Finite Group Databasebeta

← C53×#7599 #27215111

Subgroups — C53×#7599 #27215111

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#121×C53C211yesmaximal, characteristic
2C53×#1817C211yesmaximal, characteristic, complemented
3C53×#1797C211yesmaximal, characteristic, complemented
4C53×#1271C212yesmaximal, complemented
5C53×#1270C212yesmaximal, complemented
6C2×#121×C53C411yescharacteristic
7C2×C4×D4×C53—21no—
8C2×C4×D4×C53C411yescharacteristic
9C22×C42×C53C2211yescharacteristic
10C2×C4⋊C8×C53—21no—
11C2×C22⋊C8×C53—21no—
12C2×C22⋊C8×C53—21no—
13C53×#390C2211yescharacteristic
14C53×#389C2211yescharacteristic
15C53×#388—21no—
16C53×#386—21no—
17C53×#380C411yescharacteristic
18C4×C4⋊C4×C53C411yescharacteristic
19C4×C22⋊C4×C53—21no—
20C53×#336—21no—
21C53×#336C2212yes—
22C53×#336C2212yes—
23C22×D4×C53—21no—
24C22×D4×C53C2×C411yescharacteristic
25C23×C4×C53—21no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C22×C8×C53—22no—
30C22×C8×C53—22no—
31#121×C53—22no—
32#121×C53C814yescomplemented
33C4×D4×C53—44no—
34C4×D4×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53—42no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53C2311yescharacteristic
50C4⋊C8×C53—42no—
51C22⋊C8×C53—42no—
52C22⋊C8×C53—42no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56C24×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—24no—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C2×D4×C53C2×C812yes—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C2×C812yes—
84C22×C4×C53M1612yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C8×C53—42no—
89C2×C8×C53—42no—
90C2×C8×C53—42no—
91C2×C8×C53—42no—
92C2×C8×C53—42no—
93C2×C8×C53—42no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53C2×C812yes—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—41no—
105C42×C53—22no—
106C42×C53—22no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53C22⋊C812yes—
124C23×C53C2×C22⋊C411yescharacteristic
125D4×C53—42no—
126D4×C53—42no—
127D4×C53—48no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
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—84no—
156C2×C4×C53C22⋊C812yes—
157C2×C4×C53#9912yes—
158C2×C4×C53C22×C811yescharacteristic
159C2×C4×C53C2×M1611yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)12yes—
162C2×C4×C53#13711yescharacteristic
163C8×C53—82no—
164C8×C53—82no—
165#7599C5311yesmaximal, characteristic, direct product factor, Sylow
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53C2×C22⋊C811yescharacteristic
181C22×C53C2×#9911yescharacteristic
182C22×C53#41311yescharacteristic
183C22×C53#41711yescharacteristic
184C22×C53#41811yescharacteristic
185C22×C53#42111yescharacteristic
186C22×C53#42111yescharacteristic
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—41no—
196C4×C53—81no—
197C4×C53—81no—
198C4×#121C2×C5311yescharacteristic
199#1817C2×C5311yescharacteristic, complemented
200#1797C2×C5311yescharacteristic, complemented
201#1271C2×C5312yescomplemented
202#1270C2×C5312yescomplemented
203C2×C53—42no—
204C2×C53—82no—
205C2×C53#145911yescharacteristic
206C2×C53#147711yescharacteristic
207C2×C53#147811yescharacteristic
208C2×C53#151011yescharacteristic
209C2×C53#151311yescharacteristic
210C2×C53#153511yescharacteristic
211C2×C53#155711yescharacteristic
212C2×#121C4×C5311yescharacteristic
213C2×C4×D4—21no—
214C2×C4×D4C4×C5311yescharacteristic
215C22×C42C22×C5311yescharacteristic
216C2×C4⋊C8—21no—
217C2×C22⋊C8—21no—
218C2×C22⋊C8—21no—
219#390C22×C5311yescharacteristic
220#389C22×C5311yescharacteristic
221#388—21no—
222#386—21no—
223#380C4×C5311yescharacteristic
224C4×C4⋊C4C4×C5311yescharacteristic
225C4×C22⋊C4—21no—
226#336—21no—
227#336C22×C5312yes—
228#336C22×C5312yes—
229C53#759911yescharacteristic, direct product factor, Sylow
230C22×D4—21no—
231C22×D4C2×C4×C5311yescharacteristic
232C23×C4—21no—
233C23×C4C2×C4×C5311yescharacteristic
234C23×C4D4×C5311yescharacteristic
235C23×C4D4×C5311yescharacteristic
236C22×C8—22no—
237C22×C8—22no—
238#121—22no—
239#121C8×C5314yescomplemented
240C4×D4—42no—
241C4×D4—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C2×C4×C5311yescharacteristic
244C2×C4⋊C4C2×C4×C5311yescharacteristic
245C2×C4⋊C4C2×C4×C5311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—42no—
253C2×C42C2×C4×C5311yescharacteristic
254C2×C42D4×C5311yescharacteristic
255C2×C42D4×C5311yescharacteristic
256C2×C42C23×C5311yescharacteristic
257C4⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#95—21no—
261#95—21no—
262#95—21no—
263C24—21no—
264C24C22⋊C4×C5311yescharacteristic
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4C2×C8×C5312yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C22⋊C4×C5311yescharacteristic
288C22×C4C22⋊C4×C5311yescharacteristic
289C22×C4C22⋊C4×C5311yescharacteristic
290C22×C4C2×C8×C5312yes—
291C22×C4M16×C5312yes—
292C22×C4C22×C4×C5311yescharacteristic
293C22×C4C2×D4×C5311yescharacteristic
294C22×C4C2×D4×C5311yescharacteristic
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4C2×C8×C5312yes—
306C22⋊C4—24no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—42no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C22⋊C8×C5312yes—
331C23C2×C22⋊C4×C5311yescharacteristic
332D4—42no—
333D4—42no—
334D4—48no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—84no—
363C2×C4C22⋊C8×C5312yes—
364C2×C4#99×C5312yes—
365C2×C4C22×C8×C5311yescharacteristic
366C2×C4C2×M16×C5311yescharacteristic
367C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
368C2×C4AGL(1,ℤ/8)×C5312yes—
369C2×C4#137×C5311yescharacteristic
370C8—82no—
371C8—82no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×C22⋊C8×C5311yescharacteristic
387C22C2×#99×C5311yescharacteristic
388C22C53×#41311yescharacteristic
389C22C53×#41711yescharacteristic
390C22C53×#41811yescharacteristic
391C22C53×#42111yescharacteristic
392C22C53×#42111yescharacteristic
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—41no—
402C4—81no—
403C4—81no—
404C2—42no—
405C2—82no—
406C2C53×#145911yescharacteristic
407C2C53×#147711yescharacteristic
408C2C53×#147811yescharacteristic
409C2C53×#151011yescharacteristic
410C2C53×#151311yescharacteristic
411C2C53×#153511yescharacteristic
412C2C53×#155711yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC2×#121×C53C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1817C2semidirect
3. C53×#1797C2semidirect
4. C53×#1271C2semidirect
5. C53×#1270C2semidirect
32. #121×C53C8semidirect
165. #7599C53direct (N × H)
199. #1817C2×C53semidirect
200. #1797C2×C53semidirect
201. #1271C2×C53semidirect
202. #1270C2×C53semidirect
229. C53#7599direct (N × H)
239. #121C8×C53semidirect

Maximal subgroups

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

subgroupindex
C4×#121×C532
C53×#18172
C53×#17972
C53×#12712
C53×#12702
#759953