Finite Group Databasebeta

← #27294633

Subgroups — #27294633

Order 13568.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#20285365C211yesmaximal, characteristic
2#20276531C211yesmaximal, characteristic, complemented
3#20272705C212yesmaximal, complemented
4#20272699C212yesmaximal, complemented
5C53×#1747C211yesmaximal, characteristic, complemented
6#16811228—21no—
7#16811025C411yescharacteristic, complemented
8#16811000C411yescharacteristic, complemented
9#16810988—21no—
10#16810876C2212yes—
11#16810859—21no—
12#16810736C412yescomplemented
13C4×D424—21no—
14#16810444—21no—
15#16810405C2212yes—
16#16810401—21no—
17C53×#405—21no—
18C53×#405C2211yescharacteristic
19C53×#338C2211yescharacteristic
20C53×H(3,ℤ/4)—21no—
21C53×H(3,ℤ/4)C2211yescharacteristic
22C2×D424—21no—
23C2×D424C2×C412yes—
24#13133214—21no—
25#13133209—21no—
26#13133202—42no—
27#13133191—22no—
28#13133190—42no—
29C4×D212—22no—
30#13133157—22no—
31#13133151—21no—
32#13133149C2×C412yes—
33#13133139—22no—
34#13133139—22no—
35C2×M16×C53—21no—
36C2×M16×C53C2311yescharacteristic
37C22×C8×C53D411yescharacteristic
38#117×C53—21no—
39#117×C53D411yescharacteristic
40#97×C53—22no—
41#97×C53C2×C411yescharacteristic
42#97×C53D411yescharacteristic
43C4×C8×C53—22no—
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C22×C4×C53C2×D411yescharacteristic
47C2×C22⋊D53—42no—
48C2×D212—22no—
49C2×D212—42no—
50C2×C4×D53—42no—
51M16×C53—41no—
52M16×C53—41no—
53M16×C53—41no—
54C2×C8×C53—21no—
55C2×C8×C53—21no—
56C2×C8×C53—22no—
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C22×C411yescharacteristic
60C2×C8×C53C2×D411yescharacteristic
61C4⋊C4×C53—21no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—21no—
64C22⋊C4×C53—41no—
65C42×C53—22no—
66C42×C53C22⋊C411yescharacteristic
67C42×C53C22⋊C411yescharacteristic
68#11719921—42no—
69#11719921—42no—
70C4⋊Dic53—22no—
71#11719919—42no—
72D424—42no—
73D424—41no—
74D424—41no—
75#11719908—42no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77C22×D53—42no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—42no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
85C22⋊D53—162no—
86C2×Dic53—42no—
87D212—42no—
88D212—42no—
89D212—162no—
90C4×D53—162no—
91C8×C53—21no—
92C8×C53—21no—
93C8×C53—21no—
94C8×C53—42no—
95C8×C53#10411yescharacteristic
96C8×C53#10411yescharacteristic
97#11770—531nomaximal, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53#41811yescharacteristic
101C2×D53—82no—
102C2×D53—82no—
103C2×D53—82no—
104C4×C53—21no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—82no—
108C4×C53#41011yescharacteristic
109C4×C53C2×#10411yescharacteristic
110Dic53—82no—
111#3176—531no—
112#2166—531no—
113#1747D5311yescharacteristic, complemented
114#1341—532no—
115#1339—532no—
116C2×C53—21no—
117C2×C53—41no—
118C2×C53#150511yescharacteristic
119D53—162no—
120#540—1061no—
121#499—531no—
122#493—531no—
123#490—1061no—
124#469—532no—
125#466—1061no—
126#442—532no—
127C4×D8—1061no—
128#405—21no—
129#405C2×D5311yescharacteristic
130#364—1061no—
131#349—532no—
132#347—1061no—
133#338C2×D5311yescharacteristic
134H(3,ℤ/4)—21no—
135H(3,ℤ/4)C2×D5311yescharacteristic
136C53#1177011yescharacteristic, complemented, Sylow
137C2×D8—532no—
138C2×D8—1061no—
139C2×M16—21no—
140C2×M16C22×D5311yescharacteristic
141C22×C8C22⋊D5311yescharacteristic
142#127—1061no—
143#125—1061no—
144#123—2122no—
145#121—1062no—
146#121—2122no—
147C4×D4—1062no—
148#117—21no—
149#117C22⋊D5311yescharacteristic
150M32—1062no—
151#107—1061no—
152#106—532no—
153#102—1062no—
154#102—1062no—
155#97—22no—
156#97C4×D5311yescharacteristic
157#97D21211yescharacteristic
158C4×C8—22no—
159C4×C8C4×D5311yescharacteristic
160C4×C8D21211yescharacteristic
161C2×D4—1062no—
162C2×D4—2122no—
163C2×D4—2122no—
164C22×C4—2122no—
165C22×C4C2×C22⋊D5311yescharacteristic
166D8—2122no—
167D8—2121no—
168D8—2121no—
169M16—41no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8#1171992111yescharacteristic
176C2×C8#1171992111yescharacteristic
177C2×C8C2×C4×D5311yescharacteristic
178C2×C8C2×D21211yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—41no—
181C4⋊C4—1062no—
182C4⋊C4—2122no—
183C22⋊C4—21no—
184C22⋊C4—41no—
185C22⋊C4—2122no—
186C22⋊C4—2122no—
187C42—22no—
188C42#1171992111yescharacteristic
189C42#1171992111yescharacteristic
190C16—2122no—
191C23—2122no—
192C23#1313324211yescharacteristic
193D4—8482no—
194D4—8482no—
195D4—2122no—
196D4—2122no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—8482no—
203C2×C4—2122no—
204C2×C4C2×#1171992111yescharacteristic
205C2×C4#1313324211yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—42no—
210C8#1313314511yescharacteristic
211C8#1313314511yescharacteristic
212C22—21no—
213C22—41no—
214C22—4242no—
215C22—4242no—
216C22—4242no—
217C22#1681062711yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—4242no—
223C4#1681059611yescharacteristic
224C4C2×#1313314511yescharacteristic
225C2—21no—
226C2—41no—
227C2—8482no—
228C2#2027322211yescharacteristic
subgroupG/H
centerC2#20273222
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1747C2
socleC2×C53#1505
prime radical#20285365C2
cosocleC2×M16×C53C23
hypercenter#1747D53
hypocenterC53#11770
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
2. #20276531C2semidirect
3. #20272705C2semidirect
4. #20272699C2semidirect
5. C53×#1747C2semidirect
7. #16811025C4semidirect
8. #16811000C4semidirect
12. #16810736C4semidirect
113. #1747D53semidirect
136. C53#11770semidirect

Maximal subgroups

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

subgroupindex
#202853652
#202765312
#202727052
#202726992
C53×#17472
#1177053