Finite Group Databasebeta

← #27272990

Subgroups — #27272990

Order 13568.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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#20276048C211yesmaximal, characteristic
2#20276048C211yesmaximal, characteristic
3#20275961C211yesmaximal, characteristic
4#20272538C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272538C211yesmaximal, characteristic
7C53×#2726C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133148—21no—
10C2×#13133148—21no—
11C2×#13133140—21no—
12C2×#13133140—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C2×#125×C53—21no—
19C53×#398—21no—
20C53×#398—21no—
21C53×#398—21no—
22C53×#389—21no—
23C53×#389C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#389C2211yescharacteristic
26C53×#387—21no—
27C22×Dic106—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133217—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34#13133148—42no—
35#13133148—42no—
36#13133140—42no—
37#13133140—42no—
38#125×C53—42no—
39#125×C53—42no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—41no—
71C2×Dic106—22no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—81no—
94C4⋊C4×C53SD1612yes—
95C4⋊C4×C53SD1612yes—
96C42×C53—41no—
97C42×C53—42no—
98C4⋊Dic53—22no—
99C4⋊Dic53—22no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C2×#88322—22no—
103C2×#88322—22no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53#12011yescharacteristic
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53C2×SD1611yescharacteristic
137C2×C4×C53C2×SD1611yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—42no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145Dic106—42no—
146Dic106—42no—
147Dic106—81no—
148Dic106—81no—
149#88322—42no—
150#88322—42no—
151#88322—81no—
152#7768—531nomaximal, Sylow
153C22×C53#44811yescharacteristic
154C22×C53#45011yescharacteristic
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—42no—
168C4×C53—81no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—81no—
172Dic53—82no—
173#2726D5311yescharacteristic
174#2058—531no—
175#2058—531no—
176#2043—531no—
177#1271—531no—
178#1271—531no—
179#1271—531no—
180C2×C53#159911yescharacteristic
181C2×C53#160311yescharacteristic
182C2×C53#160411yescharacteristic
183C2×C53#163511yescharacteristic
184C2×C53#165111yescharacteristic
185C2×C53#165311yescharacteristic
186C2×C53#166311yescharacteristic
187C2×C4⋊Q8—531no—
188C2×#125—21no—
189C2×#106—1061no—
190C2×#106—1061no—
191C2×#103—1061no—
192C2×#103—1061no—
193#398—21no—
194#398—21no—
195#398—21no—
196#389—21no—
197#389C2×D5311yescharacteristic
198#389C2×D5311yescharacteristic
199#389C2×D5311yescharacteristic
200#387—21no—
201#340—1061no—
202#340—1061no—
203#336—531no—
204#336—531no—
205#336—531no—
206C53#776811yescharacteristic, complemented, Sylow
207C22×Q8—1061no—
208C22×C8—1061no—
209C22×C8—1061no—
210C22×C8—1061no—
211C4⋊Q8—2122no—
212#125—42no—
213#125—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C4⋊C4C22⋊D5311yescharacteristic
225C2×C4⋊C4C22⋊D5311yescharacteristic
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C42—21no—
230C2×C42C22×D5311yescharacteristic
231#106—2122no—
232#106—2122no—
233#103—2122no—
234#103—2122no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241C2×Q8—1062no—
242C2×Q8—2121no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—2121no—
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257C2×C8—1062no—
258C2×C8—1062no—
259C2×C8—2121no—
260C2×C8—2121no—
261C2×C8—2121no—
262C2×C8—2121no—
263C2×C8—2121no—
264C2×C8—2121no—
265C2×C8—2121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4#1171992312yes—
288C4⋊C4#1171992412yes—
289C42—41no—
290C42—42no—
291C23#1313318611yescharacteristic
292Q8—4241no—
293Q8—4241no—
294Q8—2122no—
295Q8—2122no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—2122no—
323C2×C4C2×#1171992411yescharacteristic
324C2×C4C2×#1171992311yescharacteristic
325C2×C4#1313323711yescharacteristic
326C2×C4#1313324311yescharacteristic
327C2×C4#1313324711yescharacteristic
328C2×C4#1313324811yescharacteristic
329C8—4241no—
330C8—2122no—
331C8—2122no—
332C22#1681076211yescharacteristic
333C22#1681077411yescharacteristic
334C22#1681078511yescharacteristic
335C22#1681079611yescharacteristic
336C22#1681080611yescharacteristic
337C22#1681080611yescharacteristic
338C22#1681081011yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—4242no—
352C2#2027365511yescharacteristic
353C2#2027367811yescharacteristic
354C2#2027368411yescharacteristic
355C2#2027386611yescharacteristic
356C2#2027397811yescharacteristic
357C2#2027399211yescharacteristic
358C2#2027404311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2726C2
socle, prime radicalC23×C53#120
hypercenter#2726D53
hypocenterC53#7768
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2726

Complements

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

NH ≅ G/Ntype
206. C53#7768semidirect

Maximal subgroups

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

subgroupindex
#202760482
#202759612
#202725382
C53×#27262
#776853