Finite Group Databasebeta

← C53×#12226 #27219738

Subgroups — C53×#12226 #27219738

Order 13568.

510 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2455C211yesmaximal, characteristic
2C53×#2062C211yesmaximal, characteristic, complemented
3C53×#1614C211yesmaximal, characteristic, complemented
4C53×#1596C211yesmaximal, characteristic
5C53×#1352C211yesmaximal, characteristic, complemented
6C53×#1343C211yesmaximal, characteristic
7C53×#1326C211yesmaximal, characteristic, complemented
8C53×#543C2211yescharacteristic
9C2×#123×C53—21no—
10C2×C4×D4×C53—21no—
11C53×#484—21no—
12C53×#450—21no—
13C53×#399C2211yescharacteristic
14C53×#394—21no—
15C53×#392C2211yescharacteristic
16C53×#389—21no—
17C53×#388—21no—
18C53×#385—21no—
19C53×#385—21no—
20C53×#385C2211yescharacteristic
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—
32#123×C53—44no—
33C22⋊Q8×C53—21no—
34C22⋊Q8×C53—21no—
35C22⋊Q8×C53—41no—
36C22⋊Q8×C53—41no—
37C4×D4×C53—44no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—21no—
54#107×C53—41no—
55#103×C53—41no—
56#102×C53—41no—
57C22⋊C8×C53—21no—
58C22⋊C8×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
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—42no—
82C22×C4×C53—42no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—42no—
86C22×C4×C53—81no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89SD16×C53—82no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—82no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C42×C53—42no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—22no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#12011yescharacteristic
127C23×C53#13511yescharacteristic
128C23×C53#13711yescharacteristic
129Q8×C53—81no—
130Q8×C53—81no—
131D4×C53—42no—
132D4×C53—44no—
133D4×C53—44no—
134D4×C53—81no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—84no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C8×C53—81no—
165C8×C53—81no—
166#12226C5311yesmaximal, characteristic, direct product factor, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—42no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#45211yescharacteristic
184C22×C53#45611yescharacteristic
185C22×C53B(4,𝔽2)11yescharacteristic
186C4×C53—42no—
187C4×C53—42no—
188C4×C53—41no—
189C4×C53—42no—
190C4×C53—41no—
191C4×C53—81no—
192C4×C53—82no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53—81no—
196#2455C2×C5311yescharacteristic
197#2062C2×C5311yescharacteristic, complemented
198#1614C2×C5311yescharacteristic, complemented
199#1596C2×C5311yescharacteristic
200#1352C2×C5311yescharacteristic, complemented
201#1343C2×C5311yescharacteristic
202#1326C2×C5311yescharacteristic, complemented
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—41no—
206C2×C53—41no—
207C2×C53—82no—
208C2×C53#160311yescharacteristic
209C2×C53#219611yescharacteristic
210C2×C53#219811yescharacteristic
211#543C22×C5311yescharacteristic
212C2×#123—21no—
213C2×C4×D4—21no—
214#484—21no—
215#450—21no—
216#399C22×C5311yescharacteristic
217#394—21no—
218#392C22×C5311yescharacteristic
219#389—21no—
220#388—21no—
221#385—21no—
222#385—21no—
223#385C22×C5311yescharacteristic
224#342—21no—
225#342C22×C5311yescharacteristic
226#328—21no—
227#328—21no—
228#323C22×C5311yescharacteristic
229#323C22×C5311yescharacteristic
230C53#1222611yescharacteristic, direct product factor, Sylow
231C22×D4—21no—
232C22×D4D4×C5311yescharacteristic
233C23×C4—21no—
234C23×C4D4×C5311yescharacteristic
235C2×SD16—41no—
236#123—44no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C4×D4—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4D4×C5311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4D4×C5311yescharacteristic
256C2×C22⋊C4C23×C5311yescharacteristic
257C2×C42—21no—
258#107—41no—
259#103—41no—
260#102—41no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95D4×C5311yescharacteristic
267#95D4×C5311yescharacteristic
268C24—21no—
269C24C2×D4×C5311yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
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—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4C2×D4×C5311yescharacteristic
292C22×C4C2×D4×C5311yescharacteristic
293SD16—82no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—42no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#120×C5311yescharacteristic
331C23#135×C5311yescharacteristic
332C23#137×C5311yescharacteristic
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—44no—
337D4—44no—
338D4—81no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C8—81no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C53×#45211yescharacteristic
387C22C53×#45611yescharacteristic
388C22C53×B(4,𝔽2)11yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2C53×#160311yescharacteristic
405C2C53×#219611yescharacteristic
406C2C53×#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×#2062C2semidirect
3. C53×#1614C2semidirect
5. C53×#1352C2semidirect
7. C53×#1326C2semidirect
166. #12226C53direct (N × H)
197. #2062C2×C53semidirect
198. #1614C2×C53semidirect
200. #1352C2×C53semidirect
202. #1326C2×C53semidirect
230. C53#12226direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24552
C53×#20622
C53×#16142
C53×#15962
C53×#13522
C53×#13432
C53×#13262
#1222653