Finite Group Databasebeta

← #28312437

Subgroups — #28312437

Order 13568.

458 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 60 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#20290988C212yesmaximal, complemented
2#20289846C214yesmaximal, complemented
3#20285165C211yesmaximal, characteristic
4#16811602—22no—
5#16811530—22no—
6#16811528C2214yescomplemented
7C2×#13133243—22no—
8#16811173C2211yescharacteristic
9#16811010—22no—
10#16810999C2212yes—
11#16810995C412yescomplemented
12C4×#11719917C411yescharacteristic, complemented
13C2×#127×C53C411yescharacteristic, complemented
14#13133296—42no—
15#13133276—24no—
16#13133275—42no—
17C2×#11719945D412yes—
18#13133243—44no—
19#13133243—44no—
20C2×#11719923—24no—
21C2×#11719922—24no—
22C2×#11719917—21no—
23#13133217C2×C411yescharacteristic
24#13133214C2311yescharacteristic
25#13133205D412yes—
26#13133191—41no—
27C4×D212—41no—
28#13133128—21no—
29C4×#88322C2×C412yes—
30C22×D4×C53—21no—
31C22×D4×C53—41no—
32#127×C53—21no—
33#127×C53—41no—
34#127×C53C2×C412yes—
35C2×C42×C53C2×C411yescharacteristic
36C24×C53—41no—
37C2×D4×C53—22no—
38C2×D4×C53—42no—
39C2×D4×C53—42no—
40C2×D4×C53—42no—
41C2×D4×C53—42no—
42C2×D4×C53—41no—
43C2×D4×C53—84no—
44C22×C4×C53—21no—
45C22×C4×C53C22⋊C411yescharacteristic
46C2×C22⋊D53—22no—
47#11719945—42no—
48#11719945—42no—
49C2×D212—41no—
50C2×D212C2×D412yes—
51C2×C4×D53—22no—
52C2×Dic106C22⋊C412yes—
53C22⋊#6157—44no—
54C4⋊#6157—82no—
55C4×#6157—82no—
56C42×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C22×C411yescharacteristic
59#11719923—48no—
60#11719922—48no—
61#11719921—42no—
62C4⋊Dic53—41no—
63#11719917—24no—
64C2×#88322—22no—
65C23×C53—42no—
66C23×C53—81no—
67C23×C53—81no—
68C23×C53—84no—
69C23×C53#9911yescharacteristic
70C22×D53—22no—
71C2×#6157—84no—
72D4×C53—44no—
73D4×C53—84no—
74D4×C53—84no—
75D4×C53—82no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—42no—
80C2×C4×C53#9911yescharacteristic
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C22⋊D53—82no—
83C2×Dic53—22no—
84D212—42no—
85D212—82no—
86C4×D53—42no—
87C4×D53—42no—
88Dic106—42no—
89#88322—24no—
90#13585—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—41no—
93C22×C53—84no—
94C22×C53—82no—
95C22×C53—162no—
96C22×C53—162no—
97C22×C53C2×#9911yescharacteristic
98C2×D53—42no—
99C2×D53—82no—
100#6157—164no—
101C4×C53—22no—
102C4×C53—21no—
103C4×C53—41no—
104C4×C53#35312yes—
105Dic53—42no—
106#3151—531no—
107#2126—532no—
108#1408—534no—
109C2×C53—21no—
110C2×C53—41no—
111C2×C53—162no—
112C2×C53C2×#35311yescharacteristic
113D53—82no—
114C2×AGL(1,ℤ/8)—1062no—
115#532—531no—
116C2×#127#615711yescharacteristic, complemented
117#496—1062no—
118#493—532no—
119#492—532no—
120#410—1062no—
121C4×M16—531no—
122#354—1062no—
123#353—534no—
124C53#1358511yescharacteristic, complemented, Sylow
125C2×#41—532no—
126C22×D4—21no—
127C22×D4—41no—
128AGL(1,ℤ/8)—2124no—
129AGL(1,ℤ/8)—2124no—
130C2×SD16—1064no—
131C2×D8—1064no—
132C2×M16—1061no—
133C4⋊Q8—531no—
134#127—21no—
135#127—41no—
136#127—531no—
137#127C2×#615712yes—
138#124—532no—
139#121—2121no—
140C4×D4—2121no—
141#117—2122no—
142C2×C42C2×#615711yescharacteristic
143#99—1064no—
144#99—2122no—
145#97—1061no—
146C4×C8—532no—
147C24—41no—
148#41—2122no—
149#41—2122no—
150C2×Q8—532no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—41no—
157C2×D4—84no—
158C2×D4—532no—
159C2×D4—1062no—
160C2×D4—2121no—
161C22×C4—21no—
162C22×C4—1062no—
163C22×C4C22⋊#615711yescharacteristic
164SD16—2128no—
165D8—2128no—
166M16—1064no—
167C2×C8—1062no—
168C4⋊C4—4242no—
169C4⋊C4—2121no—
170C22⋊C4—2122no—
171C22⋊C4—2124no—
172C42—21no—
173C42—4242no—
174C42C22⋊#615711yescharacteristic
175C42C22×#615711yescharacteristic
176C23—42no—
177C23—81no—
178C23—81no—
179C23—84no—
180C23—1062no—
181C23#1313327511yescharacteristic
182Q8—2122no—
183D4—44no—
184D4—84no—
185D4—84no—
186D4—82no—
187D4—4242no—
188D4—4242no—
189D4—2122no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—1062no—
195C2×C4—4244no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1313327511yescharacteristic
199C2×C4C2×C22⋊#615711yescharacteristic
200C8—1064no—
201C22—21no—
202C22—41no—
203C22—84no—
204C22—82no—
205C22—162no—
206C22—162no—
207C22—4242no—
208C22—2122no—
209C22C2×#1313327511yescharacteristic
210C4—22no—
211C4—21no—
212C4—41no—
213C4—8484no—
214C4—2122no—
215C4#1681152712yes—
216C2—21no—
217C2—41no—
218C2—162no—
219C2—4242no—
220C2C2×#1681152711yescharacteristic
subgroupG/H
centerC2C2×#16811527
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#127×C53C4
socleC2×C53C2×#353
prime radical#20285165C2
cosocle#13133214C23
hypercenterC2×#127#6157
hypocenterC53#13585
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290988C2semidirect
2. #20289846C2semidirect
6. #16811528C22semidirect
11. #16810995C4semidirect
12. C4×#11719917C4semidirect
13. C2×#127×C53C4semidirect
116. C2×#127#6157semidirect
124. C53#13585semidirect

Maximal subgroups

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

subgroupindex
#202909882
#202898462
#202851652
#1358553