Finite Group Databasebeta

← C53×#13360 #27220872

Subgroups — C53×#13360 #27220872

Order 13568.

702 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3021C212yesmaximal, complemented
3C53×#2122C212yesmaximal, complemented
4C53×#1794C211yesmaximal, characteristic, complemented
5C53×#1392C211yesmaximal, characteristic, complemented
6C53×#585C2211yescharacteristic
7C2×AGL(1,ℤ/8)×C53—22no—
8C53×#535C2212yes—
9C53×#482—22no—
10C53×#469—21no—
11C53×#466—21no—
12C53×#454—24no—
13C53×#453—24no—
14C53×#447—22no—
15C53×#421—22no—
16C53×#421—21no—
17C53×#416—21no—
18C53×#413C2212yes—
19C53×#410C2211yescharacteristic
20C53×#408C2211yescharacteristic
21C53×AΣL(1,𝔽16)—24no—
222−1+4×C53—41no—
232−1+4×C53D412yes—
24H(5,𝔽2)×C53—22no—
25H(5,𝔽2)×C53—21no—
26H(5,𝔽2)×C53—41no—
27C2×#41×C53—22no—
28C2×#41×C53—21no—
29C2×#41×C53—42no—
30C2×#41×C53C2311yescharacteristic
31C22×D4×C53D412yes—
32AGL(1,ℤ/8)×C53—28no—
33AGL(1,ℤ/8)×C53—44no—
34C2×SD16×C53—42no—
35C2×D8×C53—21no—
36C2×D8×C53—21no—
37C2×D8×C53—42no—
38C2×M16×C53—22no—
39C2×M16×C53D411yescharacteristic
40C22×C8×C53D411yescharacteristic
41#127×C53—22no—
42#124×C53—22no—
43#121×C53—22no—
44#121×C53—22no—
45#120×C53—22no—
46#120×C53—22no—
47#117×C53—21no—
48#117×C53D412yes—
49#107×C53—21no—
50#106×C53—21no—
51#104×C53—28no—
52#104×C53—42no—
53#103×C53—41no—
54#102×C53—22no—
55#102×C53—22no—
56#102×C53—41no—
57#101×C53—22no—
58#100×C53D414yes—
59#99×C53—22no—
60C22⋊C8×C53—22no—
61C24×C53—22no—
62#41×C53—22no—
63#41×C53—24no—
64#41×C53—24no—
65#41×C53—22no—
66#41×C53—41no—
67#41×C53—41no—
68#41×C53—42no—
69#41×C53—41no—
70#41×C53—42no—
71#41×C53—44no—
72#41×C53—42no—
73#41×C53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—42no—
77C2×Q8×C53C2×D411yescharacteristic
78C2×D4×C53—21no—
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—44no—
86C2×D4×C53—42no—
87C2×D4×C53—41no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53C2×D412yes—
91C2×D4×C53#4111yescharacteristic
92C22×C4×C53—42no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D411yescharacteristic
96SD16×C53—48no—
97D8×C53—48no—
98D8×C53—81no—
99D8×C53—81no—
100M16×C53—24no—
101M16×C53—42no—
102M16×C53—41no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—42no—
106C2×C8×C53—41no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—41no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C42×C53—22no—
114C42×C53—41no—
115C23×C53—41no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—82no—
119C23×C53—162no—
120C23×C53#12112yes—
121C23×C53#12711yescharacteristic
122Q8×C53—22no—
123Q8×C53—24no—
124Q8×C53—41no—
125Q8×C53—41no—
126D4×C53—22no—
127D4×C53—24no—
128D4×C53—42no—
129D4×C53—41no—
130D4×C53—44no—
131D4×C53—42no—
132D4×C53—44no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—162no—
138D4×C53—162no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—44no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—81no—
150C2×C4×C53#12012yes—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C8×C53—44no—
154C8×C53—41no—
155C8×C53—41no—
156#13360C5311yesmaximal, characteristic, direct product factor, Sylow
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—161no—
166C22×C53—162no—
167C22×C53—164no—
168C22×C53#39211yescharacteristic
169C4×C53—22no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—82no—
173C4×C53—161no—
174C4×C53B(4,𝔽2)11yescharacteristic
175C4×C53B(4,𝔽2)11yescharacteristic
176#3199C2×C5311yescharacteristic, complemented
177#3021C2×C5312yescomplemented
178#2122C2×C5312yescomplemented
179#1794C2×C5311yescharacteristic, complemented
180#1392C2×C5311yescharacteristic, complemented
181C2×C53—21no—
182C2×C53—41no—
183C2×C53—42no—
184C2×C53—82no—
185C2×C53—162no—
186C2×C53#202111yescharacteristic
187#585C22×C5311yescharacteristic
188C2×AGL(1,ℤ/8)—22no—
189#535C22×C5312yes—
190#482—22no—
191#469—21no—
192#466—21no—
193#454—24no—
194#453—24no—
195#447—22no—
196#421—22no—
197#421—21no—
198#416—21no—
199#413C22×C5312yes—
200#410C22×C5311yescharacteristic
201#408C22×C5311yescharacteristic
202AΣL(1,𝔽16)—24no—
203C53#1336011yescharacteristic, direct product factor, Sylow
2042−1+4—41no—
2052−1+4D4×C5312yes—
206H(5,𝔽2)—22no—
207H(5,𝔽2)—21no—
208H(5,𝔽2)—41no—
209C2×#41—22no—
210C2×#41—21no—
211C2×#41—42no—
212C2×#41C23×C5311yescharacteristic
213C22×D4D4×C5312yes—
214AGL(1,ℤ/8)—28no—
215AGL(1,ℤ/8)—44no—
216C2×SD16—42no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—42no—
220C2×M16—22no—
221C2×M16D4×C5311yescharacteristic
222C22×C8D4×C5311yescharacteristic
223#127—22no—
224#124—22no—
225#121—22no—
226#121—22no—
227#120—22no—
228#120—22no—
229#117—21no—
230#117D4×C5312yes—
231#107—21no—
232#106—21no—
233#104—28no—
234#104—42no—
235#103—41no—
236#102—22no—
237#102—22no—
238#102—41no—
239#101—22no—
240#100D4×C5314yes—
241#99—22no—
242C22⋊C8—22no—
243C24—22no—
244#41—22no—
245#41—24no—
246#41—24no—
247#41—22no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—42no—
252#41—42no—
253#41—44no—
254#41—42no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8C2×D4×C5311yescharacteristic
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4C2×D4×C5312yes—
273C2×D4#41×C5311yescharacteristic
274C22×C4—41no—
275C22×C4—42no—
276C22×C4C2×D4×C5312yes—
277C22×C4C2×D4×C5311yescharacteristic
278SD16—48no—
279D8—48no—
280D8—81no—
281D8—81no—
282M16—24no—
283M16—42no—
284M16—41no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—42no—
288C2×C8—41no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C22⋊C4—22no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C42—22no—
296C42—41no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—82no—
301C23—162no—
302C23#121×C5312yes—
303C23#127×C5311yescharacteristic
304Q8—24no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308D4—24no—
309D4—22no—
310D4—42no—
311D4—41no—
312D4—44no—
313D4—42no—
314D4—44no—
315D4—42no—
316D4—41no—
317D4—42no—
318D4—42no—
319D4—162no—
320D4—162no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4#120×C5312yes—
333C2×C4#120×C5311yescharacteristic
334C2×C4#121×C5311yescharacteristic
335C8—44no—
336C8—41no—
337C8—41no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—161no—
347C22—162no—
348C22—164no—
349C22C53×#39211yescharacteristic
350C4—22no—
351C4—21no—
352C4—42no—
353C4—82no—
354C4—161no—
355C4C53×B(4,𝔽2)11yescharacteristic
356C4C53×B(4,𝔽2)11yescharacteristic
357C2—21no—
358C2—41no—
359C2—42no—
360C2—82no—
361C2—162no—
362C2C53×#202111yescharacteristic
subgroupG/H
center, socleC2×C53#2021
commutator, Frattini, cosocleC2×#41C23×C53
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3199C2semidirect
2. C53×#3021C2semidirect
3. C53×#2122C2semidirect
4. C53×#1794C2semidirect
5. C53×#1392C2semidirect
156. #13360C53direct (N × H)
176. #3199C2×C53semidirect
177. #3021C2×C53semidirect
178. #2122C2×C53semidirect
179. #1794C2×C53semidirect
180. #1392C2×C53semidirect
203. C53#13360direct (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×#30212
C53×#21222
C53×#17942
C53×#13922
#1336053