Finite Group Databasebeta

← #27294636

Subgroups — #27294636

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#20274470C211yesmaximal, characteristic, complemented
3#20272707C212yesmaximal, complemented
4#20272701C212yesmaximal, complemented
5C53×#2166C211yesmaximal, 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#16810578—21no—
15#16810381C2211yescharacteristic
16#16810380C2211yescharacteristic
17#16810379—21no—
18C53×#405C2211yescharacteristic
19C53×#364—21no—
20C53×#349C2212yes—
21C53×#347—21no—
22C2×D424—21no—
23C2×D424C2×C412yes—
24C2×#11719917—21no—
25#13133214—21no—
26#13133209—21no—
27#13133202—42no—
28#13133191—22no—
29#13133190—42no—
30C4×D212—22no—
31#13133176—21no—
32#13133151—21no—
33#13133149C2×C412yes—
34#13133139—22no—
35#13133139—22no—
36#13133127—22no—
37C8×Dic53—22no—
38C2×M16×C53C2311yescharacteristic
39C22×C8×C53D411yescharacteristic
40#117×C53D411yescharacteristic
41M32×C53—22no—
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
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—
52C2×C8×C53—21no—
53C2×C8×C53—21no—
54C2×C8×C53C22⋊C411yescharacteristic
55C2×C8×C53C22⋊C411yescharacteristic
56C2×C8×C53C22×C411yescharacteristic
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—21no—
59C22⋊C4×C53—21no—
60C42×C53C22⋊C411yescharacteristic
61C42×C53C22⋊C411yescharacteristic
62C22⋊Dic53—41no—
63#11719921—42no—
64#11719921—42no—
65C4⋊Dic53—22no—
66C4⋊Dic53—41no—
67#11719919—42no—
68C4×Dic53—22no—
69#11719917—41no—
70#11719917—41no—
71C2×#88322—22no—
72D424—42no—
73D424—41no—
74D424—41no—
75C16×C53—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×C53C2×C22⋊C411yescharacteristic
83C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
84C22⋊D53—162no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87D212—42no—
88D212—42no—
89D212—162no—
90C4×D53—162no—
91C8×C53—21no—
92C8×C53—21no—
93C8×C53—21no—
94C8×C53#10411yescharacteristic
95C8×C53#10411yescharacteristic
96#88322—42no—
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#41011yescharacteristic
108C4×C53C2×#10411yescharacteristic
109Dic53—82no—
110Dic53—82no—
111#3176—531no—
112#2166D5311yescharacteristic, complemented
113#1747—531no—
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—1061no—
129#405C2×D5311yescharacteristic
130#364—21no—
131#349C2×D5312yes—
132#347—21no—
133#338—531no—
134H(3,ℤ/4)—531no—
135H(3,ℤ/4)—1061no—
136C53#1177011yescharacteristic, complemented, Sylow
137C2×D8—532no—
138C2×D8—1061no—
139C2×M16—1061no—
140C2×M16C22×D5311yescharacteristic
141C22×C8D21211yescharacteristic
142#127—1061no—
143#125—1061no—
144#123—2122no—
145#121—1062no—
146#121—2122no—
147C4×D4—1062no—
148#117—1061no—
149#117D21211yescharacteristic
150M32—22no—
151#107—1061no—
152#106—532no—
153#102—1062no—
154#102—1062no—
155#97—1062no—
156#97C4×D5311yescharacteristic
157#97C22⋊D5311yescharacteristic
158C4×C8—1062no—
159C4×C8C4×D5311yescharacteristic
160C4×C8C22⋊D5311yescharacteristic
161C2×D4—1062no—
162C2×D4—2122no—
163C2×D4—2122no—
164C22×C4—2122no—
165C22×C4C2×D21211yescharacteristic
166D8—2122no—
167D8—2121no—
168D8—2121no—
169M16—41no—
170M16—2121no—
171M16—2121no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1062no—
175C2×C8#1171992111yescharacteristic
176C2×C8#1171992111yescharacteristic
177C2×C8C2×C4×D5311yescharacteristic
178C2×C8C2×C22⋊D5311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—1062no—
181C4⋊C4—2121no—
182C4⋊C4—2122no—
183C22⋊C4—21no—
184C22⋊C4—2122no—
185C22⋊C4—2121no—
186C22⋊C4—2122no—
187C42—1062no—
188C42#1171992111yescharacteristic
189C42#1171992111yescharacteristic
190C16—42no—
191C23—2122no—
192C23#1313324511yescharacteristic
193D4—8482no—
194D4—8482no—
195D4—2122no—
196D4—2122no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—8482no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4C2×#1171992111yescharacteristic
205C2×C4#1313324511yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—2122no—
210C8#1313314311yescharacteristic
211C8#1313314311yescharacteristic
212C22—21no—
213C22—41no—
214C22—4242no—
215C22—4242no—
216C22—4242no—
217C22#1681062911yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—4242no—
222C4—4242no—
223C4#1681059611yescharacteristic
224C4C2×#1313314311yescharacteristic
225C2—21no—
226C2—41no—
227C2—8482no—
228C2#2027322411yescharacteristic
subgroupG/H
centerC2#20273224
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#2166C2
socleC2×C53#1505
prime radical#20285365C2
cosocleC2×M16×C53C23
hypercenter#2166D53
hypocenterC53#11770
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #20274470C2semidirect
3. #20272707C2semidirect
4. #20272701C2semidirect
5. C53×#2166C2semidirect
7. #16811025C4semidirect
8. #16811000C4semidirect
12. #16810736C4semidirect
112. #2166D53semidirect
136. C53#11770semidirect

Maximal subgroups

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

subgroupindex
#202853652
#202744702
#202727072
#202727012
C53×#21662
#1177053