Finite Group Databasebeta

← #27273036

Subgroups — #27273036

Order 13568.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#20282706C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic
3#20276040C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2043C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133148—21no—
10#16810563—21no—
11#16810562—21no—
12#16810561—21no—
13#16810521C2211yescharacteristic
14#16810521C2211yescharacteristic
15#16810510—21no—
16#16810509—21no—
17#16810504—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#103×C53—21no—
24C2×#103×C53—21no—
25C53×#389C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53D411yescharacteristic
30C2×C4⋊Dic53D411yescharacteristic
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C4×Dic53—21no—
36#13133150—42no—
37#13133148—42no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×Q8×C53—21no—
48C22×C8×C53—21no—
49C4⋊Q8×C53—42no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#103×C53—42no—
57#103×C53—42no—
58#95×C53—21no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C42×C53—41no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94#11719919—41no—
95#11719919—41no—
96#11719919—41no—
97#11719919—41no—
98C4×Dic53—81no—
99C2×#88322—22no—
100C2×#88322—22no—
101C2×#88322—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C23×C53#12011yescharacteristic
106Q8×C53—42no—
107Q8×C53—42no—
108Q8×C53—81no—
109Q8×C53—81no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53#13511yescharacteristic
126C2×C4×C53#13511yescharacteristic
127C2×C4×C53#13511yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C8×C53—81no—
144#88322—42no—
145#88322—42no—
146#7775—531nomaximal, Sylow
147C22×C53#44911yescharacteristic
148C22×C53#45211yescharacteristic
149C22×C53#45211yescharacteristic
150C22×C53#45411yescharacteristic
151C22×C53#45611yescharacteristic
152C22×C53#45611yescharacteristic
153C22×C53B(4,𝔽2)11yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—82no—
161C4×C53—81no—
162C4×C53—81no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166Dic53—81no—
167#2810—531no—
168#2058—531no—
169#2057—531no—
170#2043D5311yescharacteristic
171#1271—531no—
172#1271—531no—
173#1271—531no—
174C2×C53#159711yescharacteristic
175C2×C53#160311yescharacteristic
176C2×C53#160311yescharacteristic
177C2×C53#163811yescharacteristic
178C2×C53#165011yescharacteristic
179C2×C53#165211yescharacteristic
180C2×C53#166311yescharacteristic
181C2×C4⋊Q8C2×D5311yescharacteristic
182C2×#107—1061no—
183C2×#106—1061no—
184C2×#103—21no—
185C2×#103—21no—
186#400—1061no—
187#400—1061no—
188#400—1061no—
189#389—531no—
190#389—531no—
191#389C2×D5311yescharacteristic
192#387—1061no—
193#387—1061no—
194#387—1061no—
195#340—1061no—
196#340—1061no—
197#336—531no—
198#336—531no—
199#336C2×D5311yescharacteristic
200C53#777511yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C22×C8—21no—
203C22×C8—1061no—
204C22×C8—1061no—
205C4⋊Q8—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C42—1061no—
218C2×C42C22×D5311yescharacteristic
219#107—2122no—
220#106—2122no—
221#103—42no—
222#103—42no—
223#95—21no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4D4×D5311yescharacteristic
247C22×C4D4×D5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—1062no—
253C2×C8—1062no—
254C2×C8—2121no—
255C2×C8—2121no—
256C2×C8—2121no—
257C2×C8—2121no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C42—41no—
278C42—4241no—
279C23#1313318511yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—81no—
283Q8—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4#1313323711yescharacteristic
312C2×C4#1313323811yescharacteristic
313C2×C4#1313324011yescharacteristic
314C2×C4#1313324311yescharacteristic
315C2×C4#1313324611yescharacteristic
316C2×C4#1313324911yescharacteristic
317C8—81no—
318C8—2122no—
319C8—2122no—
320C22#1681076611yescharacteristic
321C22#1681078211yescharacteristic
322C22#1681078411yescharacteristic
323C22#1681079511yescharacteristic
324C22#1681080411yescharacteristic
325C22#1681080411yescharacteristic
326C22#1681080911yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—81no—
334C4—81no—
335C4—82no—
336C4—4241no—
337C4—4241no—
338C4—4241no—
339C4—4241no—
340C2#2027364211yescharacteristic
341C2#2027367511yescharacteristic
342C2#2027367711yescharacteristic
343C2#2027388411yescharacteristic
344C2#2027396911yescharacteristic
345C2#2027398311yescharacteristic
346C2#2027404111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2043C2
socle, prime radicalC23×C53#120
hypercenter#2043D53
hypocenterC53#7775
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
200. C53#7775semidirect

Maximal subgroups

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

subgroupindex
#202827062
#202760452
#202760402
#202725392
#202725382
#202725372
C53×#20432
#777553