Finite Group Databasebeta

← C53×#7488 #27215000

Subgroups — C53×#7488 #27215000

Order 13568.

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C53×#2132C211yesmaximal, characteristic
2C53×#1884C211yesmaximal, characteristic
3C53×#1543C211yesmaximal, characteristic
4C53×#568C411yescharacteristic
5C53×#532C2211yescharacteristic
6C53×#431—21no—
7C53×#421—21no—
8C53×#413—21no—
9C53×#410C411yescharacteristic
10C53×#356—21no—
11C53×#355C412yescomplemented
12C53×#339—21no—
13C53×#333—21no—
14C53×#331C412yescomplemented
15C2×#41×C53C2×C411yescharacteristic
16C2×#41×C53D411yescharacteristic
17#131×C53—42no—
18C2×M16×C53—21no—
19C2×M16×C53—41no—
20C2×M16×C53—41no—
21C4⋊Q8×C53D411yescharacteristic
22#127×C53C2×C411yescharacteristic
23#124×C53C2×C411yescharacteristic
24#124×C53Q811yescharacteristic
25#121×C53—41no—
26C4×D4×C53—41no—
27#117×C53—21no—
28C2×C42×C53D411yescharacteristic
29C4⋊C8×C53—22no—
30#104×C53—42no—
31#101×C53—22no—
32#100×C53—41no—
33#99×C53—41no—
34C22⋊C8×C53—41no—
35C4×C8×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C22⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4211yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51M16×C53—22no—
52M16×C53—42no—
53M16×C53—41no—
54M16×C53—42no—
55M16×C53—81no—
56C2×C8×C53—41no—
57C2×C8×C53—42no—
58C2×C8×C53—42no—
59C2×C8×C53—81no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C22⋊C4×C53—41no—
64C22⋊C4×C53—41no—
65C42×C53—21no—
66C42×C53—41no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C4⋊C411yescharacteristic
69C23×C53—21no—
70C23×C53—21no—
71C23×C53#9911yescharacteristic
72Q8×C53—41no—
73Q8×C53—41no—
74D4×C53—41no—
75D4×C53—41no—
76D4×C53—42no—
77D4×C53—82no—
78D4×C53—81no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—81no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C8×C53—42no—
94C8×C53—82no—
95C8×C53—82no—
96#7488C5311yesmaximal, characteristic, direct product factor, Sylow
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53—81no—
103C22×C53#34211yescharacteristic
104C4×C53—22no—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—161no—
110C4×C53AΣL(1,𝔽16)11yescharacteristic
111C4×C53#35211yescharacteristic
112#2132C2×C5311yescharacteristic
113#1884C2×C5311yescharacteristic
114#1543C2×C5311yescharacteristic
115C2×C53—21no—
116C2×C53—41no—
117C2×C53—81no—
118C2×C53—81no—
119C2×C53#139011yescharacteristic
120#568C4×C5311yescharacteristic
121#532C22×C5311yescharacteristic
122#431—21no—
123#421—21no—
124#413—21no—
125#410C4×C5311yescharacteristic
126#356—21no—
127#355C4×C5312yescomplemented
128#339—21no—
129#333—21no—
130#331C4×C5312yescomplemented
131C53#748811yescharacteristic, direct product factor, Sylow
132C2×#41C2×C4×C5311yescharacteristic
133C2×#41D4×C5311yescharacteristic
134#131—42no—
135C2×M16—21no—
136C2×M16—41no—
137C2×M16—41no—
138C4⋊Q8D4×C5311yescharacteristic
139#127C2×C4×C5311yescharacteristic
140#124C2×C4×C5311yescharacteristic
141#124Q8×C5311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145C2×C42D4×C5311yescharacteristic
146C4⋊C8—22no—
147#104—42no—
148#101—22no—
149#100—41no—
150#99—41no—
151C22⋊C8—41no—
152C4×C8—41no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C22⋊C4×C5311yescharacteristic
158C2×Q8C4⋊C4×C5311yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4C42×C5311yescharacteristic
163C2×D4C4⋊C4×C5311yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22⋊C4×C5311yescharacteristic
168M16—22no—
169M16—42no—
170M16—41no—
171M16—42no—
172M16—81no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—81no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42C22⋊C4×C5311yescharacteristic
185C42C4⋊C4×C5311yescharacteristic
186C23—21no—
187C23—21no—
188C23#99×C5311yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4#95×C5311yescharacteristic
209C2×C4#99×C5311yescharacteristic
210C8—42no—
211C8—82no—
212C8—82no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22C53×#34211yescharacteristic
220C4—22no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4C53×AΣL(1,𝔽16)11yescharacteristic
227C4C53×#35211yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2C53×#139011yescharacteristic
subgroupG/H
centerC4×C53AΣL(1,𝔽16)
commutatorC2×D4C42×C53
Frattini, cosocle#532C22×C53
socleC2×C53#1390
prime radicalC53×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#532 ◁ G

Complements

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

NH ≅ G/Ntype
11. C53×#355C4semidirect
14. C53×#331C4semidirect
96. #7488C53direct (N × H)
127. #355C4×C53semidirect
130. #331C4×C53semidirect
131. C53#7488direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#21322
C53×#18842
C53×#15432
#748853