Finite Group Databasebeta

← C53×#7776 #27215288

Subgroups — C53×#7776 #27215288

Order 13568.

446 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2727C211yesmaximal, characteristic
2C53×#2058C211yesmaximal, characteristic, complemented
3C53×#2043C211yesmaximal, characteristic, complemented
4C53×#2023C211yesmaximal, characteristic
5C53×#1271C211yesmaximal, characteristic, complemented
6C53×#1271C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×#126×C53—21no—
10C2×#106×C53—21no—
11C2×#103×C53—21no—
12C2×#103×C53—21no—
13C2×#102×C53—21no—
14C53×#399—21no—
15C53×#396—21no—
16C53×#394—21no—
17C53×#390C2211yescharacteristic
18C53×#389—21no—
19C53×#389C2211yescharacteristic
20C53×#389C2211yescharacteristic
21C53×#388—21no—
22C53×#340—21no—
23C53×#340—21no—
24C53×#336C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Q8×C53—21no—
28C22×D4×C53D411yescharacteristic
29C22×C8×C53—21no—
30C22×C8×C53—21no—
31C22×C8×C53—21no—
32C4⋊Q8×C53—42no—
33#126×C53—42no—
34#126×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
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×C53—21no—
52C2×C42×C53—21no—
53C2×C42×C53C2311yescharacteristic
54#106×C53—42no—
55#103×C53—42no—
56#103×C53—42no—
57#102×C53—42no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—81no—
106C4⋊C4×C53SD1612yes—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—81no—
114C22⋊C4×C53—81no—
115C42×C53—41no—
116C42×C53—42no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53#12011yescharacteristic
125Q8×C53—42no—
126Q8×C53—42no—
127Q8×C53—81no—
128Q8×C53—81no—
129D4×C53—42no—
130D4×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
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—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C8×C53—42no—
162C8×C53—81no—
163C8×C53—81no—
164#7776C5311yesmaximal, characteristic, direct product factor, Sylow
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#44911yescharacteristic
173C22×C53#45011yescharacteristic
174C22×C53#45211yescharacteristic
175C22×C53#45411yescharacteristic
176C22×C53#45511yescharacteristic
177C22×C53#45611yescharacteristic
178C22×C53B(4,𝔽2)11yescharacteristic
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—42no—
186C4×C53—82no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—81no—
191#2727C2×C5311yescharacteristic
192#2058C2×C5311yescharacteristic, complemented
193#2043C2×C5311yescharacteristic, complemented
194#2023C2×C5311yescharacteristic
195#1271C2×C5311yescharacteristic, complemented
196#1271C2×C5311yescharacteristic, complemented
197#1270C2×C5311yescharacteristic
198C2×C53—161no—
199C2×C53#159711yescharacteristic
200C2×C53#160411yescharacteristic
201C2×C53#160611yescharacteristic
202C2×C53#163511yescharacteristic
203C2×C53#164211yescharacteristic
204C2×C53#165211yescharacteristic
205C2×C53#166411yescharacteristic
206C2×C4⋊Q8C22×C5311yescharacteristic
207C2×#126—21no—
208C2×#106—21no—
209C2×#103—21no—
210C2×#103—21no—
211C2×#102—21no—
212#399—21no—
213#396—21no—
214#394—21no—
215#390C22×C5311yescharacteristic
216#389—21no—
217#389C22×C5311yescharacteristic
218#389C22×C5311yescharacteristic
219#388—21no—
220#340—21no—
221#340—21no—
222#336C22×C5311yescharacteristic
223#336C22×C5311yescharacteristic
224#336C22×C5311yescharacteristic
225C53#777611yescharacteristic, direct product factor, Sylow
226C22×Q8—21no—
227C22×D4D4×C5311yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C4⋊Q8—42no—
232#126—42no—
233#126—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C5311yescharacteristic
241C2×C4⋊C4D4×C5311yescharacteristic
242C2×C4⋊C4D4×C5311yescharacteristic
243C2×C4⋊C4D4×C5311yescharacteristic
244C2×C4⋊C4D4×C5311yescharacteristic
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×C42—21no—
252C2×C42C23×C5311yescharacteristic
253#106—42no—
254#103—42no—
255#103—42no—
256#102—42no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4C2×D4×C5311yescharacteristic
278C22×C4C2×D4×C5311yescharacteristic
279C22×C4C2×D4×C5311yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4SD16×C5312yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#120×C5311yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4C2×SD16×C5311yescharacteristic
355C2×C4#135×C5311yescharacteristic
356C2×C4#135×C5311yescharacteristic
357C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
358C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
359C2×C4#137×C5311yescharacteristic
360C8—42no—
361C8—81no—
362C8—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C53×#44911yescharacteristic
371C22C53×#45011yescharacteristic
372C22C53×#45211yescharacteristic
373C22C53×#45411yescharacteristic
374C22C53×#45511yescharacteristic
375C22C53×#45611yescharacteristic
376C22C53×B(4,𝔽2)11yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C2—161no—
390C2C53×#159711yescharacteristic
391C2C53×#160411yescharacteristic
392C2C53×#160611yescharacteristic
393C2C53×#163511yescharacteristic
394C2C53×#164211yescharacteristic
395C2C53×#165211yescharacteristic
396C2C53×#166411yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2058C2semidirect
3. C53×#2043C2semidirect
5. C53×#1271C2semidirect
6. C53×#1271C2semidirect
164. #7776C53direct (N × H)
192. #2058C2×C53semidirect
193. #2043C2×C53semidirect
195. #1271C2×C53semidirect
196. #1271C2×C53semidirect
225. C53#7776direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27272
C53×#20582
C53×#20432
C53×#20232
C53×#12712
C53×#12702
#777653