Finite Group Databasebeta

← #27351843

Subgroups — #27351843

Order 13568.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#20288525C211yesmaximal, characteristic
2#20283796C211yesmaximal, characteristic
3#20273278C216yesmaximal
4#20273213C216yesmaximal
5C53×#2351C211yesmaximal, characteristic
6Q8×Dic106C414yescomplemented
7#16811352C414yescomplemented
8#16811349—24no—
9#16811254—23no—
10#16811180C2216yes—
11#16811101—21no—
12Q8×#88322—24no—
13#16810676C2216yes—
14#16810359C2214yes—
15#16810359C22112yes—
16C2×C4×Q8×C53—21no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#389—23no—
19C53×#382C2213yes—
20C4×C4⋊C4×C53C2213yes—
21#13133217C2×C416yes—
22#13133217C2×C416yes—
23#13133216—212no—
24#13133209—26no—
25#13133208—212no—
26#13133196—26no—
27#13133196C2314yes—
28Q8×Dic53—24no—
29Q8×Dic53—24no—
30C4×Dic106—26no—
31C4×Dic106C2×C4112yes—
32#13133176—26no—
33C4⋊#88322—26no—
34C22⋊#88322C2314yes—
35C4×#88322—26no—
36C22×Q8×C53D411yescharacteristic
37C4×Q8×C53—44no—
38C4×Q8×C53C2×C414yes—
39C4×Q8×C53D414yes—
40C2×C4⋊C4×C53—23no—
41C2×C4⋊C4×C53—23no—
42C2×C4⋊C4×C53D413yes—
43C2×C42×C53—23no—
44C2×C42×C53C2313yes—
45#95×C53—23no—
46#95×C53C2311yescharacteristic
47#95×C53C2313yes—
48C2×Q8×C53C22⋊C414yes—
49C2×Q8×C53SD1614yes—
50C2×Q8×C53Q1614yes—
51C22×C4×C53—21no—
52C22×C4×C53—23no—
53C22×C4×C53C2×D413yes—
54C22×C4×C53C2×D413yes—
55C22×C4×C53C2411yescharacteristic
56C2×Dic106—26no—
57C2×Dic106C22×C414yes—
58C4⋊C4×C53—46no—
59C4⋊C4×C53—46no—
60C4⋊C4×C53C22⋊C4112yes—
61C42×C53—46no—
62C42×C53C22×C416yes—
63C42×C53C2×D416yes—
64C22⋊Dic53—24no—
65C4⋊Dic53—26no—
66C4⋊Dic53—26no—
67C4⋊Dic53—26no—
68C4⋊Dic53C22×C414yes—
69#11719919—212no—
70#11719919—24no—
71C4×Dic53—212no—
72C2×#88322—24no—
73C23×C53C22×D411yescharacteristic
74Q8×C53#103116yes—
75C2×C4×C53—22no—
76C2×C4×C53—43no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—43no—
81C2×C4×C53—43no—
82C2×C4×C53—43no—
83C2×C4×C53C2×C22⋊C416yes—
84C2×C4×C53C2×C22⋊C416yes—
85C2×C4×C53C2×SD1616yes—
86C2×C4×C53C2×Q1616yes—
87C2×C4×C53C23×C411yescharacteristic
88C2×C4×C53C22×D411yescharacteristic
89C2×Dic53—24no—
90C2×Dic53—24no—
91Dic106—44no—
92Dic106—86no—
93#88322—44no—
94#17619—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—21no—
97C22×C53C22×C22⋊C411yescharacteristic
98C22×C53C22×SD1611yescharacteristic
99C22×C53C22×Q1611yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—26no—
103C4×C53—42no—
104C4×C53—83no—
105C4×C53C2×#103112yes—
106Dic53—44no—
107Dic53—44no—
108#3502—531no—
109#2982—531no—
110#2351D5311yescharacteristic
111#1518—536no—
112#1503—536no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#288411yescharacteristic
116C2×C53C22×#10311yescharacteristic
117C2×C53#290111yescharacteristic
118Q82—534no—
119#557—534no—
120#557—1064no—
121#544—1063no—
122#533—536no—
123#519—1061no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2×D5311yescharacteristic
126C8×Q8—1064no—
127#431—536no—
128#389—23no—
129#382C2×D5313yes—
130C4×C4⋊C4C2×D5313yes—
131#328—534no—
132#328—5312no—
133C53#1761911yescharacteristic, complemented, Sylow
134C22×Q8C22⋊D5311yescharacteristic
135C4⋊Q8—536no—
136C4⋊Q8—536no—
137C4⋊Q8—10612no—
138#125—10612no—
139#125—1066no—
140C22⋊Q8—534no—
141C22⋊Q8—1066no—
142C4×Q8—44no—
143C4×Q8—5312no—
144C4×Q8—1064no—
145C4×Q8—1064no—
146C4×Q8—1066no—
147C4×Q8C4×D5314yes—
148C4×Q8D21214yes—
149#117—1066no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4—23no—
152C2×C4⋊C4C22⋊D5313yes—
153C2×C42—23no—
154C2×C42C22×D5313yes—
155C4⋊C8—1066no—
156C22⋊C8—534no—
157C4×C8—1066no—
158#95—23no—
159#95C22×D5311yescharacteristic
160#95C22×D5313yes—
161C2×Q8—534no—
162C2×Q8—1066no—
163C2×Q8#1171992114yes—
164C2×Q8#1171992314yes—
165C2×Q8#1171992514yes—
166C22×C4—21no—
167C22×C4—23no—
168C22×C4C2×C22⋊D5313yes—
169C22×C4C2×C22⋊D5313yes—
170C22×C4C23×D5311yescharacteristic
171C2×C8—1064no—
172C4⋊C4—46no—
173C4⋊C4—46no—
174C4⋊C4—534no—
175C4⋊C4—1066no—
176C4⋊C4—1066no—
177C4⋊C4—10612no—
178C4⋊C4—1066no—
179C4⋊C4—1064no—
180C4⋊C4#11719921112yes—
181C22⋊C4—1064no—
182C42—46no—
183C42—10612no—
184C42C2×C4×D5316yes—
185C42C2×D21216yes—
186C23C22×C22⋊D5311yescharacteristic
187Q8—4246no—
188Q8—2124no—
189Q8#13133140116yes—
190C2×C4—22no—
191C2×C4—43no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—43no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—1064no—
199C2×C4—1064no—
200C2×C4C2×#1171992116yes—
201C2×C4C2×#1171992116yes—
202C2×C4C2×#1171992316yes—
203C2×C4C2×#1171992516yes—
204C2×C4C22×C4×D5311yescharacteristic
205C2×C4C22×D21211yescharacteristic
206C8—2124no—
207C22—21no—
208C22—21no—
209C22C22×#1171992111yescharacteristic
210C22C22×#1171992311yescharacteristic
211C22C22×#1171992511yescharacteristic
212C4—21no—
213C4—21no—
214C4—26no—
215C4—42no—
216C4—83no—
217C4—2124no—
218C4—2124no—
219C4C2×#13133140112yes—
220C2—21no—
221C2—21no—
222C2#2028317811yescharacteristic
223C2C22×#1313314011yescharacteristic
224C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2351C2
socleC22×C53C22×C22⋊C4
prime radicalC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenter#2351D53
hypocenterC53#17619
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
6. Q8×Dic106C4semidirect
7. #16811352C4semidirect
133. C53#17619semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202837962
#202732782
#202732132
C53×#23512
#1761953