Finite Group Databasebeta

← #27272647

Subgroups — #27272647

Order 13568.

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

normal subgroups only

This group has 394 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#20283074C211yesmaximal, characteristic
2#20274857C211yesmaximal, characteristic
3#20274462C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133207—21no—
9C2×#13133127—21no—
10#16810522C411yescharacteristic
11#16810521C2211yescharacteristic
12#16810520—21no—
13#16810519C411yescharacteristic
14#16810518—21no—
15#16810492C411yescharacteristic
16#16810492C411yescharacteristic
17#16810487—21no—
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—
29#13133207—42no—
30#13133207—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46#13133127—24no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×C8×C53—21no—
51C22×C8×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C2×C8×C53—22no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C42×C53—41no—
80C4⋊Dic53—21no—
81C4⋊Dic53—21no—
82C4⋊Dic53—42no—
83C4⋊Dic53—41no—
84#11719919—42no—
85#11719919—42no—
86#11719919—42no—
87#11719919—42no—
88#11719919—41no—
89#11719919—41no—
90#11719919—42no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—41no—
100C2×#88322—22no—
101C2×#88322—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C2×#88322—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53#13111yescharacteristic
120C2×C4×C53#13111yescharacteristic
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53#10112yes—
142C2×Dic53#10412yes—
143C8×C53—42no—
144C8×C53—81no—
145#88322—42no—
146#88322—81no—
147#7703—531nomaximal, Sylow
148C22×C53#40811yescharacteristic
149C22×C53#41011yescharacteristic
150C22×C53C2×#10111yescharacteristic
151C22×C53#41811yescharacteristic
152C22×C53#41911yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C22×C53#42111yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—81no—
162C4×C53—81no—
163Dic53—22no—
164Dic53—22no—
165Dic53—41no—
166Dic53—41no—
167Dic53—81no—
168Dic53—81no—
169#2859—531no—
170#1839—531no—
171#1745—531no—
172#1271—531no—
173#1271—531no—
174#1271—531no—
175#1271D5311yescharacteristic
176C2×C53#147011yescharacteristic
177C2×C53#148411yescharacteristic
178C2×C53#149711yescharacteristic
179C2×C53#150611yescharacteristic
180C2×C53#151411yescharacteristic
181C2×C53#152811yescharacteristic
182C2×C53#155611yescharacteristic
183C2×#125—1061no—
184C2×#97—1061no—
185#389—531no—
186#389—531no—
187#389—531no—
188#389—1061no—
189#389—1061no—
190#389C2×D5311yescharacteristic
191#384—531no—
192#384—531no—
193#382—1061no—
194C43—531no—
195#336—531no—
196#336—531no—
197#336—1061no—
198#336—1061no—
199#336—1061no—
200#336C2×D5311yescharacteristic
201#336C2×D5311yescharacteristic
202C53#770311yescharacteristic, complemented, Sylow
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—1061no—
206C22×C8—1061no—
207#125—2122no—
208#125—2122no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C4×D5311yescharacteristic
220C2×C4⋊C4C4×D5311yescharacteristic
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42C22×D5311yescharacteristic
228#97—1064no—
229#95—21no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4C2×C4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4D4×D5311yescharacteristic
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—1062no—
251C2×C8—2121no—
252C2×C8—2121no—
253C2×C8—2121no—
254C2×C8—2121no—
255C2×C8—2121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—2121no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C4⋊C4—2122no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C42—41no—
273C42—1062no—
274C42—1062no—
275C42—1061no—
276C42—1061no—
277C42—1061no—
278C42—1061no—
279C42—1061no—
280C42—1061no—
281C42—2121no—
282C23C22⋊C4×D5311yescharacteristic
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—532no—
296C2×C4—532no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313322311yescharacteristic
314C2×C4#1313322511yescharacteristic
315C2×C4#1313324111yescharacteristic
316C2×C4#1313324411yescharacteristic
317C2×C4#1313324611yescharacteristic
318C2×C4#1313324911yescharacteristic
319C8—42no—
320C8—81no—
321C8—4241no—
322C8—2122no—
323C22#1681058811yescharacteristic
324C22#1681059511yescharacteristic
325C22#101×D5311yescharacteristic
326C22#1681062611yescharacteristic
327C22#1681063011yescharacteristic
328C22#104×D5311yescharacteristic
329C22#1681063811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—1062no—
339C4—1062no—
340C4—4241no—
341C4—4241no—
342C4—2121no—
343C4—2121no—
344C2#2027306911yescharacteristic
345C2#2027313011yescharacteristic
346C2#2027318611yescharacteristic
347C2#2027322511yescharacteristic
348C2#2027326011yescharacteristic
349C2#2027331511yescharacteristic
350C2#2027343011yescharacteristic
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#7703
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
202. C53#7703semidirect

Maximal subgroups

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

subgroupindex
#202830742
#202748572
#202744622
#202725392
#202725382
#202725372
C53×#12712
#770353