Finite Group Databasebeta

← C53×#10031 #27217543

Subgroups — C53×#10031 #27217543

Order 13568.

468 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3249C211yesmaximal, characteristic
2C53×#3113C212yesmaximal
3C53×#1985C211yesmaximal, characteristic
4C53×#1531C212yesmaximal
5C53×#1290C211yesmaximal, characteristic
6C53×#533C2212yes—
7C2×C4×Q8×C53C2211yescharacteristic
8C53×#483—22no—
9C53×#477—22no—
10C53×#475—22no—
11C53×#467—22no—
12C53×#462—22no—
13C53×#462—22no—
14C53×#451—22no—
15C53×#441—24no—
16C53×#432C2212yes—
17C2×C4⋊C8×C53C2211yescharacteristic
18C2×#97×C53C2211yescharacteristic
19C53×#336—22no—
20C53×#330—24no—
21C22×Q8×C53D411yescharacteristic
22C2×Q16×C53—24no—
23C2×Q16×C53—22no—
24C22×C8×C53—21no—
25C22×C8×C53D412yes—
26C4⋊Q8×C53D412yes—
27C4⋊Q8×C53D412yes—
28#125×C53—22no—
29C22⋊Q8×C53—22no—
30C22⋊Q8×C53—22no—
31C4×Q8×C53—22no—
32C4×Q8×C53—22no—
33C4×Q8×C53—22no—
34C4×Q8×C53—22no—
35#117×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C42×C53—21no—
39C2×C42×C53C2311yescharacteristic
40#107×C53—22no—
41#106×C53—24no—
42C4⋊C8×C53—22no—
43C4⋊C8×C53—22no—
44#103×C53—24no—
45#103×C53—24no—
46#103×C53—24no—
47C22⋊C8×C53—22no—
48#97×C53—22no—
49#97×C53D414yes—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—42no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53#4111yescharacteristic
59Q16×C53—48no—
60Q16×C53—82no—
61C2×C8×C53—22no—
62C2×C8×C53—24no—
63C2×C8×C53—22no—
64C2×C8×C53—41no—
65C2×C8×C53—41no—
66C2×C8×C53—42no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C42×C53—41no—
77C42×C53—42no—
78C42×C53—41no—
79C42×C53C2×D412yes—
80C42×C53C2×D412yes—
81C23×C53#12111yescharacteristic
82Q8×C53—24no—
83Q8×C53—42no—
84Q8×C53—44no—
85Q8×C53—42no—
86Q8×C53—42no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53#12012yes—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12112yes—
101C2×C4×C53#12711yescharacteristic
102C8×C53—42no—
103C8×C53—44no—
104C8×C53—42no—
105#10031C5311yesmaximal, characteristic, direct product factor, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#39211yescharacteristic
109C22×C53#47312yes—
110C4×C53—21no—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—42no—
114C4×C53—82no—
115C4×C53—82no—
116C4×C53—81no—
117C4×C53#45511yescharacteristic
118C4×C53#45611yescharacteristic
119#3249C2×C5311yescharacteristic
120#3113C2×C5312yes—
121#1985C2×C5311yescharacteristic
122#1531C2×C5312yes—
123#1290C2×C5311yescharacteristic
124C2×C53—22no—
125C2×C53#200711yescharacteristic
126C2×C53#201011yescharacteristic
127C2×C53#202011yescharacteristic
128#533C22×C5312yes—
129C2×C4×Q8C22×C5311yescharacteristic
130#483—22no—
131#477—22no—
132#475—22no—
133#467—22no—
134#462—22no—
135#462—22no—
136#451—22no—
137#441—24no—
138#432C22×C5312yes—
139C2×C4⋊C8C22×C5311yescharacteristic
140C2×#97C22×C5311yescharacteristic
141#336—22no—
142#330—24no—
143C53#1003111yescharacteristic, direct product factor, Sylow
144C22×Q8D4×C5311yescharacteristic
145C2×Q16—24no—
146C2×Q16—22no—
147C22×C8—21no—
148C22×C8D4×C5312yes—
149C4⋊Q8D4×C5312yes—
150C4⋊Q8D4×C5312yes—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158#117—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4D4×C5311yescharacteristic
161C2×C42—21no—
162C2×C42C23×C5311yescharacteristic
163#107—22no—
164#106—24no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167#103—24no—
168#103—24no—
169#103—24no—
170C22⋊C8—22no—
171#97—22no—
172#97D4×C5314yes—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4C2×D4×C5311yescharacteristic
180C22×C4C2×D4×C5311yescharacteristic
181C22×C4#41×C5311yescharacteristic
182Q16—48no—
183Q16—82no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—42no—
199C42—41no—
200C42—42no—
201C42—41no—
202C42C2×D4×C5312yes—
203C42C2×D4×C5312yes—
204C23#121×C5311yescharacteristic
205Q8—24no—
206Q8—42no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4#120×C5312yes—
222C2×C4#120×C5311yescharacteristic
223C2×C4#121×C5312yes—
224C2×C4#127×C5311yescharacteristic
225C8—42no—
226C8—44no—
227C8—42no—
228C22—21no—
229C22—21no—
230C22C53×#39211yescharacteristic
231C22C53×#47312yes—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—82no—
237C4—82no—
238C4—81no—
239C4C53×#45511yescharacteristic
240C4C53×#45611yescharacteristic
241C2—22no—
242C2C53×#200711yescharacteristic
243C2C53×#201011yescharacteristic
244C2C53×#202011yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC23×C53#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
105. #10031C53direct (N × H)
143. C53#10031direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32492
C53×#31132
C53×#19852
C53×#15312
C53×#12902
#1003153