Finite Group Databasebeta

← #27293266

Subgroups — #27293266

Order 13568.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277197C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272664C212yesmaximal
4#20272662C212yesmaximal
5C53×#1458C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8#16810676—21no—
9#16810519—21no—
10#16810519—21no—
11#16810495C411yescharacteristic
12#16810495C411yescharacteristic
13#16810495C2211yescharacteristic
14#16810495C2211yescharacteristic
15C4×#11719919—21no—
16C4×#11719919—21no—
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25#13133176—22no—
26#13133176C814yescomplemented
27C2×C4⋊Dic53—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—41no—
38C2×C4×Dic53—41no—
39C4⋊#88322—41no—
40C22⋊#88322—22no—
41C4×#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47#117×C53—41no—
48#117×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57C4×C8×C53—41no—
58C24×C53C22×C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C2×C812yes—
67C22×C4×C53M1612yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C2×C8×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53C2×C812yes—
90C22⋊Dic53C2×C812yes—
91C4⋊Dic53—24no—
92#11719919—42no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96C4×Dic53—24no—
97C4×Dic53—42no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53C22×C811yescharacteristic
107C23×C53C2×M1611yescharacteristic
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53C22⋊C812yes—
123C2×C4×C53C22⋊C812yes—
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#10112yes—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C8×C53—82no—
144#88322—82no—
145#11452—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C2×C22⋊C811yescharacteristic
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10111yescharacteristic
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53#32412yes—
164C4×C53#32412yes—
165Dic53—42no—
166Dic53—42no—
167Dic53—42no—
168Dic53—42no—
169#2296—531no—
170#1458—531no—
171#1458D5311yescharacteristic
172#1326—532no—
173#1326—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—41no—
178C2×C53C2×#32411yescharacteristic
179C2×C53#212011yescharacteristic
180C2×C53#212011yescharacteristic
181C2×#117—531no—
182C2×#117C2×D5311yescharacteristic
183C22×C42—531no—
184#431—21no—
185#431—1061no—
186#389—1061no—
187#389—1061no—
188#385—531no—
189#385—531no—
190#385—531no—
191#385—531no—
192C4×C4⋊C4—1061no—
193C4×C4⋊C4—1061no—
194#324—21no—
195#324—1061no—
196#323—532no—
197#323C2×D5312yes—
198C53#1145211yescharacteristic, complemented, Sylow
199C23×C4—531no—
200C23×C4—531no—
201C23×C4C4×D5311yescharacteristic
202#117—41no—
203#117—41no—
204#117—534no—
205#117—1062no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4D21211yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4C4×D5311yescharacteristic
216C2×C22⋊C4C22×D5311yescharacteristic
217C2×C42—1061no—
218C2×C42—1061no—
219C2×C42—1061no—
220C2×C42—2121no—
221C2×C42—2121no—
222C2×C42D21211yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—2121no—
226C22⋊C8—22no—
227C22⋊C8—1062no—
228C4×C8—41no—
229C4×C8—2121no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234C24C2×C4×D5311yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—2122no—
243C22×C4—2122no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4C8×D5312yes—
247C22×C4#1171991212yes—
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4#1171992111yescharacteristic
252C22×C4C2×D21211yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254C2×C8—42no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—1064no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—532no—
266C22⋊C4—532no—
267C22⋊C4—1061no—
268C22⋊C4—1061no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—1064no—
274C42—2122no—
275C42—2121no—
276C42—2121no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23C2×#1171992111yescharacteristic
282C23C2×C8×D5311yescharacteristic
283C23C2×#1171991211yescharacteristic
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1062no—
299C2×C4—1062no—
300C2×C4—1062no—
301C2×C4—1062no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4#1313312912yes—
315C2×C4#1313312912yes—
316C2×C4#1313313111yescharacteristic
317C2×C4#1313313111yescharacteristic
318C2×C4#1313313512yes—
319C8—82no—
320C8—4242no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22C2×#1313312911yescharacteristic
331C22C2×#1313313111yescharacteristic
332C22C2×#1313313511yescharacteristic
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2122no—
339C4—2122no—
340C4—2122no—
341C4—2122no—
342C4#1681034812yes—
343C4#1681034812yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1681034811yescharacteristic
349C2#2027635411yescharacteristic
350C2#2027635711yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11452
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
26. #13133176C8semidirect
198. C53#11452semidirect

Maximal subgroups

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

subgroupindex
#202771972
#202730292
#202726642
#202726622
C53×#14582
#1145253