Finite Group Databasebeta

← #27294596

Subgroups — #27294596

Order 13568.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280292C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272720C212yesmaximal, complemented
5C53×#1742C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic
7#16810680—21no—
8#16810553C2211yescharacteristic
9#16810548C411yescharacteristic
10#16810543C2211yescharacteristic
11#16810535C411yescharacteristic
12#16810511—21no—
13#16810507—21no—
14#16810504—21no—
15#16810496—21no—
16#16810491C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C22×M16×C53—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#343—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C4×Dic53—21no—
33C4⋊#88322—41no—
34C22⋊#88322—22no—
35#13133128—41no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C2×M16×C53—24no—
46C2×M16×C53—22no—
47C2×M16×C53—41no—
48C2×M16×C53—42no—
49C22×C8×C53—41no—
50#117×C53—41no—
51#117×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×Dic106—41no—
75C2×Dic106—42no—
76M16×C53—24no—
77M16×C53—44no—
78M16×C53—42no—
79C2×C8×C53—44no—
80C2×C8×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137Dic106—42no—
138Dic106—42no—
139C8×C53—44no—
140#88322—82no—
141#11763—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53C2×#9911yescharacteristic
150C22×C53#41711yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160Dic53—42no—
161Dic53—81no—
162Dic53—81no—
163Dic53—81no—
164#2599—531no—
165#1742D5311yescharacteristic
166#1464—531no—
167#1354—532no—
168#1346—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53#150411yescharacteristic
173C2×C53#212011yescharacteristic
174C2×C53#213511yescharacteristic
175C22×M16—21no—
176C2×C4×Q8—531no—
177C2×#117C2×D5311yescharacteristic
178#432—1061no—
179#398—531no—
180#397—531no—
181#395—531no—
182#393—531no—
183#388—1061no—
184#387—1061no—
185#387—1061no—
186#385—1061no—
187#383—531no—
188#343—21no—
189#343C2×D5312yes—
190#324—532no—
191#323—1061no—
192C53#1176311yescharacteristic, complemented, Sylow
193C22×Q8—531no—
194C23×C4C22⋊D5311yescharacteristic
195C2×M16—24no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—42no—
199C22×C8—41no—
200C4×Q8—2121no—
201C4×Q8—2121no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4C4×D5311yescharacteristic
213C2×C22⋊C4D21211yescharacteristic
214C2×C42—1061no—
215C2×C42C4×D5311yescharacteristic
216C2×C42D21211yescharacteristic
217C4⋊C8—2121no—
218C22⋊C8—1062no—
219#97—2121no—
220#95—531no—
221#95—531no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228C24#1171992111yescharacteristic
229C2×Q8—2121no—
230C2×Q8—2122no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4C2×C4×D5311yescharacteristic
244C22×C4C2×D21211yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—24no—
247M16—44no—
248M16—42no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—2122no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2121no—
255C4⋊C4—2122no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—2122no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1313314512yes—
274C23C2×#1171992111yescharacteristic
275Q8—2122no—
276Q8—2122no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
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—1062no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313313111yescharacteristic
306C2×C4#1313313111yescharacteristic
307C2×C4#1313324311yescharacteristic
308C2×C4#1313324811yescharacteristic
309C8—44no—
310C8—4242no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1313313111yescharacteristic
319C22#1681062211yescharacteristic
320C22C2×#1313314511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4241no—
330C4—4241no—
331C4—4241no—
332C4—2122no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2027321711yescharacteristic
337C2#2027635711yescharacteristic
338C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1742C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1742D53
hypocenterC53#11763
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #20272744C2semidirect
4. #20272720C2semidirect
192. C53#11763semidirect

Maximal subgroups

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

subgroupindex
#202802922
#202730492
#202727442
#202727202
C53×#17422
#1176353