Finite Group Databasebeta

← #27294634

Subgroups — #27294634

Order 13568.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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#20276531C211yesmaximal, characteristic, complemented
2#20274470C211yesmaximal, characteristic, complemented
3#20272706C212yesmaximal, complemented
4#20272700C212yesmaximal, complemented
5C53×#3176C211yesmaximal, characteristic
6#16810578—21no—
7#16810444—21no—
8#16810405C2212yes—
9#16810401—21no—
10#16810381C2211yescharacteristic
11#16810380C2211yescharacteristic
12#16810379—21no—
13C53×#540—21no—
14C53×#499C411yescharacteristic, complemented
15C53×#493C411yescharacteristic, complemented
16C53×#490—21no—
17C53×#469C2212yes—
18C53×#466—21no—
19C53×#442C412yescomplemented
20C4×D8×C53—21no—
21C53×#405C2211yescharacteristic
22C2×#11719917—21no—
23#13133176—21no—
24#13133157—22no—
25#13133127—22no—
26C8×Dic53—22no—
27C2×D8×C53—21no—
28C2×D8×C53C2×C412yes—
29C2×M16×C53C2311yescharacteristic
30C22×C8×C53D411yescharacteristic
31#127×C53—21no—
32#125×C53—21no—
33#123×C53—42no—
34#121×C53—22no—
35#121×C53—42no—
36C4×D4×C53—22no—
37#117×C53D411yescharacteristic
38#107×C53—21no—
39#106×C53C2×C412yes—
40#102×C53—22no—
41#102×C53—22no—
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C2×D4×C53—22no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C22×C4×C53—42no—
50C22×C4×C53C2×D411yescharacteristic
51D8×C53—42no—
52D8×C53—41no—
53D8×C53—41no—
54M16×C53—41no—
55C2×C8×C53—21no—
56C2×C8×C53—21no—
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C22×C411yescharacteristic
60C2×C8×C53C2×D411yescharacteristic
61C4⋊C4×C53—21no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—42no—
64C22⋊C4×C53—21no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C22⋊C411yescharacteristic
69C22⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4×Dic53—22no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—22no—
75#11719908—42no—
76C23×C53—42no—
77C23×C53AGL(1,ℤ/8)11yescharacteristic
78D4×C53—42no—
79D4×C53—42no—
80D4×C53—162no—
81D4×C53—162no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—162no—
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
90C2×Dic53—42no—
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—82no—
101C22×C53—82no—
102C22×C53—82no—
103C22×C53#41811yescharacteristic
104C4×C53—21no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—82no—
108C4×C53#41011yescharacteristic
109C4×C53C2×#10411yescharacteristic
110Dic53—82no—
111#3176D5311yescharacteristic
112#2166—531no—
113#1747—531no—
114#1341—532no—
115#1339—532no—
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—162no—
119C2×C53#150511yescharacteristic
120#540—21no—
121#499Dic5311yescharacteristic, complemented
122#493Dic5311yescharacteristic, complemented
123#490—21no—
124#469C2×D5312yes—
125#466—21no—
126#442Dic5312yescomplemented
127C4×D8—21no—
128#405—1061no—
129#405C2×D5311yescharacteristic
130#364—1061no—
131#349—532no—
132#347—1061no—
133#338—531no—
134H(3,ℤ/4)—531no—
135H(3,ℤ/4)—1061no—
136C53#1177011yescharacteristic, complemented, Sylow
137C2×D8—21no—
138C2×D8C2×Dic5312yes—
139C2×M16—1061no—
140C2×M16C22×D5311yescharacteristic
141C22×C8C22⋊D5311yescharacteristic
142#127—21no—
143#125—21no—
144#123—42no—
145#121—22no—
146#121—42no—
147C4×D4—22no—
148#117—1061no—
149#117C22⋊D5311yescharacteristic
150M32—1062no—
151#107—21no—
152#106C2×Dic5312yes—
153#102—22no—
154#102—22no—
155#97—1062no—
156#97C2×Dic5311yescharacteristic
157#97C22⋊D5311yescharacteristic
158C4×C8—1062no—
159C4×C8C2×Dic5311yescharacteristic
160C4×C8C22⋊D5311yescharacteristic
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—42no—
164C22×C4—42no—
165C22×C4C2×C22⋊D5311yescharacteristic
166D8—42no—
167D8—41no—
168D8—41no—
169M16—41no—
170M16—2121no—
171M16—2121no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1062no—
175C2×C8C22⋊Dic5311yescharacteristic
176C2×C8C22⋊Dic5311yescharacteristic
177C2×C8C22×Dic5311yescharacteristic
178C2×C8C2×C22⋊D5311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—2121no—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—2121no—
187C42—1062no—
188C42C22⋊Dic5311yescharacteristic
189C42C22⋊Dic5311yescharacteristic
190C16—2122no—
191C23—42no—
192C23#1313324311yescharacteristic
193D4—42no—
194D4—42no—
195D4—162no—
196D4—162no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—162no—
203C2×C4—2122no—
204C2×C4C2×C22⋊Dic5311yescharacteristic
205C2×C4#1313324311yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—2122no—
210C8#1313314411yescharacteristic
211C8#1313314411yescharacteristic
212C22—21no—
213C22—41no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22#1681062811yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—4242no—
223C4#1681059711yescharacteristic
224C4C2×#1313314411yescharacteristic
225C2—21no—
226C2—41no—
227C2—162no—
228C2#2027322311yescharacteristic
subgroupG/H
centerC2#20273223
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
Fitting, prime radicalC53×#3176C2
socleC2×C53#1505
cosocleC2×M16×C53C23
hypercenter#3176D53
hypocenterC53#11770
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #20276531C2semidirect
2. #20274470C2semidirect
3. #20272706C2semidirect
4. #20272700C2semidirect
14. C53×#499C4semidirect
15. C53×#493C4semidirect
19. C53×#442C4semidirect
121. #499Dic53semidirect
122. #493Dic53semidirect
126. #442Dic53semidirect
136. C53#11770semidirect

Maximal subgroups

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

subgroupindex
#202765312
#202744702
#202727062
#202727002
C53×#31762
#1177053