Finite Group Databasebeta

← C53×#12098 #27219610

Subgroups — C53×#12098 #27219610

Order 13568.

476 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2728C211yesmaximal, characteristic, complemented
2C53×#2007C211yesmaximal, characteristic, complemented
3C53×#1610C211yesmaximal, characteristic, complemented
4C53×#1598C211yesmaximal, characteristic
5C53×#1346C211yesmaximal, characteristic, complemented
6C53×#1345C211yesmaximal, characteristic, complemented
7C53×#1319C211yesmaximal, characteristic, complemented
8C2×#137×C53—21no—
9C53×#537C2211yescharacteristic
10C2×C22⋊Q8×C53—21no—
11C53×#480—21no—
12C53×#447—21no—
13C2×#104×C53—21no—
14C53×#400—21no—
15C53×#398C2211yescharacteristic
16C53×#397—21no—
17C53×#394C2211yescharacteristic
18C53×#391C2211yescharacteristic
19C53×#388—21no—
20C53×#386—21no—
21C53×#385—21no—
22C53×#343C2211yescharacteristic
23C53×#327—21no—
24C53×#327—21no—
25C53×#324C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C2×#41×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C22×D4×C53D411yescharacteristic
30C23×C4×C53—21no—
31#137×C53—42no—
32#137×C53—42no—
33C2×Q16×C53—41no—
34C2×SD16×C53—41no—
35C2×D8×C53—41no—
36C2×M16×C53—21no—
37C22⋊Q8×C53—41no—
38C22⋊Q8×C53—42no—
39C22⋊Q8×C53—42no—
40#121×C53—21no—
41#121×C53—21no—
42#121×C53—41no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53C2311yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C42×C53D411yescharacteristic
54#107×C53—41no—
55#104×C53—42no—
56#102×C53—41no—
57#102×C53—41no—
58C22⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53D411yescharacteristic
66#95×C53D411yescharacteristic
67C24×C53—21no—
68#41×C53—41no—
69#41×C53—41no—
70#41×C53—41no—
71#41×C53—41no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53D812yescomplemented
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—41no—
81C2×D4×C53—42no—
82C2×D4×C53—41no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—42no—
90C22×C4×C53—81no—
91C22×C4×C53—81no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95Q16×C53—44no—
96SD16×C53—44no—
97D8×C53—82no—
98M16×C53—22no—
99M16×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—42no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—42no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—81no—
119C42×C53—41no—
120C23×C53—22no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#12011yescharacteristic
129Q8×C53—42no—
130Q8×C53—41no—
131Q8×C53—41no—
132Q8×C53—81no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—81no—
137D4×C53—81no—
138D4×C53—81no—
139D4×C53—81no—
140D4×C53—81no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
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—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—84no—
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C8×C53—42no—
169C8×C53—81no—
170C8×C53—81no—
171#12098C5311yesmaximal, characteristic, direct product factor, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—81no—
184C22×C53#44711yescharacteristic
185C22×C53#45311yescharacteristic
186C22×C53B(4,𝔽2)11yescharacteristic
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—42no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53—81no—
197#2728C2×C5311yescharacteristic, complemented
198#2007C2×C5311yescharacteristic, complemented
199#1610C2×C5311yescharacteristic, complemented
200#1598C2×C5311yescharacteristic
201#1346C2×C5311yescharacteristic, complemented
202#1345C2×C5311yescharacteristic, complemented
203#1319C2×C5311yescharacteristic, complemented
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—82no—
207C2×C53—161no—
208C2×C53#159511yescharacteristic
209C2×C53#219711yescharacteristic
210C2×C53#220311yescharacteristic
211C2×#137—21no—
212#537C22×C5311yescharacteristic
213C2×C22⋊Q8—21no—
214#480—21no—
215#447—21no—
216C2×#104—21no—
217#400—21no—
218#398C22×C5311yescharacteristic
219#397—21no—
220#394C22×C5311yescharacteristic
221#391C22×C5311yescharacteristic
222#388—21no—
223#386—21no—
224#385—21no—
225#343C22×C5311yescharacteristic
226#327—21no—
227#327—21no—
228#324C22×C5311yescharacteristic
229#324C22×C5311yescharacteristic
230C53#1209811yescharacteristic, direct product factor, Sylow
231C2×#41—21no—
232C22×Q8D4×C5311yescharacteristic
233C22×D4D4×C5311yescharacteristic
234C23×C4—21no—
235#137—42no—
236#137—42no—
237C2×Q16—41no—
238C2×SD16—41no—
239C2×D8—41no—
240C2×M16—21no—
241C22⋊Q8—41no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244#121—21no—
245#121—21no—
246#121—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4D4×C5311yescharacteristic
251C2×C4⋊C4C23×C5311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42D4×C5311yescharacteristic
258#107—41no—
259#104—42no—
260#102—41no—
261#102—41no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95D4×C5311yescharacteristic
270#95D4×C5311yescharacteristic
271C24—21no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8D8×C5312yescomplemented
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—41no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C2×D4×C5311yescharacteristic
297C22×C4C2×D4×C5311yescharacteristic
298C22×C4C2×D4×C5311yescharacteristic
299Q16—44no—
300SD16—44no—
301D8—82no—
302M16—22no—
303M16—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—81no—
323C42—41no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#120×C5311yescharacteristic
333Q8—42no—
334Q8—41no—
335Q8—41no—
336Q8—81no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
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—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4C2×D8×C5311yescharacteristic
371C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
372C8—42no—
373C8—81no—
374C8—81no—
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22C53×#44711yescharacteristic
388C22C53×#45311yescharacteristic
389C22C53×B(4,𝔽2)11yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C2—21no—
401C2—21no—
402C2—82no—
403C2—161no—
404C2C53×#159511yescharacteristic
405C2C53×#219711yescharacteristic
406C2C53×#220311yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC53×#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2728C2semidirect
2. C53×#2007C2semidirect
3. C53×#1610C2semidirect
5. C53×#1346C2semidirect
6. C53×#1345C2semidirect
7. C53×#1319C2semidirect
76. C2×Q8×C53D8semidirect
171. #12098C53direct (N × H)
197. #2728C2×C53semidirect
198. #2007C2×C53semidirect
199. #1610C2×C53semidirect
201. #1346C2×C53semidirect
202. #1345C2×C53semidirect
203. #1319C2×C53semidirect
230. C53#12098direct (N × H)
280. C2×Q8D8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#27282
C53×#20072
C53×#16102
C53×#15982
C53×#13462
C53×#13452
C53×#13192
#1209853