Finite Group Databasebeta

← C53×#10178 #27217690

Subgroups — C53×#10178 #27217690

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3208C211yesmaximal, characteristic, complemented
2C53×#3077C211yesmaximal, characteristic, complemented
3C2×C53×#459C211yesmaximal, characteristic
4C53×#1545C211yesmaximal, characteristic, complemented
5C53×#1481C211yesmaximal, characteristic, complemented
6C53×#1460C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8C22×D8×C53—21no—
9C53×#533C2211yescharacteristic
10C2×#127×C53C2211yescharacteristic
11C2×C4×D4×C53C2211yescharacteristic
12C53×#492—21no—
13C53×#486—21no—
14C53×#478—22no—
15C53×#476—21no—
16C53×#474—21no—
17C53×#460—22no—
18C53×#459—22no—
19C53×#459—22no—
20C53×#450—22no—
21C53×#431C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C53×#428—21no—
24C53×#423C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#418—21no—
27C2×#102×C53—21no—
28C53×#331—22no—
29C53×Q8⋊C8—22no—
30C53×#323—22no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53D411yescharacteristic
35C23×C4×C53—21no—
36C2×SD16×C53—24no—
37C2×D8×C53—24no—
38C2×D8×C53—42no—
39C2×D8×C53—42no—
40C2×M16×C53—21no—
41C22×C8×C53—21no—
42C4⋊Q8×C53D411yescharacteristic
43C4⋊Q8×C53D411yescharacteristic
44#127×C53—21no—
45#127×C53—21no—
46#127×C53—21no—
47#127×C53D412yes—
48#125×C53—21no—
49C22⋊Q8×C53—21no—
50C22⋊Q8×C53—21no—
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57#117×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C42×C53C2311yescharacteristic
61#107×C53—22no—
62#106×C53—22no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53D412yes—
68C4⋊C8×C53D412yes—
69#102×C53—24no—
70#102×C53—24no—
71#102×C53—22no—
72#102×C53—22no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53D412yes—
75C4×C8×C53—21no—
76C4×C8×C53—21no—
77C24×C53—21no—
78C24×C53—41no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—82no—
97C2×D4×C53—82no—
98C2×D4×C53SD1612yescomplemented
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—82no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106SD16×C53—84no—
107D8×C53—48no—
108M16×C53—41no—
109M16×C53—41no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53—41no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C23×C53—22no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—42no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53—82no—
139C23×C53—82no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53#12111yescharacteristic
143Q8×C53—42no—
144Q8×C53—42no—
145D4×C53—22no—
146D4×C53—22no—
147D4×C53—44no—
148D4×C53—41no—
149D4×C53—42no—
150D4×C53—41no—
151D4×C53—42no—
152D4×C53—82no—
153D4×C53—82no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—82no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—82no—
163C2×C4×C53—82no—
164C2×C4×C53—81no—
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12711yescharacteristic
171C2×C4×C53C2×SD1611yescharacteristic
172C2×C4×C53#13511yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53#13711yescharacteristic
175C8×C53—42no—
176C8×C53—42no—
177C8×C53—42no—
178C8×C53—81no—
179#10178C5311yesmaximal, characteristic, direct product factor, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—42no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—82no—
188C22×C53—82no—
189C22×C53—82no—
190C22×C53—162no—
191C22×C53—162no—
192C22×C53#39211yescharacteristic
193C22×C53#46511yescharacteristic
194C22×C53#46811yescharacteristic
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53#44811yescharacteristic
202C4×C53#44911yescharacteristic
203C4×C53#45011yescharacteristic
204C4×C53#45211yescharacteristic
205C4×C53#47212yes—
206#3208C2×C5311yescharacteristic, complemented
207#3077C2×C5311yescharacteristic, complemented
208C2×#459C2×C5311yescharacteristic
209#1545C2×C5311yescharacteristic, complemented
210#1481C2×C5311yescharacteristic, complemented
211#1460C2×C5311yescharacteristic, complemented
212#1301C2×C5311yescharacteristic, complemented
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—82no—
216C2×C53—162no—
217C2×C53#200011yescharacteristic
218C2×C53#200811yescharacteristic
219C2×C53#201811yescharacteristic
220C22×D8—21no—
221#533C22×C5311yescharacteristic
222C2×#127C22×C5311yescharacteristic
223C2×C4×D4C22×C5311yescharacteristic
224#492—21no—
225#486—21no—
226#478—22no—
227#476—21no—
228#474—21no—
229#460—22no—
230#459—22no—
231#459—22no—
232#450—22no—
233#431C22×C5311yescharacteristic
234#431C22×C5311yescharacteristic
235#428—21no—
236#423C22×C5311yescharacteristic
237C2×C4⋊C8C22×C5311yescharacteristic
238#418—21no—
239C2×#102—21no—
240#331—22no—
241Q8⋊C8—22no—
242#323—22no—
243C53#1017811yescharacteristic, direct product factor, Sylow
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4D4×C5311yescharacteristic
248C23×C4—21no—
249C2×SD16—24no—
250C2×D8—24no—
251C2×D8—42no—
252C2×D8—42no—
253C2×M16—21no—
254C22×C8—21no—
255C4⋊Q8D4×C5311yescharacteristic
256C4⋊Q8D4×C5311yescharacteristic
257#127—21no—
258#127—21no—
259#127—21no—
260#127D4×C5312yes—
261#125—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117—21no—
271C2×C4⋊C4D4×C5311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C23×C5311yescharacteristic
274#107—22no—
275#106—22no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8D4×C5312yes—
281C4⋊C8D4×C5312yes—
282#102—24no—
283#102—22no—
284#102—22no—
285#102—24no—
286C22⋊C8—21no—
287C22⋊C8D4×C5312yes—
288C4×C8—21no—
289C4×C8—21no—
290C24—21no—
291C24—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4SD16×C5312yescomplemented
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4C2×D4×C5311yescharacteristic
317C22×C4C2×D4×C5311yescharacteristic
318C22×C4#41×C5311yescharacteristic
319SD16—84no—
320D8—48no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—41no—
341C42C2×D4×C5311yescharacteristic
342C42C2×D4×C5311yescharacteristic
343C42C2×D4×C5311yescharacteristic
344C42C2×D4×C5311yescharacteristic
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23—82no—
353C23—81no—
354C23—81no—
355C23#121×C5311yescharacteristic
356Q8—42no—
357Q8—42no—
358D4—22no—
359D4—22no—
360D4—44no—
361D4—41no—
362D4—42no—
363D4—41no—
364D4—42no—
365D4—82no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—82no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4#120×C5311yescharacteristic
379C2×C4#120×C5311yescharacteristic
380C2×C4#120×C5311yescharacteristic
381C2×C4#121×C5311yescharacteristic
382C2×C4#121×C5311yescharacteristic
383C2×C4#127×C5311yescharacteristic
384C2×C4C2×SD16×C5311yescharacteristic
385C2×C4#135×C5311yescharacteristic
386C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
387C2×C4#137×C5311yescharacteristic
388C8—42no—
389C8—42no—
390C8—42no—
391C8—81no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—162no—
403C22—162no—
404C22C53×#39211yescharacteristic
405C22C53×#46511yescharacteristic
406C22C53×#46811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4C53×#44811yescharacteristic
414C4C53×#44911yescharacteristic
415C4C53×#45011yescharacteristic
416C4C53×#45211yescharacteristic
417C4C53×#47212yes—
418C2—21no—
419C2—21no—
420C2—82no—
421C2—162no—
422C2C53×#200011yescharacteristic
423C2C53×#200811yescharacteristic
424C2C53×#201811yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC2×C53×#459C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3208C2semidirect
2. C53×#3077C2semidirect
4. C53×#1545C2semidirect
5. C53×#1481C2semidirect
6. C53×#1460C2semidirect
7. C53×#1301C2semidirect
98. C2×D4×C53SD16semidirect
179. #10178C53direct (N × H)
206. #3208C2×C53semidirect
207. #3077C2×C53semidirect
209. #1545C2×C53semidirect
210. #1481C2×C53semidirect
211. #1460C2×C53semidirect
212. #1301C2×C53semidirect
243. C53#10178direct (N × H)
311. C2×D4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32082
C53×#30772
C2×C53×#4592
C53×#15452
C53×#14812
C53×#14602
C53×#13012
#1017853