Finite Group Databasebeta

← C53×#7774 #27215286

Subgroups — C53×#7774 #27215286

Order 13568.

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

normal subgroups only

This group has 472 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×#2043C211yesmaximal, characteristic, complemented
3C53×#2024C211yesmaximal, characteristic, complemented
4C53×#2023C211yesmaximal, characteristic
5C53×#1272C211yesmaximal, characteristic, complemented
6C53×#1271C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×C4⋊Q8×C53C2211yescharacteristic
9C2×C22⋊Q8×C53—21no—
10C2×#103×C53—21no—
11C2×#103×C53—21no—
12C2×#103×C53—21no—
13C2×#102×C53—21no—
14C53×#399—21no—
15C53×#398—21no—
16C53×#396—21no—
17C53×#391C2211yescharacteristic
18C53×#390C2211yescharacteristic
19C53×#389C2211yescharacteristic
20C53×#386—21no—
21C53×#385—21no—
22C53×#340—21no—
23C53×#340—21no—
24C53×#336C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Q8×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C22×D4×C53D411yescharacteristic
30C23×C4×C53—21no—
31C22×C8×C53—21no—
32C22×C8×C53—21no—
33C22×C8×C53—21no—
34C4⋊Q8×C53—42no—
35C22⋊Q8×C53—42no—
36C22⋊Q8×C53—42no—
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×C53—21no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
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×C53C2311yescharacteristic
53#103×C53—42no—
54#103×C53—42no—
55#103×C53—42no—
56#102×C53—42no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C2×Q8×C53—21no—
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53SD1612yes—
72C2×D4×C53—21no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53Q1612yes—
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—41no—
83C22×C4×C53—81no—
84C22×C4×C53—81no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—81no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—81no—
117C22⋊C4×C53—81no—
118C42×C53—41no—
119C23×C53—22no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#12011yescharacteristic
127Q8×C53—42no—
128Q8×C53—42no—
129Q8×C53—42no—
130Q8×C53—81no—
131Q8×C53—81no—
132Q8×C53—81no—
133D4×C53—42no—
134D4×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
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—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—84no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53C2×SD1611yescharacteristic
161C2×C4×C53C2×Q1611yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C8×C53—81no—
167C8×C53—81no—
168C8×C53—81no—
169#7774C5311yesmaximal, characteristic, direct product factor, Sylow
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53#44911yescharacteristic
178C22×C53#45011yescharacteristic
179C22×C53#45111yescharacteristic
180C22×C53#45411yescharacteristic
181C22×C53#45511yescharacteristic
182C22×C53#45511yescharacteristic
183C22×C53B(4,𝔽2)11yescharacteristic
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—42no—
191C4×C53—82no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53—81no—
196#2732C2×C5311yescharacteristic
197#2043C2×C5311yescharacteristic, complemented
198#2024C2×C5311yescharacteristic, complemented
199#2023C2×C5311yescharacteristic
200#1272C2×C5311yescharacteristic, complemented
201#1271C2×C5311yescharacteristic, complemented
202#1270C2×C5311yescharacteristic
203C2×C53—82no—
204C2×C53#159711yescharacteristic
205C2×C53#160111yescharacteristic
206C2×C53#160211yescharacteristic
207C2×C53#163711yescharacteristic
208C2×C53#164211yescharacteristic
209C2×C53#164411yescharacteristic
210C2×C53#166211yescharacteristic
211C2×C4⋊Q8C22×C5311yescharacteristic
212C2×C22⋊Q8—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#102—21no—
217#399—21no—
218#398—21no—
219#396—21no—
220#391C22×C5311yescharacteristic
221#390C22×C5311yescharacteristic
222#389C22×C5311yescharacteristic
223#386—21no—
224#385—21no—
225#340—21no—
226#340—21no—
227#336C22×C5311yescharacteristic
228#336C22×C5311yescharacteristic
229#336C22×C5311yescharacteristic
230C53#777411yescharacteristic, direct product factor, Sylow
231C22×Q8—21no—
232C22×Q8D4×C5311yescharacteristic
233C22×D4D4×C5311yescharacteristic
234C23×C4—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C4⋊Q8—42no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4D4×C5311yescharacteristic
248C2×C4⋊C4D4×C5311yescharacteristic
249C2×C4⋊C4D4×C5311yescharacteristic
250C2×C4⋊C4D4×C5311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42C23×C5311yescharacteristic
257#103—42no—
258#103—42no—
259#103—42no—
260#102—42no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8SD16×C5312yes—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4Q16×C5312yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C2×D4×C5311yescharacteristic
290C22×C4C2×D4×C5311yescharacteristic
291C22×C4C2×D4×C5311yescharacteristic
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—81no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—41no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#120×C5311yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—41no—
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—42no—
360C2×C4—41no—
361C2×C4—84no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4C2×SD16×C5311yescharacteristic
365C2×C4C2×Q16×C5311yescharacteristic
366C2×C4#135×C5311yescharacteristic
367C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
368C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
369C2×C4#137×C5311yescharacteristic
370C8—81no—
371C8—81no—
372C8—81no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C53×#44911yescharacteristic
381C22C53×#45011yescharacteristic
382C22C53×#45111yescharacteristic
383C22C53×#45411yescharacteristic
384C22C53×#45511yescharacteristic
385C22C53×#45511yescharacteristic
386C22C53×B(4,𝔽2)11yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—82no—
400C2C53×#159711yescharacteristic
401C2C53×#160111yescharacteristic
402C2C53×#160211yescharacteristic
403C2C53×#163711yescharacteristic
404C2C53×#164211yescharacteristic
405C2C53×#164411yescharacteristic
406C2C53×#166211yescharacteristic
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×#2043C2semidirect
3. C53×#2024C2semidirect
5. C53×#1272C2semidirect
6. C53×#1271C2semidirect
169. #7774C53direct (N × H)
197. #2043C2×C53semidirect
198. #2024C2×C53semidirect
200. #1272C2×C53semidirect
201. #1271C2×C53semidirect
230. C53#7774direct (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×#20432
C53×#20242
C53×#20232
C53×#12722
C53×#12712
C53×#12702
#777453