Finite Group Databasebeta

← D53×#1711 #27371766

Subgroups — D53×#1711 #27371766

Order 13568.

802 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 179 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#487×D53C213yesmaximal, complemented
2#486×D53C213yesmaximal
3#20274328C213yesmaximal, complemented
4#20274327C211yesmaximal, characteristic, complemented
5#20274326C213yesmaximal, complemented
6C82×D53C211yesmaximal, characteristic, complemented
7C53×#1711C211yesmaximal, characteristic, complemented
8C4⋊Q8×D53C2213yes—
9#127×D53C2211yescharacteristic
10#16810975C2213yescomplemented
11#16810974C2213yescomplemented
12#16810973C2213yescomplemented
13#16810972—26no—
14#16810971C2213yes—
15#16810970C2213yes—
16#16810969C2213yes—
17#16810968—26no—
18#103×D53—26no—
19#102×D53—26no—
20C4×C8×D53C2213yes—
21C8×#88322C2213yescomplemented
22C53×#487C2213yescomplemented
23C53×#486C2213yes—
24C82×C53C2211yescharacteristic, complemented
25C2×Q8×D53—23no—
26C2×D4×D53—23no—
27C2×C8×D53D416yes—
28#13133217C2313yes—
29#13133216—23no—
30#13133216—23no—
31#13133215—23no—
32#13133214C2311yescharacteristic
33#13133213—23no—
34C4⋊C4×D53—23no—
35C4⋊C4×D53—23no—
36C42×D53C2311yescharacteristic
37#13133142—26no—
38#13133141—26no—
39#13133140—26no—
40#13133139—26no—
41#13133138—26no—
42#13133137—26no—
43C4×#88322C2313yes—
44C8×Dic53D416yes—
45C4⋊Q8×C53C2313yes—
46#127×C53C2311yescharacteristic
47#103×C53—26no—
48#102×C53—26no—
49C4×C8×C53C2313yes—
50C23×D53—41no—
51C2×Q8×C53—23no—
52C2×D4×C53—23no—
53C2×C22⋊D53—43no—
54Q8×D53—412no—
55D4×D53—412no—
56C2×D212—23no—
57C2×C4×D53—43no—
58C2×C4×D53C2×D413yes—
59C2×Dic106—23no—
60C2×Dic106—43no—
61C2×C8×C53C2×D416yes—
62C4⋊C4×C53—23no—
63C4⋊C4×C53—23no—
64C42×C53C2411yescharacteristic
65C4⋊Dic53—23no—
66C4⋊Dic53—23no—
67#11719919—43no—
68#11719919—43no—
69C4×Dic53C2×D413yes—
70C2×#88322C2×D416yes—
71C8×D53—212no—
72C23×C53—41no—
73C22×D53—41no—
74C22×D53—86no—
75C22×D53#12711yescharacteristic
76Q8×C53—46no—
77D4×C53—46no—
78C2×C4×C53—43no—
79C2×C4×C53C22×D413yes—
80C22⋊D53—86no—
81C2×Dic53—43no—
82C2×Dic53#12713yes—
83D212—46no—
84C4×D53—86no—
85C4×D53#135112yes—
86Dic106—46no—
87Dic106—86no—
88C8×C53—26no—
89#88322—26no—
90C2×#1711—531nomaximal, Sylow
91C22×C53—83no—
92C22×C53C2×#12711yescharacteristic
93C2×D53—83no—
94C2×D53—164no—
95C2×D53#49516yes—
96C4×C53—83no—
97C4×C53C2×#13516yes—
98Dic53—83no—
99Dic53#49516yes—
100C2×#487—533no—
101C2×#486—533no—
102#1711—533no—
103#1711—533no—
104#1711—531no—
105#1711D5311yescharacteristic, direct product factor
106C2×C82—531no—
107C2×C53—161no—
108C2×C53C2×#49513yes—
109D53—161no—
110D53#171114yesdirect product factor
111C2×C4⋊Q8—533no—
112C2×#127—531no—
113#487—533no—
114#487—533no—
115#487—533no—
116#487—1066no—
117#487C2×D5313yescomplemented
118#486—533no—
119#486—533no—
120#486—533no—
121#486—1066no—
122#486C2×D5313yes—
123C2×#103—1066no—
124C2×#102—1066no—
125C2×C4×C8—533no—
126C82—533no—
127C82C2×D5311yescharacteristic, complemented
128C53C2×#171111yescharacteristic, complemented, Sylow
129C22×Q8—1063no—
130C22×D4—1063no—
131C22×C8—536no—
132C4⋊Q8—533no—
133C4⋊Q8—1063no—
134C4⋊Q8—1063no—
135C4⋊Q8—1063no—
136C4⋊Q8C22×D5313yes—
137#127—531no—
138#127—1063no—
139#127C22×D5311yescharacteristic
140C2×C4⋊C4—1063no—
141C2×C4⋊C4—1063no—
142C2×C42—531no—
143#103—26no—
144#103—1066no—
145#103—1066no—
146#103—1066no—
147#102—26no—
148#102—1066no—
149#102—1066no—
150#102—1066no—
151C4×C8—536no—
152C4×C8—533no—
153C4×C8C22×D5313yes—
154C24—2121no—
155C2×Q8—23no—
156C2×Q8—1063no—
157C2×Q8—2123no—
158C2×Q8—21212no—
159C2×D4—23no—
160C2×D4—1063no—
161C2×D4—21212no—
162C2×D4—2123no—
163C22×C4—533no—
164C22×C4—2123no—
165C2×C8—536no—
166C2×C8—10612no—
167C2×C8D4×D5316yes—
168C4⋊C4—23no—
169C4⋊C4—23no—
170C4⋊C4—1063no—
171C4⋊C4—1063no—
172C4⋊C4—2123no—
173C4⋊C4—2123no—
174C42—533no—
175C42C23×D5311yescharacteristic
176C23—41no—
177C23—531no—
178C23—4246no—
179C23—2121no—
180Q8—46no—
181Q8—4246no—
182Q8—2126no—
183D4—46no—
184D4—4246no—
185D4—2126no—
186C2×C4—43no—
187C2×C4—533no—
188C2×C4—5312no—
189C2×C4—4246no—
190C2×C4—2123no—
191C2×C4C2×D4×D5313yes—
192C8—26no—
193C8—1066no—
194C22—83no—
195C22—536no—
196C22—8484no—
197C22—4243no—
198C22#127×D5311yescharacteristic
199C4—83no—
200C4—536no—
201C4—4243no—
202C4#135×D5316yes—
203C2—161no—
204C2—534no—
205C2—8481no—
206C2#495×D5313yes—
subgroupG/H
centerC22#127×D53
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1711C2
socleC22×C53C2×#127
prime radical#127×D53C22
hypercenter#1711D53
hypocenterC53C2×#1711
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #487×D53C2semidirect
3. #20274328C2semidirect
4. #20274327C2semidirect
5. #20274326C2semidirect
6. C82×D53C2semidirect
7. C53×#1711C2semidirect
10. #16810975C22semidirect
11. #16810974C22semidirect
12. #16810973C22semidirect
21. C8×#88322C22semidirect
22. C53×#487C22semidirect
24. C82×C53C22semidirect
105. #1711D53direct (N × H)
110. D53#1711direct (N × H)
117. #487C2×D53semidirect
127. C82C2×D53semidirect
128. C53C2×#1711semidirect

Maximal subgroups

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

subgroupindex
#487×D532
#486×D532
#202743282
#202743272
#202743262
C82×D532
C53×#17112
C2×#171153