Finite Group Databasebeta

← C53×#12722 #27220234

Subgroups — C53×#12722 #27220234

Order 13568.

790 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3508C211yesmaximal, characteristic
2C53×#2134C211yesmaximal, characteristic, complemented
3C53×#2128C211yesmaximal, characteristic, complemented
4C53×#2122C212yesmaximal, complemented
5C53×#1827C212yesmaximal, complemented
6C53×#559—22no—
7C53×#550—22no—
8C53×#546—24no—
9C53×#540—22no—
10C53×#535C2211yescharacteristic
11C53×#535C2211yescharacteristic
12C53×#533C411yescharacteristic, complemented
13C53×#532—22no—
14C2×#127×C53C411yescharacteristic, complemented
15C53×#518C411yescharacteristic, complemented
16C53×#518C411yescharacteristic, complemented
17C4×#41×C53C2211yescharacteristic
18C53×#413C2212yes—
19C53×#410C2212yes—
20C53×#355—22no—
21C53×#353—22no—
22C53×AΣL(1,𝔽16)—24no—
23C53×#339—22no—
24C53×H(3,ℤ/4)—22no—
25C2×#41×C53—24no—
26C2×#41×C53C2311yescharacteristic
27C22×D4×C53—22no—
28C22×D4×C53C2×C411yescharacteristic
29C22×D4×C53C2×C411yescharacteristic
30C2×M16×C53—22no—
31C4⋊Q8×C53—21no—
32#127×C53—21no—
33#127×C53—21no—
34#127×C53—22no—
35#127×C53—21no—
36#127×C53D412yes—
37#125×C53—21no—
38#124×C53—24no—
39#124×C53—21no—
40#124×C53D412yes—
41#123×C53—24no—
42#123×C53—24no—
43C22⋊Q8×C53—22no—
44#121×C53—24no—
45#121×C53—24no—
46#121×C53—24no—
47#121×C53—22no—
48#121×C53—22no—
49#120×C53—22no—
50#120×C53D414yes—
51C4×Q8×C53—22no—
52C4×Q8×C53—21no—
53C4×D4×C53—24no—
54C4×D4×C53—22no—
55C4×D4×C53—22no—
56C4×D4×C53—22no—
57C4×D4×C53—22no—
58C4×D4×C53—22no—
59C4×D4×C53—21no—
60#117×C53—22no—
61#117×C53C2×C411yescharacteristic
62#117×C53C2×C411yescharacteristic
63#117×C53C2×C411yescharacteristic
64#117×C53D412yes—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C42×C53C2×C411yescharacteristic
68C2×C42×C53D412yes—
69#101×C53—22no—
70#100×C53—22no—
71#99×C53—22no—
72#99×C53—22no—
73C24×C53—21no—
74C24×C53C22⋊C412yes—
75#41×C53—42no—
76#41×C53—88no—
77C2×Q8×C53—22no—
78C2×Q8×C53C2×D411yescharacteristic
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—24no—
82C2×D4×C53—24no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53—84no—
89C2×D4×C53C22⋊C412yes—
90C2×D4×C53C2×D411yescharacteristic
91C2×D4×C53C2×D412yes—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—24no—
95C22×C4×C53—24no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D412yes—
98M16×C53—42no—
99M16×C53—42no—
100C2×C8×C53—42no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—42no—
109C22⋊C4×C53—24no—
110C22⋊C4×C53—24no—
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53C22⋊C412yes—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53—22no—
121C42×C53—42no—
122C42×C53C22⋊C412yes—
123C42×C53C22⋊C411yescharacteristic
124C42×C53C22⋊C411yescharacteristic
125C42×C53C22⋊C412yes—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53C2×C22⋊C411yescharacteristic
133C23×C53#12012yes—
134Q8×C53—41no—
135Q8×C53—44no—
136D4×C53—42no—
137D4×C53—41no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—48no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—84no—
144D4×C53—84no—
145D4×C53—84no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53—84no—
157C2×C4×C53—84no—
158C2×C4×C53—82no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53#12012yes—
162C8×C53—82no—
163#12722C5311yesmaximal, characteristic, direct product factor, Sylow
164C22×C53—22no—
165C22×C53—21no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—162no—
173C22×C53—162no—
174C22×C53#37911yescharacteristic
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—21no—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53—162no—
183C4×C53#41012yes—
184#3508C2×C5311yescharacteristic
185#2134C2×C5311yescharacteristic, complemented
186#2128C2×C5311yescharacteristic, complemented
187#2122C2×C5312yescomplemented
188#1827C2×C5312yescomplemented
189C2×C53—21no—
190C2×C53—42no—
191C2×C53—41no—
192C2×C53—82no—
193C2×C53—82no—
194C2×C53#178211yescharacteristic
195#559—22no—
196#550—22no—
197#546—24no—
198#540—22no—
199#535C22×C5311yescharacteristic
200#535C22×C5311yescharacteristic
201#533C4×C5311yescharacteristic, complemented
202#532—22no—
203C2×#127C4×C5311yescharacteristic, complemented
204#518C4×C5311yescharacteristic, complemented
205#518C4×C5311yescharacteristic, complemented
206C4×#41C22×C5311yescharacteristic
207#413C22×C5312yes—
208#410C22×C5312yes—
209#355—22no—
210#353—22no—
211AΣL(1,𝔽16)—24no—
212#339—22no—
213H(3,ℤ/4)—22no—
214C53#1272211yescharacteristic, direct product factor, Sylow
215C2×#41—24no—
216C2×#41C23×C5311yescharacteristic
217C22×D4—22no—
218C22×D4C2×C4×C5311yescharacteristic
219C22×D4C2×C4×C5311yescharacteristic
220C2×M16—22no—
221C4⋊Q8—21no—
222#127—21no—
223#127—22no—
224#127—21no—
225#127—21no—
226#127D4×C5312yes—
227#125—21no—
228#124—24no—
229#124—21no—
230#124D4×C5312yes—
231#123—24no—
232#123—24no—
233C22⋊Q8—22no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—22no—
238#121—22no—
239#120—22no—
240#120D4×C5314yes—
241C4×Q8—21no—
242C4×Q8—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—21no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250#117—22no—
251#117C2×C4×C5311yescharacteristic
252#117C2×C4×C5311yescharacteristic
253#117C2×C4×C5311yescharacteristic
254#117D4×C5312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42C2×C4×C5311yescharacteristic
258C2×C42D4×C5312yes—
259#101—22no—
260#100—22no—
261#99—22no—
262#99—22no—
263C24—21no—
264C24C22⋊C4×C5312yes—
265#41—42no—
266#41—88no—
267C2×Q8—22no—
268C2×Q8C2×D4×C5311yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4C22⋊C4×C5312yes—
280C2×D4C2×D4×C5311yescharacteristic
281C2×D4C2×D4×C5312yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C22×C4×C5311yescharacteristic
287C22×C4C2×D4×C5312yes—
288M16—42no—
289M16—42no—
290C2×C8—42no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4C22⋊C4×C5312yes—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—42no—
312C42C22⋊C4×C5311yescharacteristic
313C42C22⋊C4×C5311yescharacteristic
314C42C22⋊C4×C5312yes—
315C42C22⋊C4×C5312yes—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23C2×C22⋊C4×C5311yescharacteristic
323C23#120×C5312yes—
324Q8—41no—
325Q8—44no—
326D4—42no—
327D4—41no—
328D4—42no—
329D4—42no—
330D4—48no—
331D4—42no—
332D4—42no—
333D4—84no—
334D4—84no—
335D4—84no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4C2×C22⋊C4×C5311yescharacteristic
350C2×C4C2×C22⋊C4×C5311yescharacteristic
351C2×C4#120×C5312yes—
352C8—82no—
353C22—22no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—162no—
362C22—162no—
363C22C53×#37911yescharacteristic
364C4—22no—
365C4—22no—
366C4—21no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—162no—
372C4C53×#41012yes—
373C2—21no—
374C2—42no—
375C2—41no—
376C2—82no—
377C2—82no—
378C2C53×#178211yescharacteristic
subgroupG/H
center, socleC2×C53#1782
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×#41C23×C53
prime radicalC53×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2134C2semidirect
3. C53×#2128C2semidirect
4. C53×#2122C2semidirect
5. C53×#1827C2semidirect
12. C53×#533C4semidirect
14. C2×#127×C53C4semidirect
15. C53×#518C4semidirect
16. C53×#518C4semidirect
163. #12722C53direct (N × H)
185. #2134C2×C53semidirect
186. #2128C2×C53semidirect
187. #2122C2×C53semidirect
188. #1827C2×C53semidirect
201. #533C4×C53semidirect
203. C2×#127C4×C53semidirect
204. #518C4×C53semidirect
205. #518C4×C53semidirect
214. C53#12722direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35082
C53×#21342
C53×#21282
C53×#21222
C53×#18272
#1272253