Finite Group Databasebeta

← C53×#9529 #27217041

Subgroups — C53×#9529 #27217041

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3109C211yesmaximal, characteristic
2C53×#2981C211yesmaximal, characteristic, complemented
3C53×#1557C211yesmaximal, characteristic, complemented
4C53×#1486C211yesmaximal, characteristic, complemented
5C53×#1466C211yesmaximal, characteristic, complemented
6C53×#1307C211yesmaximal, characteristic, complemented
7C53×#1287C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C2×C4×D4×C53C2211yescharacteristic
10C53×#481C412yes—
11C53×#476—21no—
12C53×#459—21no—
13C53×#447C412yes—
14C53×#442—22no—
15C53×#436—21no—
16C53×#435—22no—
17C8×D4×C53—21no—
18C53×#432C2211yescharacteristic
19C53×#432C2211yescharacteristic
20C53×#432C2211yescharacteristic
21C53×#431C2211yescharacteristic
22C53×#423C2211yescharacteristic
23C53×#408—22no—
24C53×#332—21no—
25C53×#331—21no—
26C53×D4⋊C8—21no—
27C53×D4⋊C8—21no—
28C53×#323—22no—
29C2×#41×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C2×D8×C53—22no—
33C2×M16×C53—22no—
34C2×M16×C53—21no—
35C22×C8×C53—22no—
36C4⋊Q8×C53D411yescharacteristic
37#127×C53D411yescharacteristic
38#124×C53—21no—
39#121×C53—21no—
40#121×C53C2×C412yes—
41C4×D4×C53—21no—
42C4×D4×C53—21no—
43C4×D4×C53—21no—
44C4×D4×C53—21no—
45C4×D4×C53—21no—
46C4×D4×C53D412yes—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C42×C53C2311yescharacteristic
50#106×C53—22no—
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57#102×C53—22no—
58#102×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63C22⋊C8×C53C2×C412yes—
64C22⋊C8×C53D412yes—
65#97×C53—21no—
66#97×C53—21no—
67#97×C53—21no—
68C4×C8×C53D411yescharacteristic
69C4×C8×C53D411yescharacteristic
70C24×C53—21no—
71#41×C53—82no—
72C2×Q8×C53—21no—
73C2×D4×C53—22no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—42no—
79C2×D4×C53C22⋊C412yes—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—81no—
85C22×C4×C53—81no—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89D8×C53—44no—
90M16×C53—82no—
91M16×C53—81no—
92C2×C8×C53—22no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—82no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53C22⋊C412yes—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—81no—
116C23×C53—81no—
117C23×C53C4×D411yescharacteristic
118Q8×C53—81no—
119D4×C53—22no—
120D4×C53—22no—
121D4×C53—42no—
122D4×C53—41no—
123D4×C53—41no—
124D4×C53—41no—
125D4×C53—41no—
126D4×C53—81no—
127D4×C53—82no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—82no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C8×C53—41no—
147C8×C53—42no—
148C8×C53—41no—
149C8×C53—42no—
150C8×C53—81no—
151C8×C53—81no—
152#9529C5311yesmaximal, characteristic, direct product factor, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#38611yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—42no—
171C4×C53—81no—
172C4×C53#47111yescharacteristic
173C4×C53#47211yescharacteristic
174#3109C2×C5311yescharacteristic
175#2981C2×C5311yescharacteristic, complemented
176#1557C2×C5311yescharacteristic, complemented
177#1486C2×C5311yescharacteristic, complemented
178#1466C2×C5311yescharacteristic, complemented
179#1307C2×C5311yescharacteristic, complemented
180#1287C2×C5311yescharacteristic, complemented
181C2×C53—41no—
182C2×C53—82no—
183C2×C53—82no—
184C2×C53#189111yescharacteristic
185C2×C53#190411yescharacteristic
186C2×C53#190711yescharacteristic
187#532C22×C5311yescharacteristic
188C2×C4×D4C22×C5311yescharacteristic
189#481C4×C5312yes—
190#476—21no—
191#459—21no—
192#447C4×C5312yes—
193#442—22no—
194#436—21no—
195#435—22no—
196C8×D4—21no—
197#432C22×C5311yescharacteristic
198#432C22×C5311yescharacteristic
199#432C22×C5311yescharacteristic
200#431C22×C5311yescharacteristic
201#423C22×C5311yescharacteristic
202#408—22no—
203#332—21no—
204#331—21no—
205D4⋊C8—21no—
206D4⋊C8—21no—
207#323—22no—
208C53#952911yescharacteristic, direct product factor, Sylow
209C2×#41—21no—
210C22×D4C2×C4×C5311yescharacteristic
211C23×C4—21no—
212C2×D8—22no—
213C2×M16—22no—
214C2×M16—21no—
215C22×C8—22no—
216C4⋊Q8D4×C5311yescharacteristic
217#127D4×C5311yescharacteristic
218#124—21no—
219#121—21no—
220#121C2×C4×C5312yes—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4D4×C5312yes—
227C2×C4⋊C4C2×C4×C5311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C42C23×C5311yescharacteristic
230#106—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237#102—22no—
238#102—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8C2×C4×C5312yes—
244C22⋊C8D4×C5312yes—
245#97—21no—
246#97—21no—
247#97—21no—
248C4×C8D4×C5311yescharacteristic
249C4×C8D4×C5311yescharacteristic
250C24—21no—
251#41—82no—
252C2×Q8—21no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—42no—
259C2×D4C22⋊C4×C5312yes—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C22×C4×C5311yescharacteristic
267C22×C4C2×D4×C5311yescharacteristic
268C22×C4#41×C5311yescharacteristic
269D8—44no—
270M16—82no—
271M16—81no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—82no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4C22⋊C4×C5312yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42C2×D4×C5311yescharacteristic
288C42C2×D4×C5311yescharacteristic
289C42C2×D4×C5311yescharacteristic
290C42#41×C5311yescharacteristic
291C23—22no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—81no—
296C23—81no—
297C23C4×D4×C5311yescharacteristic
298Q8—81no—
299D4—22no—
300D4—22no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—81no—
307D4—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C2×C22⋊C4×C5311yescharacteristic
321C2×C4C4×D4×C5311yescharacteristic
322C2×C4#120×C5311yescharacteristic
323C2×C4#121×C5311yescharacteristic
324C2×C4#121×C5311yescharacteristic
325C2×C4#123×C5311yescharacteristic
326C8—41no—
327C8—42no—
328C8—41no—
329C8—42no—
330C8—81no—
331C8—81no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22—81no—
342C22—81no—
343C22C53×#38611yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4C53×#47111yescharacteristic
352C4C53×#47211yescharacteristic
353C2—41no—
354C2—82no—
355C2—82no—
356C2C53×#189111yescharacteristic
357C2C53×#190411yescharacteristic
358C2C53×#190711yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3109C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2981C2semidirect
3. C53×#1557C2semidirect
4. C53×#1486C2semidirect
5. C53×#1466C2semidirect
6. C53×#1307C2semidirect
7. C53×#1287C2semidirect
152. #9529C53direct (N × H)
175. #2981C2×C53semidirect
176. #1557C2×C53semidirect
177. #1486C2×C53semidirect
178. #1466C2×C53semidirect
179. #1307C2×C53semidirect
180. #1287C2×C53semidirect
208. C53#9529direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31092
C53×#29812
C53×#15572
C53×#14862
C53×#14662
C53×#13072
C53×#12872
#952953