Finite Group Databasebeta

← C53×#13463 #27220975

Subgroups — C53×#13463 #27220975

Order 13568.

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

normal subgroups only

This group has 484 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×#1867C212yesmaximal
3C53×#1856C211yesmaximal, characteristic
4C53×#1393C211yesmaximal, characteristic
5C53×#1390C212yesmaximal
6C53×#518C2211yescharacteristic
7C53×#428—21no—
8C2×#100×C53C2211yescharacteristic
9C2×#99×C53C2212yes—
10C53×#400—21no—
11C53×#393—22no—
12C53×#391—21no—
13C53×#390C2211yescharacteristic
14C53×#386C2212yes—
15C53×#385—22no—
16C53×#385—22no—
17C53×#342—22no—
18C53×#339—21no—
19C22×Q8×C53—21no—
20C22×D4×C53C2311yescharacteristic
21C23×C4×C53D412yes—
22C2×M16×C53—21no—
23C4×D4×C53—41no—
24C4×D4×C53—41no—
25#117×C53—21no—
26C2×C4⋊C4×C53—21no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—42no—
29C2×C4⋊C4×C53D411yescharacteristic
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53D412yes—
35C2×C42×C53D411yescharacteristic
36#107×C53—41no—
37#106×C53—41no—
38#100×C53—21no—
39#100×C53Q812yescomplemented
40#99×C53—22no—
41#99×C53Q814yescomplemented
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—21no—
45#95×C53—22no—
46#95×C53—42no—
47C24×C53C2×D412yes—
48C2×Q8×C53—42no—
49C2×Q8×C53—81no—
50C2×Q8×C53—81no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53C2×Q811yescharacteristic
54C2×D4×C53C2×Q812yes—
55C2×D4×C53#4111yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—82no—
65C22×C4×C53C2×D411yescharacteristic
66M16×C53—22no—
67M16×C53—41no—
68C2×C8×C53—41no—
69C4⋊C4×C53—21no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—81no—
73C4⋊C4×C53—81no—
74C4⋊C4×C53—84no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—82no—
82C42×C53—41no—
83C42×C53—41no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53#12011yescharacteristic
89C23×C53C22⋊Q812yes—
90C23×C53C4⋊Q812yes—
91Q8×C53—84no—
92D4×C53—41no—
93D4×C53—42no—
94D4×C53—41no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—44no—
106C2×C4×C53—44no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—82no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53—82no—
117C2×C4×C53C22⋊Q811yescharacteristic
118C2×C4×C53C4⋊Q811yescharacteristic
119C8×C53—42no—
120#13463C5311yesmaximal, characteristic, direct product factor, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—82no—
130C22×C53#39511yescharacteristic
131C22×C53B(4,𝔽2)11yescharacteristic
132C22×C53#45811yescharacteristic
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—42no—
136C4×C53—82no—
137C4×C53—84no—
138C4×C53—81no—
139C4×C53—81no—
140C4×C53—82no—
141C4×C53—81no—
142#2804C2×C5311yescharacteristic
143#1867C2×C5312yes—
144#1856C2×C5311yescharacteristic
145#1393C2×C5311yescharacteristic
146#1390C2×C5312yes—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53—42no—
150C2×C53—42no—
151C2×C53#206211yescharacteristic
152C2×C53#219811yescharacteristic
153C2×C53#219911yescharacteristic
154#518C22×C5311yescharacteristic
155#428—21no—
156C2×#100C22×C5311yescharacteristic
157C2×#99C22×C5312yes—
158#400—21no—
159#393—22no—
160#391—21no—
161#390C22×C5311yescharacteristic
162#386C22×C5312yes—
163#385—22no—
164#385—22no—
165#342—22no—
166#339—21no—
167C53#1346311yescharacteristic, direct product factor, Sylow
168C22×Q8—21no—
169C22×D4C23×C5311yescharacteristic
170C23×C4D4×C5312yes—
171C2×M16—21no—
172C4×D4—41no—
173C4×D4—41no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4D4×C5311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4D4×C5312yes—
184C2×C42D4×C5311yescharacteristic
185#107—41no—
186#106—41no—
187#100—21no—
188#100Q8×C5312yescomplemented
189#99—22no—
190#99Q8×C5314yescomplemented
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—42no—
196C24C2×D4×C5312yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C2×Q8×C5311yescharacteristic
203C2×D4C2×Q8×C5312yes—
204C2×D4#41×C5311yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—82no—
214C22×C4C2×D4×C5311yescharacteristic
215M16—22no—
216M16—41no—
217C2×C8—41no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—84no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—82no—
231C42—41no—
232C42—41no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#120×C5311yescharacteristic
238C23C22⋊Q8×C5312yes—
239C23C4⋊Q8×C5312yes—
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4C22⋊Q8×C5311yescharacteristic
267C2×C4C4⋊Q8×C5311yescharacteristic
268C8—42no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22C53×#39511yescharacteristic
279C22C53×B(4,𝔽2)11yescharacteristic
280C22C53×#45811yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—84no—
286C4—81no—
287C4—81no—
288C4—82no—
289C4—81no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2C53×#206211yescharacteristic
295C2C53×#219811yescharacteristic
296C2C53×#219911yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C53
prime radicalC22×D4×C53C23
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
39. #100×C53Q8semidirect
41. #99×C53Q8semidirect
120. #13463C53direct (N × H)
167. C53#13463direct (N × H)
188. #100Q8×C53semidirect
190. #99Q8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28042
C53×#18672
C53×#18562
C53×#13932
C53×#13902
#1346353