Finite Group Databasebeta

← C53×#7612 #27215124

Subgroups — C53×#7612 #27215124

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2297C211yesmaximal, characteristic
2C53×#1797C211yesmaximal, characteristic, complemented
3C53×#1749C211yesmaximal, characteristic, complemented
4C53×#1272C212yesmaximal, complemented
5C53×#1270C212yesmaximal, complemented
6C22×#41×C53C411yescharacteristic
7C2×#117×C53C411yescharacteristic
8C22×C42×C53C2211yescharacteristic
9C2×C4⋊C8×C53—21no—
10C2×C22⋊C8×C53—21no—
11C2×C22⋊C8×C53—21no—
12C2×#97×C53—21no—
13C53×#391C411yescharacteristic
14C53×#391C2211yescharacteristic
15C53×#390C411yescharacteristic
16C53×#390C2211yescharacteristic
17C53×#386—21no—
18C53×#386—21no—
19C4×C22⋊C4×C53—21no—
20C4×C22⋊C4×C53—21no—
21C53×#336C2212yes—
22C53×#336C2212yes—
23C2×#41×C53—22no—
24C2×#41×C53—22no—
25C2×#41×C53—42no—
26C2×#41×C53—42no—
27C22×Q8×C53C2×C411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C22×C8×C53—22no—
35C22×C8×C53—22no—
36#117×C53—42no—
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—24no—
52C22⋊C8×C53—42no—
53#97×C53—42no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60#41×C53—48no—
61#41×C53—48no—
62C2×Q8×C53—21no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53M1612yes—
66C2×D4×C53—24no—
67C2×D4×C53—21no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—42no—
71C2×D4×C53—42no—
72C2×D4×C53M1612yes—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—24no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C2×C8×C53—24no—
97C2×C8×C53—42no—
98C2×C8×C53—42no—
99C2×C8×C53—42no—
100C2×C8×C53—42no—
101C2×C8×C53—42no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—22no—
111C42×C53—22no—
112C42×C53—41no—
113C42×C53—41no—
114C42×C53—42no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—22no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#10012yes—
129C23×C53C2×C22⋊C411yescharacteristic
130Q8×C53—42no—
131Q8×C53—42no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—48no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—44no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—48no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53#9912yes—
164C2×C4×C53#10412yes—
165C2×C4×C53C2×M1611yescharacteristic
166C2×C4×C53C2×M1611yescharacteristic
167C2×C4×C53#13512yes—
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C8×C53—44no—
171C8×C53—82no—
172#7612C5311yesmaximal, characteristic, direct product factor, Sylow
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—42no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#40711yescharacteristic
188C22×C53C2×#9911yescharacteristic
189C22×C53C2×#10011yescharacteristic
190C22×C53#41611yescharacteristic
191C22×C53#41711yescharacteristic
192C22×C53C2×#10411yescharacteristic
193C22×C53#42111yescharacteristic
194C4×C53—22no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—42no—
202C4×C53—41no—
203C4×C53—81no—
204C4×C53—81no—
205#2297C2×C5311yescharacteristic
206#1797C2×C5311yescharacteristic, complemented
207#1749C2×C5311yescharacteristic, complemented
208#1272C2×C5312yescomplemented
209#1270C2×C5312yescomplemented
210C2×C53—42no—
211C2×C53—82no—
212C2×C53#146211yescharacteristic
213C2×C53#148811yescharacteristic
214C2×C53#148911yescharacteristic
215C2×C53#150111yescharacteristic
216C2×C53#151311yescharacteristic
217C2×C53#153011yescharacteristic
218C2×C53#155111yescharacteristic
219C22×#41C4×C5311yescharacteristic
220C2×#117C4×C5311yescharacteristic
221C22×C42C22×C5311yescharacteristic
222C2×C4⋊C8—21no—
223C2×C22⋊C8—21no—
224C2×C22⋊C8—21no—
225C2×#97—21no—
226#391C4×C5311yescharacteristic
227#391C22×C5311yescharacteristic
228#390C4×C5311yescharacteristic
229#390C22×C5311yescharacteristic
230#386—21no—
231#386—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—21no—
234#336C22×C5312yes—
235#336C22×C5312yes—
236C53#761211yescharacteristic, direct product factor, Sylow
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C22×Q8C2×C4×C5311yescharacteristic
242C22×D4—21no—
243C22×D4C2×C4×C5311yescharacteristic
244C23×C4—21no—
245C23×C4C2×C4×C5311yescharacteristic
246C23×C4D4×C5311yescharacteristic
247C23×C4D4×C5311yescharacteristic
248C22×C8—22no—
249C22×C8—22no—
250#117—42no—
251C2×C4⋊C4C2×C4×C5311yescharacteristic
252C2×C4⋊C4C2×C4×C5311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42C2×C4×C5311yescharacteristic
261C2×C42D4×C5311yescharacteristic
262C2×C42D4×C5311yescharacteristic
263C2×C42C23×C5311yescharacteristic
264C4⋊C8—42no—
265C22⋊C8—24no—
266C22⋊C8—42no—
267#97—42no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24—21no—
273C24C22⋊C4×C5311yescharacteristic
274#41—48no—
275#41—48no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8M16×C5312yes—
280C2×D4—24no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4M16×C5312yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C22⋊C4×C5311yescharacteristic
305C22×C4C22⋊C4×C5311yescharacteristic
306C22×C4C22⋊C4×C5311yescharacteristic
307C22×C4C22×C4×C5311yescharacteristic
308C22×C4C2×D4×C5311yescharacteristic
309C22×C4C2×D4×C5311yescharacteristic
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—22no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#100×C5312yes—
343C23C2×C22⋊C4×C5311yescharacteristic
344Q8—42no—
345Q8—42no—
346D4—42no—
347D4—42no—
348D4—48no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4#99×C5312yes—
378C2×C4#104×C5312yes—
379C2×C4C2×M16×C5311yescharacteristic
380C2×C4C2×M16×C5311yescharacteristic
381C2×C4#135×C5312yes—
382C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
383C2×C4#137×C5311yescharacteristic
384C8—44no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C53×#40711yescharacteristic
401C22C2×#99×C5311yescharacteristic
402C22C2×#100×C5311yescharacteristic
403C22C53×#41611yescharacteristic
404C22C53×#41711yescharacteristic
405C22C2×#104×C5311yescharacteristic
406C22C53×#42111yescharacteristic
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—41no—
416C4—81no—
417C4—81no—
418C2—42no—
419C2—82no—
420C2C53×#146211yescharacteristic
421C2C53×#148811yescharacteristic
422C2C53×#148911yescharacteristic
423C2C53×#150111yescharacteristic
424C2C53×#151311yescharacteristic
425C2C53×#153011yescharacteristic
426C2C53×#155111yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×#41×C53C4
LCSG ▷ C22×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×#1797C2semidirect
3. C53×#1749C2semidirect
4. C53×#1272C2semidirect
5. C53×#1270C2semidirect
172. #7612C53direct (N × H)
206. #1797C2×C53semidirect
207. #1749C2×C53semidirect
208. #1272C2×C53semidirect
209. #1270C2×C53semidirect
236. C53#7612direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#22972
C53×#17972
C53×#17492
C53×#12722
C53×#12702
#761253