Finite Group Databasebeta

← C53×#13406 #27220918

Subgroups — C53×#13406 #27220918

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3067C211yesmaximal, characteristic
2C53×#2155C211yesmaximal, characteristic, complemented
3C53×#2143C211yesmaximal, characteristic, complemented
4C53×#2138C211yesmaximal, characteristic, complemented
5C53×#2113C211yesmaximal, characteristic, complemented
6C53×#1814C211yesmaximal, characteristic, complemented
7C53×#1381C211yesmaximal, characteristic, complemented
8C53×#575—21no—
9C2×#135×C53C2211yescharacteristic
10C2×#131×C53C2211yescharacteristic
11C2×M32×C53C2211yescharacteristic
12C22×C16×C53C2211yescharacteristic
13C53×#473—21no—
14C53×#472—21no—
15C53×#471—22no—
16C2×#108×C53C2211yescharacteristic
17C2×#108×C53C2211yescharacteristic
18C53×#427C2211yescharacteristic
19C53×#421—21no—
20C53×#413—21no—
21C53×#365—22no—
22C53×#359—22no—
23C53×#358—21no—
24C53×#357—21no—
25C53×C22⋊C16—22no—
26C53×#344—21no—
27C2×#41×C53—21no—
28C2×#41×C53D411yescharacteristic
29#137×C53—22no—
30#137×C53—41no—
31AGL(1,ℤ/8)×C53—22no—
32AGL(1,ℤ/8)×C53—41no—
33#135×C53—21no—
34#135×C53—21no—
35#135×C53—21no—
36#135×C53—21no—
37#135×C53—42no—
38C2×Q16×C53D411yescharacteristic
39C2×SD16×C53—21no—
40C2×D8×C53D411yescharacteristic
41#131×C53—21no—
42#131×C53—21no—
43#131×C53—21no—
44#131×C53—21no—
45C2×M16×C53—21no—
46C2×M16×C53—21no—
47C2×M16×C53—21no—
48C2×M16×C53D411yescharacteristic
49C22×C8×C53—21no—
50C22×C8×C53C2311yescharacteristic
51#117×C53—21no—
52M32×C53—21no—
53M32×C53—21no—
54C2×C16×C53—21no—
55C2×C16×C53—21no—
56C2×C16×C53—21no—
57C2×C16×C53D412yes—
58#108×C53—21no—
59#108×C53—21no—
60#108×C53—21no—
61#108×C53D412yes—
62#107×C53—21no—
63#106×C53—21no—
64#104×C53—42no—
65#101×C53—22no—
66#100×C53—22no—
67#41×C53—21no—
68#41×C53—21no—
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—41no—
72#41×C53—41no—
73#41×C53—41no—
74#41×C53—41no—
75C2×Q8×C53—21no—
76C2×Q8×C53SD1611yescharacteristic
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—41no—
80C2×D4×C53SD1611yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84Q16×C53—21no—
85Q16×C53—21no—
86Q16×C53—42no—
87SD16×C53—42no—
88SD16×C53—42no—
89SD16×C53—41no—
90SD16×C53—41no—
91D8×C53—21no—
92D8×C53—21no—
93D8×C53—42no—
94M16×C53—21no—
95M16×C53—22no—
96M16×C53—21no—
97M16×C53—41no—
98M16×C53—41no—
99M16×C53—41no—
100M16×C53—41no—
101M16×C53—41no—
102C2×C8×C53—21no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C4⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53—41no—
116C16×C53—22no—
117C16×C53—41no—
118C23×C53—21no—
119C23×C53—41no—
120C23×C53#13511yescharacteristic
121Q8×C53—21no—
122Q8×C53—21no—
123Q8×C53—41no—
124Q8×C53—41no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—41no—
128D4×C53—41no—
129D4×C53—41no—
130D4×C53—41no—
131D4×C53—81no—
132D4×C53—81no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12311yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C8×C53—41no—
147C8×C53—42no—
148C8×C53—81no—
149C8×C53#12111yescharacteristic
150C8×C53#12111yescharacteristic
151C8×C53#12311yescharacteristic
152C8×C53#12411yescharacteristic
153#13406C5311yesmaximal, characteristic, direct product factor, Sylow
154C22×C53—41no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—81no—
158C22×C53—161no—
159C22×C53#44911yescharacteristic
160C22×C53#46511yescharacteristic
161C22×C53#48211yescharacteristic
162C4×C53—41no—
163C4×C53—81no—
164C4×C53—161no—
165C4×C53#39411yescharacteristic
166C4×C53#45011yescharacteristic
167C4×C53#46911yescharacteristic
168C4×C53#48111yescharacteristic
169#3067C2×C5311yescharacteristic
170#2155C2×C5311yescharacteristic, complemented
171#2143C2×C5311yescharacteristic, complemented
172#2138C2×C5311yescharacteristic, complemented
173#2113C2×C5311yescharacteristic, complemented
174#1814C2×C5311yescharacteristic, complemented
175#1381C2×C5311yescharacteristic, complemented
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—81no—
180C2×C53—161no—
181C2×C53#203411yescharacteristic
182#575—21no—
183C2×#135C22×C5311yescharacteristic
184C2×#131C22×C5311yescharacteristic
185C2×M32C22×C5311yescharacteristic
186C22×C16C22×C5311yescharacteristic
187#473—21no—
188#472—21no—
189#471—22no—
190C2×#108C22×C5311yescharacteristic
191C2×#108C22×C5311yescharacteristic
192#427C22×C5311yescharacteristic
193#421—21no—
194#413—21no—
195#365—22no—
196#359—22no—
197#358—21no—
198#357—21no—
199C22⋊C16—22no—
200#344—21no—
201C53#1340611yescharacteristic, direct product factor, Sylow
202C2×#41—21no—
203C2×#41D4×C5311yescharacteristic
204#137—22no—
205#137—41no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—41no—
208#135—21no—
209#135—21no—
210#135—21no—
211#135—21no—
212#135—42no—
213C2×Q16D4×C5311yescharacteristic
214C2×SD16—21no—
215C2×D8D4×C5311yescharacteristic
216#131—21no—
217#131—21no—
218#131—21no—
219#131—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16D4×C5311yescharacteristic
224C22×C8—21no—
225C22×C8C23×C5311yescharacteristic
226#117—21no—
227M32—21no—
228M32—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16D4×C5312yes—
233#108—21no—
234#108—21no—
235#108—21no—
236#108D4×C5312yes—
237#107—21no—
238#106—21no—
239#104—42no—
240#101—22no—
241#100—22no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—21no—
246#41—41no—
247#41—41no—
248#41—41no—
249#41—41no—
250C2×Q8—21no—
251C2×Q8SD16×C5311yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4SD16×C5311yescharacteristic
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C2×D4×C5311yescharacteristic
259Q16—21no—
260Q16—21no—
261Q16—42no—
262SD16—42no—
263SD16—42no—
264SD16—41no—
265SD16—41no—
266D8—21no—
267D8—21no—
268D8—42no—
269M16—21no—
270M16—22no—
271M16—21no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—41no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8C2×D4×C5311yescharacteristic
283C2×C8C2×D4×C5311yescharacteristic
284C2×C8C2×D4×C5311yescharacteristic
285C2×C8#41×C5311yescharacteristic
286C2×C8#41×C5311yescharacteristic
287C2×C8#41×C5311yescharacteristic
288C4⋊C4—41no—
289C22⋊C4—41no—
290C42—41no—
291C16—22no—
292C16—41no—
293C23—21no—
294C23—41no—
295C23#135×C5311yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—41no—
299Q8—41no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—81no—
307D4—81no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4#120×C5311yescharacteristic
316C2×C4#121×C5311yescharacteristic
317C2×C4#123×C5311yescharacteristic
318C2×C4C2×SD16×C5311yescharacteristic
319C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
320C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
321C8—41no—
322C8—42no—
323C8—81no—
324C8#121×C5311yescharacteristic
325C8#121×C5311yescharacteristic
326C8#123×C5311yescharacteristic
327C8#124×C5311yescharacteristic
328C22—41no—
329C22—42no—
330C22—41no—
331C22—81no—
332C22—161no—
333C22C53×#44911yescharacteristic
334C22C53×#46511yescharacteristic
335C22C53×#48211yescharacteristic
336C4—41no—
337C4—81no—
338C4—161no—
339C4C53×#39411yescharacteristic
340C4C53×#45011yescharacteristic
341C4C53×#46911yescharacteristic
342C4C53×#48111yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—161no—
348C2C53×#203411yescharacteristic
subgroupG/H
centerC4×C53#469
commutator, Frattini, cosocleC22×C8C23×C53
socleC2×C53#2034
prime radicalC53×#3067C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2155C2semidirect
3. C53×#2143C2semidirect
4. C53×#2138C2semidirect
5. C53×#2113C2semidirect
6. C53×#1814C2semidirect
7. C53×#1381C2semidirect
153. #13406C53direct (N × H)
170. #2155C2×C53semidirect
171. #2143C2×C53semidirect
172. #2138C2×C53semidirect
173. #2113C2×C53semidirect
174. #1814C2×C53semidirect
175. #1381C2×C53semidirect
201. C53#13406direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30672
C53×#21552
C53×#21432
C53×#21382
C53×#21132
C53×#18142
C53×#13812
#1340653