Finite Group Databasebeta

← #27294686

Subgroups — #27294686

Order 13568.

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

normal subgroups only

This group has 414 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#20279700C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272720C214yesmaximal
4C4×#101×C53C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133215—21no—
7#16810680—21no—
8#16810553C412yes—
9#16810553C2212yes—
10#16810528—21no—
11#16810521—21no—
12#16810521—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C2×#101×C53—21no—
18C4×M16×C53—21no—
19C53×#343C2212yes—
20C22×Dic106—21no—
21#13133217—41no—
22#13133217—41no—
23#13133215—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×C4×Dic53—21no—
30C4⋊#88322—41no—
31C22⋊#88322—22no—
32#13133128—41no—
33#13133124—22no—
34#13133124—22no—
35#13133124C2×C412yes—
36C22×Q8×C53D411yescharacteristic
37C2×M16×C53—22no—
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
46#101×C53—24no—
47#97×C53—41no—
48C4×C8×C53—41no—
49C2×Q8×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—41no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C2×Dic106—42no—
62C2×Dic106—42no—
63C2×Dic106—42no—
64M16×C53—24no—
65M16×C53—42no—
66C2×C8×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C4⋊Dic53—24no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80#11719919—44no—
81#11719919—44no—
82C4×Dic53—42no—
83C2×#88322—42no—
84C23×C53C2×C22⋊C411yescharacteristic
85Q8×C53—44no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53#9911yescharacteristic
101C2×C4×C53#9911yescharacteristic
102C2×C4×C53#10412yes—
103C2×C4×C53AGL(1,ℤ/8)12yes—
104C2×Dic53—24no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110Dic106—84no—
111C8×C53—44no—
112#88322—82no—
113#11782—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53C2×#9911yescharacteristic
117C22×C53#41811yescharacteristic
118C22×C53C2×#10411yescharacteristic
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53—41no—
126C4×C53#35612yes—
127Dic53—44no—
128Dic53—82no—
129#2546—531no—
130C4×#101D5311yescharacteristic
131#1467—531no—
132#1346—534no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#150711yescharacteristic
136C2×C53#212311yescharacteristic
137C2×C53C2×#35611yescharacteristic
138C2×C4⋊Q8—531no—
139C2×C4⋊Q8—1061no—
140C2×C4×Q8C2×D5311yescharacteristic
141#432—1061no—
142C2×#101—21no—
143C4×M16—21no—
144#398—532no—
145#398—532no—
146#391—1061no—
147#389—1061no—
148#389—1061no—
149C4×C4⋊C4—531no—
150#343C2×D5312yes—
151#324—532no—
152#324—1061no—
153C53#1178211yescharacteristic, complemented, Sylow
154C22×Q8—1061no—
155C22×Q8C22⋊D5311yescharacteristic
156C2×M16—22no—
157C4⋊Q8—2121no—
158C4⋊Q8—2121no—
159C4⋊Q8—2124no—
160C4×Q8—41no—
161C4×Q8—41no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4C4×D5311yescharacteristic
168C2×C4⋊C4D21211yescharacteristic
169C2×C4⋊C4C22×D5311yescharacteristic
170C2×C42—1061no—
171C2×C42C4×D5311yescharacteristic
172C2×C42D21211yescharacteristic
173C2×C42C22⋊D5311yescharacteristic
174C4⋊C8—2121no—
175#101—24no—
176C22⋊C8—1062no—
177#97—41no—
178#97—2121no—
179C4×C8—41no—
180#95—532no—
181#95—1062no—
182#95—1062no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—2122no—
187C2×Q8—2122no—
188C2×Q8—2122no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4#1171992111yescharacteristic
192C22×C4#1171992111yescharacteristic
193C22×C4#1171992111yescharacteristic
194C22×C4#1171992111yescharacteristic
195C22×C4C2×C4×D5311yescharacteristic
196C22×C4C2×D21211yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198M16—24no—
199M16—42no—
200C2×C8—42no—
201C2×C8—2122no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—1064no—
206C4⋊C4—2122no—
207C4⋊C4—2124no—
208C4⋊C4—2121no—
209C4⋊C4—2121no—
210C4⋊C4—2124no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—2122no—
218C23C2×#1171992111yescharacteristic
219Q8—44no—
220Q8—4244no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—1064no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4#1313313111yescharacteristic
242C2×C4#1313313111yescharacteristic
243C2×C4#1313314512yes—
244C2×C4#1313324212yes—
245C8—44no—
246C8—4242no—
247C22—21no—
248C22—21no—
249C22C2×#1313313111yescharacteristic
250C22#1681062711yescharacteristic
251C22C2×#1313314511yescharacteristic
252C4—22no—
253C4—22no—
254C4—21no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—4242no—
260C4—2124no—
261C4#1681042112yes—
262C2—21no—
263C2—21no—
264C2#2027323011yescharacteristic
265C2#2027637011yescharacteristic
266C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#101D53
hypocenterC53#11782
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
153. C53#11782semidirect

Maximal subgroups

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

subgroupindex
#202797002
#202730592
#202727202
C4×#101×C532
#1178253