Finite Group Databasebeta

← #28312472

Subgroups — #28312472

Order 13568.

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

normal subgroups only

This group has 388 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
1C2×#16811528C212yesmaximal
2#20289856C214yesmaximal, complemented
3#20285090C211yesmaximal, characteristic
4C2×#13133276—22no—
5#16811528—22no—
6#16811528C2214yes—
7#16811180C411yescharacteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810988C412yescomplemented
10#16810988C2212yes—
11#16810970—22no—
12#16810629—22no—
13C4×M16×C53C411yescharacteristic, complemented
14C2×C22⋊#6157—42no—
15#13133276—24no—
16#13133276—42no—
17C22×D212—41no—
18C22×D212D412yes—
19#13133217—21no—
20#13133214—41no—
21#13133214C2×C411yescharacteristic
22#13133214D412yes—
23#13133214C2311yescharacteristic
24#13133209C2×C412yescomplemented
25#13133196—41no—
26C4×Dic106—41no—
27#13133176—21no—
28#13133139—24no—
29#13133139—24no—
30C2×M16×C53—21no—
31C2×C42×C53C2×C411yescharacteristic
32#97×C53C2×C412yescomplemented
33C4×C8×C53—21no—
34C23×D53—22no—
35C22×#6157—82no—
36C22×C4×C53—21no—
37C22×C4×C53C22⋊C411yescharacteristic
38C2×D212—22no—
39C2×D212—42no—
40C2×D212—42no—
41C2×D212—41no—
42C2×D212—42no—
43C2×D212—41no—
44C2×D212—84no—
45C2×D212C22⋊C412yes—
46C2×D212C2×D412yes—
47C2×Dic106—41no—
48C22⋊#6157—44no—
49C22⋊#6157—82no—
50M16×C53—42no—
51C2×C8×C53—22no—
52C42×C53—21no—
53C42×C53C22⋊C411yescharacteristic
54C42×C53C22×C411yescharacteristic
55C22⋊Dic53—41no—
56C4⋊Dic53—22no—
57C4⋊Dic53—41no—
58#11719919—42no—
59C4×Dic53—41no—
60C23×C53#9911yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—42no—
64C22×D53—42no—
65C22×D53—84no—
66C2×#6157—84no—
67C2×#6157—164no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—42no—
72C2×C4×C53#9911yescharacteristic
73C2×C4×C53C2×C22⋊C411yescharacteristic
74C2×Dic53—42no—
75D212—42no—
76D212—42no—
77D212—82no—
78D212—84no—
79D212—82no—
80D212—82no—
81Dic106—82no—
82C8×C53—42no—
83#13595—531nomaximal, Sylow
84C22×C53—41no—
85C22×C53#35412yes—
86C22×C53C2×#9911yescharacteristic
87C2×D53—42no—
88C2×D53—42no—
89C2×D53—84no—
90C2×D53—82no—
91C2×D53—82no—
92C2×D53—162no—
93#6157—164no—
94C4×C53—22no—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—41no—
98Dic53—82no—
99#3138—531no—
100C2×#353—532no—
101#1412—534no—
102C2×C53—21no—
103C2×C53—22no—
104C2×C53C2×#35411yescharacteristic
105D53—82no—
106D53—82no—
107#533—531no—
108C2×#127—531no—
109#490—532no—
110#490—532no—
111#486—1062no—
112#418—1062no—
113C2×#99—1062no—
114C4×M16#615711yescharacteristic, complemented
115#353—534no—
116#353—1062no—
117C53#1359511yescharacteristic, complemented, Sylow
118C22×D4—532no—
119C22×D4—2121no—
120C2×M16—21no—
121C4⋊Q8—1061no—
122#127—532no—
123#127—531no—
124#127—531no—
125#127—2121no—
126#125—532no—
127C22⋊Q8—2121no—
128C4×Q8—2121no—
129#117—1061no—
130C2×C22⋊C4—2122no—
131C2×C42C2×#615711yescharacteristic
132#102—1064no—
133#102—1064no—
134#99—1064no—
135#99—2122no—
136#97C2×#615712yescomplemented
137C4×C8—21no—
138C24—1062no—
139C2×Q8—2121no—
140C2×D4—532no—
141C2×D4—532no—
142C2×D4—1062no—
143C2×D4—4244no—
144C2×D4—2122no—
145C2×D4—2121no—
146C2×D4—2121no—
147C2×D4—2122no—
148C2×D4—2122no—
149C22×C4—21no—
150C22×C4—4242no—
151C22×C4C22⋊#615711yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—1062no—
155C4⋊C4—2122no—
156C4⋊C4—2121no—
157C22⋊C4—4242no—
158C22⋊C4—2124no—
159C22⋊C4—2121no—
160C42—21no—
161C42—2121no—
162C42C22⋊#615711yescharacteristic
163C42C22×#615711yescharacteristic
164C23—1062no—
165C23—1062no—
166C23—4244no—
167C23—2122no—
168C23—2122no—
169C23#1313327611yescharacteristic
170Q8—4242no—
171D4—4244no—
172D4—4242no—
173D4—4242no—
174D4—4242no—
175D4—2122no—
176D4—2122no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—8484no—
182C2×C4—4244no—
183C2×C4—2122no—
184C2×C4#1313327611yescharacteristic
185C2×C4C2×C22⋊#615711yescharacteristic
186C8—42no—
187C22—41no—
188C22—8482no—
189C22—4242no—
190C22—4242no—
191C22—4244no—
192C22—2122no—
193C22—2122no—
194C22#1681153012yes—
195C22C2×#1313327611yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—8484no—
201C4—4242no—
202C2—21no—
203C2—22no—
204C2—4242no—
205C2—4242no—
206C2C2×#1681153011yescharacteristic
subgroupG/H
centerC2C2×#16811530
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC4×M16×C53C4
socleC2×C53C2×#354
prime radicalC2×#13133214C22
cosocle#13133214C23
hypercenterC4×M16#6157
hypocenterC53#13595
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #20289856C2semidirect
7. #16811180C4semidirect
9. #16810988C4semidirect
13. C4×M16×C53C4semidirect
24. #13133209C2×C4semidirect
32. #97×C53C2×C4semidirect
114. C4×M16#6157semidirect
117. C53#13595semidirect
136. #97C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115282
#202898562
#202850902
#1359553