Finite Group Databasebeta

← #28298488

Subgroups — #28298488

Order 13568.

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

normal subgroups only

This group has 248 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#20290947C211yesmaximal, characteristic
2#20290471C211yesmaximal, characteristic
3#20276334C211yesmaximal, characteristic
4C2×#13133291C411yescharacteristic, complemented
5C22×#11719910C411yescharacteristic, complemented
6#16811609C411yescharacteristic
7#16811602C411yescharacteristic
8#16811586—21no—
9#16811518—21no—
10#16811518—21no—
11#16811505—21no—
12C2×#13133225C2211yescharacteristic
13M32×D53—21no—
14#16810405C411yescharacteristic, complemented
15#16810404—21no—
16C53×#349C411yescharacteristic, complemented
17D53⋊M16—21no—
18#13133296—21no—
19#13133291—41no—
20C2×#11719910—21no—
21C2×#11719910—21no—
22C2×#11719910—41no—
23#13133278C812yes—
24#13133277—21no—
25#13133276C812yes—
26#13133275—21no—
27#13133271—41no—
28C8×#6157—41no—
29C2×#11719945C2×C411yescharacteristic
30#13133225—41no—
31#13133225—41no—
32C2×#11719912D411yescharacteristic
33C2×#11719912Q811yescharacteristic
34C22×#88322C2×C411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37#13133157—21no—
38#13133156—41no—
39C16×D53—41no—
40C2×M16×C53C2×C411yescharacteristic
41M32×C53—21no—
42C22×C4×C53C4211yescharacteristic
43C2×C22⋊D53M1611yescharacteristic
44C2×C22⋊D53M1611yescharacteristic
45#11719945—41no—
46#11719945—41no—
47C2×D212C2×C811yescharacteristic
48C2×C4×D53C22⋊C411yescharacteristic
49C2×C4×D53C4⋊C411yescharacteristic
50C2×Dic106C2×C811yescharacteristic
51C22⋊#6157—22no—
52#11719940—22no—
53C4⋊#6157—41no—
54C4×#6157—41no—
55#11719936—41no—
56D53⋊C8—41no—
57M16×C53—41no—
58C2×C8×C53C22⋊C411yescharacteristic
59C2×C8×C53C4⋊C411yescharacteristic
60C2×#88322—21no—
61C2×#88322—21no—
62C2×#88322C22⋊C411yescharacteristic
63C2×#88322C4⋊C411yescharacteristic
64#11719912—41no—
65#11719912—41no—
66C8×D53—21no—
67C8×D53—21no—
68C8×D53—21no—
69C8×D53—21no—
70#11719910—41no—
71#11719910—41no—
72C16×C53—41no—
73#11719908—41no—
74C23×C53#9711yescharacteristic
75C22×D53C22⋊C811yescharacteristic
76C22×D53C4⋊C811yescharacteristic
77C2×#6157—42no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53#9511yescharacteristic
81C2×C4×C53C4×C811yescharacteristic
82C22⋊D53—41no—
83C22⋊D53—41no—
84C2×Dic53C22⋊C811yescharacteristic
85C2×Dic53C4⋊C811yescharacteristic
86D212—41no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—21no—
90C4×D53—21no—
91Dic106—41no—
92#88324—42no—
93C8×C53—21no—
94C8×C53—21no—
95#88322—21no—
96#88322#9911yescharacteristic
97#88322#9911yescharacteristic
98#7208—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53#33611yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—41no—
105C2×D53—41no—
106#6157—82no—
107C4×C53—21no—
108C4×C53#34211yescharacteristic
109C4×C53#34411yescharacteristic
110Dic53—21no—
111Dic53—21no—
112#2115—531no—
113#2113—531no—
114#1757—531no—
115C2×C53—21no—
116C2×C53—41no—
117C2×C53#128211yescharacteristic
118D53—41no—
119D53—41no—
120C2×#131—531no—
121C2×M32—531no—
122C2×M32—1061no—
123C22×C16—531no—
124#413—531no—
125#410—531no—
126#405—1061no—
127#349—531no—
128#349—1061no—
129#349#615711yescharacteristic, complemented
130C22⋊C16—1061no—
131C22⋊C16—1061no—
132#344—1061no—
133C53#720811yescharacteristic, complemented, Sylow
134C2×#41—531no—
135#131—2121no—
136#131—2121no—
137C2×M16—531no—
138C2×M16—531no—
139C2×M16—1061no—
140C2×M16C2×#615711yescharacteristic
141C22×C8—531no—
142C22×C8—531no—
143C22×C8—531no—
144#117—1061no—
145M32—21no—
146M32—1061no—
147M32—2121no—
148M32—2121no—
149C2×C16—1061no—
150C2×C16—1061no—
151C2×C16—2121no—
152C2×C16—2121no—
153#101—532no—
154#100—1061no—
155#99—532no—
156#99—1061no—
157#97—2121no—
158C4×C8—2121no—
159#41—2121no—
160#41—2121no—
161C2×Q8—531no—
162C2×D4—531no—
163C2×D4—531no—
164C2×D4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4C4×#615711yescharacteristic
168M16—41no—
169M16—1062no—
170M16—2121no—
171M16—2121no—
172M16—2121no—
173C2×C8—531no—
174C2×C8—531no—
175C2×C8—1061no—
176C2×C8—1061no—
177C2×C8—1061no—
178C2×C8—1061no—
179C2×C8—1061no—
180C2×C8—1061no—
181C2×C8—2121no—
182C2×C8C4⋊#615711yescharacteristic
183C2×C8C22⋊#615711yescharacteristic
184C4⋊C4—2121no—
185C22⋊C4—1062no—
186C42—2121no—
187C16—41no—
188C16—2121no—
189C16—2121no—
190C16—2121no—
191C23—531no—
192C23—531no—
193C23#1313327111yescharacteristic
194Q8—2121no—
195D4—2121no—
196D4—2121no—
197D4—2121no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—531no—
201C2×C4—531no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—2122no—
207C2×C4(C2×#6157)⋊C411yescharacteristic
208C2×C4C8×#615711yescharacteristic
209C8—21no—
210C8—21no—
211C8—531no—
212C8—531no—
213C8—1061no—
214C8—2122no—
215C22—21no—
216C22—41no—
217C22—1061no—
218C22—1061no—
219C22—2121no—
220C22—2121no—
221C22(C2×#6157)⋊C811yescharacteristic
222C4—21no—
223C4—1061no—
224C4—1061no—
225C4—4242no—
226C4#1681150011yescharacteristic
227C4#1681150611yescharacteristic
228C2—21no—
229C2—41no—
230C2—2121no—
231C2—2121no—
232C2#2028955611yescharacteristic
subgroupG/H
centerC4#16811500
commutatorC2×C4×C53C4×C8
FrattiniC2×M16C2×#6157
FittingC53×#349C4
socleC2×C53#1282
prime radicalC2×#11719945C2×C4
cosocleC2×#13133225C22
hypercenter#349#6157
hypocenterC53#7208
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
4. C2×#13133291C4semidirect
5. C22×#11719910C4semidirect
14. #16810405C4semidirect
16. C53×#349C4semidirect
129. #349#6157semidirect
133. C53#7208semidirect

Maximal subgroups

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

subgroupindex
#202909472
#202904712
#202763342
#720853