Finite Group Databasebeta

← C53×#8553 #27216065

Subgroups — C53×#8553 #27216065

Order 13568.

1166 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#123×C53C211yesmaximal, characteristic
2C53×#1821C212yesmaximal, complemented
3C53×#1462C212yesmaximal, complemented
4C53×#1462C212yesmaximal, complemented
5C23×D4×C53C411yescharacteristic
6C24×C4×C53C411yescharacteristic
7C2×#123×C53—24no—
8C2×#123×C53—24no—
9C2×#123×C53—24no—
10C22×C4⋊C4×C53C411yescharacteristic
11C22×C4⋊C4×C53C2211yescharacteristic
12C22×C22⋊C4×C53C411yescharacteristic
13C22×C22⋊C4×C53C2211yescharacteristic
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#407C2214yes—
16C53×#343—24no—
17C53×#323—24no—
18C53×#323—24no—
19C25×C53C2×C411yescharacteristic
20C25×C53C2×C411yescharacteristic
21C22×D4×C53—22no—
22C22×D4×C53—22no—
23C22×D4×C53—22no—
24C22×D4×C53—44no—
25C23×C4×C53—22no—
26C23×C4×C53—22no—
27C23×C4×C53—22no—
28C23×C4×C53—44no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C2×M16×C53—24no—
35#123×C53—232no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53D414yes—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53D414yes—
50C2×C22⋊C4×C53D414yes—
51C22⋊C8×C53—24no—
52C22⋊C8×C53—24no—
53C24×C53—24no—
54C24×C53—21no—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—42no—
59C24×C53—42no—
60C24×C53—42no—
61C24×C53—42no—
62C24×C53—42no—
63C24×C53C22⋊C412yes—
64C24×C53C22⋊C412yes—
65C24×C53C22×C411yescharacteristic
66C2×D4×C53—28no—
67C2×D4×C53—48no—
68C2×D4×C53—44no—
69C2×D4×C53—48no—
70C2×D4×C53—44no—
71C22×C4×C53—28no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—42no—
81C22×C4×C53—48no—
82C22×C4×C53—48no—
83C22×C4×C53—44no—
84C22×C4×C53—44no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C22⋊C412yes—
90C22×C4×C53C22⋊C412yes—
91C22×C4×C53C22⋊C412yes—
92C22×C4×C53C22⋊C412yes—
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96M16×C53—48no—
97C2×C8×C53—44no—
98C4⋊C4×C53—28no—
99C4⋊C4×C53—28no—
100C22⋊C4×C53—28no—
101C22⋊C4×C53—28no—
102C22⋊C4×C53—28no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—24no—
107C23×C53—24no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—24no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—416no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—88no—
128C23×C53#10014yes—
129C23×C53C2×C22⋊C411yescharacteristic
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53C2×C22⋊C411yescharacteristic
132D4×C53—416no—
133C2×C4×C53—24no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—416no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53#12014yes—
154C8×C53—84no—
155#8553C5311yesmaximal, characteristic, direct product factor, Sylow
156C22×C53—28no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—24no—
164C22×C53—44no—
165C22×C53—48no—
166C22×C53—44no—
167C22×C53—48no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—84no—
174C22×C53—84no—
175C22×C53—84no—
176C22×C53#37911yescharacteristic
177C22×C53#41012yes—
178C22×C53C2×#10012yes—
179C22×C53#41312yes—
180C4×C53—24no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—42no—
185C22×#123C2×C5311yescharacteristic
186#1821C2×C5312yescomplemented
187#1462C2×C5312yescomplemented
188#1462C2×C5312yescomplemented
189C2×C53—22no—
190C2×C53—22no—
191C2×C53—22no—
192C2×C53—44no—
193C2×C53—82no—
194C2×C53#178211yescharacteristic
195C2×C53#178311yescharacteristic
196C2×C53#178511yescharacteristic
197C23×D4C4×C5311yescharacteristic
198C24×C4C4×C5311yescharacteristic
199C2×#123—24no—
200C2×#123—24no—
201C2×#123—24no—
202C22×C4⋊C4C4×C5311yescharacteristic
203C22×C4⋊C4C22×C5311yescharacteristic
204C22×C22⋊C4C4×C5311yescharacteristic
205C22×C22⋊C4C22×C5311yescharacteristic
206C22×C22⋊C4C22×C5311yescharacteristic
207#407C22×C5314yes—
208#343—24no—
209#323—24no—
210#323—24no—
211C53#855311yescharacteristic, direct product factor, Sylow
212C25C2×C4×C5311yescharacteristic
213C25C2×C4×C5311yescharacteristic
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4C2×C4×C5311yescharacteristic
223C23×C4C2×C4×C5311yescharacteristic
224C23×C4C2×C4×C5311yescharacteristic
225C23×C4C2×C4×C5311yescharacteristic
226C23×C4C23×C5311yescharacteristic
227C2×M16—24no—
228#123—232no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4D4×C5314yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4D4×C5314yes—
243C2×C22⋊C4D4×C5314yes—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊C4×C5312yes—
257C24C22⋊C4×C5312yes—
258C24C22×C4×C5311yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊C4×C5312yes—
283C22×C4C22⋊C4×C5312yes—
284C22×C4C22⋊C4×C5312yes—
285C22×C4C22⋊C4×C5312yes—
286C22×C4C2×D4×C5312yes—
287C22×C4C2×D4×C5312yes—
288C22×C4C2×D4×C5312yes—
289M16—48no—
290C2×C8—44no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#100×C5314yes—
322C23C2×C22⋊C4×C5311yescharacteristic
323C23C2×C22⋊C4×C5311yescharacteristic
324C23C2×C22⋊C4×C5311yescharacteristic
325D4—416no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—416no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4#120×C5314yes—
347C8—84no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22C53×#37911yescharacteristic
369C22C53×#41012yes—
370C22C2×#100×C5312yes—
371C22C53×#41312yes—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2C53×#178211yescharacteristic
383C2C53×#178311yescharacteristic
384C2C53×#178511yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC24C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC23×D4×C53C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1821C2semidirect
3. C53×#1462C2semidirect
4. C53×#1462C2semidirect
155. #8553C53direct (N × H)
186. #1821C2×C53semidirect
187. #1462C2×C53semidirect
188. #1462C2×C53semidirect
211. C53#8553direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#123×C532
C53×#18212
C53×#14622
#855353