Finite Group Databasebeta

← C53×#13677 #27221189

Subgroups — C53×#13677 #27221189

Order 13568.

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

normal subgroups only

This group has 420 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×#3288C211yesmaximal, characteristic, complemented
2C53×#2202C211yesmaximal, characteristic, complemented
3C53×#2199C211yesmaximal, characteristic, complemented
4C53×#1655C211yesmaximal, characteristic, complemented
5C53×#1406C211yesmaximal, characteristic, complemented
6C53×#1404C211yesmaximal, characteristic, complemented
7C53×#1403C211yesmaximal, characteristic, complemented
8C53×#577—21no—
9C53×#576—21no—
10C53×H(3,𝔽4)C2211yescharacteristic, complemented
11C53×#496—21no—
12C53×#496C2211yescharacteristic
13C53×#472—21no—
14C53×#471—21no—
15C53×B(4,𝔽2)—21no—
16C53×#455C2211yescharacteristic
17C53×#454—21no—
18C53×#454C2211yescharacteristic
19C53×#453—21no—
20C53×#444—21no—
21C53×#355—21no—
22C53×#355C2211yescharacteristic, complemented
23C53×#354C2211yescharacteristic, complemented
24C53×AΣL(1,𝔽16)—21no—
25C53×#329—21no—
26C53×#329C2211yescharacteristic, complemented
272−1+4×C53D411yescharacteristic, complemented
28H(5,𝔽2)×C53—21no—
29H(5,𝔽2)×C53D411yescharacteristic
30#137×C53—21no—
31#137×C53—41no—
32AGL(1,ℤ/8)×C53—21no—
33AGL(1,ℤ/8)×C53—21no—
34AGL(1,ℤ/8)×C53—21no—
35AGL(1,ℤ/8)×C53—41no—
36AGL(1,ℤ/8)×C53—41no—
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×SD16×C53—21no—
43C2×SD16×C53—41no—
44C2×SD16×C53—41no—
45C2×D8×C53—21no—
46C2×D8×C53—41no—
47C2×D8×C53—41no—
48#131×C53—21no—
49#131×C53—21no—
50#127×C53D411yescharacteristic, complemented
51#124×C53—21no—
52#124×C53—21no—
53#124×C53—41no—
54#124×C53C2311yescharacteristic
55#120×C53—21no—
56#120×C53—41no—
57#120×C53—41no—
58#108×C53—21no—
59#104×C53—21no—
60#104×C53—21no—
61#104×C53—41no—
62#101×C53—21no—
63#101×C53D411yescharacteristic, complemented
64#100×C53—21no—
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—
78C2×Q8×C53—21no—
79C2×Q8×C53—41no—
80C2×Q8×C53C2×D411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—81no—
91C2×D4×C53—81no—
92C2×D4×C53C2×D411yescharacteristic
93Q16×C53—21no—
94Q16×C53—41no—
95SD16×C53—21no—
96SD16×C53—21no—
97SD16×C53—41no—
98SD16×C53—41no—
99SD16×C53—41no—
100SD16×C53—41no—
101SD16×C53—41no—
102SD16×C53—41no—
103SD16×C53—82no—
104SD16×C53—81no—
105D8×C53—21no—
106D8×C53—41no—
107D8×C53—41no—
108D8×C53—41no—
109D8×C53—41no—
110D8×C53—41no—
111D8×C53—82no—
112D8×C53—81no—
113M16×C53—21no—
114M16×C53—21no—
115M16×C53—41no—
116M16×C53—41no—
117M16×C53—41no—
118C2×C8×C53—21no—
119C2×C8×C53—41no—
120C2×C8×C53—41no—
121C2×C8×C53—41no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—81no—
129C42×C53—21no—
130C42×C53C2×D411yescharacteristic
131C23×C53—21no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—161no—
138Q8×C53—21no—
139Q8×C53—21no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—81no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—81no—
152D4×C53—81no—
153D4×C53—81no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—81no—
157D4×C53—81no—
158D4×C53—161no—
159D4×C53—162no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53#12011yescharacteristic
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—41no—
173C8×C53—41no—
174C8×C53—42no—
175C8×C53—81no—
176#13677C5311yesmaximal, characteristic, direct product factor, Sylow
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—82no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—161no—
187C22×C53—161no—
188C22×C53—161no—
189C22×C53—321no—
190C22×C53B(4,𝔽2)11yescharacteristic
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—81no—
197C4×C53—161no—
198#3288C2×C5311yescharacteristic, complemented
199#2202C2×C5311yescharacteristic, complemented
200#2199C2×C5311yescharacteristic, complemented
201#1655C2×C5311yescharacteristic, complemented
202#1406C2×C5311yescharacteristic, complemented
203#1404C2×C5311yescharacteristic, complemented
204#1403C2×C5311yescharacteristic, complemented
205C2×C53—21no—
206C2×C53—81no—
207C2×C53—81no—
208C2×C53—81no—
209C2×C53—161no—
210C2×C53—161no—
211C2×C53#219611yescharacteristic
212#577—21no—
213#576—21no—
214H(3,𝔽4)C22×C5311yescharacteristic, complemented
215#496—21no—
216#496C22×C5311yescharacteristic
217#472—21no—
218#471—21no—
219B(4,𝔽2)—21no—
220#455C22×C5311yescharacteristic
221#454—21no—
222#454C22×C5311yescharacteristic
223#453—21no—
224#444—21no—
225#355—21no—
226#355C22×C5311yescharacteristic, complemented
227#354C22×C5311yescharacteristic, complemented
228AΣL(1,𝔽16)—21no—
229#329—21no—
230#329C22×C5311yescharacteristic, complemented
231C53#1367711yescharacteristic, direct product factor, Sylow
2322−1+4D4×C5311yescharacteristic, complemented
233H(5,𝔽2)—21no—
234H(5,𝔽2)D4×C5311yescharacteristic
235#137—21no—
236#137—41no—
237AGL(1,ℤ/8)—21no—
238AGL(1,ℤ/8)—21no—
239AGL(1,ℤ/8)—21no—
240AGL(1,ℤ/8)—41no—
241AGL(1,ℤ/8)—41no—
242AGL(1,ℤ/8)—41no—
243AGL(1,ℤ/8)—41no—
244#135—21no—
245#135—41no—
246#135—41no—
247C2×SD16—21no—
248C2×SD16—41no—
249C2×SD16—41no—
250C2×D8—21no—
251C2×D8—41no—
252C2×D8—41no—
253#131—21no—
254#131—21no—
255#127D4×C5311yescharacteristic, complemented
256#124—21no—
257#124—21no—
258#124—41no—
259#124C23×C5311yescharacteristic
260#120—21no—
261#120—41no—
262#120—41no—
263#108—21no—
264#104—21no—
265#104—21no—
266#104—41no—
267#101—21no—
268#101D4×C5311yescharacteristic, complemented
269#100—21no—
270#100D4×C5311yescharacteristic
271#99—21no—
272#99—41no—
273#97—21no—
274#97D4×C5311yescharacteristic, complemented
275C24—21no—
276#41—21no—
277#41—21no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8C2×D4×C5311yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4C2×D4×C5311yescharacteristic
298Q16—21no—
299Q16—41no—
300SD16—21no—
301SD16—21no—
302SD16—41no—
303SD16—41no—
304SD16—41no—
305SD16—41no—
306SD16—41no—
307SD16—41no—
308SD16—82no—
309SD16—81no—
310D8—21no—
311D8—41no—
312D8—41no—
313D8—41no—
314D8—41no—
315D8—41no—
316D8—82no—
317D8—81no—
318M16—21no—
319M16—21no—
320M16—41no—
321M16—41no—
322M16—41no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C42—21no—
335C42C2×D4×C5311yescharacteristic
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—81no—
341C23—81no—
342C23—161no—
343Q8—21no—
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—81no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—161no—
364D4—162no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#120×C5311yescharacteristic
375C8—21no—
376C8—21no—
377C8—41no—
378C8—41no—
379C8—42no—
380C8—81no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—321no—
394C22C53×B(4,𝔽2)11yescharacteristic
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—161no—
402C2—21no—
403C2—81no—
404C2—81no—
405C2—81no—
406C2—161no—
407C2—161no—
408C2C53×#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×#3288C2semidirect
2. C53×#2202C2semidirect
3. C53×#2199C2semidirect
4. C53×#1655C2semidirect
5. C53×#1406C2semidirect
6. C53×#1404C2semidirect
7. C53×#1403C2semidirect
10. C53×H(3,𝔽4)C22semidirect
22. C53×#355C22semidirect
23. C53×#354C22semidirect
26. C53×#329C22semidirect
27. 2−1+4×C53D4semidirect
50. #127×C53D4semidirect
63. #101×C53D4semidirect
69. #97×C53D4semidirect
176. #13677C53direct (N × H)
198. #3288C2×C53semidirect
199. #2202C2×C53semidirect
200. #2199C2×C53semidirect
201. #1655C2×C53semidirect
202. #1406C2×C53semidirect
203. #1404C2×C53semidirect
204. #1403C2×C53semidirect
214. H(3,𝔽4)C22×C53semidirect
226. #355C22×C53semidirect
227. #354C22×C53semidirect
230. #329C22×C53semidirect
231. C53#13677direct (N × H)
232. 2−1+4D4×C53semidirect
255. #127D4×C53semidirect
268. #101D4×C53semidirect
274. #97D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32882
C53×#22022
C53×#21992
C53×#16552
C53×#14062
C53×#14042
C53×#14032
#1367753