Finite Group Databasebeta

← C53×#9531 #27217043

Subgroups — C53×#9531 #27217043

Order 13568.

412 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3117C211yesmaximal, characteristic
2C53×#2967C211yesmaximal, characteristic
3C53×#1535C211yesmaximal, characteristic
4C53×#1491C211yesmaximal, characteristic, complemented
5C53×#1467C211yesmaximal, characteristic, complemented
6C53×#1308C211yesmaximal, characteristic, complemented
7C53×#1291C211yesmaximal, characteristic, complemented
8C53×#529C2211yescharacteristic
9C2×C4×Q8×C53C2211yescharacteristic
10C53×#484C411yescharacteristic
11C53×#481C411yescharacteristic
12C53×#479—21no—
13C53×#464—21no—
14C53×#451C411yescharacteristic
15C53×#448C411yescharacteristic
16C53×#446—21no—
17C53×#441—21no—
18C53×#440—21no—
19C53×#436—21no—
20C53×#435—21no—
21C53×#435—21no—
22C53×#433—21no—
23C53×#432C2211yescharacteristic
24C53×#432C2211yescharacteristic
25C53×#432C2211yescharacteristic
26C53×#432C2211yescharacteristic
27C53×#405—21no—
28C4×M16×C53C2211yescharacteristic
29C53×#330—21no—
30C53×#329—21no—
31C53×Q8⋊C8—21no—
32C53×Q8⋊C8—21no—
33C53×#324—21no—
34C53×#324—21no—
35C22×Q8×C53C2×C411yescharacteristic
36C2×Q16×C53—21no—
37C2×SD16×C53—21no—
38C2×M16×C53—21no—
39C2×M16×C53—21no—
40C22×C8×C53—21no—
41#126×C53—21no—
42#125×C53D411yescharacteristic
43#124×C53D411yescharacteristic
44#123×C53—21no—
45C22⋊Q8×C53C2×C411yescharacteristic
46#121×C53C2×C411yescharacteristic
47C4×Q8×C53—21no—
48C4×Q8×C53—21no—
49C4×Q8×C53—21no—
50C4×Q8×C53—21no—
51C4×Q8×C53D411yescharacteristic
52C4×D4×C53—21no—
53C4×D4×C53D411yescharacteristic
54#117×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59#107×C53—21no—
60#106×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67#103×C53—21no—
68#103×C53—21no—
69#103×C53—21no—
70#102×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53C2×C411yescharacteristic
76C22⋊C8×C53C2×C411yescharacteristic
77C22⋊C8×C53D411yescharacteristic
78C22⋊C8×C53D411yescharacteristic
79#97×C53—21no—
80#97×C53—21no—
81#97×C53—21no—
82#97×C53—21no—
83#97×C53—21no—
84#97×C53—21no—
85C4×C8×C53—21no—
86C4×C8×C53D411yescharacteristic
87C4×C8×C53D411yescharacteristic
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53—42no—
91C2×Q8×C53C22⋊C411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53C22⋊C411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100Q16×C53—42no—
101SD16×C53—42no—
102M16×C53—42no—
103M16×C53—42no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—42no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53C22⋊C411yescharacteristic
122C4⋊C4×C53C22⋊C411yescharacteristic
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C42×C53—21no—
129C42×C53—41no—
130C42×C53—41no—
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C23×C53—21no—
136C23×C53C4×D411yescharacteristic
137Q8×C53—22no—
138Q8×C53—22no—
139Q8×C53—41no—
140Q8×C53—41no—
141Q8×C53—41no—
142D4×C53—41no—
143D4×C53—81no—
144C2×C4×C53—22no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—81no—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12311yescharacteristic
163C8×C53—22no—
164C8×C53—41no—
165C8×C53—41no—
166C8×C53—41no—
167C8×C53—41no—
168C8×C53—41no—
169C8×C53—81no—
170C8×C53—81no—
171#9531C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53#38611yescharacteristic
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—42no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—81no—
190#3117C2×C5311yescharacteristic
191#2967C2×C5311yescharacteristic
192#1535C2×C5311yescharacteristic
193#1491C2×C5311yescharacteristic, complemented
194#1467C2×C5311yescharacteristic, complemented
195#1308C2×C5311yescharacteristic, complemented
196#1291C2×C5311yescharacteristic, complemented
197C2×C53—41no—
198C2×C53—81no—
199C2×C53#189111yescharacteristic
200C2×C53#190211yescharacteristic
201C2×C53#190611yescharacteristic
202#529C22×C5311yescharacteristic
203C2×C4×Q8C22×C5311yescharacteristic
204#484C4×C5311yescharacteristic
205#481C4×C5311yescharacteristic
206#479—21no—
207#464—21no—
208#451C4×C5311yescharacteristic
209#448C4×C5311yescharacteristic
210#446—21no—
211#441—21no—
212#440—21no—
213#436—21no—
214#435—21no—
215#435—21no—
216#433—21no—
217#432C22×C5311yescharacteristic
218#432C22×C5311yescharacteristic
219#432C22×C5311yescharacteristic
220#432C22×C5311yescharacteristic
221#405—21no—
222C4×M16C22×C5311yescharacteristic
223#330—21no—
224#329—21no—
225Q8⋊C8—21no—
226Q8⋊C8—21no—
227#324—21no—
228#324—21no—
229C53#953111yescharacteristic, direct product factor, Sylow
230C22×Q8C2×C4×C5311yescharacteristic
231C2×Q16—21no—
232C2×SD16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C22×C8—21no—
236#126—21no—
237#125D4×C5311yescharacteristic
238#124D4×C5311yescharacteristic
239#123—21no—
240C22⋊Q8C2×C4×C5311yescharacteristic
241#121C2×C4×C5311yescharacteristic
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8D4×C5311yescharacteristic
247C4×D4—21no—
248C4×D4D4×C5311yescharacteristic
249#117—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×C4×C5311yescharacteristic
252C2×C42—21no—
253C2×C42C23×C5311yescharacteristic
254#107—21no—
255#106—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262#103—21no—
263#103—21no—
264#103—21no—
265#102—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8C2×C4×C5311yescharacteristic
271C22⋊C8C2×C4×C5311yescharacteristic
272C22⋊C8D4×C5311yescharacteristic
273C22⋊C8D4×C5311yescharacteristic
274#97—21no—
275#97—21no—
276#97—21no—
277#97—21no—
278#97—21no—
279#97—21no—
280C4×C8—21no—
281C4×C8D4×C5311yescharacteristic
282C4×C8D4×C5311yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8C22⋊C4×C5311yescharacteristic
287C2×D4—21no—
288C2×D4C22⋊C4×C5311yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4C22×C4×C5311yescharacteristic
293C22×C4C2×D4×C5311yescharacteristic
294C22×C4#41×C5311yescharacteristic
295Q16—42no—
296SD16—42no—
297M16—42no—
298M16—42no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4C22⋊C4×C5311yescharacteristic
317C4⋊C4C22⋊C4×C5311yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—41no—
325C42—41no—
326C42C2×D4×C5311yescharacteristic
327C42C2×D4×C5311yescharacteristic
328C42C2×D4×C5311yescharacteristic
329C42#41×C5311yescharacteristic
330C23—21no—
331C23C4×D4×C5311yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337D4—41no—
338D4—81no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C5311yescharacteristic
353C2×C4C4×D4×C5311yescharacteristic
354C2×C4#120×C5311yescharacteristic
355C2×C4#121×C5311yescharacteristic
356C2×C4#121×C5311yescharacteristic
357C2×C4#123×C5311yescharacteristic
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—41no—
364C8—81no—
365C8—81no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C53×#38611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C2—41no—
385C2—81no—
386C2C53×#189111yescharacteristic
387C2C53×#190211yescharacteristic
388C2C53×#190611yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radical#121×C53C2×C4
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
4. C53×#1491C2semidirect
5. C53×#1467C2semidirect
6. C53×#1308C2semidirect
7. C53×#1291C2semidirect
171. #9531C53direct (N × H)
193. #1491C2×C53semidirect
194. #1467C2×C53semidirect
195. #1308C2×C53semidirect
196. #1291C2×C53semidirect
229. C53#9531direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31172
C53×#29672
C53×#15352
C53×#14912
C53×#14672
C53×#13082
C53×#12912
#953153