Finite Group Databasebeta

← C53×#7497 #27215009

Subgroups — C53×#7497 #27215009

Order 13568.

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

normal subgroups only

This group has 290 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×#2126C211yesmaximal, characteristic
3C53×#1962C211yesmaximal, characteristic
4C53×#532C2211yescharacteristic
5C53×#423C411yescharacteristic, complemented
6C2×#104×C53—21no—
7C2×#104×C53—21no—
8C53×#413—21no—
9C53×#410—21no—
10C43×C53C411yescharacteristic, complemented
11C53×#356C412yescomplemented
12C53×#355—21no—
13C53×#354—21no—
14C53×#353C412yescomplemented
15C2×#41×C53D411yescharacteristic
16C2×#41×C53D411yescharacteristic
17C2×M16×C53—21no—
18C2×M16×C53—41no—
19C4⋊Q8×C53C2×C411yescharacteristic
20#127×C53C2×C411yescharacteristic
21#124×C53D411yescharacteristic
22#124×C53Q811yescharacteristic
23#121×C53—41no—
24C4×D4×C53—41no—
25#117×C53—41no—
26C2×C42×C53—21no—
27C2×C42×C53—41no—
28C2×C42×C53C2×C411yescharacteristic
29C4⋊C8×C53—41no—
30#104×C53—42no—
31#104×C53—42no—
32#101×C53—22no—
33#100×C53—41no—
34#99×C53—22no—
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×C53C22⋊C411yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52M16×C53—42no—
53M16×C53—41no—
54M16×C53—41no—
55M16×C53—81no—
56C2×C8×C53—41no—
57C2×C8×C53—81no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—81no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C42×C53—21no—
64C42×C53—22no—
65C42×C53—41no—
66C42×C53—44no—
67C42×C53—81no—
68C42×C53C4211yescharacteristic
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—82no—
78D4×C53—81no—
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—42no—
88C2×C4×C53—44no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—82no—
94C2×C4×C53#9511yescharacteristic
95C2×C4×C53#9911yescharacteristic
96C8×C53—82no—
97C8×C53—81no—
98#7497C5311yesmaximal, characteristic, direct product factor, Sylow
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—81no—
104C22×C53—81no—
105C22×C53#34211yescharacteristic
106C4×C53—22no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—44no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—162no—
113C4×C53#35311yescharacteristic
114C4×C53#35411yescharacteristic
115#2132C2×C5311yescharacteristic
116#2126C2×C5311yescharacteristic
117#1962C2×C5311yescharacteristic
118C2×C53—21no—
119C2×C53—41no—
120C2×C53—81no—
121C2×C53—81no—
122C2×C53#139311yescharacteristic
123#532C22×C5311yescharacteristic
124#423C4×C5311yescharacteristic, complemented
125C2×#104—21no—
126C2×#104—21no—
127#413—21no—
128#410—21no—
129C43C4×C5311yescharacteristic, complemented
130#356C4×C5312yescomplemented
131#355—21no—
132#354—21no—
133#353C4×C5312yescomplemented
134C53#749711yescharacteristic, direct product factor, Sylow
135C2×#41D4×C5311yescharacteristic
136C2×#41D4×C5311yescharacteristic
137C2×M16—21no—
138C2×M16—41no—
139C4⋊Q8C2×C4×C5311yescharacteristic
140#127C2×C4×C5311yescharacteristic
141#124D4×C5311yescharacteristic
142#124Q8×C5311yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—41no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42C2×C4×C5311yescharacteristic
149C4⋊C8—41no—
150#104—42no—
151#104—42no—
152#101—22no—
153#100—41no—
154#99—22no—
155#99—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8C22⋊C4×C5311yescharacteristic
161C2×Q8C4⋊C4×C5311yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C22⋊C4×C5311yescharacteristic
166C2×D4C4⋊C4×C5311yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4C22⋊C4×C5311yescharacteristic
172M16—42no—
173M16—41no—
174M16—41no—
175M16—81no—
176C2×C8—41no—
177C2×C8—81no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—44no—
187C42—81no—
188C42C42×C5311yescharacteristic
189C42C4⋊C4×C5311yescharacteristic
190C23—21no—
191C23—21no—
192C23#99×C5311yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4#95×C5311yescharacteristic
215C2×C4#99×C5311yescharacteristic
216C8—82no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22C53×#34211yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—44no—
229C4—41no—
230C4—41no—
231C4—162no—
232C4C53×#35311yescharacteristic
233C4C53×#35411yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2C53×#139311yescharacteristic
subgroupG/H
centerC4×C53#353
commutatorC42C42×C53
Frattini, cosocle#532C22×C53
socleC2×C53#1393
prime radicalC53×#532C22
LCSG ▷ C42 ▷ 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
5. C53×#423C4semidirect
10. C43×C53C4semidirect
11. C53×#356C4semidirect
14. C53×#353C4semidirect
98. #7497C53direct (N × H)
124. #423C4×C53semidirect
129. C43C4×C53semidirect
130. #356C4×C53semidirect
133. #353C4×C53semidirect
134. C53#7497direct (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×#21262
C53×#19622
#749753