Finite Group Databasebeta

← C53×#8770 #27216282

Subgroups — C53×#8770 #27216282

Order 13568.

426 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2357C211yesmaximal, characteristic
2C53×#1800C211yesmaximal, characteristic
3C53×#1460C211yesmaximal, characteristic
4C53×#1303C212yesmaximal, complemented
5C53×#1286C212yesmaximal, complemented
6C2×C4×D4×C53C2211yescharacteristic
7C53×#431C2212yes—
8C53×#423C2212yes—
9C2×#100×C53—21no—
10C53×#390—21no—
11C53×#388—22no—
12C53×#382C2211yescharacteristic
13C53×#381C412yes—
14C4×C4⋊C4×C53—21no—
15C4×C4⋊C4×C53C2211yescharacteristic
16C4×C22⋊C4×C53C412yes—
17C53×#343—21no—
18C53×#323—22no—
19C22×D4×C53D411yescharacteristic
20C23×C4×C53C2×C412yes—
21C2×M16×C53—22no—
22C4×D4×C53—22no—
23C4×D4×C53—22no—
24C2×C4⋊C4×C53—21no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53D411yescharacteristic
27C2×C22⋊C4×C53—22no—
28C2×C22⋊C4×C53—22no—
29C2×C22⋊C4×C53D412yes—
30C2×C42×C53—22no—
31C2×C42×C53—21no—
32C2×C42×C53C2311yescharacteristic
33C4⋊C8×C53—22no—
34C4⋊C8×C53—22no—
35C4⋊C8×C53—22no—
36#100×C53—42no—
37C22⋊C8×C53—22no—
38C4×C8×C53—22no—
39#95×C53—22no—
40#95×C53—21no—
41#95×C53C2×C411yescharacteristic
42#95×C53C2×C412yes—
43#95×C53C2×C411yescharacteristic
44C24×C53C22⋊C412yes—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—82no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59M16×C53—42no—
60M16×C53—42no—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C22⋊C4×C53—24no—
67C22⋊C4×C53—44no—
68C22⋊C4×C53—82no—
69C42×C53—41no—
70C42×C53—41no—
71C42×C53—82no—
72C42×C53#4112yes—
73C42×C53#4112yes—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80D4×C53—42no—
81D4×C53—42no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—82no—
96C2×C4×C53—82no—
97C2×C4×C53#11711yescharacteristic
98C2×C4×C53#11711yescharacteristic
99C2×C4×C53#12312yes—
100C2×C4×C53#12312yes—
101C2×C4×C53#13512yes—
102C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C53#13711yescharacteristic
104C8×C53—82no—
105C8×C53—82no—
106#8770C5311yesmaximal, characteristic, direct product factor, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—82no—
113C22×C53—82no—
114C22×C53#38111yescharacteristic
115C22×C53#41611yescharacteristic
116C22×C53#41711yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—81no—
123C4×C53—82no—
124C4×C53#48212yes—
125C4×C53#48512yes—
126#2357C2×C5311yescharacteristic
127#1800C2×C5311yescharacteristic
128#1460C2×C5311yescharacteristic
129#1303C2×C5312yescomplemented
130#1286C2×C5312yescomplemented
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#182511yescharacteristic
135C2×C53#182711yescharacteristic
136C2×C53#182911yescharacteristic
137C2×C4×D4C22×C5311yescharacteristic
138#431C22×C5312yes—
139#423C22×C5312yes—
140C2×#100—21no—
141#390—21no—
142#388—22no—
143#382C22×C5311yescharacteristic
144#381C4×C5312yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C22×C5311yescharacteristic
147C4×C22⋊C4C4×C5312yes—
148#343—21no—
149#323—22no—
150C53#877011yescharacteristic, direct product factor, Sylow
151C22×D4D4×C5311yescharacteristic
152C23×C4C2×C4×C5312yes—
153C2×M16—22no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C5311yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4D4×C5312yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C23×C5311yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168#100—42no—
169C22⋊C8—22no—
170C4×C8—22no—
171#95—21no—
172#95—22no—
173#95C2×C4×C5311yescharacteristic
174#95C2×C4×C5312yes—
175#95C2×C4×C5311yescharacteristic
176C24C22⋊C4×C5312yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊C4×C5312yes—
188C22×C4C22×C4×C5311yescharacteristic
189C22×C4C2×D4×C5311yescharacteristic
190C22×C4C2×D4×C5311yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—42no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C42—41no—
202C42—41no—
203C42—82no—
204C42#41×C5312yes—
205C42#41×C5312yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊C4×C5311yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#117×C5311yescharacteristic
230C2×C4#117×C5311yescharacteristic
231C2×C4#123×C5312yes—
232C2×C4#123×C5312yes—
233C2×C4#135×C5312yes—
234C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
235C2×C4#137×C5311yescharacteristic
236C8—82no—
237C8—82no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22C53×#38111yescharacteristic
246C22C53×#41611yescharacteristic
247C22C53×#41711yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4C53×#48212yes—
256C4C53×#48512yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2C53×#182511yescharacteristic
261C2C53×#182711yescharacteristic
262C2C53×#182911yescharacteristic
subgroupG/H
center, socleC22×C53#381
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1303C2semidirect
5. C53×#1286C2semidirect
106. #8770C53direct (N × H)
129. #1303C2×C53semidirect
130. #1286C2×C53semidirect
150. C53#8770direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#23572
C53×#18002
C53×#14602
C53×#13032
C53×#12862
#877053