Finite Group Databasebeta

← C53×#9302 #27216814

Subgroups — C53×#9302 #27216814

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3227C211yesmaximal, characteristic
2C53×#2331C211yesmaximal, characteristic, complemented
3C53×#1963C211yesmaximal, characteristic
4C53×#1549C211yesmaximal, characteristic, complemented
5C53×#1517C211yesmaximal, characteristic
6C53×#1286C211yesmaximal, characteristic, complemented
7C53×#1274C211yesmaximal, characteristic, complemented
8C53×#532C2211yescharacteristic
9C2×C4⋊Q8×C53C2211yescharacteristic
10C53×#498C412yescomplemented
11C53×#486C412yescomplemented
12C53×#481—22no—
13C53×#476—22no—
14C53×#431C2211yescharacteristic
15C53×#431C2211yescharacteristic
16C53×#423C2211yescharacteristic
17C2×#104×C53—21no—
18C2×#104×C53—21no—
19C53×#391—22no—
20C53×#389—21no—
21C53×#387—22no—
22C53×#382C2211yescharacteristic
23C4×C4⋊C4×C53—21no—
24C53×#376C2211yescharacteristic
25C53×#332—22no—
26C53×#327—22no—
27C2×#41×C53—21no—
28C22×Q8×C53—21no—
29C2×M16×C53—21no—
30C4⋊Q8×C53—21no—
31C4⋊Q8×C53—21no—
32C4⋊Q8×C53C2×C411yescharacteristic
33C4⋊Q8×C53C2×C412yes—
34C4⋊Q8×C53D412yes—
35#127×C53C2×C411yescharacteristic
36#124×C53—21no—
37#121×C53—21no—
38#121×C53—21no—
39C4×D4×C53—21no—
40C4×D4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53D412yes—
45C2×C42×C53—22no—
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49#106×C53—24no—
50C4⋊C8×C53—21no—
51C4⋊C8×C53—21no—
52C4⋊C8×C53—21no—
53C4⋊C8×C53—21no—
54#104×C53—42no—
55#104×C53—42no—
56#102×C53—24no—
57C22⋊C8×C53—21no—
58C22⋊C8×C53—21no—
59C4×C8×C53—21no—
60C4×C8×C53C2×C412yes—
61#95×C53—22no—
62#95×C53—22no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67#41×C53—41no—
68#41×C53—41no—
69#41×C53—41no—
70#41×C53—41no—
71C2×Q8×C53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—21no—
74C2×Q8×C53—82no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—41no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87M16×C53—41no—
88M16×C53—41no—
89C2×C8×C53—22no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—82no—
100C4⋊C4×C53C22⋊C414yes—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C42×C53—21no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—82no—
107C42×C53SD1612yes—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112C23×C53—41no—
113C23×C53#12011yescharacteristic
114Q8×C53—41no—
115Q8×C53—41no—
116Q8×C53—82no—
117Q8×C53—82no—
118D4×C53—42no—
119D4×C53—41no—
120D4×C53—41no—
121D4×C53—81no—
122D4×C53—81no—
123D4×C53—81no—
124C2×C4×C53—24no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53#13711yescharacteristic
151C8×C53—42no—
152C8×C53—81no—
153C8×C53—81no—
154#9302C5311yesmaximal, characteristic, direct product factor, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#38611yescharacteristic
161C22×C53#44811yescharacteristic
162C22×C53#44911yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—44no—
169C4×C53—82no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53C4×SD1611yescharacteristic
173C4×C53#44011yescharacteristic
174C4×C53#46011yescharacteristic
175C4×C53#46411yescharacteristic
176#3227C2×C5311yescharacteristic
177#2331C2×C5311yescharacteristic, complemented
178#1963C2×C5311yescharacteristic
179#1549C2×C5311yescharacteristic, complemented
180#1517C2×C5311yescharacteristic
181#1286C2×C5311yescharacteristic, complemented
182#1274C2×C5311yescharacteristic, complemented
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—161no—
186C2×C53#188011yescharacteristic
187C2×C53#190411yescharacteristic
188C2×C53#191211yescharacteristic
189#532C22×C5311yescharacteristic
190C2×C4⋊Q8C22×C5311yescharacteristic
191#498C4×C5312yescomplemented
192#486C4×C5312yescomplemented
193#481—22no—
194#476—22no—
195#431C22×C5311yescharacteristic
196#431C22×C5311yescharacteristic
197#423C22×C5311yescharacteristic
198C2×#104—21no—
199C2×#104—21no—
200#391—22no—
201#389—21no—
202#387—22no—
203#382C22×C5311yescharacteristic
204C4×C4⋊C4—21no—
205#376C22×C5311yescharacteristic
206#332—22no—
207#327—22no—
208C53#930211yescharacteristic, direct product factor, Sylow
209C2×#41—21no—
210C22×Q8—21no—
211C2×M16—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C2×C4×C5311yescharacteristic
215C4⋊Q8C2×C4×C5312yes—
216C4⋊Q8D4×C5312yes—
217#127C2×C4×C5311yescharacteristic
218#124—21no—
219#121—21no—
220#121—21no—
221C4×D4—21no—
222C4×D4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4D4×C5312yes—
227C2×C42—22no—
228C2×C42D4×C5311yescharacteristic
229C2×C42D4×C5311yescharacteristic
230C2×C42C23×C5311yescharacteristic
231#106—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#104—42no—
237#104—42no—
238#102—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C4×C8—21no—
242C4×C8C2×C4×C5312yes—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95D4×C5312yes—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—82no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4C2×D4×C5311yescharacteristic
267C22×C4C2×D4×C5311yescharacteristic
268C22×C4C2×D4×C5311yescharacteristic
269M16—41no—
270M16—41no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—82no—
282C4⋊C4C22⋊C4×C5314yes—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42SD16×C5312yes—
290C42C22×C4×C5311yescharacteristic
291C42C2×D4×C5311yescharacteristic
292C42#41×C5311yescharacteristic
293C42#41×C5311yescharacteristic
294C23—41no—
295C23#120×C5311yescharacteristic
296Q8—41no—
297Q8—41no—
298Q8—82no—
299Q8—82no—
300D4—42no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—81no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊C4×C5311yescharacteristic
324C2×C4C4×D4×C5311yescharacteristic
325C2×C4C4×D4×C5311yescharacteristic
326C2×C4#121×C5311yescharacteristic
327C2×C4#121×C5311yescharacteristic
328C2×C4#123×C5311yescharacteristic
329C2×C4C2×SD16×C5311yescharacteristic
330C2×C4#135×C5311yescharacteristic
331C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
332C2×C4#137×C5311yescharacteristic
333C8—42no—
334C8—81no—
335C8—81no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C53×#38611yescharacteristic
342C22C53×#44811yescharacteristic
343C22C53×#44911yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4C4×SD16×C5311yescharacteristic
354C4C53×#44011yescharacteristic
355C4C53×#46011yescharacteristic
356C4C53×#46411yescharacteristic
357C2—21no—
358C2—21no—
359C2—161no—
360C2C53×#188011yescharacteristic
361C2C53×#190411yescharacteristic
362C2C53×#191211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C42 ▷ 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×#2331C2semidirect
4. C53×#1549C2semidirect
6. C53×#1286C2semidirect
7. C53×#1274C2semidirect
10. C53×#498C4semidirect
11. C53×#486C4semidirect
154. #9302C53direct (N × H)
177. #2331C2×C53semidirect
179. #1549C2×C53semidirect
181. #1286C2×C53semidirect
182. #1274C2×C53semidirect
191. #498C4×C53semidirect
192. #486C4×C53semidirect
208. C53#9302direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32272
C53×#23312
C53×#19632
C53×#15492
C53×#15172
C53×#12862
C53×#12742
#930253