Finite Group Databasebeta

← #28298484

Subgroups — #28298484

Order 13568.

248 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 62 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#20290952C211yesmaximal, characteristic
2#20290470C211yesmaximal, characteristic
3#20276328C211yesmaximal, characteristic
4D53⋊M32—21no—
5#16811609C411yescharacteristic
6#16811602C411yescharacteristic
7#16811586—21no—
8#16811520—21no—
9#16811519C412yescomplemented
10#16811505—21no—
11C2×#13133223C2211yescharacteristic
12C2×#13133157C411yescharacteristic, complemented
13#16810402—21no—
14#16810402—21no—
15C22×C16×C53C411yescharacteristic, complemented
16D53⋊M16—21no—
17#13133296—21no—
18#13133292—41no—
19#13133291—22no—
20D53⋊C16—41no—
21#13133278—21no—
22#13133277C812yes—
23#13133276—21no—
24#13133275C812yes—
25#13133271—41no—
26C8×#6157—41no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—41no—
29#13133223—41no—
30C2×#11719917C2×C411yescharacteristic
31C2×#11719912D411yescharacteristic
32C2×#11719912Q811yescharacteristic
33C2×C8×D53D411yescharacteristic
34C2×C8×D53D411yescharacteristic
35#13133157—41no—
36C2×#11719908—21no—
37C22×C8×C53C2×C411yescharacteristic
38C2×C16×C53—21no—
39C2×C16×C53—41no—
40C22×C4×C53C4211yescharacteristic
41C2×C22⋊D53C2×C811yescharacteristic
42C2×C22⋊D53C2×C811yescharacteristic
43#11719945—41no—
44#11719945—41no—
45C2×D212M1611yescharacteristic
46C2×C4×D53C22⋊C411yescharacteristic
47C2×C4×D53C4⋊C411yescharacteristic
48C2×Dic106M1611yescharacteristic
49C22⋊#6157—22no—
50#11719940—22no—
51C4⋊#6157—41no—
52C4×#6157—41no—
53#11719936—41no—
54D53⋊C8—41no—
55C2×C8×C53—21no—
56C2×C8×C53—21no—
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C4⋊C411yescharacteristic
59#11719917—41no—
60C2×#88322C22⋊C411yescharacteristic
61C2×#88322C4⋊C411yescharacteristic
62#11719912—41no—
63#11719912—41no—
64C8×D53—21no—
65C8×D53—21no—
66C8×D53—21no—
67C8×D53—21no—
68#11719910—42no—
69C16×C53—41no—
70#11719908—41no—
71C23×C53C4×C811yescharacteristic
72C22×D53C22⋊C811yescharacteristic
73C22×D53C4⋊C811yescharacteristic
74C2×#6157—42no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53#9511yescharacteristic
78C2×C4×C53#9711yescharacteristic
79C22⋊D53—41no—
80C22⋊D53—41no—
81C2×Dic53C22⋊C811yescharacteristic
82C2×Dic53C4⋊C811yescharacteristic
83D212—41no—
84C4×D53—21no—
85C4×D53—21no—
86C4×D53—21no—
87C4×D53—21no—
88Dic106—41no—
89#88324—42no—
90C8×C53—21no—
91C8×C53#9911yescharacteristic
92C8×C53#9911yescharacteristic
93#88322—21no—
94#88322—21no—
95#7207—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—41no—
98C22×C53#33611yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—41no—
102C2×D53—41no—
103#6157—82no—
104C4×C53—21no—
105C4×C53#34211yescharacteristic
106C4×C53#34411yescharacteristic
107Dic53—21no—
108Dic53—21no—
109#2115—531no—
110#2113—531no—
111#1757—531no—
112C2×C53—21no—
113C2×C53—41no—
114C2×C53#128211yescharacteristic
115D53—41no—
116D53—41no—
117C2×#131—531no—
118C2×M32—531no—
119C2×M32—1061no—
120C22×C16#615711yescharacteristic, complemented
121#413—531no—
122#410—531no—
123#405—1061no—
124#349—532no—
125#349—1061no—
126C22⋊C16—1061no—
127C22⋊C16—1061no—
128#344—1061no—
129C53#720711yescharacteristic, complemented, Sylow
130C2×#41—531no—
131#131—2121no—
132#131—2121no—
133C2×M16—531no—
134C2×M16—531no—
135C2×M16—531no—
136C2×M16—1061no—
137C22×C8—531no—
138C22×C8—531no—
139C22×C8C2×#615711yescharacteristic
140#117—1061no—
141M32—1062no—
142M32—2121no—
143M32—2121no—
144C2×C16—21no—
145C2×C16—41no—
146C2×C16—1061no—
147C2×C16—2121no—
148#101—1061no—
149#100—532no—
150#99—532no—
151#99—1061no—
152#97—2121no—
153C4×C8—2121no—
154#41—2121no—
155#41—2121no—
156C2×Q8—531no—
157C2×D4—531no—
158C2×D4—531no—
159C2×D4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4C4×#615711yescharacteristic
163M16—1062no—
164M16—2121no—
165M16—2121no—
166M16—2121no—
167M16—2121no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—531no—
171C2×C8—531no—
172C2×C8—1061no—
173C2×C8—1061no—
174C2×C8—1061no—
175C2×C8—1061no—
176C2×C8—2121no—
177C2×C8C4⋊#615711yescharacteristic
178C2×C8C22⋊#615711yescharacteristic
179C4⋊C4—2121no—
180C22⋊C4—1062no—
181C42—2121no—
182C16—41no—
183C16—2122no—
184C16—2121no—
185C23—531no—
186C23—531no—
187C23C8×#615711yescharacteristic
188Q8—2121no—
189D4—2121no—
190D4—2121no—
191D4—2121no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—2122no—
201C2×C4(C2×#6157)⋊C411yescharacteristic
202C2×C4#1313327111yescharacteristic
203C8—21no—
204C8—1061no—
205C8—1061no—
206C8—2122no—
207C8#1313327511yescharacteristic
208C8#1313327611yescharacteristic
209C22—21no—
210C22—41no—
211C22—1061no—
212C22—1061no—
213C22—2121no—
214C22—2121no—
215C22(C2×#6157)⋊C811yescharacteristic
216C4—21no—
217C4—1061no—
218C4—1061no—
219C4—4242no—
220C4#1681150111yescharacteristic
221C4#1681150511yescharacteristic
222C2—21no—
223C2—41no—
224C2—2121no—
225C2—2121no—
226C2#2028955711yescharacteristic
subgroupG/H
centerC8#13133275
commutatorC23×C53C4×C8
FrattiniC22×C8C2×#6157
FittingC22×C16×C53C4
socleC2×C53#1282
prime radicalC2×#11719945C2×C4
cosocleC2×#13133223C22
hypercenterC22×C16#6157
hypocenterC53#7207
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ C22×C16

Complements

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

NH ≅ G/Ntype
9. #16811519C4semidirect
12. C2×#13133157C4semidirect
15. C22×C16×C53C4semidirect
120. C22×C16#6157semidirect
129. C53#7207semidirect

Maximal subgroups

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

subgroupindex
#202909522
#202904702
#202763282
#720753