Finite Group Databasebeta

← C53×#7752 #27215264

Subgroups — C53×#7752 #27215264

Order 13568.

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

normal subgroups only

This group has 574 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×#2682C211yesmaximal, characteristic
2C53×#2057C211yesmaximal, characteristic, complemented
3C53×#2023C211yesmaximal, characteristic, complemented
4C53×#2013C211yesmaximal, characteristic, complemented
5C53×#1271C211yesmaximal, characteristic, complemented
6C53×#1270C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic, complemented
8C22×SD16×C53—21no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×#123×C53—21no—
11C2×#121×C53—21no—
12C2×#107×C53—21no—
13C2×#103×C53—21no—
14C2×#102×C53—21no—
15C53×#396—21no—
16C53×#394—21no—
17C53×#392—21no—
18C53×#390C2211yescharacteristic
19C53×#390C2211yescharacteristic
20C53×#389C2211yescharacteristic
21C53×#385—21no—
22C53×#340—21no—
23C53×#340—21no—
24C53×#336C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Q8×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53—21no—
30C22×D4×C53D411yescharacteristic
31C22×D4×C53D411yescharacteristic
32C23×C4×C53—21no—
33C2×SD16×C53—42no—
34C2×SD16×C53—42no—
35C2×SD16×C53—42no—
36C22×C8×C53—21no—
37C22×C8×C53—21no—
38C22×C8×C53—21no—
39C4⋊Q8×C53—42no—
40#123×C53—42no—
41#123×C53—42no—
42#121×C53—42no—
43#121×C53—42no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C42×C53C2311yescharacteristic
60#107×C53—42no—
61#103×C53—42no—
62#102×C53—42no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53—21no—
66C24×C53—21no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—42no—
83C2×D4×C53—81no—
84C2×D4×C53—81no—
85C2×D4×C53—81no—
86C2×D4×C53D812yes—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—41no—
94C22×C4×C53—81no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98SD16×C53—48no—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53SD1612yescomplemented
120C22⋊C4×C53—42no—
121C22⋊C4×C53—42no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—81no—
131C42×C53—41no—
132C23×C53—22no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53#12011yescharacteristic
147Q8×C53—42no—
148Q8×C53—42no—
149Q8×C53—81no—
150Q8×C53—81no—
151D4×C53—42no—
152D4×C53—42no—
153D4×C53—81no—
154D4×C53—81no—
155D4×C53—84no—
156D4×C53—84no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—84no—
177C2×C4×C53—81no—
178C2×C4×C53—81no—
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53C2×SD1611yescharacteristic
181C2×C4×C53#13511yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53#13711yescharacteristic
185C8×C53—42no—
186C8×C53—42no—
187C8×C53—81no—
188#7752C5311yesmaximal, characteristic, direct product factor, Sylow
189C22×C53—42no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53#44711yescharacteristic
204C22×C53#45011yescharacteristic
205C22×C53#45211yescharacteristic
206C22×C53#45311yescharacteristic
207C22×C53#45511yescharacteristic
208C22×C53#45611yescharacteristic
209C22×C53B(4,𝔽2)11yescharacteristic
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—42no—
217C4×C53—81no—
218C4×C53—82no—
219C4×C53—81no—
220C4×C53—81no—
221#2682C2×C5311yescharacteristic
222#2057C2×C5311yescharacteristic, complemented
223#2023C2×C5311yescharacteristic, complemented
224#2013C2×C5311yescharacteristic, complemented
225#1271C2×C5311yescharacteristic, complemented
226#1270C2×C5311yescharacteristic, complemented
227#1270C2×C5311yescharacteristic, complemented
228C2×C53—82no—
229C2×C53—161no—
230C2×C53#159511yescharacteristic
231C2×C53#160411yescharacteristic
232C2×C53#160611yescharacteristic
233C2×C53#163111yescharacteristic
234C2×C53#163911yescharacteristic
235C2×C53#164711yescharacteristic
236C2×C53#166111yescharacteristic
237C22×SD16—21no—
238C2×C4⋊Q8C22×C5311yescharacteristic
239C2×#123—21no—
240C2×#121—21no—
241C2×#107—21no—
242C2×#103—21no—
243C2×#102—21no—
244#396—21no—
245#394—21no—
246#392—21no—
247#390C22×C5311yescharacteristic
248#390C22×C5311yescharacteristic
249#389C22×C5311yescharacteristic
250#385—21no—
251#340—21no—
252#340—21no—
253#336C22×C5311yescharacteristic
254#336C22×C5311yescharacteristic
255#336C22×C5311yescharacteristic
256C53#775211yescharacteristic, direct product factor, Sylow
257C22×Q8—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4D4×C5311yescharacteristic
261C22×D4D4×C5311yescharacteristic
262C23×C4—21no—
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16—42no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C4⋊Q8—42no—
270#123—42no—
271#123—42no—
272#121—42no—
273#121—42no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4D4×C5311yescharacteristic
279C2×C4⋊C4D4×C5311yescharacteristic
280C2×C4⋊C4D4×C5311yescharacteristic
281C2×C4⋊C4D4×C5311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42C23×C5311yescharacteristic
290#107—42no—
291#103—42no—
292#102—42no—
293#95—21no—
294#95—21no—
295C24—21no—
296C24—21no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4D8×C5312yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4C2×D4×C5311yescharacteristic
326C22×C4C2×D4×C5311yescharacteristic
327C22×C4C2×D4×C5311yescharacteristic
328SD16—48no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4SD16×C5312yescomplemented
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—81no—
361C42—41no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23#120×C5311yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—42no—
382D4—42no—
383D4—81no—
384D4—81no—
385D4—84no—
386D4—84no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C2×D8×C5311yescharacteristic
410C2×C4C2×SD16×C5311yescharacteristic
411C2×C4#135×C5311yescharacteristic
412C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
413C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
414C2×C4#137×C5311yescharacteristic
415C8—42no—
416C8—42no—
417C8—81no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C53×#44711yescharacteristic
433C22C53×#45011yescharacteristic
434C22C53×#45211yescharacteristic
435C22C53×#45311yescharacteristic
436C22C53×#45511yescharacteristic
437C22C53×#45611yescharacteristic
438C22C53×B(4,𝔽2)11yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—81no—
450C2—82no—
451C2—161no—
452C2C53×#159511yescharacteristic
453C2C53×#160411yescharacteristic
454C2C53×#160611yescharacteristic
455C2C53×#163111yescharacteristic
456C2C53×#163911yescharacteristic
457C2C53×#164711yescharacteristic
458C2C53×#166111yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutator, Frattini, cosocleC2×C42C23×C53
prime radicalC53×#2682C2
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×#2057C2semidirect
3. C53×#2023C2semidirect
4. C53×#2013C2semidirect
5. C53×#1271C2semidirect
6. C53×#1270C2semidirect
7. C53×#1270C2semidirect
119. C4⋊C4×C53SD16semidirect
188. #7752C53direct (N × H)
222. #2057C2×C53semidirect
223. #2023C2×C53semidirect
224. #2013C2×C53semidirect
225. #1271C2×C53semidirect
226. #1270C2×C53semidirect
227. #1270C2×C53semidirect
256. C53#7752direct (N × H)
349. C4⋊C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26822
C53×#20572
C53×#20232
C53×#20132
C53×#12712
C53×#12702
#775253