Finite Group Databasebeta

← C53×#7496 #27215008

Subgroups — C53×#7496 #27215008

Order 13568.

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

normal subgroups only

This group has 274 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×#1560C211yesmaximal, characteristic
4C53×#568C411yescharacteristic
5C53×#532C2211yescharacteristic
6C53×#423—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—
21C2×M16×C53—41no—
22C4⋊Q8×C53D411yescharacteristic
23#127×C53C2×C411yescharacteristic
24#124×C53C2×C411yescharacteristic
25#124×C53Q811yescharacteristic
26#121×C53—41no—
27C4×D4×C53—41no—
28#117×C53—21no—
29C2×C42×C53D411yescharacteristic
30C4⋊C8×C53—22no—
31C4⋊C8×C53—41no—
32#104×C53—42no—
33#101×C53—22no—
34#100×C53—41no—
35#99×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—41no—
53M16×C53—42no—
54M16×C53—42no—
55M16×C53—82no—
56M16×C53—81no—
57C2×C8×C53—41no—
58C2×C8×C53—42no—
59C2×C8×C53—42no—
60C2×C8×C53—81no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—41no—
64C22⋊C4×C53—41no—
65C22⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—41no—
68C42×C53C22⋊C411yescharacteristic
69C42×C53C4⋊C411yescharacteristic
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73Q8×C53—41no—
74Q8×C53—41no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—42no—
78D4×C53—81no—
79D4×C53—81no—
80D4×C53—81no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—81no—
94C2×C4×C53#9511yescharacteristic
95C2×C4×C53#9911yescharacteristic
96C8×C53—42no—
97C8×C53—82no—
98C8×C53—82no—
99#7496C5311yesmaximal, characteristic, direct product factor, Sylow
100C22×C53—21no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—81no—
105C22×C53—81no—
106C22×C53#34211yescharacteristic
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—161no—
114C4×C53AΣL(1,𝔽16)11yescharacteristic
115C4×C53#35211yescharacteristic
116#2132C2×C5311yescharacteristic
117#1884C2×C5311yescharacteristic
118#1560C2×C5311yescharacteristic
119C2×C53—21no—
120C2×C53—41no—
121C2×C53—81no—
122C2×C53—81no—
123C2×C53#139011yescharacteristic
124#568C4×C5311yescharacteristic
125#532C22×C5311yescharacteristic
126#423—21no—
127#421—21no—
128#413—21no—
129#410C4×C5311yescharacteristic
130#356—21no—
131#355C4×C5312yescomplemented
132#339—21no—
133#333—21no—
134#331C4×C5312yescomplemented
135C53#749611yescharacteristic, direct product factor, Sylow
136C2×#41C2×C4×C5311yescharacteristic
137C2×#41D4×C5311yescharacteristic
138#131—42no—
139C2×M16—21no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—41no—
143C4⋊Q8D4×C5311yescharacteristic
144#127C2×C4×C5311yescharacteristic
145#124C2×C4×C5311yescharacteristic
146#124Q8×C5311yescharacteristic
147#121—41no—
148C4×D4—41no—
149#117—21no—
150C2×C42D4×C5311yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—41no—
153#104—42no—
154#101—22no—
155#100—41no—
156#99—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161C2×Q8C22⋊C4×C5311yescharacteristic
162C2×Q8C4⋊C4×C5311yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4C42×C5311yescharacteristic
167C2×D4C4⋊C4×C5311yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4C22⋊C4×C5311yescharacteristic
172M16—22no—
173M16—41no—
174M16—42no—
175M16—42no—
176M16—82no—
177M16—81no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—81no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C42—21no—
188C42—41no—
189C42C22⋊C4×C5311yescharacteristic
190C42C4⋊C4×C5311yescharacteristic
191C23—21no—
192C23—21no—
193C23#99×C5311yescharacteristic
194Q8—41no—
195Q8—41no—
196D4—41no—
197D4—41no—
198D4—42no—
199D4—81no—
200D4—81no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4#95×C5311yescharacteristic
216C2×C4#99×C5311yescharacteristic
217C8—42no—
218C8—82no—
219C8—82no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22C53×#34211yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—161no—
234C4C53×AΣL(1,𝔽16)11yescharacteristic
235C4C53×#35211yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—81no—
240C2C53×#139011yescharacteristic
subgroupG/H
center, socleC2×C53#1390
commutatorC2×D4C42×C53
Frattini, cosocle#532C22×C53
prime radicalC53×#532C22
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×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
99. #7496C53direct (N × H)
131. #355C4×C53semidirect
134. #331C4×C53semidirect
135. C53#7496direct (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×#15602
#749653