Finite Group Databasebeta

← C53×#7760 #27215272

Subgroups — C53×#7760 #27215272

Order 13568.

424 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2809C211yesmaximal, characteristic
2C53×#2058C212yesmaximal
3C53×#2016C211yesmaximal, characteristic
4C53×#1272C211yesmaximal, characteristic
5C53×#1271C212yesmaximal
6C22×Q16×C53—21no—
7C2×C4⋊Q8×C53C2211yescharacteristic
8C2×#106×C53—22no—
9C2×#103×C53—21no—
10C53×#400—21no—
11C53×#398—22no—
12C53×#391—21no—
13C53×#391C2211yescharacteristic
14C53×#389C2212yes—
15C53×#387—22no—
16C53×#340—22no—
17C53×#336C2211yescharacteristic
18C53×#336C2212yes—
19C22×Q8×C53—21no—
20C22×Q8×C53D411yescharacteristic
21C2×Q16×C53—42no—
22C2×Q16×C53—42no—
23C2×Q16×C53—42no—
24C22×C8×C53—21no—
25C22×C8×C53—22no—
26C4⋊Q8×C53—42no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D411yescharacteristic
34C2×C42×C53—21no—
35C2×C42×C53C2311yescharacteristic
36#106×C53—44no—
37#103×C53—42no—
38#95×C53—21no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53SD1612yes—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—41no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60Q16×C53—48no—
61C2×C8×C53—22no—
62C2×C8×C53—24no—
63C2×C8×C53—42no—
64C2×C8×C53—41no—
65C2×C8×C53—41no—
66C2×C8×C53—42no—
67C4⋊C4×C53—24no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—81no—
76C4⋊C4×C53—81no—
77C42×C53—41no—
78C42×C53—81no—
79C23×C53#12011yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82Q8×C53—42no—
83Q8×C53—81no—
84Q8×C53—81no—
85Q8×C53—81no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—81no—
103C2×C4×C53—81no—
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53#13512yes—
106C2×C4×C53#13711yescharacteristic
107C2×C4×C53#13712yes—
108C8×C53—44no—
109C8×C53—42no—
110#7760C5311yesmaximal, characteristic, direct product factor, Sylow
111C22×C53#44811yescharacteristic
112C22×C53#45212yes—
113C22×C53#45311yescharacteristic
114C22×C53#45612yes—
115C22×C53B(4,𝔽2)11yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53—82no—
122C4×C53—82no—
123C4×C53—82no—
124C4×C53—81no—
125#2809C2×C5311yescharacteristic
126#2058C2×C5312yes—
127#2016C2×C5311yescharacteristic
128#1272C2×C5311yescharacteristic
129#1271C2×C5312yes—
130C2×C53#159611yescharacteristic
131C2×C53#160312yes—
132C2×C53#163411yescharacteristic
133C2×C53#165312yes—
134C2×C53#166011yescharacteristic
135C22×Q16—21no—
136C2×C4⋊Q8C22×C5311yescharacteristic
137C2×#106—22no—
138C2×#103—21no—
139#400—21no—
140#398—22no—
141#391—21no—
142#391C22×C5311yescharacteristic
143#389C22×C5312yes—
144#387—22no—
145#340—22no—
146#336C22×C5312yes—
147#336C22×C5311yescharacteristic
148C53#776011yescharacteristic, direct product factor, Sylow
149C22×Q8—21no—
150C22×Q8D4×C5311yescharacteristic
151C2×Q16—42no—
152C2×Q16—42no—
153C2×Q16—42no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4D4×C5312yes—
162C2×C4⋊C4D4×C5312yes—
163C2×C4⋊C4D4×C5311yescharacteristic
164C2×C42—21no—
165C2×C42C23×C5311yescharacteristic
166#106—44no—
167#103—42no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8SD16×C5312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4C2×D4×C5311yescharacteristic
189C22×C4C2×D4×C5312yes—
190Q16—48no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C42—41no—
208C42—81no—
209C23#120×C5311yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4C2×SD16×C5311yescharacteristic
235C2×C4#135×C5312yes—
236C2×C4#137×C5311yescharacteristic
237C2×C4#137×C5312yes—
238C8—44no—
239C8—42no—
240C22C53×#44811yescharacteristic
241C22C53×#45212yes—
242C22C53×#45311yescharacteristic
243C22C53×#45612yes—
244C22C53×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—81no—
251C4—82no—
252C4—82no—
253C4—82no—
254C2C53×#159611yescharacteristic
255C2C53×#160312yes—
256C2C53×#163411yescharacteristic
257C2C53×#165312yes—
258C2C53×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
110. #7760C53direct (N × H)
148. C53#7760direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28092
C53×#20582
C53×#20162
C53×#12722
C53×#12712
#776053