Finite Group Databasebeta

← #28298575

Subgroups — #28298575

Order 13568.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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
1D53⋊(C22⋊C16)C211yesmaximal, characteristic
2#20290591C211yesmaximal, characteristic
3#20274752C211yesmaximal, characteristic
4C2×D53⋊C16C412yescomplemented
5C2×#13133285—21no—
6#16811565—21no—
7#16811518—21no—
8C22⋊#11719910—21no—
9#16811497C412yes—
10C22×C8×D53C2211yescharacteristic
11C2×#13133152—21no—
12#16810586—21no—
13#16810388C411yescharacteristic, complemented
14C53×#341C411yescharacteristic, complemented
15C2×C4⋊#6157—22no—
16C2×C22⋊#6157—41no—
17D53⋊C16—42no—
18C2×#11719910—22no—
19#13133285—42no—
20(C2×#6157)⋊C4—41no—
21C22×C4×D53D411yescharacteristic
22C2×#11719917—21no—
23C22×#88322D411yescharacteristic
24C2×C8×D53—42no—
25C2×C8×D53C2×C411yescharacteristic
26C2×C8×D53C2×C411yescharacteristic
27C2×C8×D53D411yescharacteristic
28C2×C8×D53Q811yescharacteristic
29#13133152—42no—
30#13133129—41no—
31#13133129—41no—
32C2×M16×C53—21no—
33C22×C8×C53C2×C411yescharacteristic
34C23×D53SD1611yescharacteristic
35C22×#6157—42no—
36C22×C4×C53C22⋊C411yescharacteristic
37C22×Dic53Q1611yescharacteristic
38C2×C4×D53—41no—
39C2×C4×D53—41no—
40C2×C4×D53C22⋊C411yescharacteristic
41C2×C4×D53C4⋊C411yescharacteristic
42C2×C4×D53D811yescharacteristic
43C2×C4×D53SD1611yescharacteristic
44C22⋊#6157—82no—
45C4⋊#6157—42no—
46C4⋊#6157—42no—
47M16×C53—41no—
48M16×C53—41no—
49C2×C8×C53—41no—
50C2×C8×C53—41no—
51C2×C8×C53C4211yescharacteristic
52C2×C8×C53C4⋊C411yescharacteristic
53#11719917—41no—
54#11719917—41no—
55C2×#88322—22no—
56C2×#88322—41no—
57C2×#88322C22⋊C411yescharacteristic
58C2×#88322C4⋊C411yescharacteristic
59C8×D53—22no—
60C8×D53—22no—
61C8×D53—41no—
62C8×D53—41no—
63#11719910—42no—
64C23×C53#10311yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—41no—
68C22×D53#10211yescharacteristic
69C22×D53#10611yescharacteristic
70C2×#6157—82no—
71C2×#6157—82no—
72C2×#6157—82no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53#9511yescharacteristic
76C2×C4×C53#10211yescharacteristic
77C2×Dic53—41no—
78C2×Dic53#10311yescharacteristic
79C2×Dic53#10711yescharacteristic
80C4×D53—21no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—21no—
84C4×D53—41no—
85C4×D53—41no—
86C8×C53—21no—
87C8×C53—21no—
88C8×C53—81no—
89#88322—21no—
90#88322—81no—
91#88322#9912yes—
92#7232—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53#34011yescharacteristic
97C2×D53—21no—
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—44no—
102C2×D53#35912yes—
103C2×D53#36512yes—
104#6157—162no—
105C4×C53—21no—
106C4×C53H(3,ℤ/4)11yescharacteristic
107C4×C53#34211yescharacteristic
108Dic53—21no—
109Dic53—21no—
110C2×C22⋊C16—531no—
111#1810—531no—
112#1807—531no—
113C2×C53—41no—
114C2×C53#128911yescharacteristic
115C2×C53#138211yescharacteristic
116C2×C53#138311yescharacteristic
117D53—22no—
118D53—22no—
119C23×C8—531no—
120C22×C16—532no—
121C2×#108—1061no—
122C2×#106—1061no—
123#407—1061no—
124#380—1061no—
125C22⋊C16—1061no—
126C22⋊C16—1061no—
127#341—531no—
128#341#615711yescharacteristic, complemented
129#340—532no—
130C53#723211yescharacteristic, complemented, Sylow
131C23×C4—531no—
132C2×M16—21no—
133C2×M16—1061no—
134C22×C8—531no—
135C22×C8—531no—
136C22×C8—531no—
137C22×C8—531no—
138C22×C8—531no—
139C22×C8—2122no—
140C22×C8C2×#615711yescharacteristic
141C2×C4⋊C4—1062no—
142C2×C22⋊C4—2121no—
143C2×C16—1062no—
144C2×C16—2122no—
145#108—2122no—
146#106—2122no—
147C22⋊C8—2121no—
148C22⋊C8—2121no—
149#95—2121no—
150C24—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—2121no—
157C22×C4—2121no—
158C22×C4—2122no—
159C22×C4C22⋊#615711yescharacteristic
160M16—41no—
161M16—41no—
162M16—2121no—
163M16—2121no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—531no—
167C2×C8—531no—
168C2×C8—1062no—
169C2×C8—1062no—
170C2×C8—1062no—
171C2×C8—2121no—
172C2×C8—2121no—
173C2×C8—2121no—
174C2×C8C4×#615711yescharacteristic
175C2×C8C4⋊#615711yescharacteristic
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C22⋊C4—4242no—
179C16—2122no—
180C23—531no—
181C23—531no—
182C23—1062no—
183C23—1062no—
184C23—2121no—
185C23#1313328011yescharacteristic
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—4242no—
195C2×C4—4242no—
196C2×C4—4242no—
197C2×C4—2121no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4(C2×#6157)⋊C411yescharacteristic
201C2×C4#1313327911yescharacteristic
202C8—21no—
203C8—21no—
204C8—81no—
205C8—532no—
206C8—1061no—
207C8—4241no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—532no—
212C22—532no—
213C22—1061no—
214C22—1061no—
215C22—1061no—
216C22—1061no—
217C22—2124no—
218C22#1681149611yescharacteristic
219C4—21no—
220C4—8482no—
221C4—1061no—
222C4—1061no—
223C4#1681149011yescharacteristic
224C4#1681150011yescharacteristic
225C2—41no—
226C2—1062no—
227C2—1062no—
228C2#2028957711yescharacteristic
229C2#2028977811yescharacteristic
230C2#2028978111yescharacteristic
subgroupG/H
centerC22#16811496
commutatorC2×C8×C53C42
FrattiniC22×C8C2×#6157
FittingC53×#341C4
socleC22×C53#340
prime radicalC23×D53SD16
cosocleC22×C8×D53C22
hypercenter#341#6157
hypocenterC53#7232
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #341

Complements

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

NH ≅ G/Ntype
4. C2×D53⋊C16C4semidirect
13. #16810388C4semidirect
14. C53×#341C4semidirect
128. #341#6157semidirect
130. C53#7232semidirect

Maximal subgroups

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

subgroupindex
D53⋊(C22⋊C16)2
#202905912
#202747522
#723253