Finite Group Databasebeta

← C53×#10359 #27217871

Subgroups — C53×#10359 #27217871

Order 13568.

400 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2844C211yesmaximal, characteristic
2C53×#1993C212yesmaximal
3C53×#1960C211yesmaximal, characteristic
4C53×#1295C211yesmaximal, characteristic
5C53×#1295C212yesmaximal
6C2×C4⋊Q8×C53C2211yescharacteristic
7C2×#103×C53—21no—
8C2×#103×C53—21no—
9C2×#97×C53C2211yescharacteristic
10C2×#97×C53C2212yes—
11C53×#400—22no—
12C53×#400—21no—
13C53×#395—22no—
14C53×#395—21no—
15C53×#382C2211yescharacteristic
16C53×#382C2212yes—
17C53×#340—22no—
18C53×#340—22no—
19C22×Q8×C53—21no—
20C22×Q8×C53—21no—
21C22×C8×C53—22no—
22C22×C8×C53—21no—
23C4⋊Q8×C53—21no—
24C4⋊Q8×C53—22no—
25C4⋊Q8×C53Q812yescomplemented
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53D412yes—
30C2×C4⋊C4×C53D412yes—
31C2×C4⋊C4×C53D411yescharacteristic
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C42×C53C2311yescharacteristic
34#103×C53—42no—
35#103×C53—42no—
36#97×C53—22no—
37#97×C53—22no—
38#97×C53—22no—
39#95×C53—22no—
40#95×C53—21no—
41#95×C53—21no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—22no—
45C2×Q8×C53—22no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—81no—
52C2×Q8×C53—82no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—41no—
58C22×C4×C53C2×D412yes—
59C22×C4×C53C2×D411yescharacteristic
60C2×C8×C53—22no—
61C2×C8×C53—24no—
62C2×C8×C53—41no—
63C2×C8×C53—42no—
64C2×C8×C53—41no—
65C2×C8×C53—42no—
66C4⋊C4×C53—24no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—81no—
73C4⋊C4×C53—82no—
74C42×C53C2×Q811yescharacteristic
75C42×C53#4111yescharacteristic
76C42×C53#4112yes—
77C23×C53#12011yescharacteristic
78Q8×C53—42no—
79Q8×C53—42no—
80Q8×C53—81no—
81Q8×C53—81no—
82Q8×C53—84no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—81no—
99C2×C4×C53—81no—
100C2×C4×C53C22⋊Q811yescharacteristic
101C2×C4×C53C22⋊Q812yes—
102C2×C4×C53#12411yescharacteristic
103C2×C4×C53#12412yes—
104C8×C53—42no—
105C8×C53—44no—
106#10359C5311yesmaximal, characteristic, direct product factor, Sylow
107C22×C53#39311yescharacteristic
108C22×C53#45311yescharacteristic
109C22×C53#45411yescharacteristic
110C22×C53#45512yes—
111C22×C53#45612yes—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—81no—
117C4×C53—82no—
118C4×C53—81no—
119C4×C53—82no—
120C4×C53—82no—
121#2844C2×C5311yescharacteristic
122#1993C2×C5312yes—
123#1960C2×C5311yescharacteristic
124#1295C2×C5311yescharacteristic
125#1295C2×C5312yes—
126C2×C53#165911yescharacteristic
127C2×C53#166011yescharacteristic
128C2×C53#166412yes—
129C2×C53#202711yescharacteristic
130C2×C53#202812yes—
131C2×C4⋊Q8C22×C5311yescharacteristic
132C2×#103—21no—
133C2×#103—21no—
134C2×#97C22×C5311yescharacteristic
135C2×#97C22×C5312yes—
136#400—22no—
137#400—21no—
138#395—22no—
139#395—21no—
140#382C22×C5311yescharacteristic
141#382C22×C5312yes—
142#340—22no—
143#340—22no—
144C53#1035911yescharacteristic, direct product factor, Sylow
145C22×Q8—21no—
146C22×Q8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8Q8×C5312yescomplemented
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4D4×C5312yes—
156C2×C4⋊C4D4×C5312yes—
157C2×C4⋊C4D4×C5311yescharacteristic
158C2×C4⋊C4D4×C5311yescharacteristic
159C2×C42C23×C5311yescharacteristic
160#103—42no—
161#103—42no—
162#97—22no—
163#97—22no—
164#97—22no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—82no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4C2×D4×C5312yes—
185C22×C4C2×D4×C5311yescharacteristic
186C2×C8—22no—
187C2×C8—24no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—82no—
200C42C2×Q8×C5311yescharacteristic
201C42#41×C5312yes—
202C42#41×C5311yescharacteristic
203C23#120×C5311yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—81no—
208Q8—84no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4C22⋊Q8×C5312yes—
227C2×C4C22⋊Q8×C5311yescharacteristic
228C2×C4#124×C5312yes—
229C2×C4#124×C5311yescharacteristic
230C8—42no—
231C8—44no—
232C22C53×#39311yescharacteristic
233C22C53×#45311yescharacteristic
234C22C53×#45411yescharacteristic
235C22C53×#45512yes—
236C22C53×#45612yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—82no—
246C2C53×#165911yescharacteristic
247C2C53×#166011yescharacteristic
248C2C53×#166412yes—
249C2C53×#202711yescharacteristic
250C2C53×#202812yes—
subgroupG/H
center, socle, prime radicalC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
25. C4⋊Q8×C53Q8semidirect
106. #10359C53direct (N × H)
144. C53#10359direct (N × H)
151. C4⋊Q8Q8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28442
C53×#19932
C53×#19602
C53×#12952
#1035953