Finite Group Databasebeta

← C53×#10387 #27217899

Subgroups — C53×#10387 #27217899

Order 13568.

414 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2607C211yesmaximal, characteristic
2C53×#1826C212yesmaximal
3C53×#1820C211yesmaximal, characteristic
4C53×#1306C211yesmaximal, characteristic
5C53×#1284C212yesmaximal
6C2×#117×C53C2211yescharacteristic
7C53×#407C2212yes—
8C53×#399—21no—
9C53×#393—21no—
10C53×#391—22no—
11C53×#387—22no—
12C53×#381C2211yescharacteristic
13C53×#381C2211yescharacteristic
14C4×C22⋊C4×C53C2212yes—
15C53×#342—21no—
16C53×#342—21no—
17C53×#339—22no—
18C53×#339—22no—
19C22×Q8×C53—21no—
20C23×C4×C53C2311yescharacteristic
21C2×M16×C53—22no—
22#117×C53—22no—
23#117×C53—22no—
24C2×C4⋊C4×C53—22no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53Q812yescomplemented
27C2×C22⋊C4×C53—21no—
28C2×C22⋊C4×C53—22no—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53D411yescharacteristic
31C2×C22⋊C4×C53D411yescharacteristic
32C2×C42×C53—21no—
33C2×C42×C53D412yes—
34C2×C42×C53D412yes—
35C22⋊C8×C53—22no—
36C22⋊C8×C53—22no—
37#95×C53—21no—
38#95×C53—22no—
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53—22no—
43C24×C53C2×D411yescharacteristic
44C2×Q8×C53—44no—
45C2×Q8×C53—81no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×Q811yescharacteristic
56C22×C4×C53#4112yes—
57C22×C4×C53#4111yescharacteristic
58M16×C53—42no—
59M16×C53—42no—
60C2×C8×C53—42no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—44no—
63C4⋊C4×C53—81no—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—41no—
66C22⋊C4×C53—41no—
67C22⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—41no—
70C22⋊C4×C53—41no—
71C42×C53—42no—
72C42×C53—42no—
73C42×C53—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—42no—
77C23×C53#12011yescharacteristic
78C23×C53C22⋊Q811yescharacteristic
79C23×C53#12411yescharacteristic
80Q8×C53—84no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—24no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—82no—
98C2×C4×C53—82no—
99C2×C4×C53—81no—
100C2×C4×C53C22⋊Q812yes—
101C2×C4×C53#12412yes—
102C8×C53—82no—
103#10387C5311yesmaximal, characteristic, direct product factor, Sylow
104C22×C53—22no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53#39311yescharacteristic
112C22×C53#45811yescharacteristic
113C22×C53#45811yescharacteristic
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—44no—
117C4×C53—82no—
118C4×C53—81no—
119C4×C53—81no—
120C4×C53—82no—
121C4×C53—161no—
122#2607C2×C5311yescharacteristic
123#1826C2×C5312yes—
124#1820C2×C5311yescharacteristic
125#1306C2×C5311yescharacteristic
126#1284C2×C5312yes—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53—41no—
130C2×C53—41no—
131C2×C53#203011yescharacteristic
132C2×C53#203112yes—
133C2×#117C22×C5311yescharacteristic
134#407C22×C5312yes—
135#399—21no—
136#393—21no—
137#391—22no—
138#387—22no—
139#381C22×C5311yescharacteristic
140#381C22×C5311yescharacteristic
141C4×C22⋊C4C22×C5312yes—
142#342—21no—
143#342—21no—
144#339—22no—
145#339—22no—
146C53#1038711yescharacteristic, direct product factor, Sylow
147C22×Q8—21no—
148C23×C4C23×C5311yescharacteristic
149C2×M16—22no—
150#117—22no—
151#117—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4Q8×C5312yescomplemented
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4D4×C5311yescharacteristic
159C2×C22⋊C4D4×C5311yescharacteristic
160C2×C42—21no—
161C2×C42D4×C5312yes—
162C2×C42D4×C5312yes—
163C22⋊C8—22no—
164C22⋊C8—22no—
165#95—21no—
166#95—22no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C24C2×D4×C5311yescharacteristic
172C2×Q8—44no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4C2×D4×C5312yes—
183C22×C4C2×Q8×C5311yescharacteristic
184C22×C4#41×C5312yes—
185C22×C4#41×C5311yescharacteristic
186M16—42no—
187M16—42no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—81no—
192C22⋊C4—44no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—42no—
200C42—42no—
201C42—42no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#120×C5311yescharacteristic
206C23C22⋊Q8×C5311yescharacteristic
207C23#124×C5311yescharacteristic
208Q8—84no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4C22⋊Q8×C5312yes—
229C2×C4#124×C5312yes—
230C8—82no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22C53×#39311yescharacteristic
239C22C53×#45811yescharacteristic
240C22C53×#45811yescharacteristic
241C4—22no—
242C4—22no—
243C4—44no—
244C4—82no—
245C4—81no—
246C4—81no—
247C4—82no—
248C4—161no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2C53×#203011yescharacteristic
254C2C53×#203112yes—
subgroupG/H
center, socleC22×C53#393
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
26. C2×C4⋊C4×C53Q8semidirect
103. #10387C53direct (N × H)
146. C53#10387direct (N × H)
154. C2×C4⋊C4Q8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26072
C53×#18262
C53×#18202
C53×#13062
C53×#12842
#1038753