Finite Group Databasebeta

← #27272438

Subgroups — #27272438

Order 13568.

370 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 95 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#20283101C211yesmaximal, characteristic
2#20274857C211yesmaximal, characteristic
3#20274459C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×C8×Dic53—21no—
9#16810522C411yescharacteristic
10#16810521C2211yescharacteristic
11#16810519C411yescharacteristic
12#16810509—21no—
13#16810508—21no—
14#16810505—21no—
15#16810504—21no—
16#16810488C411yescharacteristic
17#16810488C411yescharacteristic
18C42×Dic53C2211yescharacteristic
19#16810378C2211yescharacteristic
20#16810378C2211yescharacteristic
21#16810377—21no—
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C8×Dic53—24no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×C8×C53—21no—
49C22×C8×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53C2311yescharacteristic
54#95×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C2×C8×C53—22no—
67C2×C8×C53—41no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C42×C53—41no—
78C4⋊Dic53—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53—42no—
81C4⋊Dic53—41no—
82#11719919—42no—
83#11719919—41no—
84#11719919—41no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—41no—
94C2×#88322—22no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#13111yescharacteristic
114C2×C4×C53#13111yescharacteristic
115C2×C4×C53#13511yescharacteristic
116C2×C4×C53#13511yescharacteristic
117C2×C4×C53#13511yescharacteristic
118C2×C4×C53#13511yescharacteristic
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53#10112yes—
136C2×Dic53#10412yes—
137C8×C53—42no—
138C8×C53—81no—
139#88322—42no—
140#88322—81no—
141#7657—531nomaximal, Sylow
142C22×C53#40811yescharacteristic
143C22×C53#41011yescharacteristic
144C22×C53C2×#10111yescharacteristic
145C22×C53#41611yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53C2×#10411yescharacteristic
148C22×C53#42111yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—81no—
156C4×C53—81no—
157Dic53—22no—
158Dic53—22no—
159Dic53—41no—
160Dic53—41no—
161Dic53—81no—
162Dic53—81no—
163#2864—531no—
164#1839—531no—
165#1744—531no—
166#1271—531no—
167#1271—531no—
168#1271—531no—
169#1271D5311yescharacteristic
170C2×C53#147011yescharacteristic
171C2×C53#148211yescharacteristic
172C2×C53#149411yescharacteristic
173C2×C53#150011yescharacteristic
174C2×C53#152111yescharacteristic
175C2×C53#153111yescharacteristic
176C2×C53#154411yescharacteristic
177C2×C4×C8—1061no—
178#389—531no—
179#389—531no—
180#389—531no—
181#389C2×D5311yescharacteristic
182#387—1061no—
183#387—1061no—
184#387—1061no—
185#387—1061no—
186#382—531no—
187#382—531no—
188C43—531no—
189#336—531no—
190#336—531no—
191#336—1061no—
192#336—1061no—
193#336—1061no—
194#336C2×D5311yescharacteristic
195#336C2×D5311yescharacteristic
196C53#765711yescharacteristic, complemented, Sylow
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—1061no—
200C22×C8—1061no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C4×D5311yescharacteristic
208C2×C4⋊C4C4×D5311yescharacteristic
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42C22×D5311yescharacteristic
216C4×C8—1064no—
217#95—21no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4C2×C4×D5311yescharacteristic
234C22×C4D4×D5311yescharacteristic
235C22×C4D4×D5311yescharacteristic
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—1062no—
243C2×C8—2121no—
244C2×C8—2121no—
245C2×C8—2121no—
246C2×C8—2121no—
247C2×C8—2121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C42—41no—
261C42—1062no—
262C42—1062no—
263C42—1061no—
264C42—1061no—
265C42—1061no—
266C42—1061no—
267C42—1061no—
268C42—1061no—
269C42—2121no—
270C23C22⋊C4×D5311yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313322311yescharacteristic
302C2×C4#1313322511yescharacteristic
303C2×C4#1313323611yescharacteristic
304C2×C4#1313323811yescharacteristic
305C2×C4#1313323811yescharacteristic
306C2×C4#1313324011yescharacteristic
307C8—42no—
308C8—81no—
309C8—4241no—
310C8—2122no—
311C22#1681058811yescharacteristic
312C22#1681059511yescharacteristic
313C22#101×D5311yescharacteristic
314C22#1681061611yescharacteristic
315C22#1681061811yescharacteristic
316C22#104×D5311yescharacteristic
317C22#1681063811yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—81no—
325C4—81no—
326C4—1062no—
327C4—1062no—
328C4—4241no—
329C4—4241no—
330C4—2121no—
331C4—2121no—
332C2#2027306911yescharacteristic
333C2#2027312011yescharacteristic
334C2#2027317411yescharacteristic
335C2#2027319811yescharacteristic
336C2#2027329111yescharacteristic
337C2#2027332811yescharacteristic
338C2#2027337811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7657
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
196. C53#7657semidirect

Maximal subgroups

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

subgroupindex
#202831012
#202748572
#202744592
#202725392
#202725382
#202725372
C53×#12712
#765753