Finite Group Databasebeta

← C53×#13678 #27221190

Subgroups — C53×#13678 #27221190

Order 13568.

400 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3289C211yesmaximal, characteristic, complemented
2C53×#2202C211yesmaximal, characteristic, complemented
3C53×#2199C211yesmaximal, characteristic, complemented
4C53×#1659C211yesmaximal, characteristic, complemented
5C53×#1407C211yesmaximal, characteristic, complemented
6C53×#1404C211yesmaximal, characteristic, complemented
7C53×#1403C211yesmaximal, characteristic, complemented
8C53×#578—21no—
9C53×#577—21no—
10C53×H(3,𝔽4)C2211yescharacteristic
11C53×#497C2211yescharacteristic
12C53×#489—21no—
13C53×#473—21no—
14C53×#471—21no—
15C53×B(4,𝔽2)—21no—
16C53×#456—21no—
17C53×#455C2211yescharacteristic
18C53×#454—21no—
19C53×#454C2211yescharacteristic
20C53×#444—21no—
21C53×#355—21no—
22C53×#355C2211yescharacteristic
23C53×#354C2211yescharacteristic
24C53×AΣL(1,𝔽16)—21no—
25C53×#329—21no—
26C53×#329C2211yescharacteristic
272−1+4×C53—21no—
282−1+4×C53D411yescharacteristic, complemented
29H(5,𝔽2)×C53D411yescharacteristic
30#137×C53—21no—
31#137×C53—21no—
32#137×C53—21no—
33#137×C53—41no—
34#137×C53—41no—
35#137×C53—41no—
36AGL(1,ℤ/8)×C53—21no—
37AGL(1,ℤ/8)×C53—41no—
38AGL(1,ℤ/8)×C53—41no—
39#135×C53—21no—
40#135×C53—41no—
41#135×C53—41no—
42C2×Q16×C53—21no—
43C2×Q16×C53—41no—
44C2×SD16×C53—21no—
45C2×SD16×C53—41no—
46#131×C53—21no—
47#131×C53—21no—
48C4⋊Q8×C53D411yescharacteristic, complemented
49#124×C53—21no—
50#124×C53—21no—
51#124×C53—41no—
52#124×C53C2311yescharacteristic
53#120×C53—21no—
54#120×C53—41no—
55#120×C53—41no—
56#108×C53—21no—
57#104×C53—21no—
58#104×C53—21no—
59#104×C53—41no—
60#103×C53—41no—
61#102×C53—41no—
62#101×C53—21no—
63#101×C53—21no—
64#101×C53D411yescharacteristic, complemented
65#100×C53D411yescharacteristic
66#99×C53—21no—
67#99×C53—41no—
68#97×C53—21no—
69#97×C53D411yescharacteristic, complemented
70C24×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—41no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—81no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×Q8×C53C2×D411yescharacteristic
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—81no—
91C2×D4×C53C2×D411yescharacteristic
92Q16×C53—21no—
93Q16×C53—41no—
94Q16×C53—41no—
95Q16×C53—41no—
96Q16×C53—41no—
97Q16×C53—41no—
98SD16×C53—21no—
99SD16×C53—21no—
100SD16×C53—41no—
101SD16×C53—41no—
102SD16×C53—41no—
103SD16×C53—41no—
104SD16×C53—41no—
105SD16×C53—41no—
106SD16×C53—81no—
107D8×C53—21no—
108D8×C53—41no—
109D8×C53—81no—
110M16×C53—21no—
111M16×C53—21no—
112M16×C53—41no—
113M16×C53—41no—
114M16×C53—41no—
115C2×C8×C53—21no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—81no—
127C42×C53—21no—
128C42×C53C2×D411yescharacteristic
129C23×C53—21no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—81no—
133C23×C53—161no—
134Q8×C53—21no—
135Q8×C53—21no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—81no—
141Q8×C53—81no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152D4×C53—161no—
153D4×C53—162no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53#12011yescharacteristic
166C8×C53—21no—
167C8×C53—21no—
168C8×C53—41no—
169C8×C53—41no—
170C8×C53—81no—
171C8×C53—81no—
172#13678C5311yesmaximal, characteristic, direct product factor, Sylow
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—82no—
180C22×C53—81no—
181C22×C53—161no—
182C22×C53—161no—
183C22×C53B(4,𝔽2)11yescharacteristic
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—161no—
192#3289C2×C5311yescharacteristic, complemented
193#2202C2×C5311yescharacteristic, complemented
194#2199C2×C5311yescharacteristic, complemented
195#1659C2×C5311yescharacteristic, complemented
196#1407C2×C5311yescharacteristic, complemented
197#1404C2×C5311yescharacteristic, complemented
198#1403C2×C5311yescharacteristic, complemented
199C2×C53—21no—
200C2×C53—81no—
201C2×C53—81no—
202C2×C53—81no—
203C2×C53—161no—
204C2×C53#219611yescharacteristic
205#578—21no—
206#577—21no—
207H(3,𝔽4)C22×C5311yescharacteristic
208#497C22×C5311yescharacteristic
209#489—21no—
210#473—21no—
211#471—21no—
212B(4,𝔽2)—21no—
213#456—21no—
214#455C22×C5311yescharacteristic
215#454—21no—
216#454C22×C5311yescharacteristic
217#444—21no—
218#355—21no—
219#355C22×C5311yescharacteristic
220#354C22×C5311yescharacteristic
221AΣL(1,𝔽16)—21no—
222#329—21no—
223#329C22×C5311yescharacteristic
224C53#1367811yescharacteristic, direct product factor, Sylow
2252−1+4—21no—
2262−1+4D4×C5311yescharacteristic, complemented
227H(5,𝔽2)D4×C5311yescharacteristic
228#137—21no—
229#137—21no—
230#137—21no—
231#137—41no—
232#137—41no—
233#137—41no—
234AGL(1,ℤ/8)—21no—
235AGL(1,ℤ/8)—41no—
236AGL(1,ℤ/8)—41no—
237#135—21no—
238#135—41no—
239#135—41no—
240C2×Q16—21no—
241C2×Q16—41no—
242C2×SD16—21no—
243C2×SD16—41no—
244#131—21no—
245#131—21no—
246C4⋊Q8D4×C5311yescharacteristic, complemented
247#124—21no—
248#124—21no—
249#124—41no—
250#124C23×C5311yescharacteristic
251#120—21no—
252#120—41no—
253#120—41no—
254#108—21no—
255#104—21no—
256#104—21no—
257#104—41no—
258#103—41no—
259#102—41no—
260#101—21no—
261#101—21no—
262#101D4×C5311yescharacteristic, complemented
263#100D4×C5311yescharacteristic
264#99—21no—
265#99—41no—
266#97—21no—
267#97D4×C5311yescharacteristic, complemented
268C24—21no—
269#41—21no—
270#41—21no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—81no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8C2×D4×C5311yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C2×D4C2×D4×C5311yescharacteristic
290Q16—21no—
291Q16—41no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295Q16—41no—
296SD16—21no—
297SD16—21no—
298SD16—41no—
299SD16—41no—
300SD16—41no—
301SD16—41no—
302SD16—41no—
303SD16—41no—
304SD16—81no—
305D8—21no—
306D8—41no—
307D8—81no—
308M16—21no—
309M16—21no—
310M16—41no—
311M16—41no—
312M16—41no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—41no—
318C22⋊C4—21no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—81no—
325C42—21no—
326C42C2×D4×C5311yescharacteristic
327C23—21no—
328C23—41no—
329C23—41no—
330C23—81no—
331C23—161no—
332Q8—21no—
333Q8—21no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—81no—
339Q8—81no—
340D4—21no—
341D4—21no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—161no—
351D4—162no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4#120×C5311yescharacteristic
364C8—21no—
365C8—21no—
366C8—41no—
367C8—41no—
368C8—81no—
369C8—81no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—82no—
377C22—81no—
378C22—161no—
379C22—161no—
380C22C53×B(4,𝔽2)11yescharacteristic
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—161no—
389C2—21no—
390C2—81no—
391C2—81no—
392C2—81no—
393C2—161no—
394C2C53×#219611yescharacteristic
subgroupG/H
center, socleC2×C53#2196
commutator, Frattini, cosocle#124C23×C53
LCSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ #124×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3289C2semidirect
2. C53×#2202C2semidirect
3. C53×#2199C2semidirect
4. C53×#1659C2semidirect
5. C53×#1407C2semidirect
6. C53×#1404C2semidirect
7. C53×#1403C2semidirect
28. 2−1+4×C53D4semidirect
48. C4⋊Q8×C53D4semidirect
64. #101×C53D4semidirect
69. #97×C53D4semidirect
172. #13678C53direct (N × H)
192. #3289C2×C53semidirect
193. #2202C2×C53semidirect
194. #2199C2×C53semidirect
195. #1659C2×C53semidirect
196. #1407C2×C53semidirect
197. #1404C2×C53semidirect
198. #1403C2×C53semidirect
224. C53#13678direct (N × H)
226. 2−1+4D4×C53semidirect
246. C4⋊Q8D4×C53semidirect
262. #101D4×C53semidirect
267. #97D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32892
C53×#22022
C53×#21992
C53×#16592
C53×#14072
C53×#14042
C53×#14032
#1367853