic
112C2×C53#153512yes—
113C2×C53#153711yescharacteristic
114C2×C53C4×#10111yescharacteristic
115C2×C53#175912yes—
116C2×#125—531no—
117C2×#97—532no—
118C2×#97C2×D5312yes—
119#389—532no—
120#387—1062no—
121#382C2×D5311yescharacteristic
122C4×C4⋊C4—532no—
123C4×C4⋊C4—1062no—
124C43—531no—
125#336—1062no—
126#336—1062no—
127#330—538no—
128C53#822511yescharacteristic, complemented, Sylow
129C22×C8—22no—
130C22×C8—1062no—
131#125—534no—
132#125—1061no—
133#125—1061no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—1062no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4C4×D5312yes—
138C2×C42—531no—
139C2×C42—531no—
140C2×C42—532no—
141C2×C42—532no—
142C2×C42—1062no—
143C2×C42C22×D5311yescharacteristic
144#97—22no—
145#97—534no—
146#97—1062no—
147#97C4×D5314yes—
148#95—22no—
149#95—1062no—
150#95—1062no—
151C22×C4—22no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—532no—
155C22×C4—1062no—
156C22×C4C2×C4×D5311yescharacteristic
157C22×C4D4×D5312yes—
158C2×C8—24no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—1064no—
162C2×C8—2122no—
163C2×C8—2122no—
164C4⋊C4—42no—
165C4⋊C4—534no—
166C4⋊C4—1062no—
167C4⋊C4—1064no—
168C4⋊C4—1064no—
169C4⋊C4—2124no—
170C42—1062no—
171C42—2121no—
172C42—2122no—
173C42—2122no—
174C42—2124no—
175C42C2×C4×D5312yes—
176C42#1171994711yescharacteristic
177C42#1171994911yescharacteristic
178C23C22⋊C4×D5311yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—532no—
187C2×C4—1062no—
188C2×C4—1061no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1064no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4C42×D5311yescharacteristic
197C2×C4#1313317511yescharacteristic
198C2×C4#1313317712yes—
199C2×C4#1313317912yes—
200C8—44no—
201C8—2124no—
202C22#1681047011yescharacteristic
203C22#101×D5311yescharacteristic
204C22#1681060411yescharacteristic
205C22#1681063812yes—
206C22#1681063812yes—
207C4—21no—
208C4—21no—
209C4—22no—
210C4—22no—
211C4—82no—
212C4—1062no—
213C4—2124no—
214C4—2122no—
215C4—2121no—
216C2#2027334011yescharacteristic
217C2#2027334312yes—
218C2#2027335111yescharacteristic
219C2#2027451011yescharacteristic
220C2#2027452412yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1295D53
hypocenterC53#8225
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
15. #16810364C4semidirect
128. C53#8225semidirect

Maximal subgroups

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

subgroupindex
#202772702
#202744632
C2×#168103642
#202725962
C53×#12952
#822553