Finite Group Databasebeta

← #27272952

Subgroups — #27272952

Order 13568.

416 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 85 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#20281667C211yesmaximal, characteristic
2#20276040C211yesmaximal, characteristic
3#20276040C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272538C211yesmaximal, characteristic
7C53×#2043C211yesmaximal, characteristic
8C2×#13133207—21no—
9C2×#13133150—21no—
10C2×#13133150—21no—
11#16810555—21no—
12#16810553—21no—
13#16810553—21no—
14#16810522—21no—
15#16810521C2211yescharacteristic
16#16810521C2211yescharacteristic
17#16810508—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—
29#13133207—42no—
30#13133207—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C4×Dic53—21no—
42#13133150—42no—
43#13133150—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53—21no—
50C22×C8×C53—21no—
51C4⋊Q8×C53—42no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#103×C53—42no—
59#103×C53—42no—
60#95×C53—21no—
61C2×Q8×C53—22no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53D812yes—
89C42×C53—41no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—81no—
97C4⋊Dic53Q1612yes—
98#11719919—42no—
99#11719919—42no—
100#11719919—41no—
101#11719919—41no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—42no—
105C2×#88322—22no—
106C2×#88322—22no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53#12011yescharacteristic
112Q8×C53—42no—
113Q8×C53—42no—
114Q8×C53—81no—
115Q8×C53—81no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53C2×D811yescharacteristic
132C2×C4×C53C2×Q1611yescharacteristic
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×C4×C53#13711yescharacteristic
137C2×Dic53—22no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C8×C53—81no—
150#88322—42no—
151#88322—42no—
152#7763—531nomaximal, Sylow
153C22×C53#44711yescharacteristic
154C22×C53#45111yescharacteristic
155C22×C53#45211yescharacteristic
156C22×C53#45411yescharacteristic
157C22×C53#45611yescharacteristic
158C22×C53#45611yescharacteristic
159C22×C53B(4,𝔽2)11yescharacteristic
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—42no—
167C4×C53—82no—
168C4×C53—81no—
169Dic53—42no—
170Dic53—81no—
171Dic53—81no—
172Dic53—81no—
173#2726—531no—
174#2057—531no—
175#2057—531no—
176#2043D5311yescharacteristic
177#1271—531no—
178#1271—531no—
179#1271—531no—
180C2×C53#159811yescharacteristic
181C2×C53#160311yescharacteristic
182C2×C53#160511yescharacteristic
183C2×C53#163611yescharacteristic
184C2×C53#164711yescharacteristic
185C2×C53#164911yescharacteristic
186C2×C53#166311yescharacteristic
187C2×C4⋊Q8C2×D5311yescharacteristic
188C2×#125—1061no—
189C2×#107—1061no—
190C2×#107—1061no—
191C2×#103—21no—
192C2×#103—21no—
193#398—1061no—
194#398—1061no—
195#398—1061no—
196#389—531no—
197#389—531no—
198#389—1061no—
199#389C2×D5311yescharacteristic
200#387—1061no—
201#340—1061no—
202#340—1061no—
203#336—531no—
204#336—531no—
205#336C2×D5311yescharacteristic
206C53#776311yescharacteristic, complemented, Sylow
207C22×Q8—21no—
208C22×C8—21no—
209C22×C8—1061no—
210C22×C8—1061no—
211C4⋊Q8—42no—
212#125—2122no—
213#125—2122no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C42—1061no—
230C2×C42C22×D5311yescharacteristic
231#107—2122no—
232#107—2122no—
233#103—42no—
234#103—42no—
235#95—21no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4D4×D5311yescharacteristic
255C22×C4D4×D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—1062no—
261C2×C8—1062no—
262C2×C8—2121no—
263C2×C8—2121no—
264C2×C8—2121no—
265C2×C8—2121no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—532no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—4241no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4#1171992212yes—
289C42—41no—
290C42—2122no—
291C23#1313318511yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—81no—
295Q8—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—1062no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4C2×#1171992211yescharacteristic
324C2×C4Q16×D5311yescharacteristic
325C2×C4#1313323811yescharacteristic
326C2×C4#1313324211yescharacteristic
327C2×C4#1313324611yescharacteristic
328C2×C4#1313324911yescharacteristic
329C8—81no—
330C8—2122no—
331C8—2122no—
332C22#1681075611yescharacteristic
333C22#1681077711yescharacteristic
334C22#1681078211yescharacteristic
335C22#1681079511yescharacteristic
336C22#1681080411yescharacteristic
337C22#1681080411yescharacteristic
338C22#1681080911yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—82no—
348C4—4241no—
349C4—4241no—
350C4—4241no—
351C4—2122no—
352C2#2027364911yescharacteristic
353C2#2027367511yescharacteristic
354C2#2027368711yescharacteristic
355C2#2027387011yescharacteristic
356C2#2027394811yescharacteristic
357C2#2027396211yescharacteristic
358C2#2027404111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2043C2
socle, prime radicalC23×C53#120
hypercenter#2043D53
hypocenterC53#7763
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
206. C53#7763semidirect

Maximal subgroups

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

subgroupindex
#202816672
#202760402
#202725392
#202725382
C53×#20432
#776353