Finite Group Databasebeta

← #27294302

Subgroups — #27294302

Order 13568.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#20280465C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic, complemented
3#20272717C214yesmaximal, complemented
4C53×#1742C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic
6C2×#13133191—21no—
7#16810676—21no—
8#16810540—21no—
9#16810540C2212yes—
10#16810532—21no—
11#16810532C412yes—
12#16810525—21no—
13#16810489C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C22×M16×C53—21no—
17C2×#117×C53C2211yescharacteristic
18C53×#343—21no—
19C53×#343C2212yes—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C22×C22⋊D53—22no—
24#13133214—41no—
25#13133214—41no—
26#13133191—42no—
27#13133191—42no—
28C2×C4⋊Dic53—21no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C4⋊#88322—41no—
33C22⋊#88322—22no—
34C4×#88322—41no—
35#13133124—21no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C2×M16×C53—24no—
39C2×M16×C53—22no—
40C2×M16×C53—41no—
41C2×M16×C53—42no—
42C22×C8×C53—41no—
43#117×C53—41no—
44#117×C53—41no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53C2311yescharacteristic
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C23×D53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53D812yes—
60C22×C4×C53SD1612yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C2×C22⋊D53—44no—
65C2×C22⋊D53—82no—
66C2×C22⋊D53—82no—
67C22×Dic53—22no—
68C2×D212—42no—
69C2×D212—42no—
70C2×D212—41no—
71C2×D212—41no—
72C2×D212—42no—
73C2×D212—42no—
74C2×D212—82no—
75C2×D212—82no—
76M16×C53—24no—
77M16×C53—44no—
78M16×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—44no—
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—
89#11719921—42no—
90#11719921—42no—
91#11719921—82no—
92#11719921—82no—
93C4⋊Dic53—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53#10212yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—42no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—42no—
105C22×D53—42no—
106C22×D53—42no—
107C22×D53—42no—
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×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#10212yes—
125C2×C4×C53#10412yes—
126C2×C4×C53C2×D811yescharacteristic
127C2×C4×C53C2×SD1611yescharacteristic
128C22⋊D53—88no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132D212—84no—
133D212—84no—
134D212—82no—
135D212—82no—
136C8×C53—44no—
137#88322—82no—
138#11699—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—22no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—81no—
146C22×C53C2×#9911yescharacteristic
147C22×C53C2×#10211yescharacteristic
148C22×C53C2×#10411yescharacteristic
149C2×D53—82no—
150C2×D53—82no—
151C2×D53—82no—
152C2×D53—82no—
153C2×D53—82no—
154C2×D53—82no—
155C2×D53—82no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53#32712yes—
163C4×C53#32712yes—
164Dic53—82no—
165#2619—531no—
166#1742D5311yescharacteristic, complemented
167#1458—531no—
168#1345—534no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53C2×#32711yescharacteristic
173C2×C53#212211yescharacteristic
174C2×C53#213411yescharacteristic
175D53—162no—
176C22×M16—21no—
177C2×#127—531no—
178C2×#121—1061no—
179C2×#117C2×D5311yescharacteristic
180#431—1061no—
181#394—532no—
182#394—1061no—
183#392—532no—
184#392—1061no—
185#390—1061no—
186#382—531no—
187#343—21no—
188#343C2×D5312yes—
189#324—532no—
190#323—1061no—
191C53#1169911yescharacteristic, complemented, Sylow
192C22×D4—532no—
193C22×D4—1062no—
194C22×D4—1061no—
195C22×D4—1061no—
196C23×C4C22⋊D5311yescharacteristic
197C2×M16—24no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—42no—
201C22×C8—41no—
202#127—2121no—
203#127—2121no—
204#121—2122no—
205#121—2122no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C4⋊C4C22×D5311yescharacteristic
211C2×C22⋊C4—1062no—
212C2×C22⋊C4—1062no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4C4×D5311yescharacteristic
215C2×C22⋊C4D21211yescharacteristic
216C2×C42C4×D5311yescharacteristic
217C2×C42D21211yescharacteristic
218C4⋊C8—2121no—
219C22⋊C8—1062no—
220C4×C8—2121no—
221#95—532no—
222#95—1061no—
223C24—1062no—
224C24#1171992111yescharacteristic
225C2×D4—4242no—
226C2×D4—4242no—
227C2×D4—4242no—
228C2×D4—4242no—
229C2×D4—2122no—
230C2×D4—2122no—
231C2×D4—2124no—
232C2×D4—2121no—
233C2×D4—2121no—
234C2×D4—2122no—
235C2×D4—2122no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—1062no—
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992212yes—
244C22×C4#1171992412yes—
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4C2×D21211yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—24no—
249M16—44no—
250M16—42no—
251C2×C8—42no—
252C2×C8—44no—
253C2×C8—2122no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2122no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—4242no—
260C22⋊C4—4242no—
261C22⋊C4—2122no—
262C22⋊C4—2122no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—2122no—
272C23—2122no—
273C23—2122no—
274C23—2122no—
275C23—2122no—
276C23—2122no—
277C23—2122no—
278C23#1313313712yes—
279C23C2×#1171992111yescharacteristic
280D4—4244no—
281D4—4244no—
282D4—4248no—
283D4—4242no—
284D4—4242no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4#1313313111yescharacteristic
303C2×C4#1313313111yescharacteristic
304C2×C4#1313313712yes—
305C2×C4#1313314512yes—
306C2×C4C2×#1171992211yescharacteristic
307C2×C4C2×#1171992411yescharacteristic
308C8—44no—
309C8—4242no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—4242no—
318C22—4242no—
319C22—4242no—
320C22—4242no—
321C22—4242no—
322C22—4242no—
323C22—4242no—
324C22C2×#1313313111yescharacteristic
325C22C2×#1313313711yescharacteristic
326C22C2×#1313314511yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4242no—
334C4#1681035612yes—
335C4#1681035612yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—8482no—
340C2C2×#1681035611yescharacteristic
341C2#2027636611yescharacteristic
342C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1742C2
socleC22×C53C2×#99
prime radical#20280465C2
cosocleC2×C4⋊C4×C53C23
hypercenter#1742D53
hypocenterC53#11699
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
2. #20273029C2semidirect
3. #20272717C2semidirect
4. C53×#1742C2semidirect
166. #1742D53semidirect
191. C53#11699semidirect

Maximal subgroups

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

subgroupindex
#202804652
#202730292
#202727172
C53×#17422
#1169953