Finite Group Databasebeta

← #27273405

Subgroups — #27273405

Order 13568.

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

normal subgroups only

This group has 322 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#20276277C211yesmaximal, characteristic
3#20276277C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2100C211yesmaximal, characteristic
8#16810563—21no—
9#16810562—21no—
10#16810561—21no—
11#16810521C2211yescharacteristic
12#16810521C2211yescharacteristic
13#16810510—21no—
14#16810509—21no—
15#16810504—21no—
16#16810385—21no—
17#16810385—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C53×#389C2211yescharacteristic
23C53×#384C2211yescharacteristic
24C53×#340—21no—
25C53×#340—21no—
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#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C22×C8×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53#95×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—41no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C2×C8×C53—41no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—21no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C42×C53—41no—
76C4⋊Dic53—21no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82#11719919—41no—
83#11719919—41no—
84#11719919—41no—
85#11719919—41no—
86C4×Dic53—81no—
87C2×#88322—41no—
88C2×#88322—41no—
89C2×#88322—41no—
90C2×#88322—41no—
91C2×#88322—41no—
92C2×#88322—41no—
93C23×C53#12011yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—81no—
107C2×C4×C53—81no—
108C2×C4×C53—81no—
109C2×C4×C53#13511yescharacteristic
110C2×C4×C53#13511yescharacteristic
111C2×C4×C53#13511yescharacteristic
112C2×C4×C53#13711yescharacteristic
113C2×C4×C53#13711yescharacteristic
114C2×C4×C53#13711yescharacteristic
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C8×C53—81no—
128#88322—81no—
129#88322—81no—
130#7832—531nomaximal, Sylow
131C22×C53#45211yescharacteristic
132C22×C53#45211yescharacteristic
133C22×C53#45211yescharacteristic
134C22×C53#45411yescharacteristic
135C22×C53#45411yescharacteristic
136C22×C53#45411yescharacteristic
137C22×C53#45811yescharacteristic
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—81no—
145C4×C53—81no—
146C4×C53—161no—
147Dic53—81no—
148Dic53—81no—
149Dic53—81no—
150Dic53—81no—
151#2810—531no—
152#2100—531no—
153#2100—531no—
154#2100D5311yescharacteristic
155#1271—531no—
156#1271—531no—
157#1271—531no—
158C2×C53#161511yescharacteristic
159C2×C53#161511yescharacteristic
160C2×C53#161511yescharacteristic
161C2×C53#163811yescharacteristic
162C2×C53#163811yescharacteristic
163C2×C53#163811yescharacteristic
164C2×C53#165811yescharacteristic
165#400—1061no—
166#400—1061no—
167#400—1061no—
168#389—531no—
169#389—531no—
170#389C2×D5311yescharacteristic
171#387—1061no—
172#387—1061no—
173#387—1061no—
174#384C2×D5311yescharacteristic
175#340—21no—
176#340—21no—
177#340—1061no—
178#340—1061no—
179#340—1061no—
180#340—1061no—
181#336—531no—
182#336—531no—
183#336C2×D5311yescharacteristic
184C53#783211yescharacteristic, complemented, Sylow
185C22×C8—21no—
186C22×C8—1061no—
187C22×C8—1061no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4C22⋊D5311yescharacteristic
199C2×C4⋊C4C22⋊D5311yescharacteristic
200C2×C42—1061no—
201C2×C42C22×D5311yescharacteristic
202#95—21no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4D4×D5311yescharacteristic
220C22×C4D4×D5311yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—2121no—
226C2×C8—2121no—
227C2×C8—2121no—
228C2×C8—2121no—
229C2×C8—2121no—
230C2×C8—2121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—1061no—
240C4⋊C4—1061no—
241C4⋊C4—1061no—
242C4⋊C4—1061no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C4⋊C4—2121no—
249C42—41no—
250C42—4241no—
251C23#1313318511yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313323711yescharacteristic
280C2×C4#1313323811yescharacteristic
281C2×C4#1313324011yescharacteristic
282C2×C4#1313324611yescharacteristic
283C2×C4#1313324811yescharacteristic
284C2×C4#1313324911yescharacteristic
285C8—81no—
286C8—4241no—
287C8—4241no—
288C22#1681078211yescharacteristic
289C22#1681078311yescharacteristic
290C22#1681078411yescharacteristic
291C22#1681079411yescharacteristic
292C22#1681079411yescharacteristic
293C22#1681079511yescharacteristic
294C22#1681081211yescharacteristic
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—81no—
302C4—81no—
303C4—161no—
304C4—4241no—
305C4—4241no—
306C4—4241no—
307C4—4241no—
308C2#2027374311yescharacteristic
309C2#2027374411yescharacteristic
310C2#2027374511yescharacteristic
311C2#2027388411yescharacteristic
312C2#2027388511yescharacteristic
313C2#2027388611yescharacteristic
314C2#2027401611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2100C2
socle, prime radicalC23×C53#120
hypercenter#2100D53
hypocenterC53#7832
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

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

NH ≅ G/Ntype
184. C53#7832semidirect

Maximal subgroups

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

subgroupindex
#202827062
#202762772
#202725392
#202725382
#202725372
C53×#21002
#783253