Finite Group Databasebeta

← C53×#10333 #27217845

Subgroups — C53×#10333 #27217845

Order 13568.

914 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3195C211yesmaximal, characteristic, complemented
2C53×#2435C212yesmaximal
3C53×#1783C211yesmaximal, characteristic
4C53×#1508C212yesmaximal, complemented
5C53×#1297C211yesmaximal, characteristic, complemented
6C23×D4×C53C2211yescharacteristic
7C2×#124×C53—22no—
8C2×C22⋊Q8×C53C2212yes—
9C53×#483—22no—
10C53×#470—22no—
11C53×#451—22no—
12C2×#100×C53—22no—
13C53×#407C2211yescharacteristic
14C53×#407C2211yescharacteristic
15C53×#394—24no—
16C53×#390—22no—
17C53×#379—24no—
18C4×C22⋊C4×C53C2212yes—
19C53×#328—24no—
20C25×C53D412yes—
21C22×Q8×C53D412yes—
22C22×D4×C53—22no—
23C22×D4×C53—21no—
24C22×D4×C53—22no—
25C22×D4×C53—21no—
26C22×D4×C53—44no—
27C23×C4×C53C2311yescharacteristic
28C2×Q16×C53—22no—
29C2×M16×C53—21no—
30C2×M16×C53D412yes—
31#124×C53—48no—
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—22no—
34C22⋊Q8×C53—22no—
35C22⋊Q8×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D412yes—
38C2×C22⋊C4×C53—28no—
39C2×C22⋊C4×C53—28no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53—22no—
43#106×C53—22no—
44#103×C53—24no—
45#100×C53—44no—
46C22⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48#95×C53D414yes—
49C24×C53—24no—
50C24×C53—22no—
51C24×C53—44no—
52C24×C53—48no—
53C24×C53C2×D411yescharacteristic
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—44no—
57C2×D4×C53—24no—
58C2×D4×C53—28no—
59C2×D4×C53—44no—
60C2×D4×C53—42no—
61C2×D4×C53—48no—
62C2×D4×C53—44no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
66C22×C4×C53—24no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—42no—
70C22×C4×C53C2×D412yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53#4111yescharacteristic
74Q16×C53—82no—
75M16×C53—42no—
76M16×C53—42no—
77C2×C8×C53—22no—
78C2×C8×C53—41no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C22⋊C4×C53—44no—
84C22⋊C4×C53—416no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—88no—
88C42×C53—44no—
89C23×C53—24no—
90C23×C53—28no—
91C23×C53—24no—
92C23×C53—41no—
93C23×C53—44no—
94C23×C53—42no—
95C23×C53—48no—
96C23×C53—42no—
97C23×C53—48no—
98C23×C53—84no—
99C23×C53—816no—
100C23×C53#12012yes—
101C23×C53#12111yescharacteristic
102Q8×C53—44no—
103Q8×C53—82no—
104D4×C53—44no—
105D4×C53—48no—
106D4×C53—44no—
107C2×C4×C53—24no—
108C2×C4×C53—22no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—44no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—44no—
117C2×C4×C53—44no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12112yes—
120C2×C4×C53#12711yescharacteristic
121C8×C53—42no—
122C8×C53—81no—
123#10333C5311yesmaximal, characteristic, direct product factor, Sylow
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—21no—
128C22×C53—44no—
129C22×C53—416no—
130C22×C53—44no—
131C22×C53—88no—
132C22×C53—84no—
133C22×C53—84no—
134C22×C53—84no—
135C22×C53#39211yescharacteristic
136C22×C53#45514yes—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—84no—
140C4×C53—82no—
141C4×C53—82no—
142C4×C53B(4,𝔽2)12yes—
143#3195C2×C5311yescharacteristic, complemented
144#2435C2×C5312yes—
145#1783C2×C5311yescharacteristic
146#1508C2×C5312yescomplemented
147#1297C2×C5311yescharacteristic, complemented
148C2×C53—24no—
149C2×C53—41no—
150C2×C53—84no—
151C2×C53#200412yes—
152C2×C53#202111yescharacteristic
153C23×D4C22×C5311yescharacteristic
154C2×#124—22no—
155C2×C22⋊Q8C22×C5312yes—
156#483—22no—
157#470—22no—
158#451—22no—
159C2×#100—22no—
160#407C22×C5311yescharacteristic
161#407C22×C5311yescharacteristic
162#394—24no—
163#390—22no—
164#379—24no—
165C4×C22⋊C4C22×C5312yes—
166#328—24no—
167C53#1033311yescharacteristic, direct product factor, Sylow
168C25D4×C5312yes—
169C22×Q8D4×C5312yes—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—44no—
175C23×C4C23×C5311yescharacteristic
176C2×Q16—22no—
177C2×M16—21no—
178C2×M16D4×C5312yes—
179#124—48no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4D4×C5312yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—28no—
190C2×C42—22no—
191#106—22no—
192#103—24no—
193#100—44no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95D4×C5314yes—
197C24—24no—
198C24—22no—
199C24—44no—
200C24—48no—
201C24C2×D4×C5311yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—44no—
205C2×D4—24no—
206C2×D4—28no—
207C2×D4—44no—
208C2×D4—42no—
209C2×D4—48no—
210C2×D4—44no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4C2×D4×C5312yes—
219C22×C4C2×D4×C5311yescharacteristic
220C22×C4C2×D4×C5312yes—
221C22×C4#41×C5311yescharacteristic
222Q16—82no—
223M16—42no—
224M16—42no—
225C2×C8—22no—
226C2×C8—41no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C22⋊C4—416no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—88no—
236C42—44no—
237C23—24no—
238C23—28no—
239C23—24no—
240C23—41no—
241C23—44no—
242C23—42no—
243C23—48no—
244C23—42no—
245C23—48no—
246C23—84no—
247C23—816no—
248C23#120×C5312yes—
249C23#121×C5311yescharacteristic
250Q8—44no—
251Q8—82no—
252D4—44no—
253D4—48no—
254D4—44no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4#120×C5311yescharacteristic
267C2×C4#121×C5312yes—
268C2×C4#127×C5311yescharacteristic
269C8—42no—
270C8—81no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—44no—
276C22—416no—
277C22—44no—
278C22—88no—
279C22—84no—
280C22—84no—
281C22—84no—
282C22C53×#39211yescharacteristic
283C22C53×#45514yes—
284C4—21no—
285C4—22no—
286C4—84no—
287C4—82no—
288C4—82no—
289C4C53×B(4,𝔽2)12yes—
290C2—24no—
291C2—41no—
292C2—84no—
293C2C53×#200412yes—
294C2C53×#202111yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC23×D4×C53C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3195C2semidirect
4. C53×#1508C2semidirect
5. C53×#1297C2semidirect
123. #10333C53direct (N × H)
143. #3195C2×C53semidirect
146. #1508C2×C53semidirect
147. #1297C2×C53semidirect
167. C53#10333direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31952
C53×#24352
C53×#17832
C53×#15082
C53×#12972
#1033353