Finite Group Databasebeta

← #27294684

Subgroups — #27294684

Order 13568.

482 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279714C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic, complemented
3#20272719C214yesmaximal, complemented
4C53×#1467C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic
6C2×#13133206—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810540C412yes—
10#16810540C2212yes—
11#16810525—21no—
12#16810525—21no—
13#16810524—21no—
14C4×C4⋊Dic53C411yescharacteristic
15#16810392C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C2×#11719917—22no—
24#13133214—41no—
25#13133214—41no—
26#13133206—44no—
27C2×C4⋊Dic53—21no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×C4×Dic53—21no—
33#13133136—24no—
34#13133128—41no—
35C4×#88322—41no—
36#13133124C2×C412yes—
37C22×Q8×C53D411yescharacteristic
38C4×Q8×C53—41no—
39C4×Q8×C53—41no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C4⋊C8×C53—41no—
47C22⋊C8×C53—22no—
48#97×C53—41no—
49C23×D53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—41no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—22no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212—42no—
65C2×D212—42no—
66C2×D212—41no—
67C2×D212—41no—
68C2×D212—42no—
69C2×D212—82no—
70C2×C8×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80#11719921—44no—
81#11719921—44no—
82#11719921—82no—
83#11719921—82no—
84C4⋊Dic53—42no—
85C4×Dic53—42no—
86#11719917—24no—
87#11719917—42no—
88C2×#88322—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97Q8×C53—44no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53#10412yes—
115C2×C4×C53AGL(1,ℤ/8)12yes—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119D212—82no—
120D212—84no—
121D212—84no—
122D212—82no—
123C8×C53—82no—
124#88322—44no—
125#11781—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53C2×#9911yescharacteristic
129C22×C53#41811yescharacteristic
130C22×C53C2×#10411yescharacteristic
131C2×D53—82no—
132C2×D53—82no—
133C2×D53—82no—
134C2×D53—82no—
135C2×D53—82no—
136C2×D53—82no—
137C2×D53—82no—
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53#35512yes—
146Dic53—82no—
147#2548—531no—
148C4×#101—531no—
149#1467D5311yescharacteristic, complemented
150#1345—534no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#150711yescharacteristic
154C2×C53#212211yescharacteristic
155C2×C53C2×#35511yescharacteristic
156D53—162no—
157C2×#127—531no—
158C2×#124—1061no—
159C2×C4×Q8C2×D5311yescharacteristic
160#432—21no—
161C2×#101—1061no—
162C4×M16—1061no—
163#394—532no—
164#394—532no—
165#390—1061no—
166#390—1061no—
167#390—1061no—
168C4×C4⋊C4—531no—
169#343—532no—
170#324—21no—
171#324C2×D5312yes—
172C53#1178111yescharacteristic, complemented, Sylow
173C22×Q8D21211yescharacteristic
174C22×D4—532no—
175C22×D4—1061no—
176C22×D4—1061no—
177C2×M16—1062no—
178#127—2121no—
179#127—2121no—
180#124—2124no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C4×D5311yescharacteristic
185C2×C4⋊C4C22⋊D5311yescharacteristic
186C2×C4⋊C4C22×D5311yescharacteristic
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C42—1061no—
192C2×C42C4×D5311yescharacteristic
193C2×C42D21211yescharacteristic
194C2×C42C22⋊D5311yescharacteristic
195C4⋊C8—41no—
196#101—1064no—
197C22⋊C8—22no—
198#97—41no—
199#97—2121no—
200C4×C8—2121no—
201#95—532no—
202C24—1062no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—4242no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2122no—
210C2×D4—2122no—
211C2×D4—2122no—
212C2×D4—2121no—
213C2×D4—2121no—
214C2×D4—2122no—
215C22×C4—1062no—
216C22×C4#1171992111yescharacteristic
217C22×C4#1171992111yescharacteristic
218C22×C4#1171992111yescharacteristic
219C22×C4#1171992111yescharacteristic
220C22×C4C2×C4×D5311yescharacteristic
221C22×C4C2×D21211yescharacteristic
222C22×C4C2×C22⋊D5311yescharacteristic
223M16—1064no—
224M16—2122no—
225C2×C8—42no—
226C2×C8—2122no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—2122no—
231C22⋊C4—4242no—
232C22⋊C4—4242no—
233C22⋊C4—2124no—
234C22⋊C4—2124no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—2122no—
242C23—2122no—
243C23—2122no—
244C23—2122no—
245C23—2122no—
246C23—2122no—
247C23—2122no—
248C23—2122no—
249C23C2×#1171992111yescharacteristic
250Q8—44no—
251D4—4244no—
252D4—4242no—
253D4—4244no—
254D4—4242no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4#1313313111yescharacteristic
273C2×C4#1313313111yescharacteristic
274C2×C4#1313314312yes—
275C2×C4#1313324512yes—
276C8—82no—
277C8—2124no—
278C22—21no—
279C22—21no—
280C22—4242no—
281C22—4242no—
282C22—4242no—
283C22—4242no—
284C22—4242no—
285C22—4242no—
286C22—4242no—
287C22C2×#1313313111yescharacteristic
288C22#1681062911yescharacteristic
289C22C2×#1313314311yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—4242no—
298C4#1681041912yes—
299C2—21no—
300C2—21no—
301C2—8482no—
302C2#2027323211yescharacteristic
303C2#2027636411yescharacteristic
304C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11781
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C4×#13133136C2semidirect
3. #20272719C2semidirect
4. C53×#1467C2semidirect
149. #1467D53semidirect
172. C53#11781semidirect

Maximal subgroups

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

subgroupindex
#202797142
C4×#131331362
#202727192
C53×#14672
#1178153