Finite Group Databasebeta

← C53×#7492 #27215004

Subgroups — C53×#7492 #27215004

Order 13568.

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

normal subgroups only

This group has 282 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×#1525C211yesmaximal, characteristic
4C53×#568C411yescharacteristic
5C53×#532C2211yescharacteristic
6C53×#421—21no—
7C53×#413—21no—
8C53×#410C411yescharacteristic
9C4×M16×C53—21no—
10C53×#356C412yescomplemented
11C53×#355—21no—
12C53×#339—21no—
13C53×#329—21no—
14C53×#329C412yescomplemented
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×C53C2×C411yescharacteristic
23#127×C53Q811yescharacteristic
24#124×C53C2×C411yescharacteristic
25#124×C53D411yescharacteristic
26#121×C53—41no—
27C4×D4×C53—41no—
28#117×C53—21no—
29C2×C42×C53D411yescharacteristic
30#104×C53—42no—
31#101×C53—22no—
32#100×C53—41no—
33#99×C53—41no—
34#97×C53—22no—
35C4×C8×C53—41no—
36#41×C53—41no—
37#41×C53—41no—
38#41×C53—41no—
39#41×C53—41no—
40C2×Q8×C53C4211yescharacteristic
41C2×Q8×C53C22⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53C4⋊C411yescharacteristic
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—44no—
53M16×C53—41no—
54M16×C53—42no—
55M16×C53—42no—
56M16×C53—81no—
57C2×C8×C53—42no—
58C2×C8×C53—41no—
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—82no—
79D4×C53—81no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—81no—
92C2×C4×C53#9511yescharacteristic
93C2×C4×C53#9911yescharacteristic
94C8×C53—44no—
95C8×C53—42no—
96C8×C53—82no—
97#7492C5311yesmaximal, characteristic, direct product factor, Sylow
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53#34211yescharacteristic
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—161no—
111C4×C53AΣL(1,𝔽16)11yescharacteristic
112C4×C53#35211yescharacteristic
113#2132C2×C5311yescharacteristic
114#1884C2×C5311yescharacteristic
115#1525C2×C5311yescharacteristic
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—81no—
119C2×C53—81no—
120C2×C53#139011yescharacteristic
121#568C4×C5311yescharacteristic
122#532C22×C5311yescharacteristic
123#421—21no—
124#413—21no—
125#410C4×C5311yescharacteristic
126C4×M16—21no—
127#356C4×C5312yescomplemented
128#355—21no—
129#339—21no—
130#329—21no—
131#329C4×C5312yescomplemented
132C53#749211yescharacteristic, direct product factor, Sylow
133C2×#41C2×C4×C5311yescharacteristic
134C2×#41D4×C5311yescharacteristic
135#131—42no—
136C2×M16—21no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C4⋊Q8C2×C4×C5311yescharacteristic
141#127Q8×C5311yescharacteristic
142#124C2×C4×C5311yescharacteristic
143#124D4×C5311yescharacteristic
144#121—41no—
145C4×D4—41no—
146#117—21no—
147C2×C42D4×C5311yescharacteristic
148#104—42no—
149#101—22no—
150#100—41no—
151#99—41no—
152#97—22no—
153C4×C8—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8C42×C5311yescharacteristic
159C2×Q8C22⋊C4×C5311yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C4⋊C4×C5311yescharacteristic
164C2×D4C4⋊C4×C5311yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C22⋊C4×C5311yescharacteristic
169M16—22no—
170M16—44no—
171M16—41no—
172M16—42no—
173M16—42no—
174M16—81no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—81no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42C22⋊C4×C5311yescharacteristic
187C42C4⋊C4×C5311yescharacteristic
188C23—21no—
189C23—21no—
190C23#99×C5311yescharacteristic
191Q8—41no—
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—42no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4#95×C5311yescharacteristic
211C2×C4#99×C5311yescharacteristic
212C8—44no—
213C8—42no—
214C8—82no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22C53×#34211yescharacteristic
222C4—22no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—161no—
228C4C53×AΣL(1,𝔽16)11yescharacteristic
229C4C53×#35211yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2C53×#139011yescharacteristic
subgroupG/H
centerC4×C53AΣL(1,𝔽16)
commutatorC2×Q8C42×C53
Frattini, cosocle#532C22×C53
socleC2×C53#1390
prime radicalC53×#532C22
LCSG ▷ C2×Q8 ▷ 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
10. C53×#356C4semidirect
14. C53×#329C4semidirect
97. #7492C53direct (N × H)
127. #356C4×C53semidirect
131. #329C4×C53semidirect
132. C53#7492direct (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×#15252
#749253