Finite Group Databasebeta

← C53×#13679 #27221191

Subgroups — C53×#13679 #27221191

Order 13568.

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

normal subgroups only

This group has 386 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
2C53×#2203C211yesmaximal, characteristic, complemented
3C53×#2201C211yesmaximal, characteristic, complemented
4C53×#1657C211yesmaximal, characteristic, complemented
5C53×#1406C211yesmaximal, characteristic, complemented
6C53×#1404C211yesmaximal, characteristic, complemented
7C53×#1403C211yesmaximal, characteristic, complemented
8C53×#577—21no—
9C53×#576—21no—
10C53×#563C2211yescharacteristic, complemented
11C53×#496C2211yescharacteristic
12C53×#490—21no—
13C53×#472—21no—
14C53×#471—21no—
15C53×#458—21no—
16C53×#456—21no—
17C53×#455C2211yescharacteristic
18C53×#454C2211yescharacteristic
19C53×#453—21no—
20C53×#444—21no—
21C53×#356—21no—
22C53×#355C2211yescharacteristic, complemented
23C53×#354C2211yescharacteristic, complemented
24C53×#352—21no—
25C53×#330—21no—
26C53×#329C2211yescharacteristic, complemented
272−1+4×C53D411yescharacteristic
28H(5,𝔽2)×C53—21no—
29H(5,𝔽2)×C53D411yescharacteristic
30#137×C53—21no—
31#137×C53—41no—
32#137×C53—41no—
33AGL(1,ℤ/8)×C53—21no—
34AGL(1,ℤ/8)×C53—21no—
35AGL(1,ℤ/8)×C53—21no—
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×D8×C53—21no—
45C2×D8×C53—41no—
46#131×C53—21no—
47#131×C53—21no—
48C4⋊Q8×C53—21no—
49#127×C53D411yescharacteristic
50#126×C53—41no—
51#125×C53—21no—
52#124×C53C2311yescharacteristic
53#123×C53—21no—
54#123×C53—41no—
55C22⋊Q8×C53—41no—
56#108×C53—21no—
57#104×C53—21no—
58#104×C53—21no—
59#104×C53—41no—
60#102×C53—41no—
61#102×C53—41no—
62#101×C53—21no—
63#101×C53D411yescharacteristic
64#100×C53—21no—
65#100×C53D411yescharacteristic
66#99×C53—21no—
67#99×C53—41no—
68#97×C53—21no—
69#97×C53D411yescharacteristic
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—41no—
73#41×C53—41no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77C2×Q8×C53—21no—
78C2×Q8×C53—41no—
79C2×Q8×C53C2×D411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—81no—
87C2×D4×C53C2×D411yescharacteristic
88C22×C4×C53—21no—
89Q16×C53—21no—
90Q16×C53—41no—
91Q16×C53—81no—
92SD16×C53—21no—
93SD16×C53—21no—
94SD16×C53—41no—
95SD16×C53—41no—
96SD16×C53—41no—
97SD16×C53—41no—
98SD16×C53—41no—
99SD16×C53—41no—
100SD16×C53—81no—
101D8×C53—21no—
102D8×C53—41no—
103D8×C53—41no—
104D8×C53—41no—
105D8×C53—41no—
106D8×C53—41no—
107M16×C53—21no—
108M16×C53—21no—
109M16×C53—41no—
110M16×C53—41no—
111M16×C53—41no—
112C2×C8×C53—21no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—81no—
122C22⋊C4×C53—21no—
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—
133Q8×C53—21no—
134Q8×C53—21no—
135Q8×C53—41no—
136Q8×C53—41no—
137Q8×C53—81no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149D4×C53—82no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53#12011yescharacteristic
162C8×C53—21no—
163C8×C53—21no—
164C8×C53—41no—
165C8×C53—41no—
166C8×C53—81no—
167C8×C53—81no—
168#13679C5311yesmaximal, characteristic, direct product factor, Sylow
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—161no—
175C22×C53—161no—
176C22×C53—321no—
177C22×C53B(4,𝔽2)11yescharacteristic
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53—161no—
186#3288C2×C5311yescharacteristic
187#2203C2×C5311yescharacteristic, complemented
188#2201C2×C5311yescharacteristic, complemented
189#1657C2×C5311yescharacteristic, complemented
190#1406C2×C5311yescharacteristic, complemented
191#1404C2×C5311yescharacteristic, complemented
192#1403C2×C5311yescharacteristic, complemented
193C2×C53—21no—
194C2×C53—81no—
195C2×C53—81no—
196C2×C53—81no—
197C2×C53—161no—
198C2×C53#219611yescharacteristic
199#577—21no—
200#576—21no—
201#563C22×C5311yescharacteristic, complemented
202#496C22×C5311yescharacteristic
203#490—21no—
204#472—21no—
205#471—21no—
206#458—21no—
207#456—21no—
208#455C22×C5311yescharacteristic
209#454C22×C5311yescharacteristic
210#453—21no—
211#444—21no—
212#356—21no—
213#355C22×C5311yescharacteristic, complemented
214#354C22×C5311yescharacteristic, complemented
215#352—21no—
216#330—21no—
217#329C22×C5311yescharacteristic, complemented
218C53#1367911yescharacteristic, direct product factor, Sylow
2192−1+4D4×C5311yescharacteristic
220H(5,𝔽2)—21no—
221H(5,𝔽2)D4×C5311yescharacteristic
222#137—21no—
223#137—41no—
224#137—41no—
225AGL(1,ℤ/8)—21no—
226AGL(1,ℤ/8)—21no—
227AGL(1,ℤ/8)—21no—
228AGL(1,ℤ/8)—41no—
229AGL(1,ℤ/8)—41no—
230AGL(1,ℤ/8)—41no—
231#135—21no—
232#135—41no—
233#135—41no—
234C2×SD16—21no—
235C2×SD16—41no—
236C2×D8—21no—
237C2×D8—41no—
238#131—21no—
239#131—21no—
240C4⋊Q8—21no—
241#127D4×C5311yescharacteristic
242#126—41no—
243#125—21no—
244#124C23×C5311yescharacteristic
245#123—21no—
246#123—41no—
247C22⋊Q8—41no—
248#108—21no—
249#104—21no—
250#104—21no—
251#104—41no—
252#102—41no—
253#102—41no—
254#101—21no—
255#101D4×C5311yescharacteristic
256#100—21no—
257#100D4×C5311yescharacteristic
258#99—21no—
259#99—41no—
260#97—21no—
261#97D4×C5311yescharacteristic
262#41—21no—
263#41—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8C2×D4×C5311yescharacteristic
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—81no—
279C2×D4C2×D4×C5311yescharacteristic
280C22×C4—21no—
281Q16—21no—
282Q16—41no—
283Q16—81no—
284SD16—21no—
285SD16—21no—
286SD16—41no—
287SD16—41no—
288SD16—41no—
289SD16—41no—
290SD16—41no—
291SD16—41no—
292SD16—81no—
293D8—21no—
294D8—41no—
295D8—41no—
296D8—41no—
297D8—41no—
298D8—41no—
299M16—21no—
300M16—21no—
301M16—41no—
302M16—41no—
303M16—41no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C22⋊C4—21no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C42—21no—
320C42C2×D4×C5311yescharacteristic
321C23—21no—
322C23—41no—
323C23—41no—
324C23—81no—
325Q8—21no—
326Q8—21no—
327Q8—41no—
328Q8—41no—
329Q8—81no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#120×C5311yescharacteristic
354C8—21no—
355C8—21no—
356C8—41no—
357C8—41no—
358C8—81no—
359C8—81no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—81no—
365C22—161no—
366C22—161no—
367C22—321no—
368C22C53×B(4,𝔽2)11yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—161no—
377C2—21no—
378C2—81no—
379C2—81no—
380C2—81no—
381C2—161no—
382C2C53×#219611yescharacteristic
subgroupG/H
center, socleC2×C53#2196
commutator, Frattini, cosocle#124C23×C53
prime radicalC53×#3288C2
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
2. C53×#2203C2semidirect
3. C53×#2201C2semidirect
4. C53×#1657C2semidirect
5. C53×#1406C2semidirect
6. C53×#1404C2semidirect
7. C53×#1403C2semidirect
10. C53×#563C22semidirect
22. C53×#355C22semidirect
23. C53×#354C22semidirect
26. C53×#329C22semidirect
168. #13679C53direct (N × H)
187. #2203C2×C53semidirect
188. #2201C2×C53semidirect
189. #1657C2×C53semidirect
190. #1406C2×C53semidirect
191. #1404C2×C53semidirect
192. #1403C2×C53semidirect
201. #563C22×C53semidirect
213. #355C22×C53semidirect
214. #354C22×C53semidirect
217. #329C22×C53semidirect
218. C53#13679direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32882
C53×#22032
C53×#22012
C53×#16572
C53×#14062
C53×#14042
C53×#14032
#1367953