Finite Group Databasebeta

← C53×#7801 #27215313

Subgroups — C53×#7801 #27215313

Order 13568.

446 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2732C211yesmaximal, characteristic
2C53×#2096C211yesmaximal, characteristic, complemented
3C53×#2084C211yesmaximal, characteristic, complemented
4C53×#2071C211yesmaximal, characteristic
5C53×#1272C211yesmaximal, characteristic, complemented
6C53×#1271C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×C22⋊Q8×C53—21no—
9C2×#106×C53—21no—
10C2×#106×C53—21no—
11C2×#103×C53—21no—
12C2×#102×C53—21no—
13C53×#399—21no—
14C53×#398—21no—
15C53×#396—21no—
16C53×#391C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#389C2211yescharacteristic
19C53×#386—21no—
20C53×#385—21no—
21C53×#384C2211yescharacteristic
22C53×#340—21no—
23C53×#340—21no—
24C53×#336C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Q8×C53D411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C22×C8×C53—21no—
31C22×C8×C53—21no—
32C22×C8×C53—21no—
33C22⋊Q8×C53—42no—
34C22⋊Q8×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×C53—21no—
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×C42×C53C2311yescharacteristic
52#106×C53—42no—
53#106×C53—42no—
54#103×C53—42no—
55#102×C53—42no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60C24×C53—21no—
61C2×Q8×C53—21no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C2×Q8×C53SD1612yes—
65C2×D4×C53—21no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53SD1612yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—41no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×C8×C53—22no—
82C2×C8×C53—22no—
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—21no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
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—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—81no—
112C22⋊C4×C53—81no—
113C42×C53—41no—
114C23×C53—22no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53#12011yescharacteristic
122Q8×C53—42no—
123Q8×C53—81no—
124D4×C53—42no—
125D4×C53—81no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
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—84no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53C2×SD1611yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C8×C53—42no—
158C8×C53—42no—
159C8×C53—81no—
160#7801C5311yesmaximal, characteristic, direct product factor, Sylow
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#44811yescharacteristic
169C22×C53#44911yescharacteristic
170C22×C53#45011yescharacteristic
171C22×C53#45311yescharacteristic
172C22×C53#45511yescharacteristic
173C22×C53#45611yescharacteristic
174C22×C53#45811yescharacteristic
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—42no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53—161no—
187#2732C2×C5311yescharacteristic
188#2096C2×C5311yescharacteristic, complemented
189#2084C2×C5311yescharacteristic, complemented
190#2071C2×C5311yescharacteristic
191#1272C2×C5311yescharacteristic, complemented
192#1271C2×C5311yescharacteristic, complemented
193#1270C2×C5311yescharacteristic
194C2×C53—82no—
195C2×C53#160811yescharacteristic
196C2×C53#160911yescharacteristic
197C2×C53#161311yescharacteristic
198C2×C53#162211yescharacteristic
199C2×C53#164211yescharacteristic
200C2×C53#165111yescharacteristic
201C2×C53#166111yescharacteristic
202C2×C22⋊Q8—21no—
203C2×#106—21no—
204C2×#106—21no—
205C2×#103—21no—
206C2×#102—21no—
207#399—21no—
208#398—21no—
209#396—21no—
210#391C22×C5311yescharacteristic
211#390C22×C5311yescharacteristic
212#389C22×C5311yescharacteristic
213#386—21no—
214#385—21no—
215#384C22×C5311yescharacteristic
216#340—21no—
217#340—21no—
218#336C22×C5311yescharacteristic
219#336C22×C5311yescharacteristic
220#336C22×C5311yescharacteristic
221C53#780111yescharacteristic, direct product factor, Sylow
222C22×Q8D4×C5311yescharacteristic
223C22×D4D4×C5311yescharacteristic
224C23×C4—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4D4×C5311yescharacteristic
238C2×C4⋊C4D4×C5311yescharacteristic
239C2×C4⋊C4D4×C5311yescharacteristic
240C2×C4⋊C4D4×C5311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42C23×C5311yescharacteristic
247#106—42no—
248#106—42no—
249#103—42no—
250#102—42no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8SD16×C5312yes—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4SD16×C5312yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C2×D4×C5311yescharacteristic
274C22×C4C2×D4×C5311yescharacteristic
275C22×C4C2×D4×C5311yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—41no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23#120×C5311yescharacteristic
317Q8—42no—
318Q8—81no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4C2×SD16×C5311yescharacteristic
347C2×C4C2×SD16×C5311yescharacteristic
348C2×C4#135×C5311yescharacteristic
349C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
350C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
351C2×C4#137×C5311yescharacteristic
352C8—42no—
353C8—42no—
354C8—81no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C53×#44811yescharacteristic
363C22C53×#44911yescharacteristic
364C22C53×#45011yescharacteristic
365C22C53×#45311yescharacteristic
366C22C53×#45511yescharacteristic
367C22C53×#45611yescharacteristic
368C22C53×#45811yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—161no—
381C2—82no—
382C2C53×#160811yescharacteristic
383C2C53×#160911yescharacteristic
384C2C53×#161311yescharacteristic
385C2C53×#162211yescharacteristic
386C2C53×#164211yescharacteristic
387C2C53×#165111yescharacteristic
388C2C53×#166111yescharacteristic
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×#2096C2semidirect
3. C53×#2084C2semidirect
5. C53×#1272C2semidirect
6. C53×#1271C2semidirect
160. #7801C53direct (N × H)
188. #2096C2×C53semidirect
189. #2084C2×C53semidirect
191. #1272C2×C53semidirect
192. #1271C2×C53semidirect
221. C53#7801direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27322
C53×#20962
C53×#20842
C53×#20712
C53×#12722
C53×#12712
C53×#12702
#780153