Finite Group Databasebeta

← C53×#8371 #27215883

Subgroups — C53×#8371 #27215883

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#552C211yesmaximal, characteristic
2C4×#100×C53C211yesmaximal, characteristic
3C53×#1528C212yesmaximal, complemented
4C53×#1491C212yesmaximal, complemented
5C53×#1466C211yesmaximal, characteristic
6C53×#552—24no—
7C53×#552—24no—
8C2×#126×C53—21no—
9C2×#125×C53—21no—
10C2×#125×C53C2211yescharacteristic
11C2×#123×C53—21no—
12C2×C22⋊Q8×C53C411yescharacteristic
13C2×C22⋊Q8×C53C411yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C2×#117×C53—21no—
17C2×#117×C53C411yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C53×#432C2212yes—
20C2×#100×C53—21no—
21C4×M16×C53C2212yes—
22C53×#343—21no—
23C53×#330—24no—
24C53×Q8⋊C8—24no—
25C53×#323—21no—
26C53×#323—21no—
27C22×Q8×C53C2×C411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C2×M16×C53—22no—
33#126×C53—28no—
34#125×C53—28no—
35#125×C53—22no—
36#125×C53D414yes—
37#123×C53—28no—
38C22⋊Q8×C53—22no—
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C4×Q8×C53—22no—
43C4×Q8×C53D414yes—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46#117×C53—28no—
47#117×C53—22no—
48#117×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—22no—
70#100×C53—24no—
71C22⋊C8×C53—22no—
72#97×C53—22no—
73#97×C53—22no—
74C4×C8×C53—22no—
75C24×C53C22⋊C411yescharacteristic
76C24×C53C22⋊C411yescharacteristic
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×Q8×C53C22⋊C412yes—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—41no—
83C22×C4×C53—24no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101M16×C53—24no—
102M16×C53—42no—
103C2×C8×C53—42no—
104C2×C8×C53—42no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53C22⋊C412yes—
116C4⋊C4×C53C22⋊C412yes—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C42×C53—24no—
124C42×C53—24no—
125C42×C53C2×D412yes—
126C42×C53C2×D412yes—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53#10412yes—
137C23×C53C2×C22⋊C411yescharacteristic
138Q8×C53—24no—
139Q8×C53—42no—
140D4×C53—44no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—48no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53#12012yes—
167C2×C4×C53#12012yes—
168C8×C53—44no—
169C8×C53—82no—
170#8371C5311yesmaximal, characteristic, direct product factor, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#37911yescharacteristic
184C22×C53C2×#10411yescharacteristic
185C22×C53#42111yescharacteristic
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—22no—
189C4×C53—42no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—42no—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53#41312yes—
196C2×#552C2×C5311yescharacteristic
197C4×#100C2×C5311yescharacteristic
198#1528C2×C5312yescomplemented
199#1491C2×C5312yescomplemented
200#1466C2×C5311yescharacteristic
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—42no—
204C2×C53—81no—
205C2×C53#178511yescharacteristic
206C2×C53#178911yescharacteristic
207C2×C53#179611yescharacteristic
208#552—24no—
209#552—24no—
210C2×#126—21no—
211C2×#125—21no—
212C2×#125C22×C5311yescharacteristic
213C2×#123—21no—
214C2×C22⋊Q8C4×C5311yescharacteristic
215C2×C22⋊Q8C4×C5311yescharacteristic
216C2×C4×Q8C22×C5311yescharacteristic
217C2×C4×D4C22×C5311yescharacteristic
218C2×#117—21no—
219C2×#117C4×C5311yescharacteristic
220C22×C4⋊C4C4×C5311yescharacteristic
221#432C22×C5312yes—
222C2×#100—21no—
223C4×M16C22×C5312yes—
224#343—21no—
225#330—24no—
226Q8⋊C8—24no—
227#323—21no—
228#323—21no—
229C53#837111yescharacteristic, direct product factor, Sylow
230C22×Q8C2×C4×C5311yescharacteristic
231C22×D4D4×C5311yescharacteristic
232C23×C4—21no—
233C23×C4C2×C4×C5311yescharacteristic
234C23×C4C2×C4×C5311yescharacteristic
235C2×M16—22no—
236#126—28no—
237#125—28no—
238#125—22no—
239#125D4×C5314yes—
240#123—28no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C4×Q8—22no—
246C4×Q8D4×C5314yes—
247C4×D4—22no—
248C4×D4—22no—
249#117—28no—
250#117—22no—
251#117—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×C4×C5311yescharacteristic
261C2×C4⋊C4C2×C4×C5311yescharacteristic
262C2×C4⋊C4C2×C4×C5311yescharacteristic
263C2×C4⋊C4D4×C5311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4D4×C5311yescharacteristic
268C2×C22⋊C4D4×C5311yescharacteristic
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C23×C5311yescharacteristic
272C4⋊C8—22no—
273#100—24no—
274C22⋊C8—22no—
275#97—22no—
276#97—22no—
277C4×C8—22no—
278C24C22⋊C4×C5311yescharacteristic
279C24C22⋊C4×C5311yescharacteristic
280C2×Q8—21no—
281C2×Q8—42no—
282C2×Q8C22⋊C4×C5312yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C22⋊C4×C5311yescharacteristic
300C22×C4C22⋊C4×C5311yescharacteristic
301C22×C4C22×C4×C5311yescharacteristic
302C22×C4C2×D4×C5311yescharacteristic
303C22×C4C2×D4×C5311yescharacteristic
304M16—24no—
305M16—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4C22⋊C4×C5312yes—
319C4⋊C4C22⋊C4×C5312yes—
320C4⋊C4C22⋊C4×C5312yes—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C42—24no—
327C42—24no—
328C42C2×D4×C5312yes—
329C42C2×D4×C5312yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#104×C5312yes—
340C23C2×C22⋊C4×C5311yescharacteristic
341Q8—24no—
342Q8—42no—
343D4—44no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4C2×C22⋊C4×C5311yescharacteristic
368C2×C4C2×C22⋊C4×C5311yescharacteristic
369C2×C4#120×C5312yes—
370C2×C4#120×C5312yes—
371C8—44no—
372C8—82no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22C53×#37911yescharacteristic
386C22C2×#104×C5311yescharacteristic
387C22C53×#42111yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4C53×#41312yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—81no—
402C2C53×#178511yescharacteristic
403C2C53×#178911yescharacteristic
404C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
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
3. C53×#1528C2semidirect
4. C53×#1491C2semidirect
170. #8371C53direct (N × H)
198. #1528C2×C53semidirect
199. #1491C2×C53semidirect
229. C53#8371direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5522
C4×#100×C532
C53×#15282
C53×#14912
C53×#14662
#837153