Finite Group Databasebeta

← C53×#7821 #27215333

Subgroups — C53×#7821 #27215333

Order 13568.

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

normal subgroups only

This group has 422 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×#2804C211yesmaximal, characteristic
2C53×#2093C211yesmaximal, characteristic
3C53×#2072C212yesmaximal, complemented
4C53×#1272C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C2×#107×C53—21no—
7C2×#103×C53—22no—
8C2×#102×C53—21no—
9C53×#400—21no—
10C53×#393—22no—
11C53×#391C2212yes—
12C53×#390C2211yescharacteristic
13C53×#386—21no—
14C53×#385—22no—
15C53×#384C2211yescharacteristic
16C53×#340—22no—
17C53×#336C2212yes—
18C53×#336C2211yescharacteristic
19C22×Q8×C53D412yes—
20C22×D4×C53D411yescharacteristic
21C23×C4×C53—21no—
22C22×C8×C53—21no—
23C22×C8×C53—22no—
24C2×C4⋊C4×C53—21no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53D412yes—
28C2×C4⋊C4×C53D411yescharacteristic
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C42×C53C2311yescharacteristic
33#107×C53—42no—
34#103×C53—44no—
35#102×C53—42no—
36#95×C53—22no—
37#95×C53—21no—
38#95×C53—22no—
39#95×C53—22no—
40C24×C53—21no—
41C2×Q8×C53—22no—
42C2×Q8×C53—22no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53Q1612yes—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—41no—
53C22×C4×C53—81no—
54C22×C4×C53—82no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D412yes—
57C2×C8×C53—22no—
58C2×C8×C53—41no—
59C2×C8×C53—41no—
60C2×C8×C53—42no—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—81no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C22⋊C4×C53—82no—
73C22⋊C4×C53—81no—
74C42×C53—41no—
75C23×C53—22no—
76C23×C53—42no—
77C23×C53—41no—
78C23×C53—42no—
79C23×C53—41no—
80C23×C53#12011yescharacteristic
81Q8×C53—44no—
82Q8×C53—82no—
83D4×C53—42no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—84no—
98C2×C4×C53—82no—
99C2×C4×C53—81no—
100C2×C4×C53C2×Q1611yescharacteristic
101C2×C4×C53#13512yes—
102C2×C4×C53AGL(1,ℤ/8)12yes—
103C2×C4×C53#13711yescharacteristic
104C8×C53—42no—
105C8×C53—82no—
106#7821C5311yesmaximal, characteristic, direct product factor, Sylow
107C22×C53—42no—
108C22×C53—44no—
109C22×C53—82no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53#44912yes—
113C22×C53#45111yescharacteristic
114C22×C53#45311yescharacteristic
115C22×C53#45512yes—
116C22×C53#45811yescharacteristic
117C4×C53—22no—
118C4×C53—22no—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—82no—
122C4×C53—81no—
123C4×C53—82no—
124C4×C53—161no—
125#2804C2×C5311yescharacteristic
126#2093C2×C5311yescharacteristic
127#2072C2×C5312yescomplemented
128#1272C2×C5312yescomplemented
129#1270C2×C5311yescharacteristic
130C2×C53—82no—
131C2×C53#161212yes—
132C2×C53#161411yescharacteristic
133C2×C53#162011yescharacteristic
134C2×C53#164412yes—
135C2×C53#165911yescharacteristic
136C2×#107—21no—
137C2×#103—22no—
138C2×#102—21no—
139#400—21no—
140#393—22no—
141#391C22×C5312yes—
142#390C22×C5311yescharacteristic
143#386—21no—
144#385—22no—
145#384C22×C5311yescharacteristic
146#340—22no—
147#336C22×C5312yes—
148#336C22×C5311yescharacteristic
149C53#782111yescharacteristic, direct product factor, Sylow
150C22×Q8D4×C5312yes—
151C22×D4D4×C5311yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C5312yes—
159C2×C4⋊C4D4×C5311yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C23×C5311yescharacteristic
164#107—42no—
165#103—44no—
166#102—42no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171C24—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Q16×C5312yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—81no—
185C22×C4—82no—
186C22×C4C2×D4×C5311yescharacteristic
187C22×C4C2×D4×C5312yes—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#120×C5311yescharacteristic
212Q8—44no—
213Q8—82no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4C2×Q16×C5311yescharacteristic
232C2×C4#135×C5312yes—
233C2×C4AGL(1,ℤ/8)×C5312yes—
234C2×C4#137×C5311yescharacteristic
235C8—42no—
236C8—82no—
237C22—42no—
238C22—44no—
239C22—81no—
240C22—82no—
241C22—81no—
242C22C53×#44912yes—
243C22C53×#45111yescharacteristic
244C22C53×#45311yescharacteristic
245C22C53×#45512yes—
246C22C53×#45811yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—82no—
253C4—81no—
254C4—161no—
255C2—82no—
256C2C53×#161212yes—
257C2C53×#161411yescharacteristic
258C2C53×#162011yescharacteristic
259C2C53×#164412yes—
260C2C53×#165911yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
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
3. C53×#2072C2semidirect
4. C53×#1272C2semidirect
106. #7821C53direct (N × H)
127. #2072C2×C53semidirect
128. #1272C2×C53semidirect
149. C53#7821direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28042
C53×#20932
C53×#20722
C53×#12722
C53×#12702
#782153