Finite Group Databasebeta

← C53×#13680 #27221192

Subgroups — C53×#13680 #27221192

Order 13568.

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

normal subgroups only

This group has 378 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
2C53×#2203C211yesmaximal, characteristic, complemented
3C53×#2201C211yesmaximal, characteristic, complemented
4C53×#1660C211yesmaximal, characteristic, complemented
5C53×#1407C211yesmaximal, characteristic, complemented
6C53×#1404C211yesmaximal, characteristic, complemented
7C53×#1403C211yesmaximal, characteristic, complemented
8C53×#578—21no—
9C53×#577—21no—
10C53×#563C2211yescharacteristic
11C53×#501—21no—
12C53×#497C2211yescharacteristic
13C53×#473—21no—
14C53×#471—21no—
15C53×#458—21no—
16C53×#456—21no—
17C53×#456—21no—
18C53×#455C2211yescharacteristic
19C53×#454C2211yescharacteristic
20C53×#444—21no—
21C53×#356—21no—
22C53×#355C2211yescharacteristic
23C53×#354C2211yescharacteristic
24C53×#352—21no—
25C53×#330—21no—
26C53×#329C2211yescharacteristic
272−1+4×C53—21no—
282−1+4×C53D411yescharacteristic
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—
36#137×C53—41no—
37AGL(1,ℤ/8)×C53—21no—
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×C53—21no—
49C4⋊Q8×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#107×C53—41no—
58#106×C53—41no—
59#104×C53—21no—
60#104×C53—21no—
61#104×C53—41no—
62#101×C53—21no—
63#101×C53—21no—
64#101×C53D411yescharacteristic
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—
77#41×C53—81no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53C2×D411yescharacteristic
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53C2×D411yescharacteristic
87C22×C4×C53—21no—
88Q16×C53—21no—
89Q16×C53—41no—
90Q16×C53—41no—
91Q16×C53—41no—
92Q16×C53—41no—
93Q16×C53—41no—
94Q16×C53—81no—
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—81no—
104D8×C53—21no—
105D8×C53—41no—
106M16×C53—21no—
107M16×C53—21no—
108M16×C53—41no—
109M16×C53—41no—
110M16×C53—41no—
111C2×C8×C53—21no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—41no—
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—
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—41no—
134Q8×C53—41no—
135Q8×C53—41no—
136Q8×C53—41no—
137Q8×C53—81no—
138Q8×C53—81no—
139D4×C53—21no—
140D4×C53—21no—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—81no—
146D4×C53—81no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53#12011yescharacteristic
161C8×C53—21no—
162C8×C53—21no—
163C8×C53—41no—
164C8×C53—41no—
165C8×C53—42no—
166C8×C53—81no—
167#13680C5311yesmaximal, characteristic, direct product factor, Sylow
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53—161no—
173C22×C53B(4,𝔽2)11yescharacteristic
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—81no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C53—161no—
183#3289C2×C5311yescharacteristic
184#2203C2×C5311yescharacteristic, complemented
185#2201C2×C5311yescharacteristic, complemented
186#1660C2×C5311yescharacteristic, complemented
187#1407C2×C5311yescharacteristic, complemented
188#1404C2×C5311yescharacteristic, complemented
189#1403C2×C5311yescharacteristic, complemented
190C2×C53—21no—
191C2×C53—81no—
192C2×C53—81no—
193C2×C53—81no—
194C2×C53#219611yescharacteristic
195#578—21no—
196#577—21no—
197#563C22×C5311yescharacteristic
198#501—21no—
199#497C22×C5311yescharacteristic
200#473—21no—
201#471—21no—
202#458—21no—
203#456—21no—
204#456—21no—
205#455C22×C5311yescharacteristic
206#454C22×C5311yescharacteristic
207#444—21no—
208#356—21no—
209#355C22×C5311yescharacteristic
210#354C22×C5311yescharacteristic
211#352—21no—
212#330—21no—
213#329C22×C5311yescharacteristic
214C53#1368011yescharacteristic, direct product factor, Sylow
2152−1+4—21no—
2162−1+4D4×C5311yescharacteristic
217H(5,𝔽2)D4×C5311yescharacteristic
218#137—21no—
219#137—21no—
220#137—21no—
221#137—41no—
222#137—41no—
223#137—41no—
224#137—41no—
225AGL(1,ℤ/8)—21no—
226AGL(1,ℤ/8)—41no—
227#135—21no—
228#135—41no—
229#135—41no—
230C2×Q16—21no—
231C2×Q16—41no—
232C2×SD16—21no—
233C2×SD16—41no—
234#131—21no—
235#131—21no—
236C4⋊Q8—21no—
237C4⋊Q8D4×C5311yescharacteristic
238#126—41no—
239#125—21no—
240#124C23×C5311yescharacteristic
241#123—21no—
242#123—41no—
243C22⋊Q8—41no—
244#108—21no—
245#107—41no—
246#106—41no—
247#104—21no—
248#104—21no—
249#104—41no—
250#101—21no—
251#101—21no—
252#101D4×C5311yescharacteristic
253#100D4×C5311yescharacteristic
254#99—21no—
255#99—41no—
256#97—21no—
257#97D4×C5311yescharacteristic
258#41—21no—
259#41—21no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—81no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8C2×D4×C5311yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4C2×D4×C5311yescharacteristic
275C22×C4—21no—
276Q16—21no—
277Q16—41no—
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282Q16—81no—
283SD16—21no—
284SD16—21no—
285SD16—41no—
286SD16—41no—
287SD16—41no—
288SD16—41no—
289SD16—41no—
290SD16—41no—
291SD16—81no—
292D8—21no—
293D8—41no—
294M16—21no—
295M16—21no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—21no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C22⋊C4—21no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C42—21no—
316C42C2×D4×C5311yescharacteristic
317C23—21no—
318C23—41no—
319Q8—21no—
320Q8—21no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—81no—
326Q8—81no—
327D4—21no—
328D4—21no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4#120×C5311yescharacteristic
349C8—21no—
350C8—21no—
351C8—41no—
352C8—41no—
353C8—42no—
354C8—81no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—161no—
360C22C53×B(4,𝔽2)11yescharacteristic
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—161no—
370C2—21no—
371C2—81no—
372C2—81no—
373C2—81no—
374C2C53×#219611yescharacteristic
subgroupG/H
center, socleC2×C53#2196
commutator, Frattini, cosocle#124C23×C53
prime radicalC53×#3289C2
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×#1660C2semidirect
5. C53×#1407C2semidirect
6. C53×#1404C2semidirect
7. C53×#1403C2semidirect
167. #13680C53direct (N × H)
184. #2203C2×C53semidirect
185. #2201C2×C53semidirect
186. #1660C2×C53semidirect
187. #1407C2×C53semidirect
188. #1404C2×C53semidirect
189. #1403C2×C53semidirect
214. C53#13680direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32892
C53×#22032
C53×#22012
C53×#16602
C53×#14072
C53×#14042
C53×#14032
#1368053