Finite Group Databasebeta

← C53×#12108 #27219620

Subgroups — C53×#12108 #27219620

Order 13568.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3199C211yesmaximal, characteristic, complemented
2C53×#2184C212yesmaximal, complemented
3C53×#2018C211yesmaximal, characteristic, complemented
4C53×#1349C212yesmaximal, complemented
5C53×#1315C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C53—21no—
7C22×D8×C53C2211yescharacteristic, complemented
8C53×#535—21no—
9C2×D16×C53—22no—
10C53×#482—22no—
11C53×#469—21no—
12C53×#466C2212yes—
13C53×#447—22no—
14C2×#100×C53—21no—
15C53×#408C2211yescharacteristic
16C53×#357—22no—
17C53×C22⋊C16C2212yescomplemented
18C53×#341C2211yescharacteristic, complemented
19C2×#41×C53D411yescharacteristic
20C22×D4×C53—21no—
21C22×D4×C53—21no—
22AGL(1,ℤ/8)×C53—42no—
23AGL(1,ℤ/8)×C53—42no—
24C2×SD16×C53—41no—
25C2×D8×C53—21no—
26C2×D8×C53—22no—
27C2×D8×C53—42no—
28C2×D8×C53—41no—
29C2×D8×C53D412yes—
30C2×M16×C53—21no—
31C2×M16×C53D411yescharacteristic
32C22×C8×C53C2311yescharacteristic
33#127×C53—41no—
34#124×C53—41no—
35#121×C53—22no—
36#121×C53—22no—
37#120×C53—41no—
38#120×C53—41no—
39#117×C53—21no—
40D16×C53—44no—
41C2×C16×C53—22no—
42#107×C53D412yes—
43#106×C53—21no—
44#102×C53—22no—
45#102×C53—22no—
46#100×C53—42no—
47C22⋊C8×C53—22no—
48C24×C53—41no—
49C24×C53—41no—
50#41×C53—41no—
51#41×C53—41no—
52C2×Q8×C53D811yescharacteristic
53C2×D4×C53—22no—
54C2×D4×C53—21no—
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—42no—
61C2×D4×C53—42no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—81no—
65C2×D4×C53—81no—
66C2×D4×C53D811yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53C2×D411yescharacteristic
69SD16×C53—44no—
70D8×C53—24no—
71D8×C53—42no—
72D8×C53—42no—
73D8×C53—44no—
74D8×C53—81no—
75M16×C53—22no—
76M16×C53—41no—
77M16×C53—41no—
78C2×C8×C53—21no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53C2×D412yes—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C42×C53—41no—
87C16×C53—42no—
88C23×C53—21no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—81no—
92C23×C53—82no—
93C23×C53—81no—
94C23×C53—162no—
95C23×C53—162no—
96C23×C53AGL(1,ℤ/8)11yescharacteristic
97Q8×C53—41no—
98D4×C53—41no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—42no—
102D4×C53—42no—
103D4×C53—81no—
104D4×C53—81no—
105D4×C53—82no—
106D4×C53—81no—
107D4×C53—162no—
108D4×C53—162no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53C2×D811yescharacteristic
117C8×C53—22no—
118C8×C53—42no—
119C8×C53—41no—
120#12108C5311yesmaximal, characteristic, direct product factor, Sylow
121C22×C53—21no—
122C22×C53—22no—
123C22×C53—41no—
124C22×C53—42no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53—82no—
129C22×C53—162no—
130C22×C53—162no—
131C22×C53—164no—
132C22×C53—162no—
133C22×C53#44711yescharacteristic
134C4×C53—21no—
135C4×C53—41no—
136C4×C53—82no—
137C4×C53#45311yescharacteristic
138C4×C53B(4,𝔽2)11yescharacteristic
139#3199C2×C5311yescharacteristic, complemented
140#2184C2×C5312yescomplemented
141#2018C2×C5311yescharacteristic, complemented
142#1349C2×C5312yescomplemented
143#1315C2×C5311yescharacteristic, complemented
144C2×C53—41no—
145C2×C53—81no—
146C2×C53—162no—
147C2×C53—162no—
148C2×C53#159511yescharacteristic
149C2×C53#219212yes—
150C2×AGL(1,ℤ/8)—21no—
151C22×D8C22×C5311yescharacteristic, complemented
152#535—21no—
153C2×D16—22no—
154#482—22no—
155#469—21no—
156#466C22×C5312yes—
157#447—22no—
158C2×#100—21no—
159#408C22×C5311yescharacteristic
160#357—22no—
161C22⋊C16C22×C5312yescomplemented
162#341C22×C5311yescharacteristic, complemented
163C53#1210811yescharacteristic, direct product factor, Sylow
164C2×#41D4×C5311yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167AGL(1,ℤ/8)—42no—
168AGL(1,ℤ/8)—42no—
169C2×SD16—41no—
170C2×D8—21no—
171C2×D8—22no—
172C2×D8—42no—
173C2×D8—41no—
174C2×D8D4×C5312yes—
175C2×M16—21no—
176C2×M16D4×C5311yescharacteristic
177C22×C8C23×C5311yescharacteristic
178#127—41no—
179#124—41no—
180#121—22no—
181#121—22no—
182#120—41no—
183#120—41no—
184#117—21no—
185D16—44no—
186C2×C16—22no—
187#107D4×C5312yes—
188#106—21no—
189#102—22no—
190#102—22no—
191#100—42no—
192C22⋊C8—22no—
193C24—41no—
194C24—41no—
195#41—41no—
196#41—41no—
197C2×Q8D8×C5311yescharacteristic
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4D8×C5311yescharacteristic
212C22×C4—21no—
213C22×C4C2×D4×C5311yescharacteristic
214SD16—44no—
215D8—24no—
216D8—42no—
217D8—42no—
218D8—44no—
219D8—81no—
220M16—22no—
221M16—41no—
222M16—41no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8C2×D4×C5312yes—
227C4⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C42—41no—
232C16—42no—
233C23—21no—
234C23—42no—
235C23—42no—
236C23—81no—
237C23—82no—
238C23—81no—
239C23—162no—
240C23—162no—
241C23AGL(1,ℤ/8)×C5311yescharacteristic
242Q8—41no—
243D4—41no—
244D4—42no—
245D4—42no—
246D4—42no—
247D4—42no—
248D4—81no—
249D4—81no—
250D4—82no—
251D4—81no—
252D4—162no—
253D4—162no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4#120×C5311yescharacteristic
261C2×C4C2×D8×C5311yescharacteristic
262C8—22no—
263C8—42no—
264C8—41no—
265C22—22no—
266C22—21no—
267C22—42no—
268C22—41no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—162no—
274C22—162no—
275C22—164no—
276C22—162no—
277C22C53×#44711yescharacteristic
278C4—21no—
279C4—41no—
280C4—82no—
281C4C53×#45311yescharacteristic
282C4C53×B(4,𝔽2)11yescharacteristic
283C2—41no—
284C2—81no—
285C2—162no—
286C2—162no—
287C2C53×#159511yescharacteristic
288C2C53×#219212yes—
subgroupG/H
center, socleC22×C53#447
commutator, Frattini, cosocleC22×C8C23×C53
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C53×#408 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3199C2semidirect
2. C53×#2184C2semidirect
3. C53×#2018C2semidirect
4. C53×#1349C2semidirect
5. C53×#1315C2semidirect
7. C22×D8×C53C22semidirect
17. C53×C22⋊C16C22semidirect
18. C53×#341C22semidirect
120. #12108C53direct (N × H)
139. #3199C2×C53semidirect
140. #2184C2×C53semidirect
141. #2018C2×C53semidirect
142. #1349C2×C53semidirect
143. #1315C2×C53semidirect
151. C22×D8C22×C53semidirect
161. C22⋊C16C22×C53semidirect
162. #341C22×C53semidirect
163. C53#12108direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31992
C53×#21842
C53×#20182
C53×#13492
C53×#13152
#1210853