Finite Group Databasebeta

← C53×#9251 #27216763

Subgroups — C53×#9251 #27216763

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#474C211yesmaximal, characteristic
2D4×M16×C53C211yesmaximal, characteristic
3C53×#1966C211yesmaximal, characteristic, complemented
4C53×#1546C211yesmaximal, characteristic, complemented
5C53×#1486C211yesmaximal, characteristic
6C53×#1292C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C22×M16×C53—21no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×C4×D4×C53C2211yescharacteristic
11C53×#474—21no—
12C53×#474—21no—
13C53×#474—21no—
14C53×#474—21no—
15C53×#436—21no—
16C53×#435—22no—
17C8×D4×C53—21no—
18C53×#431C2211yescharacteristic
19C2×#107×C53C412yes—
20C53×#423C2211yescharacteristic
21C53×#423C2211yescharacteristic
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×#102×C53C412yes—
24C2×#101×C53—21no—
25C2×#101×C53—21no—
26C53×#407—21no—
27C4×M16×C53C2211yescharacteristic
28C53×#332—22no—
29C53×D4⋊C8—21no—
30C53×D4⋊C8—21no—
31C22×Q8×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53—21no—
34C2×M16×C53—22no—
35C2×M16×C53—21no—
36C2×M16×C53—21no—
37C2×M16×C53—42no—
38C2×M16×C53—42no—
39C2×M16×C53D411yescharacteristic
40C2×M16×C53D411yescharacteristic
41C22×C8×C53—22no—
42C22×C8×C53C2×C412yes—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4×D4×C53—21no—
48C4×D4×C53—21no—
49C4×D4×C53D411yescharacteristic
50C4×D4×C53D411yescharacteristic
51C4×D4×C53D411yescharacteristic
52C4×D4×C53D411yescharacteristic
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C412yes—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C2×C42×C53C2311yescharacteristic
58#107×C53—22no—
59#107×C53—22no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67#102×C53—22no—
68#102×C53—22no—
69#101×C53—42no—
70#101×C53—42no—
71C22⋊C8×C53—22no—
72C22⋊C8×C53—22no—
73C22⋊C8×C53—21no—
74#97×C53—21no—
75C4×C8×C53—21no—
76C4×C8×C53—21no—
77C24×C53—21no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×D4×C53—22no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53C22⋊C411yescharacteristic
88C2×D4×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102M16×C53—21no—
103M16×C53—21no—
104M16×C53—21no—
105M16×C53—44no—
106M16×C53—41no—
107M16×C53—41no—
108M16×C53—41no—
109M16×C53—41no—
110M16×C53D812yes—
111C2×C8×C53—21no—
112C2×C8×C53—22no—
113C2×C8×C53—21no—
114C2×C8×C53—22no—
115C2×C8×C53—44no—
116C2×C8×C53—42no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53C22⋊C411yescharacteristic
125C4⋊C4×C53C22⋊C411yescharacteristic
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53#4111yescharacteristic
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—42no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—81no—
140C23×C53C4×D411yescharacteristic
141Q8×C53—81no—
142Q8×C53—81no—
143Q8×C53—81no—
144Q8×C53—81no—
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—21no—
148D4×C53—42no—
149D4×C53#10112yescomplemented
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
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—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53#13511yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C8×C53—22no—
179C8×C53—42no—
180C8×C53—42no—
181C8×C53—41no—
182C8×C53—81no—
183C8×C53—81no—
184#9251C5311yesmaximal, characteristic, direct product factor, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53#38611yescharacteristic
196C22×C53C4×D811yescharacteristic
197C22×C53#44211yescharacteristic
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—82no—
204C4×C53C2×#10111yescharacteristic
205C4×C53#41311yescharacteristic
206C4×C53#44711yescharacteristic
207C4×C53#44911yescharacteristic
208C4×C53#45911yescharacteristic
209C4×C53#46311yescharacteristic
210C2×#474C2×C5311yescharacteristic
211D4×M16C2×C5311yescharacteristic
212#1966C2×C5311yescharacteristic, complemented
213#1546C2×C5311yescharacteristic, complemented
214#1486C2×C5311yescharacteristic
215#1292C2×C5311yescharacteristic, complemented
216#1283C2×C5311yescharacteristic, complemented
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53—81no—
221C2×C53#187911yescharacteristic
222C2×C53#190111yescharacteristic
223C2×C53#190211yescharacteristic
224C22×M16—21no—
225C2×C4⋊Q8C22×C5311yescharacteristic
226C2×C4×D4C22×C5311yescharacteristic
227#474—21no—
228#474—21no—
229#474—21no—
230#474—21no—
231#436—21no—
232#435—22no—
233C8×D4—21no—
234#431C22×C5311yescharacteristic
235C2×#107C4×C5312yes—
236#423C22×C5311yescharacteristic
237#423C22×C5311yescharacteristic
238C2×C4⋊C8C22×C5311yescharacteristic
239C2×#102C4×C5312yes—
240C2×#101—21no—
241C2×#101—21no—
242#407—21no—
243C4×M16C22×C5311yescharacteristic
244#332—22no—
245D4⋊C8—21no—
246D4⋊C8—21no—
247C53#925111yescharacteristic, direct product factor, Sylow
248C22×Q8—21no—
249C22×D4C2×C4×C5311yescharacteristic
250C23×C4—21no—
251C2×M16—22no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—42no—
255C2×M16—42no—
256C2×M16D4×C5311yescharacteristic
257C2×M16D4×C5311yescharacteristic
258C22×C8—22no—
259C22×C8C2×C4×C5312yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4D4×C5311yescharacteristic
267C4×D4D4×C5311yescharacteristic
268C4×D4D4×C5311yescharacteristic
269C4×D4D4×C5311yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4C2×C4×C5312yes—
272C2×C4⋊C4C2×C4×C5311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C42C23×C5311yescharacteristic
275#107—22no—
276#107—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8D4×C5312yes—
284#102—22no—
285#102—22no—
286#101—42no—
287#101—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—21no—
291#97—21no—
292C4×C8—21no—
293C4×C8—21no—
294C24—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4C22⋊C4×C5311yescharacteristic
305C2×D4C22⋊C4×C5311yescharacteristic
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C22×C4×C5311yescharacteristic
317C22×C4C2×D4×C5311yescharacteristic
318C22×C4#41×C5311yescharacteristic
319M16—21no—
320M16—21no—
321M16—21no—
322M16—44no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16D8×C5312yes—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4C22⋊C4×C5311yescharacteristic
342C4⋊C4C22⋊C4×C5311yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C42C2×D4×C5311yescharacteristic
348C42C2×D4×C5311yescharacteristic
349C42C2×D4×C5311yescharacteristic
350C42#41×C5311yescharacteristic
351C23—21no—
352C23—21no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23C4×D4×C5311yescharacteristic
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4#101×C5312yescomplemented
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4C2×C22⋊C4×C5311yescharacteristic
386C2×C4C4×D4×C5311yescharacteristic
387C2×C4#120×C5311yescharacteristic
388C2×C4#121×C5311yescharacteristic
389C2×C4#121×C5311yescharacteristic
390C2×C4#123×C5311yescharacteristic
391C2×C4C2×D8×C5311yescharacteristic
392C2×C4#135×C5311yescharacteristic
393C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
394C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
395C8—22no—
396C8—42no—
397C8—42no—
398C8—41no—
399C8—81no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—82no—
410C22—82no—
411C22C53×#38611yescharacteristic
412C22C4×D8×C5311yescharacteristic
413C22C53×#44211yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4C2×#101×C5311yescharacteristic
421C4C53×#41311yescharacteristic
422C4C53×#44711yescharacteristic
423C4C53×#44911yescharacteristic
424C4C53×#45911yescharacteristic
425C4C53×#46311yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—81no—
430C2C53×#187911yescharacteristic
431C2C53×#190111yescharacteristic
432C2C53×#190211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×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
3. C53×#1966C2semidirect
4. C53×#1546C2semidirect
6. C53×#1292C2semidirect
7. C53×#1283C2semidirect
149. D4×C53#101semidirect
184. #9251C53direct (N × H)
212. #1966C2×C53semidirect
213. #1546C2×C53semidirect
215. #1292C2×C53semidirect
216. #1283C2×C53semidirect
247. C53#9251direct (N × H)
366. D4#101×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4742
D4×M16×C532
C53×#19662
C53×#15462
C53×#14862
C53×#12922
C53×#12832
#925153