Finite Group Databasebeta

← C53×#12760 #27220272

Subgroups — C53×#12760 #27220272

Order 13568.

876 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×C53×#559C211yesmaximal, characteristic
2C2×C53×#353C211yesmaximal, characteristic
3C2×C53×#352C212yesmaximal, complemented
4C53×#1890C212yesmaximal, complemented
5C53×#1887C211yesmaximal, characteristic
6C53×#559—22no—
7C53×#559—24no—
8C53×#559C414yescomplemented
9C2×#127×C53C2211yescharacteristic
10C2×#124×C53—22no—
11C2×#123×C53—21no—
12C2×#123×C53C2212yes—
13C53×#518C2211yescharacteristic
14C2×C4×Q8×C53—21no—
15C2×C4×D4×C53—22no—
16C53×#418—21no—
17C2×#100×C53C2211yescharacteristic
18C2×#99×C53C2212yes—
19C53×#381—22no—
20C53×#353—22no—
21C53×#352—24no—
22C53×#342—22no—
23C53×#339—21no—
24C22×Q8×C53—21no—
25C22×D4×C53—22no—
26C22×D4×C53D411yescharacteristic
27C22×D4×C53C2311yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53D412yes—
30C2×M16×C53—21no—
31#127×C53—22no—
32#127×C53C2×C412yes—
33#127×C53D412yes—
34#124×C53—28no—
35#124×C53—28no—
36#123×C53—24no—
37#123×C53—24no—
38#123×C53—22no—
39#123×C53—22no—
40#123×C53C2×C414yes—
41#123×C53D414yes—
42C4×Q8×C53—24no—
43C4×Q8×C53—24no—
44C4×D4×C53—28no—
45C4×D4×C53—28no—
46C4×D4×C53—42no—
47#117×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53D412yes—
56C2×C42×C53—22no—
57C2×C42×C53D411yescharacteristic
58#102×C53—42no—
59#100×C53—22no—
60#99×C53—22no—
61#99×C53—22no—
62#95×C53—44no—
63C24×C53—21no—
64C24×C53C2×D412yes—
65C2×Q8×C53—24no—
66C2×Q8×C53—44no—
67C2×D4×C53—24no—
68C2×D4×C53—24no—
69C2×D4×C53—21no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—48no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53C22⋊C412yes—
76C2×D4×C53C22×C411yescharacteristic
77C2×D4×C53C2×D411yescharacteristic
78C2×D4×C53C2×D412yes—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—24no—
83C22×C4×C53—22no—
84C22×C4×C53—24no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—81no—
89C22×C4×C53—81no—
90C22×C4×C53C22⋊C414yes—
91C22×C4×C53C2×D411yescharacteristic
92M16×C53—41no—
93M16×C53—41no—
94C2×C8×C53—41no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—22no—
99C22⋊C4×C53—28no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53C22⋊C414yes—
109C42×C53—28no—
110C42×C53—21no—
111C42×C53—41no—
112C42×C53C22⋊C412yes—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—44no—
117C23×C53—44no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53C2×C22⋊C412yes—
121C23×C53#12012yes—
122C23×C53#12011yescharacteristic
123Q8×C53—48no—
124D4×C53—41no—
125D4×C53—42no—
126D4×C53—41no—
127D4×C53—42no—
128D4×C53—48no—
129D4×C53—48no—
130D4×C53—81no—
131C2×C4×C53—24no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—44no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—44no—
141C2×C4×C53—42no—
142C2×C4×C53—82no—
143C2×C4×C53—84no—
144C2×C4×C53—84no—
145C2×C4×C53—82no—
146C2×C4×C53—84no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C8×C53—81no—
150#12760C5311yesmaximal, characteristic, direct product factor, Sylow
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—44no—
157C22×C53—44no—
158C22×C53—42no—
159C22×C53—82no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53#37911yescharacteristic
163C22×C53B(4,𝔽2)11yescharacteristic
164C22×C53#45811yescharacteristic
165C4×C53—24no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—44no—
169C4×C53—44no—
170C4×C53—82no—
171C4×C53—162no—
172C2×#559C2×C5311yescharacteristic
173C2×#353C2×C5311yescharacteristic
174C2×#352C2×C5312yescomplemented
175#1890C2×C5312yescomplemented
176#1887C2×C5311yescharacteristic
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—42no—
180C2×C53—42no—
181C2×C53—82no—
182C2×C53#179011yescharacteristic
183C2×C53#219812yes—
184#559—22no—
185#559—24no—
186#559C4×C5314yescomplemented
187C2×#127C22×C5311yescharacteristic
188C2×#124—22no—
189C2×#123—21no—
190C2×#123C22×C5312yes—
191#518C22×C5311yescharacteristic
192C2×C4×Q8—21no—
193C2×C4×D4—22no—
194#418—21no—
195C2×#100C22×C5311yescharacteristic
196C2×#99C22×C5312yes—
197#381—22no—
198#353—22no—
199#352—24no—
200#342—22no—
201#339—21no—
202C53#1276011yescharacteristic, direct product factor, Sylow
203C22×Q8—21no—
204C22×D4—22no—
205C22×D4D4×C5311yescharacteristic
206C22×D4C23×C5311yescharacteristic
207C23×C4—21no—
208C23×C4D4×C5312yes—
209C2×M16—21no—
210#127—22no—
211#127C2×C4×C5312yes—
212#127D4×C5312yes—
213#124—28no—
214#124—28no—
215#123—24no—
216#123—22no—
217#123—22no—
218#123—24no—
219#123C2×C4×C5314yes—
220#123D4×C5314yes—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—42no—
226#117—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4D4×C5312yes—
235C2×C42—22no—
236C2×C42D4×C5311yescharacteristic
237#102—42no—
238#100—22no—
239#99—22no—
240#99—22no—
241#95—44no—
242C24—21no—
243C24C2×D4×C5312yes—
244C2×Q8—24no—
245C2×Q8—44no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊C4×C5312yes—
255C2×D4C22×C4×C5311yescharacteristic
256C2×D4C2×D4×C5311yescharacteristic
257C2×D4C2×D4×C5312yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C22⋊C4×C5314yes—
270C22×C4C2×D4×C5311yescharacteristic
271M16—41no—
272M16—41no—
273C2×C8—41no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4C22⋊C4×C5314yes—
288C42—28no—
289C42—21no—
290C42—41no—
291C42C22⋊C4×C5312yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—44no—
296C23—44no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C5312yes—
300C23#120×C5311yescharacteristic
301C23#120×C5312yes—
302Q8—48no—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—81no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—84no—
326C2×C4C2×C22⋊C4×C5311yescharacteristic
327C2×C4#120×C5311yescharacteristic
328C8—81no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22C53×#37911yescharacteristic
341C22C53×B(4,𝔽2)11yescharacteristic
342C22C53×#45811yescharacteristic
343C4—24no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—82no—
349C4—162no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2C53×#179011yescharacteristic
356C2C53×#219812yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×#127×C53C22
LCSG ▷ C2×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
3. C2×C53×#352C2semidirect
4. C53×#1890C2semidirect
8. C53×#559C4semidirect
150. #12760C53direct (N × H)
174. C2×#352C2×C53semidirect
175. #1890C2×C53semidirect
186. #559C4×C53semidirect
202. C53#12760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5592
C2×C53×#3532
C2×C53×#3522
C53×#18902
C53×#18872
#1276053