Finite Group Databasebeta

← #28298562

Subgroups — #28298562

Order 13568.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#20290946C211yesmaximal, characteristic
2#20290598C211yesmaximal, characteristic
3#20274764C211yesmaximal, characteristic
4C2×#13133292C411yescharacteristic, complemented
5C2×D53⋊C16C411yescharacteristic, complemented
6#16811633—21no—
7#16811602—21no—
8#16811518—21no—
9C22⋊#11719910—21no—
10#16811491C412yes—
11C2×#13133225C2211yescharacteristic
12C2×#13133152—21no—
13#16810605—21no—
14#16810381C411yescharacteristic, complemented
15C53×#338C411yescharacteristic, complemented
16#13133296—22no—
17#13133292—41no—
18D53⋊C16—41no—
19C2×#11719910—21no—
20C2×#11719910—21no—
21#13133286—41no—
22#13133285—41no—
23#13133276—41no—
24#13133275—41no—
25C2×#11719945D411yescharacteristic
26#13133225—41no—
27#13133225—41no—
28C2×#11719917—21no—
29C2×#11719912C2×C411yescharacteristic
30C2×#11719912Q811yescharacteristic
31C22×#88322D411yescharacteristic
32C2×C8×D53C2×C411yescharacteristic
33C2×C8×D53D411yescharacteristic
34#13133152—41no—
35#13133152—41no—
36#13133135—41no—
37#13133133—41no—
38C2×M16×C53—21no—
39C2×M16×C53C2×C411yescharacteristic
40C22×C4×C53C22⋊C411yescharacteristic
41C2×C22⋊D53SD1611yescharacteristic
42C2×C22⋊D53Q1611yescharacteristic
43#11719945—41no—
44#11719945—41no—
45C2×D212D811yescharacteristic
46C2×C4×D53C22⋊C411yescharacteristic
47C2×C4×D53C4⋊C411yescharacteristic
48C2×Dic106SD1611yescharacteristic
49C22⋊#6157—42no—
50C4⋊#6157—42no—
51C4×#6157—42no—
52M16×C53—41no—
53M16×C53—41no—
54M16×C53—41no—
55C2×C8×C53—41no—
56C2×C8×C53C4211yescharacteristic
57C2×C8×C53C4⋊C411yescharacteristic
58#11719917—41no—
59#11719917—41no—
60C2×#88322—21no—
61C2×#88322—21no—
62C2×#88322—41no—
63C2×#88322C22⋊C411yescharacteristic
64C2×#88322C4⋊C411yescharacteristic
65#11719912—41no—
66#11719912—41no—
67C8×D53—21no—
68C8×D53—21no—
69C8×D53—21no—
70C8×D53—21no—
71#11719910—41no—
72#11719910—41no—
73C23×C53#10311yescharacteristic
74C22×D53#10211yescharacteristic
75C22×D53#10711yescharacteristic
76C2×#6157—82no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53#9511yescharacteristic
80C2×C4×C53#10211yescharacteristic
81C22⋊D53—41no—
82C22⋊D53—41no—
83C2×Dic53#10311yescharacteristic
84C2×Dic53#10611yescharacteristic
85D212—41no—
86C4×D53—21no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—21no—
90Dic106—41no—
91C8×C53—21no—
92C8×C53—21no—
93C8×C53—81no—
94#88322—21no—
95#88322—81no—
96#88322#9911yescharacteristic
97#88322#9911yescharacteristic
98#7227—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53#34011yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—41no—
105C2×D53—41no—
106#6157—162no—
107C4×C53—21no—
108C4×C53H(3,ℤ/4)11yescharacteristic
109C4×C53#34211yescharacteristic
110Dic53—21no—
111Dic53—21no—
112#2113—531no—
113#1813—531no—
114#1812—531no—
115C2×C53—21no—
116C2×C53—41no—
117C2×C53#128911yescharacteristic
118D53—41no—
119D53—41no—
120C2×#131—531no—
121C2×M32—531no—
122C22×C16—531no—
123C2×#108—1061no—
124#427—1061no—
125#413—1061no—
126#410—1061no—
127C22⋊C16—1061no—
128C22⋊C16—1061no—
129#338—531no—
130#338#615711yescharacteristic, complemented
131H(3,ℤ/4)—532no—
132C53#722711yescharacteristic, complemented, Sylow
133C2×#41—531no—
134#131—2121no—
135#131—2121no—
136C2×M16—21no—
137C2×M16—531no—
138C2×M16—531no—
139C2×M16—1061no—
140C2×M16C2×#615711yescharacteristic
141C22×C8—531no—
142C22×C8—531no—
143C22×C8—531no—
144#117—1062no—
145M32—2121no—
146C2×C16—1061no—
147C2×C16—1061no—
148C2×C16—2121no—
149#108—2121no—
150#108—2121no—
151#107—2121no—
152#106—2121no—
153#101—2121no—
154#100—2121no—
155#99—2121no—
156#99—2121no—
157#41—2121no—
158#41—2121no—
159C2×Q8—531no—
160C2×D4—531no—
161C2×D4—531no—
162C2×D4—531no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4C22⋊#615711yescharacteristic
166M16—41no—
167M16—41no—
168M16—41no—
169M16—2121no—
170M16—2121no—
171M16—2121no—
172M16—2121no—
173C2×C8—41no—
174C2×C8—531no—
175C2×C8—531no—
176C2×C8—1061no—
177C2×C8—1061no—
178C2×C8—1061no—
179C2×C8—1061no—
180C2×C8—1061no—
181C2×C8—1061no—
182C2×C8—2121no—
183C2×C8C4×#615711yescharacteristic
184C2×C8C4⋊#615711yescharacteristic
185C4⋊C4—2122no—
186C22⋊C4—2122no—
187C42—2122no—
188C16—2121no—
189C16—2121no—
190C23—531no—
191C23—531no—
192C23#1313328011yescharacteristic
193Q8—2121no—
194D4—2121no—
195D4—2121no—
196D4—2121no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—531no—
200C2×C4—531no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—4242no—
206C2×C4(C2×#6157)⋊C411yescharacteristic
207C2×C4#1313327911yescharacteristic
208C8—21no—
209C8—21no—
210C8—81no—
211C8—531no—
212C8—531no—
213C8—1061no—
214C8—4241no—
215C22—21no—
216C22—41no—
217C22—1061no—
218C22—1061no—
219C22—2121no—
220C22—2121no—
221C22#1681149611yescharacteristic
222C4—21no—
223C4—8482no—
224C4—1061no—
225C4—1061no—
226C4#1681149011yescharacteristic
227C4#1681150011yescharacteristic
228C2—21no—
229C2—41no—
230C2—2121no—
231C2—2121no—
232C2#2028957711yescharacteristic
subgroupG/H
centerC4#16811490
commutatorC2×C8×C53C42
FrattiniC2×M16C2×#6157
FittingC53×#338C4
socleC2×C53#1289
prime radicalC2×#11719945D4
cosocleC2×#13133225C22
hypercenter#338#6157
hypocenterC53#7227
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ #338

Complements

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

NH ≅ G/Ntype
4. C2×#13133292C4semidirect
5. C2×D53⋊C16C4semidirect
14. #16810381C4semidirect
15. C53×#338C4semidirect
130. #338#6157semidirect
132. C53#7227semidirect

Maximal subgroups

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

subgroupindex
#202909462
#202905982
#202747642
#722753