Finite Group Databasebeta

← #27272934

Subgroups — #27272934

Order 13568.

424 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 81 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#20282700C211yesmaximal, characteristic
2#20276045C211yesmaximal, characteristic
3#20275815C211yesmaximal, characteristic
4#20272540C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272538C211yesmaximal, characteristic
7C53×#2058C211yesmaximal, characteristic
8C22×#11719925—21no—
9C2×#13133148—21no—
10C2×#13133141—21no—
11#16810565—21no—
12#16810554—21no—
13#16810553—21no—
14#16810529C2211yescharacteristic
15#16810527—21no—
16#16810521C2211yescharacteristic
17#16810509—21no—
18#16810504—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#106×C53—21no—
24C53×#389C2211yescharacteristic
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C2×#11719925—42no—
29C2×#11719925—42no—
30C2×#11719925—42no—
31C22×#88322—21no—
32C22×#88322—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C4×Dic53—21no—
40#13133148—42no—
41#13133141—42no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×Q8×C53—21no—
51C22×C8×C53—21no—
52C4⋊Q8×C53—42no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#106×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×Dic106—21no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106SD1612yes—
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C42×C53—41no—
94#11719925—48no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99#11719919—41no—
100#11719919—41no—
101#11719919—81no—
102#11719919—81no—
103C4×Dic53—81no—
104C2×#88322—22no—
105C2×#88322—22no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C23×C53#12011yescharacteristic
111Q8×C53—42no—
112Q8×C53—42no—
113Q8×C53—81no—
114Q8×C53—81no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
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—42no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53C2×SD1611yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53#13511yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—41no—
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—
148Dic106—42no—
149Dic106—81no—
150C8×C53—42no—
151#88322—42no—
152#88322—42no—
153#7760—531nomaximal, Sylow
154C22×C53#44811yescharacteristic
155C22×C53#45211yescharacteristic
156C22×C53#45211yescharacteristic
157C22×C53#45311yescharacteristic
158C22×C53#45611yescharacteristic
159C22×C53#45611yescharacteristic
160C22×C53B(4,𝔽2)11yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—82no—
168C4×C53—81no—
169C4×C53—81no—
170Dic53—42no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174#2809—531no—
175#2058—531no—
176#2058D5311yescharacteristic
177#2016—531no—
178#1272—531no—
179#1271—531no—
180#1271—531no—
181C2×C53#159611yescharacteristic
182C2×C53#160311yescharacteristic
183C2×C53#160311yescharacteristic
184C2×C53#163411yescharacteristic
185C2×C53#165311yescharacteristic
186C2×C53#165311yescharacteristic
187C2×C53#166011yescharacteristic
188C22×Q16—1061no—
189C2×C4⋊Q8C2×D5311yescharacteristic
190C2×#106—21no—
191C2×#106—1061no—
192C2×#103—1061no—
193#400—1061no—
194#398—1061no—
195#398—1061no—
196#391—531no—
197#391—1061no—
198#389—531no—
199#389C2×D5311yescharacteristic
200#387—1061no—
201#387—1061no—
202#340—21no—
203#340—1061no—
204#336—531no—
205#336—531no—
206#336C2×D5311yescharacteristic
207C53#776011yescharacteristic, complemented, Sylow
208C22×Q8—21no—
209C22×Q8—531no—
210C2×Q16—2122no—
211C2×Q16—2122no—
212C2×Q16—2122no—
213C22×C8—21no—
214C22×C8—1061no—
215C22×C8—1061no—
216C4⋊Q8—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C42—1061no—
229C2×C42C22×D5311yescharacteristic
230#106—42no—
231#106—2122no—
232#103—2122no—
233#95—21no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—532no—
249C2×Q8—1061no—
250C2×Q8—2121no—
251C2×Q8—2121no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4D4×D5311yescharacteristic
260C22×C4D4×D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262Q16—2128no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—1062no—
267C2×C8—1062no—
268C2×C8—2121no—
269C2×C8—2121no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—4241no—
284C4⋊C4—4241no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C42—41no—
290C42—4241no—
291C23#1313318511yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—81no—
295Q8—81no—
296Q8—4241no—
297Q8—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—1062no—
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×C4—2121no—
324C2×C4—2121no—
325C2×C4SD16×D5311yescharacteristic
326C2×C4#1313323711yescharacteristic
327C2×C4#1313323811yescharacteristic
328C2×C4#1313324611yescharacteristic
329C2×C4#1313324611yescharacteristic
330C2×C4#1313324811yescharacteristic
331C8—42no—
332C8—2122no—
333C8—2122no—
334C22#1681076011yescharacteristic
335C22#1681078211yescharacteristic
336C22#1681078311yescharacteristic
337C22#1681078911yescharacteristic
338C22#1681080411yescharacteristic
339C22#1681080511yescharacteristic
340C22#1681080911yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—82no—
350C4—4241no—
351C4—4241no—
352C4—4241no—
353C4—2122no—
354C2#2027363611yescharacteristic
355C2#2027367511yescharacteristic
356C2#2027367611yescharacteristic
357C2#2027385911yescharacteristic
358C2#2027398911yescharacteristic
359C2#2027399111yescharacteristic
360C2#2027402511yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2058C2
socle, prime radicalC23×C53#120
hypercenter#2058D53
hypocenterC53#7760
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
207. C53#7760semidirect

Maximal subgroups

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

subgroupindex
#202827002
#202760452
#202758152
#202725402
#202725392
#202725382
C53×#20582
#776053