Finite Group Databasebeta

← #27272360

Subgroups — #27272360

Order 13568.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277286C211yesmaximal, characteristic
2#20274462C211yesmaximal, characteristic
3#20274462C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic
9C2×#13133182—21no—
10C2×#13133127—21no—
11C2×#13133127—21no—
12#16810527—21no—
13#16810521C2211yescharacteristic
14#16810518—21no—
15#16810492C411yescharacteristic
16C4⋊C4×Dic53C411yescharacteristic
17C4⋊C4×Dic53C411yescharacteristic
18#16810474—21no—
19C42×Dic53C2211yescharacteristic
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30#13133216—22no—
31#13133216—22no—
32#13133182—42no—
33#13133182—42no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133127—24no—
49#13133127—24no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C2×Dic106—24no—
72C2×Dic106—42no—
73C2×Dic106—42no—
74C2×C8×C53—22no—
75C2×C8×C53—22no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53M1612yes—
85C42×C53—41no—
86C4⋊Dic53—21no—
87C4⋊Dic53—41no—
88C4⋊Dic53M1612yes—
89#11719919—24no—
90#11719919—42no—
91#11719919—41no—
92#11719919—41no—
93#11719919—42no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—22no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53—42no—
104C4×Dic53—42no—
105C2×#88322—22no—
106C2×#88322—22no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53C2×C22⋊C411yescharacteristic
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—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53C2×M1611yescharacteristic
125C2×C4×C53C2×M1611yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C53#13711yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—22no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53#10112yes—
146C2×Dic53#10412yes—
147C2×Dic53#10412yes—
148Dic106—48no—
149C8×C53—42no—
150C8×C53—42no—
151#88322—42no—
152#88322—42no—
153#7638—531nomaximal, Sylow
154C22×C53#40711yescharacteristic
155C22×C53#41011yescharacteristic
156C22×C53C2×#10111yescharacteristic
157C22×C53#41811yescharacteristic
158C22×C53#41911yescharacteristic
159C22×C53C2×#10411yescharacteristic
160C22×C53C2×#10411yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—81no—
169Dic53—22no—
170Dic53—22no—
171Dic53—22no—
172Dic53—42no—
173Dic53—41no—
174Dic53—81no—
175C4×C4⋊Q8—531no—
176#1745—531no—
177#1745—531no—
178#1271—531no—
179#1271—531no—
180#1271—531no—
181#1271D5311yescharacteristic
182C2×C53#146711yescharacteristic
183C2×C53#149011yescharacteristic
184C2×C53#149111yescharacteristic
185C2×C53#150511yescharacteristic
186C2×C53#150611yescharacteristic
187C2×C53#152611yescharacteristic
188C2×C53#155611yescharacteristic
189C2×C4⋊Q8—531no—
190C2×C4×Q8—1061no—
191C2×#97—1061no—
192C2×#97—1061no—
193#391—1061no—
194#389—531no—
195#389—1061no—
196#389C2×D5311yescharacteristic
197#384—531no—
198C4×C4⋊C4—531no—
199C4×C4⋊C4—531no—
200C4×C4⋊C4—1061no—
201C43—531no—
202#336—531no—
203#336—531no—
204#336—1061no—
205#336—1061no—
206#336C2×D5311yescharacteristic
207#336C2×D5311yescharacteristic
208C53#763811yescharacteristic, complemented, Sylow
209C22×Q8—1061no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—1061no—
213C22×C8—1061no—
214C4⋊Q8—1062no—
215C4⋊Q8—1062no—
216C4×Q8—2122no—
217C4×Q8—2122no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4C4×D5311yescharacteristic
226C2×C4⋊C4C4×D5311yescharacteristic
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42C22×D5311yescharacteristic
236#97—1064no—
237#97—1064no—
238#95—21no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C2×Q8—1064no—
243C2×Q8—2122no—
244C2×Q8—2122no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4C2×C4×D5311yescharacteristic
254C22×C4D4×D5311yescharacteristic
255C22×C4D4×D5311yescharacteristic
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—1062no—
263C2×C8—1062no—
264C2×C8—2121no—
265C2×C8—2121no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—532no—
273C4⋊C4—1064no—
274C4⋊C4—1061no—
275C4⋊C4—2121no—
276C4⋊C4—2122no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4#1171991212yes—
281C42—41no—
282C42—1062no—
283C42—1062no—
284C42—1062no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42—1061no—
291C42—2122no—
292C42—2122no—
293C23C22⋊C4×D5311yescharacteristic
294Q8—2128no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—532no—
308C2×C4—532no—
309C2×C4—532no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1062no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4M16×D5311yescharacteristic
326C2×C4C2×#1171991211yescharacteristic
327C2×C4#1313324111yescharacteristic
328C2×C4#1313324411yescharacteristic
329C2×C4#1313324611yescharacteristic
330C2×C4#1313324911yescharacteristic
331C8—42no—
332C8—42no—
333C8—2122no—
334C8—2122no—
335C22#1681058511yescharacteristic
336C22#1681059511yescharacteristic
337C22#101×D5311yescharacteristic
338C22#1681062611yescharacteristic
339C22#1681063011yescharacteristic
340C22#104×D5311yescharacteristic
341C22#104×D5311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—81no—
350C4—1062no—
351C4—1062no—
352C4—1062no—
353C4—4241no—
354C4—2122no—
355C4—2121no—
356C2#2027305711yescharacteristic
357C2#2027315611yescharacteristic
358C2#2027316011yescharacteristic
359C2#2027322111yescharacteristic
360C2#2027322511yescharacteristic
361C2#2027330811yescharacteristic
362C2#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#7638
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
208. C53#7638semidirect

Maximal subgroups

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

subgroupindex
#202772862
#202744622
#202725392
#202725382
#202725372
C53×#12712
#763853