Finite Group Databasebeta

← C53×#12204 #27219716

Subgroups — C53×#12204 #27219716

Order 13568.

474 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2600C211yesmaximal, characteristic
2C53×#2030C211yesmaximal, characteristic, complemented
3C53×#1608C211yesmaximal, characteristic, complemented
4C53×#1596C211yesmaximal, characteristic
5C53×#1344C211yesmaximal, characteristic, complemented
6C53×#1343C211yesmaximal, characteristic
7C53×#1317C211yesmaximal, characteristic, complemented
8C53×#543C2211yescharacteristic
9C2×C4×D4×C53—21no—
10C53×#483—21no—
11C53×#450—21no—
12C53×#397C2211yescharacteristic
13C53×#394—21no—
14C53×#392C2211yescharacteristic
15C53×#388—21no—
16C53×#387—21no—
17C53×#386—21no—
18C53×#385—21no—
19C53×#381—21no—
20C53×#381C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2211yescharacteristic
23C53×#328—21no—
24C53×#328—21no—
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×D4×C53—21no—
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53D411yescharacteristic
31C2×SD16×C53—41no—
32C22⋊Q8×C53—21no—
33C22⋊Q8×C53—21no—
34C22⋊Q8×C53—41no—
35C22⋊Q8×C53—41no—
36C4×D4×C53—42no—
37C4×D4×C53—42no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53C2311yescharacteristic
50C2×C42×C53—21no—
51#106×C53—41no—
52#103×C53—41no—
53#102×C53—41no—
54C22⋊C8×C53—21no—
55C22⋊C8×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—41no—
61#95×C53—41no—
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64C24×C53—21no—
65C24×C53C2×D411yescharacteristic
66C2×Q8×C53—21no—
67C2×Q8×C53—21no—
68C2×D4×C53—42no—
69C2×D4×C53—41no—
70C2×D4×C53—42no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—42no—
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—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—81no—
87C22×C4×C53—81no—
88C22×C4×C53SD1612yescomplemented
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91SD16×C53—82no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—81no—
113C42×C53—42no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53#12011yescharacteristic
126C23×C53C2×SD1611yescharacteristic
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128Q8×C53—81no—
129Q8×C53—81no—
130D4×C53—42no—
131D4×C53—44no—
132D4×C53—44no—
133D4×C53—81no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
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—41no—
156C2×C4×C53—81no—
157C2×C4×C53—84no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C8×C53—81no—
164C8×C53—81no—
165#12204C5311yesmaximal, characteristic, direct product factor, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—42no—
173C22×C53—42no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—42no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#45011yescharacteristic
183C22×C53#45511yescharacteristic
184C22×C53B(4,𝔽2)11yescharacteristic
185C4×C53—42no—
186C4×C53—42no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—161no—
195#2600C2×C5311yescharacteristic
196#2030C2×C5311yescharacteristic, complemented
197#1608C2×C5311yescharacteristic, complemented
198#1596C2×C5311yescharacteristic
199#1344C2×C5311yescharacteristic, complemented
200#1343C2×C5311yescharacteristic
201#1317C2×C5311yescharacteristic, complemented
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—41no—
205C2×C53—41no—
206C2×C53—82no—
207C2×C53#160111yescharacteristic
208C2×C53#219711yescharacteristic
209C2×C53#219811yescharacteristic
210#543C22×C5311yescharacteristic
211C2×C4×D4—21no—
212#483—21no—
213#450—21no—
214#397C22×C5311yescharacteristic
215#394—21no—
216#392C22×C5311yescharacteristic
217#388—21no—
218#387—21no—
219#386—21no—
220#385—21no—
221#381—21no—
222#381C22×C5311yescharacteristic
223#342—21no—
224#342C22×C5311yescharacteristic
225#328—21no—
226#328—21no—
227#323C22×C5311yescharacteristic
228#323C22×C5311yescharacteristic
229C53#1220411yescharacteristic, direct product factor, Sylow
230C22×D4—21no—
231C22×D4D4×C5311yescharacteristic
232C23×C4—21no—
233C23×C4D4×C5311yescharacteristic
234C2×SD16—41no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239C4×D4—42no—
240C4×D4—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4D4×C5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4D4×C5311yescharacteristic
252C2×C22⋊C4C23×C5311yescharacteristic
253C2×C42—21no—
254#106—41no—
255#103—41no—
256#102—41no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265#95D4×C5311yescharacteristic
266#95D4×C5311yescharacteristic
267C24—21no—
268C24C2×D4×C5311yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4SD16×C5312yescomplemented
292C22×C4C2×D4×C5311yescharacteristic
293C22×C4C2×D4×C5311yescharacteristic
294SD16—82no—
295C2×C8—41no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C42—42no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#120×C5311yescharacteristic
329C23C2×SD16×C5311yescharacteristic
330C23AGL(1,ℤ/8)×C5311yescharacteristic
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—44no—
335D4—44no—
336D4—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C53×#45011yescharacteristic
385C22C53×#45511yescharacteristic
386C22C53×B(4,𝔽2)11yescharacteristic
387C4—42no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—161no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—82no—
402C2C53×#160111yescharacteristic
403C2C53×#219711yescharacteristic
404C2C53×#219811yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2030C2semidirect
3. C53×#1608C2semidirect
5. C53×#1344C2semidirect
7. C53×#1317C2semidirect
88. C22×C4×C53SD16semidirect
165. #12204C53direct (N × H)
196. #2030C2×C53semidirect
197. #1608C2×C53semidirect
199. #1344C2×C53semidirect
201. #1317C2×C53semidirect
229. C53#12204direct (N × H)
291. C22×C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26002
C53×#20302
C53×#16082
C53×#15962
C53×#13442
C53×#13432
C53×#13172
#1220453